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• Guide to Experimental Design | Overview, Steps, & Examples

Guide to Experimental Design | Overview, 5 steps & Examples

Published on December 3, 2019 by Rebecca Bevans . Revised on June 21, 2023.

Experiments are used to study causal relationships . You manipulate one or more independent variables and measure their effect on one or more dependent variables.

Experimental design create a set of procedures to systematically test a hypothesis . A good experimental design requires a strong understanding of the system you are studying.

There are five key steps in designing an experiment:

• Consider your variables and how they are related
• Write a specific, testable hypothesis
• Design experimental treatments to manipulate your independent variable
• Assign subjects to groups, either between-subjects or within-subjects
• Plan how you will measure your dependent variable

For valid conclusions, you also need to select a representative sample and control any  extraneous variables that might influence your results. If random assignment of participants to control and treatment groups is impossible, unethical, or highly difficult, consider an observational study instead. This minimizes several types of research bias, particularly sampling bias , survivorship bias , and attrition bias as time passes.

Table of contents

Step 1: define your variables, step 2: write your hypothesis, step 3: design your experimental treatments, step 4: assign your subjects to treatment groups, step 5: measure your dependent variable, other interesting articles, frequently asked questions about experiments.

You should begin with a specific research question . We will work with two research question examples, one from health sciences and one from ecology:

To translate your research question into an experimental hypothesis, you need to define the main variables and make predictions about how they are related.

Start by simply listing the independent and dependent variables .

Then you need to think about possible extraneous and confounding variables and consider how you might control  them in your experiment.

Finally, you can put these variables together into a diagram. Use arrows to show the possible relationships between variables and include signs to show the expected direction of the relationships.

Here we predict that increasing temperature will increase soil respiration and decrease soil moisture, while decreasing soil moisture will lead to decreased soil respiration.

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Now that you have a strong conceptual understanding of the system you are studying, you should be able to write a specific, testable hypothesis that addresses your research question.

The next steps will describe how to design a controlled experiment . In a controlled experiment, you must be able to:

• Systematically and precisely manipulate the independent variable(s).
• Precisely measure the dependent variable(s).
• Control any potential confounding variables.

If your study system doesn’t match these criteria, there are other types of research you can use to answer your research question.

How you manipulate the independent variable can affect the experiment’s external validity – that is, the extent to which the results can be generalized and applied to the broader world.

First, you may need to decide how widely to vary your independent variable.

• just slightly above the natural range for your study region.
• over a wider range of temperatures to mimic future warming.
• over an extreme range that is beyond any possible natural variation.

Second, you may need to choose how finely to vary your independent variable. Sometimes this choice is made for you by your experimental system, but often you will need to decide, and this will affect how much you can infer from your results.

• a categorical variable : either as binary (yes/no) or as levels of a factor (no phone use, low phone use, high phone use).
• a continuous variable (minutes of phone use measured every night).

How you apply your experimental treatments to your test subjects is crucial for obtaining valid and reliable results.

First, you need to consider the study size : how many individuals will be included in the experiment? In general, the more subjects you include, the greater your experiment’s statistical power , which determines how much confidence you can have in your results.

Then you need to randomly assign your subjects to treatment groups . Each group receives a different level of the treatment (e.g. no phone use, low phone use, high phone use).

You should also include a control group , which receives no treatment. The control group tells us what would have happened to your test subjects without any experimental intervention.

When assigning your subjects to groups, there are two main choices you need to make:

• A completely randomized design vs a randomized block design .
• A between-subjects design vs a within-subjects design .

Randomization

An experiment can be completely randomized or randomized within blocks (aka strata):

• In a completely randomized design , every subject is assigned to a treatment group at random.
• In a randomized block design (aka stratified random design), subjects are first grouped according to a characteristic they share, and then randomly assigned to treatments within those groups.

Sometimes randomization isn’t practical or ethical , so researchers create partially-random or even non-random designs. An experimental design where treatments aren’t randomly assigned is called a quasi-experimental design .

Between-subjects vs. within-subjects

In a between-subjects design (also known as an independent measures design or classic ANOVA design), individuals receive only one of the possible levels of an experimental treatment.

In medical or social research, you might also use matched pairs within your between-subjects design to make sure that each treatment group contains the same variety of test subjects in the same proportions.

In a within-subjects design (also known as a repeated measures design), every individual receives each of the experimental treatments consecutively, and their responses to each treatment are measured.

Within-subjects or repeated measures can also refer to an experimental design where an effect emerges over time, and individual responses are measured over time in order to measure this effect as it emerges.

Counterbalancing (randomizing or reversing the order of treatments among subjects) is often used in within-subjects designs to ensure that the order of treatment application doesn’t influence the results of the experiment.

Finally, you need to decide how you’ll collect data on your dependent variable outcomes. You should aim for reliable and valid measurements that minimize research bias or error.

Some variables, like temperature, can be objectively measured with scientific instruments. Others may need to be operationalized to turn them into measurable observations.

• Ask participants to record what time they go to sleep and get up each day.
• Ask participants to wear a sleep tracker.

How precisely you measure your dependent variable also affects the kinds of statistical analysis you can use on your data.

Experiments are always context-dependent, and a good experimental design will take into account all of the unique considerations of your study system to produce information that is both valid and relevant to your research question.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Student’s  t -distribution
• Normal distribution
• Null and Alternative Hypotheses
• Chi square tests
• Confidence interval
• Cluster sampling
• Stratified sampling
• Data cleansing
• Reproducibility vs Replicability
• Peer review
• Likert scale

Research bias

• Implicit bias
• Framing effect
• Cognitive bias
• Placebo effect
• Hawthorne effect
• Hindsight bias
• Affect heuristic

Experimental design means planning a set of procedures to investigate a relationship between variables . To design a controlled experiment, you need:

• A testable hypothesis
• At least one independent variable that can be precisely manipulated
• At least one dependent variable that can be precisely measured

When designing the experiment, you decide:

• How you will manipulate the variable(s)
• How you will control for any potential confounding variables
• How many subjects or samples will be included in the study
• How subjects will be assigned to treatment levels

Experimental design is essential to the internal and external validity of your experiment.

The key difference between observational studies and experimental designs is that a well-done observational study does not influence the responses of participants, while experiments do have some sort of treatment condition applied to at least some participants by random assignment .

A confounding variable , also called a confounder or confounding factor, is a third variable in a study examining a potential cause-and-effect relationship.

A confounding variable is related to both the supposed cause and the supposed effect of the study. It can be difficult to separate the true effect of the independent variable from the effect of the confounding variable.

In your research design , it’s important to identify potential confounding variables and plan how you will reduce their impact.

In a between-subjects design , every participant experiences only one condition, and researchers assess group differences between participants in various conditions.

In a within-subjects design , each participant experiences all conditions, and researchers test the same participants repeatedly for differences between conditions.

The word “between” means that you’re comparing different conditions between groups, while the word “within” means you’re comparing different conditions within the same group.

An experimental group, also known as a treatment group, receives the treatment whose effect researchers wish to study, whereas a control group does not. They should be identical in all other ways.

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Home » Experimental Design – Types, Methods, Guide

Experimental Design – Types, Methods, Guide

Table of Contents

Experimental Design

Experimental design is a process of planning and conducting scientific experiments to investigate a hypothesis or research question. It involves carefully designing an experiment that can test the hypothesis, and controlling for other variables that may influence the results.

Experimental design typically includes identifying the variables that will be manipulated or measured, defining the sample or population to be studied, selecting an appropriate method of sampling, choosing a method for data collection and analysis, and determining the appropriate statistical tests to use.

Types of Experimental Design

Here are the different types of experimental design:

Completely Randomized Design

In this design, participants are randomly assigned to one of two or more groups, and each group is exposed to a different treatment or condition.

Randomized Block Design

This design involves dividing participants into blocks based on a specific characteristic, such as age or gender, and then randomly assigning participants within each block to one of two or more treatment groups.

Factorial Design

In a factorial design, participants are randomly assigned to one of several groups, each of which receives a different combination of two or more independent variables.

Repeated Measures Design

In this design, each participant is exposed to all of the different treatments or conditions, either in a random order or in a predetermined order.

Crossover Design

This design involves randomly assigning participants to one of two or more treatment groups, with each group receiving one treatment during the first phase of the study and then switching to a different treatment during the second phase.

Split-plot Design

In this design, the researcher manipulates one or more variables at different levels and uses a randomized block design to control for other variables.

Nested Design

This design involves grouping participants within larger units, such as schools or households, and then randomly assigning these units to different treatment groups.

Laboratory Experiment

Laboratory experiments are conducted under controlled conditions, which allows for greater precision and accuracy. However, because laboratory conditions are not always representative of real-world conditions, the results of these experiments may not be generalizable to the population at large.

Field Experiment

Field experiments are conducted in naturalistic settings and allow for more realistic observations. However, because field experiments are not as controlled as laboratory experiments, they may be subject to more sources of error.

Experimental Design Methods

Experimental design methods refer to the techniques and procedures used to design and conduct experiments in scientific research. Here are some common experimental design methods:

Randomization

This involves randomly assigning participants to different groups or treatments to ensure that any observed differences between groups are due to the treatment and not to other factors.

Control Group

The use of a control group is an important experimental design method that involves having a group of participants that do not receive the treatment or intervention being studied. The control group is used as a baseline to compare the effects of the treatment group.

Blinding involves keeping participants, researchers, or both unaware of which treatment group participants are in, in order to reduce the risk of bias in the results.

Counterbalancing

This involves systematically varying the order in which participants receive treatments or interventions in order to control for order effects.

Replication

Replication involves conducting the same experiment with different samples or under different conditions to increase the reliability and validity of the results.

This experimental design method involves manipulating multiple independent variables simultaneously to investigate their combined effects on the dependent variable.

This involves dividing participants into subgroups or blocks based on specific characteristics, such as age or gender, in order to reduce the risk of confounding variables.

Data Collection Method

Experimental design data collection methods are techniques and procedures used to collect data in experimental research. Here are some common experimental design data collection methods:

Direct Observation

This method involves observing and recording the behavior or phenomenon of interest in real time. It may involve the use of structured or unstructured observation, and may be conducted in a laboratory or naturalistic setting.

Self-report Measures

Self-report measures involve asking participants to report their thoughts, feelings, or behaviors using questionnaires, surveys, or interviews. These measures may be administered in person or online.

Behavioral Measures

Behavioral measures involve measuring participants’ behavior directly, such as through reaction time tasks or performance tests. These measures may be administered using specialized equipment or software.

Physiological Measures

Physiological measures involve measuring participants’ physiological responses, such as heart rate, blood pressure, or brain activity, using specialized equipment. These measures may be invasive or non-invasive, and may be administered in a laboratory or clinical setting.

Archival Data

Archival data involves using existing records or data, such as medical records, administrative records, or historical documents, as a source of information. These data may be collected from public or private sources.

Computerized Measures

Computerized measures involve using software or computer programs to collect data on participants’ behavior or responses. These measures may include reaction time tasks, cognitive tests, or other types of computer-based assessments.

Video Recording

Video recording involves recording participants’ behavior or interactions using cameras or other recording equipment. This method can be used to capture detailed information about participants’ behavior or to analyze social interactions.

Data Analysis Method

Experimental design data analysis methods refer to the statistical techniques and procedures used to analyze data collected in experimental research. Here are some common experimental design data analysis methods:

Descriptive Statistics

Descriptive statistics are used to summarize and describe the data collected in the study. This includes measures such as mean, median, mode, range, and standard deviation.

Inferential Statistics

Inferential statistics are used to make inferences or generalizations about a larger population based on the data collected in the study. This includes hypothesis testing and estimation.

Analysis of Variance (ANOVA)

ANOVA is a statistical technique used to compare means across two or more groups in order to determine whether there are significant differences between the groups. There are several types of ANOVA, including one-way ANOVA, two-way ANOVA, and repeated measures ANOVA.

Regression Analysis

Regression analysis is used to model the relationship between two or more variables in order to determine the strength and direction of the relationship. There are several types of regression analysis, including linear regression, logistic regression, and multiple regression.

Factor Analysis

Factor analysis is used to identify underlying factors or dimensions in a set of variables. This can be used to reduce the complexity of the data and identify patterns in the data.

Structural Equation Modeling (SEM)

SEM is a statistical technique used to model complex relationships between variables. It can be used to test complex theories and models of causality.

Cluster Analysis

Cluster analysis is used to group similar cases or observations together based on similarities or differences in their characteristics.

Time Series Analysis

Time series analysis is used to analyze data collected over time in order to identify trends, patterns, or changes in the data.

Multilevel Modeling

Multilevel modeling is used to analyze data that is nested within multiple levels, such as students nested within schools or employees nested within companies.

Applications of Experimental Design 

Experimental design is a versatile research methodology that can be applied in many fields. Here are some applications of experimental design:

• Medical Research: Experimental design is commonly used to test new treatments or medications for various medical conditions. This includes clinical trials to evaluate the safety and effectiveness of new drugs or medical devices.
• Agriculture : Experimental design is used to test new crop varieties, fertilizers, and other agricultural practices. This includes randomized field trials to evaluate the effects of different treatments on crop yield, quality, and pest resistance.
• Environmental science: Experimental design is used to study the effects of environmental factors, such as pollution or climate change, on ecosystems and wildlife. This includes controlled experiments to study the effects of pollutants on plant growth or animal behavior.
• Psychology : Experimental design is used to study human behavior and cognitive processes. This includes experiments to test the effects of different interventions, such as therapy or medication, on mental health outcomes.
• Engineering : Experimental design is used to test new materials, designs, and manufacturing processes in engineering applications. This includes laboratory experiments to test the strength and durability of new materials, or field experiments to test the performance of new technologies.
• Education : Experimental design is used to evaluate the effectiveness of teaching methods, educational interventions, and programs. This includes randomized controlled trials to compare different teaching methods or evaluate the impact of educational programs on student outcomes.
• Marketing : Experimental design is used to test the effectiveness of marketing campaigns, pricing strategies, and product designs. This includes experiments to test the impact of different marketing messages or pricing schemes on consumer behavior.

Examples of Experimental Design 

Here are some examples of experimental design in different fields:

• Example in Medical research : A study that investigates the effectiveness of a new drug treatment for a particular condition. Patients are randomly assigned to either a treatment group or a control group, with the treatment group receiving the new drug and the control group receiving a placebo. The outcomes, such as improvement in symptoms or side effects, are measured and compared between the two groups.
• Example in Education research: A study that examines the impact of a new teaching method on student learning outcomes. Students are randomly assigned to either a group that receives the new teaching method or a group that receives the traditional teaching method. Student achievement is measured before and after the intervention, and the results are compared between the two groups.
• Example in Environmental science: A study that tests the effectiveness of a new method for reducing pollution in a river. Two sections of the river are selected, with one section treated with the new method and the other section left untreated. The water quality is measured before and after the intervention, and the results are compared between the two sections.
• Example in Marketing research: A study that investigates the impact of a new advertising campaign on consumer behavior. Participants are randomly assigned to either a group that is exposed to the new campaign or a group that is not. Their behavior, such as purchasing or product awareness, is measured and compared between the two groups.
• Example in Social psychology: A study that examines the effect of a new social intervention on reducing prejudice towards a marginalized group. Participants are randomly assigned to either a group that receives the intervention or a control group that does not. Their attitudes and behavior towards the marginalized group are measured before and after the intervention, and the results are compared between the two groups.

When to use Experimental Research Design 

Experimental research design should be used when a researcher wants to establish a cause-and-effect relationship between variables. It is particularly useful when studying the impact of an intervention or treatment on a particular outcome.

Here are some situations where experimental research design may be appropriate:

• When studying the effects of a new drug or medical treatment: Experimental research design is commonly used in medical research to test the effectiveness and safety of new drugs or medical treatments. By randomly assigning patients to treatment and control groups, researchers can determine whether the treatment is effective in improving health outcomes.
• When evaluating the effectiveness of an educational intervention: An experimental research design can be used to evaluate the impact of a new teaching method or educational program on student learning outcomes. By randomly assigning students to treatment and control groups, researchers can determine whether the intervention is effective in improving academic performance.
• When testing the effectiveness of a marketing campaign: An experimental research design can be used to test the effectiveness of different marketing messages or strategies. By randomly assigning participants to treatment and control groups, researchers can determine whether the marketing campaign is effective in changing consumer behavior.
• When studying the effects of an environmental intervention: Experimental research design can be used to study the impact of environmental interventions, such as pollution reduction programs or conservation efforts. By randomly assigning locations or areas to treatment and control groups, researchers can determine whether the intervention is effective in improving environmental outcomes.
• When testing the effects of a new technology: An experimental research design can be used to test the effectiveness and safety of new technologies or engineering designs. By randomly assigning participants or locations to treatment and control groups, researchers can determine whether the new technology is effective in achieving its intended purpose.

How to Conduct Experimental Research

Here are the steps to conduct Experimental Research:

• Identify a Research Question : Start by identifying a research question that you want to answer through the experiment. The question should be clear, specific, and testable.
• Develop a Hypothesis: Based on your research question, develop a hypothesis that predicts the relationship between the independent and dependent variables. The hypothesis should be clear and testable.
• Design the Experiment : Determine the type of experimental design you will use, such as a between-subjects design or a within-subjects design. Also, decide on the experimental conditions, such as the number of independent variables, the levels of the independent variable, and the dependent variable to be measured.
• Select Participants: Select the participants who will take part in the experiment. They should be representative of the population you are interested in studying.
• Randomly Assign Participants to Groups: If you are using a between-subjects design, randomly assign participants to groups to control for individual differences.
• Conduct the Experiment : Conduct the experiment by manipulating the independent variable(s) and measuring the dependent variable(s) across the different conditions.
• Analyze the Data: Analyze the data using appropriate statistical methods to determine if there is a significant effect of the independent variable(s) on the dependent variable(s).
• Draw Conclusions: Based on the data analysis, draw conclusions about the relationship between the independent and dependent variables. If the results support the hypothesis, then it is accepted. If the results do not support the hypothesis, then it is rejected.
• Communicate the Results: Finally, communicate the results of the experiment through a research report or presentation. Include the purpose of the study, the methods used, the results obtained, and the conclusions drawn.

Purpose of Experimental Design 

The purpose of experimental design is to control and manipulate one or more independent variables to determine their effect on a dependent variable. Experimental design allows researchers to systematically investigate causal relationships between variables, and to establish cause-and-effect relationships between the independent and dependent variables. Through experimental design, researchers can test hypotheses and make inferences about the population from which the sample was drawn.

Experimental design provides a structured approach to designing and conducting experiments, ensuring that the results are reliable and valid. By carefully controlling for extraneous variables that may affect the outcome of the study, experimental design allows researchers to isolate the effect of the independent variable(s) on the dependent variable(s), and to minimize the influence of other factors that may confound the results.

Experimental design also allows researchers to generalize their findings to the larger population from which the sample was drawn. By randomly selecting participants and using statistical techniques to analyze the data, researchers can make inferences about the larger population with a high degree of confidence.

Overall, the purpose of experimental design is to provide a rigorous, systematic, and scientific method for testing hypotheses and establishing cause-and-effect relationships between variables. Experimental design is a powerful tool for advancing scientific knowledge and informing evidence-based practice in various fields, including psychology, biology, medicine, engineering, and social sciences.

Advantages of Experimental Design 

Experimental design offers several advantages in research. Here are some of the main advantages:

• Control over extraneous variables: Experimental design allows researchers to control for extraneous variables that may affect the outcome of the study. By manipulating the independent variable and holding all other variables constant, researchers can isolate the effect of the independent variable on the dependent variable.
• Establishing causality: Experimental design allows researchers to establish causality by manipulating the independent variable and observing its effect on the dependent variable. This allows researchers to determine whether changes in the independent variable cause changes in the dependent variable.
• Replication : Experimental design allows researchers to replicate their experiments to ensure that the findings are consistent and reliable. Replication is important for establishing the validity and generalizability of the findings.
• Random assignment: Experimental design often involves randomly assigning participants to conditions. This helps to ensure that individual differences between participants are evenly distributed across conditions, which increases the internal validity of the study.
• Precision : Experimental design allows researchers to measure variables with precision, which can increase the accuracy and reliability of the data.
• Generalizability : If the study is well-designed, experimental design can increase the generalizability of the findings. By controlling for extraneous variables and using random assignment, researchers can increase the likelihood that the findings will apply to other populations and contexts.

Limitations of Experimental Design

Experimental design has some limitations that researchers should be aware of. Here are some of the main limitations:

• Artificiality : Experimental design often involves creating artificial situations that may not reflect real-world situations. This can limit the external validity of the findings, or the extent to which the findings can be generalized to real-world settings.
• Ethical concerns: Some experimental designs may raise ethical concerns, particularly if they involve manipulating variables that could cause harm to participants or if they involve deception.
• Participant bias : Participants in experimental studies may modify their behavior in response to the experiment, which can lead to participant bias.
• Limited generalizability: The conditions of the experiment may not reflect the complexities of real-world situations. As a result, the findings may not be applicable to all populations and contexts.
• Cost and time : Experimental design can be expensive and time-consuming, particularly if the experiment requires specialized equipment or if the sample size is large.
• Researcher bias : Researchers may unintentionally bias the results of the experiment if they have expectations or preferences for certain outcomes.
• Lack of feasibility : Experimental design may not be feasible in some cases, particularly if the research question involves variables that cannot be manipulated or controlled.

About the author

Muhammad Hassan

Researcher, Academic Writer, Web developer

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Full Factorial Design: Comprehensive Guide for Optimal Experimentation

June 4th, 2024

The full factorial design distinguishes itself as a robust, enlightening experimentation approach across industries.

At its core, this systematic method examines multiple metrics’ collective effects on an outcome simultaneously.

Considering all factor level combinations furnishes holistic comprehension beyond individual impacts—illuminating intricate relationships shaping complex systems’ behaviors.

The factorial design’s strength lies in realistically emulating dynamics’ nuances where variables interact intricately nonlinearly. Accounting for interplays guards against oversimplification. This casts light on underlying realities profoundly, priming informed resolutions and refinement pursuits.

From manufacturing to research frontiers, ponder occasions where exhaustively investigating factor relations meaningfully addresses specific problems or opportunities.

Key Highlights

• A comprehensive exploration of factor effects and interactions
• Robust methodology for process understanding and optimization
• Applications across diverse industries, including manufacturing, pharmaceuticals, and marketing
• Rigorous statistical analysis techniques, such as ANOVA and regression modeling
• Insights into factor-level optimization for enhanced process performance
• Evaluation of alternative designs, including fractional factorial and response surface methodologies
• Best practices for experimental design, data collection, and analysis

Introduction to Full Factorial Design

The full factorial design stands out as a comprehensive and robust approach, enabling researchers and practitioners to unlock valuable insights and drive meaningful change across diverse industries.

The full factorial design is a systematic way to investigate the effects of multiple factors on a response variable simultaneously.

By considering all possible combinations of factor levels, this experimental strategy provides a holistic understanding of not only the individual factor effects but also the intricate interactions that can shape outcomes in complex systems.

A full factorial design is an experimental design that considers the effects of multiple factors simultaneously on a response variable.

It involves manipulating all possible combinations of the levels of each factor, enabling researchers to determine the main effects of individual factors as well as their interactions statistically.

This comprehensive approach ensures that no potential interaction is overlooked, providing a complete picture of the system under investigation.

The key benefits of employing a full factorial design include:

• Main effects : Researchers can identify which factors have the most significant impact on the response variable, allowing them to focus their efforts on the most influential factors.
• Interaction effects : By accounting for interactions between factors, full factorial designs reveal how the effect of one factor depends on the level of another factor, providing insights into the complex relationships within the system.
• Optimization: With a comprehensive understanding of the main effects and interactions, researchers can estimate the optimal settings for the independent variables, leading to the best possible outcome for the response variable.

The versatility of the full factorial design makes it a valuable tool across various industries, including:

• Manufacturing : Optimizing processes, improving product quality, and reducing defects by identifying key variables and their interactions.
• Pharmaceuticals : Formulating and developing drugs by assessing factors such as excipient concentrations, drug particle size, and processing conditions on bioavailability, stability, and release profiles.
• Marketing: Optimizing promotional strategies by evaluating the effects of factors like ad content, media channels, target audience segments, and pricing on consumer response.

Fundamentals of Design of Experiments (DOE)

Before delving into the intricacies of full factorial design, it is essential to understand the fundamental principles of Design of Experiments (DOE) , a systematic approach to investigating the relationships between input variables (factors) and output variables (responses) .

DOE provides a structured framework for planning, executing, and analyzing experiments, ensuring reliable and insightful results.

Understanding Factors and Levels

In a DOE study, the variables that are manipulated or controlled by the experimenter are known as independent variables or factors.

These can be further classified into:

• Numerical factors : Variables that can take on a range of numerical values, such as temperature, pressure, or time.
• Categorical factors : Variables that have distinct, non-numerical levels, such as material type or production method.

The outcome or characteristic of interest that is measured and analyzed in an experiment is referred to as the response variable or dependent variable.

Common examples include product yield, strength, purity, or customer satisfaction.

Each independent variable in a DOE study can be set at different levels or values.

The choice of factor levels is crucial, as it determines the range of conditions under which the experiment is conducted.

Carefully selecting factor levels ensures that the study captures the relevant region of interest and provides meaningful insights into the system’s behavior.

Principles of DOE

Replication refers to the practice of repeating the same experimental run multiple times under identical conditions.

This allows researchers to estimate the inherent variability in the experimental process and ensures the reliability of the results by providing a measure of experimental error.

Randomization is the process of randomly assigning experimental runs to different factor level combinations.

This helps to mitigate the potential impact of nuisance variables and ensures that any observed effects can be attributed to the factors under investigation, rather than uncontrolled sources of variation .

Blocking is a technique used to account for known sources of variability in an experiment, such as differences in equipment, operators, or environmental conditions.

By grouping experimental runs into homogeneous blocks, researchers can isolate and quantify the effects of these nuisance variables, ensuring more precise estimates of the factor effects.

Types of Full Factorial Designs

Full factorial designs can be classified into different types based on the number of levels for each factor and the nature of the factors themselves.

Understanding the various types of full factorial designs is crucial for selecting the appropriate experimental strategy to address specific research questions or process optimization objectives.

2-Level Full Factorial Design

The 2-level full factorial design, where each factor has two levels (typically labeled as “low” and “high”), is commonly employed in screening experiments.

These experiments aim to identify the most significant factors influencing the response variable, allowing researchers to focus their efforts on the most promising factors in subsequent, more in-depth investigations.

By evaluating the main effects and interactions in a 2-level full factorial design, researchers can determine which factors have a statistically significant impact on the response variable.

This information is invaluable in prioritizing factors for further optimization or confirming their negligible influence, thereby streamlining the overall experimental process.

3-Level Full Factorial Design

Unlike the 2-level design, which assumes a linear relationship between factors and the response variable, the 3-level full factorial design allows for the investigation of quadratic effects.

These nonlinear effects can be important in scenarios where the response variable exhibits curvature or a peak/valley behavior within the explored factor ranges.

By incorporating three levels for each factor, researchers can model the curvature in the response surface more accurately.

This enhanced understanding of the system’s behavior enables more precise optimization and provides insights into potential optimal operating regions or factor level combinations.

Mixed-Level Full Factorial Design

In many real-world applications, experiments may involve a combination of categorical factors (e.g., material type, production method) and continuous factors (e.g., temperature, pressure).

The mixed-level full factorial design accommodates this scenario by allowing researchers to investigate the effects of both types of factors simultaneously, providing a comprehensive understanding of the system.

Analyzing Full Factorial Design Experiments

Once the experimental data has been collected, the next step is to analyze the results to gain insights into the main effects, interactions, and optimal factor level combinations.

Several statistical techniques are employed in the analysis of full factorial experiments, each serving a specific purpose and providing valuable information for process understanding and optimization.

Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA) is a powerful statistical tool used to determine the significance of main effects (individual factor effects) and interaction effects (combined effects of multiple factors) on the response variable.

By partitioning the total variability in the data into components attributable to each factor and their interactions, ANOVA enables researchers to identify the most influential factors and their relationships.

ANOVA also provides a framework for hypothesis testing, allowing researchers to assess whether the observed effects are statistically significant or simply due to random variability.

This rigorous approach ensures that conclusions drawn from the experimental data are statistically valid and reliable.

Regression Analysis

Regression analysis is another essential tool in the analysis of full factorial experiments.

It involves fitting a mathematical model to the experimental data , relating the response variable to the independent variables (factors) and their interactions.

This model can be used to predict the response variable for any combination of factor levels within the experimental region.

Once a suitable regression model has been obtained, researchers can employ optimization techniques to identify the factor level combinations that maximize or minimize the response variable.

These techniques often involve solving the regression equation subject to relevant constraints, enabling the determination of optimal operating conditions for the process under investigation.

Graphical Analysis & Full Factorial Design

Graphical analysis is a powerful tool for visualizing and interpreting the results of full factorial experiments.

Interaction plots are particularly useful for examining the presence and nature of interactions between factors.

These plots display the response variable as a function of one factor at different levels of another factor, allowing researchers to identify and understand complex relationships within the system.

Main effects plots , on the other hand, illustrate the individual impact of each factor on the response variable, providing a visual representation of the main effects.

These plots can aid in quickly identifying the most influential factors and assessing the relative importance of each factor in the experimental domain.

Advantages and Limitations of Full Factorial Design

While the full factorial design offers numerous advantages in terms of comprehensiveness and insight, it is important to recognize its limitations and potential drawbacks.

Understanding both the strengths and limitations of this experimental approach is crucial for making informed decisions and optimizing the trade-offs between resource allocation and the desired level of process understanding.

Advantages of Full Factorial Design

One of the primary advantages of the full factorial design is its ability to provide comprehensive insights into the system under investigation.

By considering all possible factor combinations, researchers can obtain a complete picture of the main effects, interactions, and potential curvature in the response surface, leading to a thorough understanding of the process dynamics.

Unlike some experimental designs that may overlook or confound interactions, the full factorial design explicitly accounts for interactions between factors.

This capability is particularly valuable in complex systems where the effect of one factor may depend on the level of another factor, allowing researchers to unravel these intricate relationships and optimize processes accordingly.

With a comprehensive understanding of the main effects and interactions, full factorial experiments enable researchers to estimate the optimal settings for the independent variables, leading to the best possible outcome for the response variable.

This optimization potential is invaluable in various industries, where process efficiency , product quality, and cost-effectiveness are paramount.

Limitations of Full Factorial Design

One of the primary limitations of the full factorial design is its resource-intensive nature.

As the number of factors and levels increases, the number of experimental runs required grows exponentially, leading to higher costs, longer experimental durations, and greater logistical challenges.

Related to the resource-intensive aspect, full factorial designs often require large sample sizes to ensure statistical validity and reliable estimates of main effects and interactions.

This can be particularly challenging in situations where resources are limited or experimental conditions are difficult to replicate.

The comprehensiveness of the full factorial design can also lead to an overwhelming amount of data, especially when dealing with numerous factors and levels.

Analyzing and interpreting such large datasets can be a daunting task, requiring advanced statistical techniques and computational resources.

Alternative Designs and Extensions

While the full factorial design is a powerful and comprehensive experimental strategy, some alternative designs and extensions can be employed depending on the specific requirements and constraints of the research or industrial application.

These alternative approaches can offer trade-offs between experimental complexity, resource requirements, and the level of information obtained.

Fractional Factorial Designs

Fractional factorial designs are a class of experimental designs that involve studying only a carefully chosen fraction of the full factorial design.

By sacrificing the ability to estimate certain higher-order interactions, fractional factorial designs can significantly reduce the number of experimental runs required, making them more resource-efficient.

Fractional factorial designs are particularly useful in screening experiments, where the primary goal is to identify the most influential factors before conducting more detailed investigations.

These designs can help researchers prioritize their efforts and allocate resources more effectively.

Response Surface Methodology

Response Surface Methodology (RSM) is a collection of statistical techniques used to model and optimize processes with multiple input variables.

The Central Composite Design (CCD) is a widely used RSM design that combines a factorial design with additional axial and center points, allowing for the estimation of quadratic effects and potential curvature in the response surface.

Another popular RSM design is the Box-Behnken Design, which is a spherical, rotatable, or nearly rotatable design.

This design is particularly efficient for exploring quadratic response surfaces and optimizing processes with three or more factors.

Unlike the Central Composite Design, the Box-Behnken Design does not include any points at the vertices of the cubic region defined by the upper and lower limits of the factors.

Taguchi Methods

Developed by Genichi Taguchi, the Taguchi methods are a set of techniques for robust parameter design and quality improvement.

One of the key elements of the Taguchi approach is the use of orthogonal arrays, which are a special class of fractional factorial designs.

Orthogonal arrays allow for the simultaneous investigation of multiple factors with a minimal number of experimental runs, making them an attractive option when resources are limited.

The Taguchi methods emphasize the concept of robust parameter design, which aims to identify factor-level combinations that minimize the variability in the response variable while achieving the desired target value.

This approach is particularly valuable in manufacturing and product development, where robustness to environmental and operational variations is critical for maintaining consistent performance and quality.

The full factorial design stands as a potent experimental strategy.

Considering every factor combination furnishes holistic comprehension of effects, interplays, and nonlinearities.

This lights pathways empowering better comprehension and optimization across varied sectors.

Advancing capabilities and statistical techniques envision expanding factorial applications.

Machine learning may analyze data efficiently while adaptive designs responsively calibrate based on real-time insights.

Additionally, sustainability priorities could drive factor prioritizations maximizing resource optimization and lessening environmental impacts.

Full factorials offer rigorous yet flexible methods unlocking enriched wisdom from intricate systems.

While exhaustive, generated learnings inspire noteworthy refinements throughout performance, quality, and workflows.

By following best practices, judiciously leveraging other designs, and tracking innovations, researchers and specialists harness factorials’ full gifts in driving innovations remarkably within respective realms.

Their gifts in illuminating intricate relations deserve recognition and prudent application wherever circumstances permit comprehensive experimentation.

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Response, Factor, and Level - Three Experimental Definitions You Might be Getting Wrong

June 24, 2020, 11:59 p.m..

When talking about experimental designs do you ever get bewildered by the terms response , factor , and level ? If so you are not alone. However, getting the definition of these terms is absolutely critical to ensure that the experimental design is not ruined in the initial planning and execution phases.

In this article, I will present definitions for these terms, synonyms, and point out ways to avoid getting it wrong. Along the way we will also define discrete, continuous, categorical, binary, and ordinal - some other commonly misused terms.

Examples of response variables could be the number of scoops of ice cream sold, the height of a corn plant, the patient's pain score after surgery, or the score on a standardized knowledge test.

Responses can be either continuous or discrete.

• Binary: The response is one of only two possible values. This might be for example, getting a disease or not getting it.
• Ordinal: The response is on an ordered scale. This might be for example score on a 1 to 5 triage scale.
• Categorical: The response is one of several categories such as eye color, type of pet, or type of antiobiotic given.

Continuous responses are somewhat easier to conceptualize. Continous responses take a value that theoretically could take on any possible value in a range of portion of the number line. An example would be height, blood pressure, temperature.

Factors can also be called independent variables, explanatory variables, manipulator variables, or risk factors.

Examples of factors may be an antibiotic given to a patient, a teaching session given to a student, the type of triage system being used, or the age of a person taking a driving test.

Why all the Fuss?

Why is it so important to know all these tedious definitions? Choosing the right statistical test to analyze your data depends directly on knowing the response, factors, and levels for the experimental data. Choosing this statistical test correctly means sailing through the power calculation, study design, data collection, and analysis. Getting it wrong can mean hours and hours of needless work and - in the worst case - an experiment that is unsalvageable.

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Experimental Design - Independent, Dependent, and Controlled Variables

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Scientific experiments are meant to show cause and effect of a phenomena (relationships in nature).  The “ variables ” are any factor, trait, or condition that can be changed in the experiment and that can have an effect on the outcome of the experiment.

An experiment can have three kinds of variables: i ndependent, dependent, and controlled .

• The independent variable is one single factor that is changed by the scientist followed by observation to watch for changes. It is important that there is just one independent variable, so that results are not confusing.
• The dependent variable is the factor that changes as a result of the change to the independent variable.
• The controlled variables (or constant variables) are factors that the scientist wants to remain constant if the experiment is to show accurate results. To be able to measure results, each of the variables must be able to be measured.

For example, let’s design an experiment with two plants sitting in the sun side by side. The controlled variables (or constants) are that at the beginning of the experiment, the plants are the same size, get the same amount of sunlight, experience the same ambient temperature and are in the same amount and consistency of soil (the weight of the soil and container should be measured before the plants are added). The independent variable is that one plant is getting watered (1 cup of water) every day and one plant is getting watered (1 cup of water) once a week. The dependent variables are the changes in the two plants that the scientist observes over time.

Can you describe the dependent variable that may result from this experiment? After four weeks, the dependent variable may be that one plant is taller, heavier and more developed than the other. These results can be recorded and graphed by measuring and comparing both plants’ height, weight (removing the weight of the soil and container recorded beforehand) and a comparison of observable foliage.

Using What You Learned: Design another experiment using the two plants, but change the independent variable. Can you describe the dependent variable that may result from this new experiment?

Think of another simple experiment and name the independent, dependent, and controlled variables. Use the graphic organizer included in the PDF below to organize your experiment's variables.

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1.1.2 - explanatory & response variables.

In some research studies one variable is used to predict or explain differences in another variable. In those cases, the  explanatory variable  is used to predict or explain differences in the  response variable . In an experimental study, the explanatory variable is the variable that is manipulated by the researcher. 

Also known as the independent  or  predictor variable , it explains variations in the response variable; in an experimental study, it is manipulated by the researcher

Also known as the  dependent  or  outcome variable,  its value is predicted or its variation is explained by the explanatory variable; in an experimental study, this is the outcome that is measured following manipulation of the explanatory variable

Example: Panda Fertility Treatments Section  

A team of veterinarians wants to compare the effectiveness of two fertility treatments for pandas in captivity. The two treatments are in-vitro fertilization and male fertility medications. This experiment has one  explanatory variable : type of fertility treatment. The  response variable  is a measure of fertility rate.

Example: Public Speaking Approaches Section  

A public speaking teacher has developed a new lesson that she believes decreases student anxiety in public speaking situations more than the old lesson. She designs an experiment to test if her new lesson works better than the old lesson. Public speaking students are randomly assigned to receive either the new or old lesson; their anxiety levels during a variety of public speaking experiences are measured.  This experiment has one  explanatory variable : the lesson received. The  response variable  is anxiety level.

Example: Coffee Bean Origin Section  

A researcher believes that the origin of the beans used to make a cup of coffee affects hyperactivity. He wants to compare coffee from three different regions: Africa, South America, and Mexico. The  explanatory variable is the origin of coffee bean; this has three levels: Africa, South America, and Mexico. The  response variable  is hyperactivity level.

Example: Height & Age Section  

A group of middle school students wants to know if they can use height to predict age. They take a random sample of 50 people at their school, both students and teachers, and record each individual's height and age. This is an observational study. The students want to use height to predict age so the  explanatory variable  is height and the  response variable  is age.

Example: Grade & Height Section  

Research question:  Do fourth graders tend to be taller than third graders?

This is an observational study. The researcher wants to use grade level to explain differences in height. The  explanatory variable  is grade level. The  response variable  is height. 

Module 1: Sampling and Data

Components of experimental design, learning outcomes.

• For a given scenario, identify the explanatory variable, response variable, treatments, experimental units, lurking variables and control group
• Explain how blinding could be used in the design of an experiment

Does aspirin reduce the risk of heart attacks? Is one brand of fertilizer more effective at growing roses than another? Is fatigue as dangerous to a driver as the influence of alcohol? Questions like these are answered using randomized experiments. In this module, you will learn important aspects of experimental design. Proper study design ensures the production of reliable, accurate data.

The purpose of an experiment is to investigate the relationship between two variables. When one variable causes change in another, we call the first variable the explanatory variable . The affected variable is called the response variable . In a randomized experiment, the researcher manipulates values of the explanatory variable and measures the resulting changes in the response variable. The different values of the explanatory variable are called treatments . An experimental unit is a single object or individual to be measured.

The following video explains the difference between collecting data from observations and collecting data from experiments.

Let’s say you want to investigate the effectiveness of vitamin E in preventing disease. You recruit a group of subjects and ask them if they regularly take vitamin E. You notice that the subjects who take vitamin E exhibit better health on average than those who do not. Does this prove that vitamin E is effective in disease prevention? It does not. There are many differences between the two groups compared in addition to vitamin E consumption. People who take vitamin E regularly often take other steps to improve their health: exercise, diet, other vitamin supplements, choosing not to smoke. Any one of these factors could be influencing health. As described, this study does not prove that vitamin E is the key to disease prevention.

Additional variables that can cloud a study are called lurking variables . In order to prove that the explanatory variable is causing a change in the response variable, it is necessary to isolate the explanatory variable. The researcher must design her experiment in such a way that there is only one difference between groups being compared: the planned treatments. This is accomplished by the random assignment of experimental units to treatment groups. When subjects are assigned treatments randomly, all of the potential lurking variables are spread equally among the groups. At this point the only difference between groups is the one imposed by the researcher. Different outcomes measured in the response variable, therefore, must be a direct result of the different treatments. In this way, an experiment can prove a cause-and-effect connection between the explanatory and response variables.

The power of suggestion can have an important influence on the outcome of an experiment. Studies have shown that the expectation of the study participant can be as important as the actual medication. In one study of performance-enhancing drugs, researchers noted:

Results showed that believing one had taken the substance resulted in [ performance ] times almost as fast as those associated with consuming the drug itself. In contrast, taking the drug without knowledge yielded no significant performance increment. 1

When participation in a study prompts a physical response from a participant, it is difficult to isolate the effects of the explanatory variable. To counter the power of suggestion, researchers set aside one treatment group as a control group . This group is given a placebo treatment — a treatment that cannot influence the response variable. The control group helps researchers balance the effects of being in an experiment with the effects of the active treatments. Of course, if you are participating in a study and you know that you are receiving a pill which contains no actual medication, then the power of suggestion is no longer a factor.

Blinding in a randomized experiment preserves the power of suggestion. When a person involved in a research study is blinded, he does not know who is receiving the active treatment(s) and who is receiving the placebo treatment. A double-blind experiment is one in which both the subjects and the researchers involved with the subjects are blinded.

Researchers want to investigate whether taking aspirin regularly reduces the risk of heart attack. Four hundred men between the ages of 50 and 84 are recruited as participants. The men are divided randomly into two groups: one group will take aspirin, and the other group will take a placebo. Each man takes one pill each day for three years, but he does not know whether he is taking aspirin or the placebo. At the end of the study, researchers count the number of men in each group who have had heart attacks.

Identify the following values for this study: population, sample, experimental units, explanatory variable, response variable, treatments.

The Smell & Taste Treatment and Research Foundation conducted a study to investigate whether smell can affect learning. Subjects completed mazes multiple times while wearing masks. They completed the pencil and paper mazes three times wearing floral-scented masks, and three times with unscented masks. Participants were assigned at random to wear the floral mask during the first three trials or during the last three trials. For each trial, researchers recorded the time it took to complete the maze and the subject’s impression of the mask’s scent: positive, negative, or neutral.

• Describe the explanatory and response variables in this study.
• What are the treatments?
• Identify any lurking variables that could interfere with this study.
• Is it possible to use blinding in this study?
• The explanatory variable is scent, and the response variable is the time it takes to complete the maze.
• There are two treatments: a floral-scented mask and an unscented mask.
• All subjects experienced both treatments. The order of treatments was randomly assigned so there were no differences between the treatment groups. Random assignment eliminates the problem of lurking variables.
• Subjects will clearly know whether they can smell flowers or not, so subjects cannot be blinded in this study. Researchers timing the mazes can be blinded, though. The researcher who is observing a subject will not know which mask is being worn.

A researcher wants to study the effects of birth order on personality. Explain why this study could not be conducted as a randomized experiment. What is the main problem in a study that cannot be designed as a randomized experiment?

You are concerned about the effects of texting on driving performance. Design a study to test the response time of drivers while texting and while driving only. How many seconds does it take for a driver to respond when a leading car hits the brakes?

• Describe the explanatory and response variables in the study.
• What should you consider when selecting participants?
• Your research partner wants to divide participants randomly into two groups: one to drive without distraction and one to text and drive simultaneously. Is this a good idea? Why or why not?
• How can blinding be used in this study?
• Experimental Design and Ethics. Provided by : OpenStax. Located at : https://openstax.org/books/introductory-statistics/pages/1-4-experimental-design-and-ethics . License : CC BY: Attribution . License Terms : Access for free at https://openstax.org/books/introductory-statistics/pages/1-introduction
• Introductory Statistics. Authored by : Barbara Illowsky, Susan Dean. Provided by : Open Stax. Located at : https://openstax.org/books/introductory-statistics/pages/1-introduction . License : CC BY: Attribution . License Terms : Access for free at https://openstax.org/books/introductory-statistics/pages/1-introduction
• Observational Studies and Experiments. Authored by : ProfessorMcComb. Located at : https://www.youtube.com/watch?v=J_O7ibkX8Ik . License : All Rights Reserved . License Terms : Standard YouTube License

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Experimental Design

• First Online: 12 October 2022

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• Edward B. Magrab 2  

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In this chapter, the terms used in experimental design are introduced: response variable, factor, extraneous variable, level, treatment, blocking variable, replication, contrasts, and effects. The relations needed to analyze a one-factor experiment, a randomized complete block design, a two-factor experiment, and a 2 k -factorial experiment are derived. For these experiments, an analysis of variance is used to determine the factors that are most influential in determining its output and, where appropriate, whether the factors interact.

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When μ i = μ j , i  ≠  j , we see from Eq. ( 4.10 ) that μ  +  τ i  =  μ  +  τ j  →  τ i  =  τ j . Then, Eq. ( 4.11 ) becomes \( \sum \limits_{i=1}^a{n}_i{\tau}_i={n}_T{\tau}_i=0; \) therefore, τ i must equal zero for all i .

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Magrab, E.B. (2022). Experimental Design. In: Engineering Statistics. Springer, Cham. https://doi.org/10.1007/978-3-031-05010-7_4

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Statistical Design and Analysis of Biological Experiments

Chapter 1 principles of experimental design, 1.1 introduction.

The validity of conclusions drawn from a statistical analysis crucially hinges on the manner in which the data are acquired, and even the most sophisticated analysis will not rescue a flawed experiment. Planning an experiment and thinking about the details of data acquisition is so important for a successful analysis that R. A. Fisher—who single-handedly invented many of the experimental design techniques we are about to discuss—famously wrote

To call in the statistician after the experiment is done may be no more than asking him to perform a post-mortem examination: he may be able to say what the experiment died of. ( Fisher 1938 )

(Statistical) design of experiments provides the principles and methods for planning experiments and tailoring the data acquisition to an intended analysis. Design and analysis of an experiment are best considered as two aspects of the same enterprise: the goals of the analysis strongly inform an appropriate design, and the implemented design determines the possible analyses.

The primary aim of designing experiments is to ensure that valid statistical and scientific conclusions can be drawn that withstand the scrutiny of a determined skeptic. Good experimental design also considers that resources are used efficiently, and that estimates are sufficiently precise and hypothesis tests adequately powered. It protects our conclusions by excluding alternative interpretations or rendering them implausible. Three main pillars of experimental design are randomization , replication , and blocking , and we will flesh out their effects on the subsequent analysis as well as their implementation in an experimental design.

An experimental design is always tailored towards predefined (primary) analyses and an efficient analysis and unambiguous interpretation of the experimental data is often straightforward from a good design. This does not prevent us from doing additional analyses of interesting observations after the data are acquired, but these analyses can be subjected to more severe criticisms and conclusions are more tentative.

In this chapter, we provide the wider context for using experiments in a larger research enterprise and informally introduce the main statistical ideas of experimental design. We use a comparison of two samples as our main example to study how design choices affect an analysis, but postpone a formal quantitative analysis to the next chapters.

1.2 A Cautionary Tale

For illustrating some of the issues arising in the interplay of experimental design and analysis, we consider a simple example. We are interested in comparing the enzyme levels measured in processed blood samples from laboratory mice, when the sample processing is done either with a kit from a vendor A, or a kit from a competitor B. For this, we take 20 mice and randomly select 10 of them for sample preparation with kit A, while the blood samples of the remaining 10 mice are prepared with kit B. The experiment is illustrated in Figure 1.1 A and the resulting data are given in Table 1.1 .

One option for comparing the two kits is to look at the difference in average enzyme levels, and we find an average level of 10.32 for vendor A and 10.66 for vendor B. We would like to interpret their difference of -0.34 as the difference due to the two preparation kits and conclude whether the two kits give equal results or if measurements based on one kit are systematically different from those based on the other kit.

Such interpretation, however, is only valid if the two groups of mice and their measurements are identical in all aspects except the sample preparation kit. If we use one strain of mice for kit A and another strain for kit B, any difference might also be attributed to inherent differences between the strains. Similarly, if the measurements using kit B were conducted much later than those using kit A, any observed difference might be attributed to changes in, e.g., mice selected, batches of chemicals used, device calibration, or any number of other influences. None of these competing explanations for an observed difference can be excluded from the given data alone, but good experimental design allows us to render them (almost) arbitrarily implausible.

A second aspect for our analysis is the inherent uncertainty in our calculated difference: if we repeat the experiment, the observed difference will change each time, and this will be more pronounced for a smaller number of mice, among others. If we do not use a sufficient number of mice in our experiment, the uncertainty associated with the observed difference might be too large, such that random fluctuations become a plausible explanation for the observed difference. Systematic differences between the two kits, of practically relevant magnitude in either direction, might then be compatible with the data, and we can draw no reliable conclusions from our experiment.

In each case, the statistical analysis—no matter how clever—was doomed before the experiment was even started, while simple ideas from statistical design of experiments would have provided correct and robust results with interpretable conclusions.

1.3 The Language of Experimental Design

By an experiment we understand an investigation where the researcher has full control over selecting and altering the experimental conditions of interest, and we only consider investigations of this type. The selected experimental conditions are called treatments . An experiment is comparative if the responses to several treatments are to be compared or contrasted. The experimental units are the smallest subdivision of the experimental material to which a treatment can be assigned. All experimental units given the same treatment constitute a treatment group . Especially in biology, we often compare treatments to a control group to which some standard experimental conditions are applied; a typical example is using a placebo for the control group, and different drugs for the other treatment groups.

The values observed are called responses and are measured on the response units ; these are often identical to the experimental units but need not be. Multiple experimental units are sometimes combined into groupings or blocks , such as mice grouped by litter, or samples grouped by batches of chemicals used for their preparation. More generally, we call any grouping of the experimental material (even with group size one) a unit .

In our example, we selected the mice, used a single sample per mouse, deliberately chose the two specific vendors, and had full control over which kit to assign to which mouse. In other words, the two kits are the treatments and the mice are the experimental units. We took the measured enzyme level of a single sample from a mouse as our response, and samples are therefore the response units. The resulting experiment is comparative, because we contrast the enzyme levels between the two treatment groups.

Figure 1.1: Three designs to determine the difference between two preparation kits A and B based on four mice. A: One sample per mouse. Comparison between averages of samples with same kit. B: Two samples per mouse treated with the same kit. Comparison between averages of mice with same kit requires averaging responses for each mouse first. C: Two samples per mouse each treated with different kit. Comparison between two samples of each mouse, with differences averaged.

In this example, we can coalesce experimental and response units, because we have a single response per mouse and cannot distinguish a sample from a mouse in the analysis, as illustrated in Figure 1.1 A for four mice. Responses from mice with the same kit are averaged, and the kit difference is the difference between these two averages.

By contrast, if we take two samples per mouse and use the same kit for both samples, then the mice are still the experimental units, but each mouse now groups the two response units associated with it. Now, responses from the same mouse are first averaged, and these averages are used to calculate the difference between kits; even though eight measurements are available, this difference is still based on only four mice (Figure 1.1 B).

If we take two samples per mouse, but apply each kit to one of the two samples, then the samples are both the experimental and response units, while the mice are blocks that group the samples. Now, we calculate the difference between kits for each mouse, and then average these differences (Figure 1.1 C).

If we only use one kit and determine the average enzyme level, then this investigation is still an experiment, but is not comparative.

To summarize, the design of an experiment determines the logical structure of the experiment ; it consists of (i) a set of treatments (the two kits); (ii) a specification of the experimental units (animals, cell lines, samples) (the mice in Figure 1.1 A,B and the samples in Figure 1.1 C); (iii) a procedure for assigning treatments to units; and (iv) a specification of the response units and the quantity to be measured as a response (the samples and associated enzyme levels).

1.4 Experiment Validity

Before we embark on the more technical aspects of experimental design, we discuss three components for evaluating an experiment’s validity: construct validity , internal validity , and external validity . These criteria are well-established in areas such as educational and psychological research, and have more recently been discussed for animal research ( Würbel 2017 ) where experiments are increasingly scrutinized for their scientific rationale and their design and intended analyses.

1.4.1 Construct Validity

Construct validity concerns the choice of the experimental system for answering our research question. Is the system even capable of providing a relevant answer to the question?

Studying the mechanisms of a particular disease, for example, might require careful choice of an appropriate animal model that shows a disease phenotype and is accessible to experimental interventions. If the animal model is a proxy for drug development for humans, biological mechanisms must be sufficiently similar between animal and human physiologies.

Another important aspect of the construct is the quantity that we intend to measure (the measurand ), and its relation to the quantity or property we are interested in. For example, we might measure the concentration of the same chemical compound once in a blood sample and once in a highly purified sample, and these constitute two different measurands, whose values might not be comparable. Often, the quantity of interest (e.g., liver function) is not directly measurable (or even quantifiable) and we measure a biomarker instead. For example, pre-clinical and clinical investigations may use concentrations of proteins or counts of specific cell types from blood samples, such as the CD4+ cell count used as a biomarker for immune system function.

1.4.2 Internal Validity

The internal validity of an experiment concerns the soundness of the scientific rationale, statistical properties such as precision of estimates, and the measures taken against risk of bias. It refers to the validity of claims within the context of the experiment. Statistical design of experiments plays a prominent role in ensuring internal validity, and we briefly discuss the main ideas before providing the technical details and an application to our example in the subsequent sections.

Scientific Rationale and Research Question

The scientific rationale of a study is (usually) not immediately a statistical question. Translating a scientific question into a quantitative comparison amenable to statistical analysis is no small task and often requires careful consideration. It is a substantial, if non-statistical, benefit of using experimental design that we are forced to formulate a precise-enough research question and decide on the main analyses required for answering it before we conduct the experiment. For example, the question: is there a difference between placebo and drug? is insufficiently precise for planning a statistical analysis and determine an adequate experimental design. What exactly is the drug treatment? What should the drug’s concentration be and how is it administered? How do we make sure that the placebo group is comparable to the drug group in all other aspects? What do we measure and what do we mean by “difference?” A shift in average response, a fold-change, change in response before and after treatment?

The scientific rationale also enters the choice of a potential control group to which we compare responses. The quote

The deep, fundamental question in statistical analysis is ‘Compared to what?’ ( Tufte 1997 )

highlights the importance of this choice.

There are almost never enough resources to answer all relevant scientific questions. We therefore define a few questions of highest interest, and the main purpose of the experiment is answering these questions in the primary analysis . This intended analysis drives the experimental design to ensure relevant estimates can be calculated and have sufficient precision, and tests are adequately powered. This does not preclude us from conducting additional secondary analyses and exploratory analyses , but we are not willing to enlarge the experiment to ensure that strong conclusions can also be drawn from these analyses.

Risk of Bias

Experimental bias is a systematic difference in response between experimental units in addition to the difference caused by the treatments. The experimental units in the different groups are then not equal in all aspects other than the treatment applied to them. We saw several examples in Section 1.2 .

Minimizing the risk of bias is crucial for internal validity and we look at some common measures to eliminate or reduce different types of bias in Section 1.5 .

Precision and Effect Size

Another aspect of internal validity is the precision of estimates and the expected effect sizes. Is the experimental setup, in principle, able to detect a difference of relevant magnitude? Experimental design offers several methods for answering this question based on the expected heterogeneity of samples, the measurement error, and other sources of variation: power analysis is a technique for determining the number of samples required to reliably detect a relevant effect size and provide estimates of sufficient precision. More samples yield more precision and more power, but we have to be careful that replication is done at the right level: simply measuring a biological sample multiple times as in Figure 1.1 B yields more measured values, but is pseudo-replication for analyses. Replication should also ensure that the statistical uncertainties of estimates can be gauged from the data of the experiment itself, without additional untestable assumptions. Finally, the technique of blocking , shown in Figure 1.1 C, can remove a substantial proportion of the variation and thereby increase power and precision if we find a way to apply it.

1.4.3 External Validity

The external validity of an experiment concerns its replicability and the generalizability of inferences. An experiment is replicable if its results can be confirmed by an independent new experiment, preferably by a different lab and researcher. Experimental conditions in the replicate experiment usually differ from the original experiment, which provides evidence that the observed effects are robust to such changes. A much weaker condition on an experiment is reproducibility , the property that an independent researcher draws equivalent conclusions based on the data from this particular experiment, using the same analysis techniques. Reproducibility requires publishing the raw data, details on the experimental protocol, and a description of the statistical analyses, preferably with accompanying source code. Many scientific journals subscribe to reporting guidelines to ensure reproducibility and these are also helpful for planning an experiment.

A main threat to replicability and generalizability are too tightly controlled experimental conditions, when inferences only hold for a specific lab under the very specific conditions of the original experiment. Introducing systematic heterogeneity and using multi-center studies effectively broadens the experimental conditions and therefore the inferences for which internal validity is available.

For systematic heterogeneity , experimental conditions are systematically altered in addition to the treatments, and treatment differences estimated for each condition. For example, we might split the experimental material into several batches and use a different day of analysis, sample preparation, batch of buffer, measurement device, and lab technician for each batch. A more general inference is then possible if effect size, effect direction, and precision are comparable between the batches, indicating that the treatment differences are stable over the different conditions.

In multi-center experiments , the same experiment is conducted in several different labs and the results compared and merged. Multi-center approaches are very common in clinical trials and often necessary to reach the required number of patient enrollments.

Generalizability of randomized controlled trials in medicine and animal studies can suffer from overly restrictive eligibility criteria. In clinical trials, patients are often included or excluded based on co-medications and co-morbidities, and the resulting sample of eligible patients might no longer be representative of the patient population. For example, Travers et al. ( 2007 ) used the eligibility criteria of 17 random controlled trials of asthma treatments and found that out of 749 patients, only a median of 6% (45 patients) would be eligible for an asthma-related randomized controlled trial. This puts a question mark on the relevance of the trials’ findings for asthma patients in general.

1.5 Reducing the Risk of Bias

1.5.1 randomization of treatment allocation.

If systematic differences other than the treatment exist between our treatment groups, then the effect of the treatment is confounded with these other differences and our estimates of treatment effects might be biased.

We remove such unwanted systematic differences from our treatment comparisons by randomizing the allocation of treatments to experimental units. In a completely randomized design , each experimental unit has the same chance of being subjected to any of the treatments, and any differences between the experimental units other than the treatments are distributed over the treatment groups. Importantly, randomization is the only method that also protects our experiment against unknown sources of bias: we do not need to know all or even any of the potential differences and yet their impact is eliminated from the treatment comparisons by random treatment allocation.

Randomization has two effects: (i) differences unrelated to treatment become part of the ‘statistical noise’ rendering the treatment groups more similar; and (ii) the systematic differences are thereby eliminated as sources of bias from the treatment comparison.

Randomization transforms systematic variation into random variation.

In our example, a proper randomization would select 10 out of our 20 mice fully at random, such that the probability of any one mouse being picked is 1/20. These ten mice are then assigned to kit A, and the remaining mice to kit B. This allocation is entirely independent of the treatments and of any properties of the mice.

To ensure random treatment allocation, some kind of random process needs to be employed. This can be as simple as shuffling a pack of 10 red and 10 black cards or using a software-based random number generator. Randomization is slightly more difficult if the number of experimental units is not known at the start of the experiment, such as when patients are recruited for an ongoing clinical trial (sometimes called rolling recruitment ), and we want to have reasonable balance between the treatment groups at each stage of the trial.

Seemingly random assignments “by hand” are usually no less complicated than fully random assignments, but are always inferior. If surprising results ensue from the experiment, such assignments are subject to unanswerable criticism and suspicion of unwanted bias. Even worse are systematic allocations; they can only remove bias from known causes, and immediately raise red flags under the slightest scrutiny.

The Problem of Undesired Assignments

Even with a fully random treatment allocation procedure, we might end up with an undesirable allocation. For our example, the treatment group of kit A might—just by chance—contain mice that are all bigger or more active than those in the other treatment group. Statistical orthodoxy recommends using the design nevertheless, because only full randomization guarantees valid estimates of residual variance and unbiased estimates of effects. This argument, however, concerns the long-run properties of the procedure and seems of little help in this specific situation. Why should we care if the randomization yields correct estimates under replication of the experiment, if the particular experiment is jeopardized?

Another solution is to create a list of all possible allocations that we would accept and randomly choose one of these allocations for our experiment. The analysis should then reflect this restriction in the possible randomizations, which often renders this approach difficult to implement.

The most pragmatic method is to reject highly undesirable designs and compute a new randomization ( Cox 1958 ) . Undesirable allocations are unlikely to arise for large sample sizes, and we might accept a small bias in estimation for small sample sizes, when uncertainty in the estimated treatment effect is already high. In this approach, whenever we reject a particular outcome, we must also be willing to reject the outcome if we permute the treatment level labels. If we reject eight big and two small mice for kit A, then we must also reject two big and eight small mice. We must also be transparent and report a rejected allocation, so that critics may come to their own conclusions about potential biases and their remedies.

1.5.2 Blinding

Bias in treatment comparisons is also introduced if treatment allocation is random, but responses cannot be measured entirely objectively, or if knowledge of the assigned treatment affects the response. In clinical trials, for example, patients might react differently when they know to be on a placebo treatment, an effect known as cognitive bias . In animal experiments, caretakers might report more abnormal behavior for animals on a more severe treatment. Cognitive bias can be eliminated by concealing the treatment allocation from technicians or participants of a clinical trial, a technique called single-blinding .

If response measures are partially based on professional judgement (such as a clinical scale), patient or physician might unconsciously report lower scores for a placebo treatment, a phenomenon known as observer bias . Its removal requires double blinding , where treatment allocations are additionally concealed from the experimentalist.

Blinding requires randomized treatment allocation to begin with and substantial effort might be needed to implement it. Drug companies, for example, have to go to great lengths to ensure that a placebo looks, tastes, and feels similar enough to the actual drug. Additionally, blinding is often done by coding the treatment conditions and samples, and effect sizes and statistical significance are calculated before the code is revealed.

In clinical trials, double-blinding creates a conflict of interest. The attending physicians do not know which patient received which treatment, and thus accumulation of side-effects cannot be linked to any treatment. For this reason, clinical trials have a data monitoring committee not involved in the final analysis, that performs intermediate analyses of efficacy and safety at predefined intervals. If severe problems are detected, the committee might recommend altering or aborting the trial. The same might happen if one treatment already shows overwhelming evidence of superiority, such that it becomes unethical to withhold this treatment from the other patients.

1.5.3 Analysis Plan and Registration

An often overlooked source of bias has been termed the researcher degrees of freedom or garden of forking paths in the data analysis. For any set of data, there are many different options for its analysis: some results might be considered outliers and discarded, assumptions are made on error distributions and appropriate test statistics, different covariates might be included into a regression model. Often, multiple hypotheses are investigated and tested, and analyses are done separately on various (overlapping) subgroups. Hypotheses formed after looking at the data require additional care in their interpretation; almost never will \(p\) -values for these ad hoc or post hoc hypotheses be statistically justifiable. Many different measured response variables invite fishing expeditions , where patterns in the data are sought without an underlying hypothesis. Only reporting those sub-analyses that gave ‘interesting’ findings invariably leads to biased conclusions and is called cherry-picking or \(p\) -hacking (or much less flattering names).

The statistical analysis is always part of a larger scientific argument and we should consider the necessary computations in relation to building our scientific argument about the interpretation of the data. In addition to the statistical calculations, this interpretation requires substantial subject-matter knowledge and includes (many) non-statistical arguments. Two quotes highlight that experiment and analysis are a means to an end and not the end in itself.

There is a boundary in data interpretation beyond which formulas and quantitative decision procedures do not go, where judgment and style enter. ( Abelson 1995 )

Often, perfectly reasonable people come to perfectly reasonable decisions or conclusions based on nonstatistical evidence. Statistical analysis is a tool with which we support reasoning. It is not a goal in itself. ( Bailar III 1981 )

There is often a grey area between exploiting researcher degrees of freedom to arrive at a desired conclusion, and creative yet informed analyses of data. One way to navigate this area is to distinguish between exploratory studies and confirmatory studies . The former have no clearly stated scientific question, but are used to generate interesting hypotheses by identifying potential associations or effects that are then further investigated. Conclusions from these studies are very tentative and must be reported honestly as such. In contrast, standards are much higher for confirmatory studies, which investigate a specific predefined scientific question. Analysis plans and pre-registration of an experiment are accepted means for demonstrating lack of bias due to researcher degrees of freedom, and separating primary from secondary analyses allows emphasizing the main goals of the study.

Analysis Plan

The analysis plan is written before conducting the experiment and details the measurands and estimands, the hypotheses to be tested together with a power and sample size calculation, a discussion of relevant effect sizes, detection and handling of outliers and missing data, as well as steps for data normalization such as transformations and baseline corrections. If a regression model is required, its factors and covariates are outlined. Particularly in biology, handling measurements below the limit of quantification and saturation effects require careful consideration.

In the context of clinical trials, the problem of estimands has become a recent focus of attention. An estimand is the target of a statistical estimation procedure, for example the true average difference in enzyme levels between the two preparation kits. A main problem in many studies are post-randomization events that can change the estimand, even if the estimation procedure remains the same. For example, if kit B fails to produce usable samples for measurement in five out of ten cases because the enzyme level was too low, while kit A could handle these enzyme levels perfectly fine, then this might severely exaggerate the observed difference between the two kits. Similar problems arise in drug trials, when some patients stop taking one of the drugs due to side-effects or other complications.

Registration

Registration of experiments is an even more severe measure used in conjunction with an analysis plan and is becoming standard in clinical trials. Here, information about the trial, including the analysis plan, procedure to recruit patients, and stopping criteria, are registered in a public database. Publications based on the trial then refer to this registration, such that reviewers and readers can compare what the researchers intended to do and what they actually did. Similar portals for pre-clinical and translational research are also available.

1.6 Notes and Summary

The problem of measurements and measurands is further discussed for statistics in Hand ( 1996 ) and specifically for biological experiments in Coxon, Longstaff, and Burns ( 2019 ) . A general review of methods for handling missing data is Dong and Peng ( 2013 ) . The different roles of randomization are emphasized in Cox ( 2009 ) .

Two well-known reporting guidelines are the ARRIVE guidelines for animal research ( Kilkenny et al. 2010 ) and the CONSORT guidelines for clinical trials ( Moher et al. 2010 ) . Guidelines describing the minimal information required for reproducing experimental results have been developed for many types of experimental techniques, including microarrays (MIAME), RNA sequencing (MINSEQE), metabolomics (MSI) and proteomics (MIAPE) experiments; the FAIRSHARE initiative provides a more comprehensive collection ( Sansone et al. 2019 ) .

The problems of experimental design in animal experiments and particularly translation research are discussed in Couzin-Frankel ( 2013 ) . Multi-center studies are now considered for these investigations, and using a second laboratory already increases reproducibility substantially ( Richter et al. 2010 ; Richter 2017 ; Voelkl et al. 2018 ; Karp 2018 ) and allows standardizing the treatment effects ( Kafkafi et al. 2017 ) . First attempts are reported of using designs similar to clinical trials ( Llovera and Liesz 2016 ) . Exploratory-confirmatory research and external validity for animal studies is discussed in Kimmelman, Mogil, and Dirnagl ( 2014 ) and Pound and Ritskes-Hoitinga ( 2018 ) . Further information on pilot studies is found in Moore et al. ( 2011 ) , Sim ( 2019 ) , and Thabane et al. ( 2010 ) .

The deliberate use of statistical analyses and their interpretation for supporting a larger argument was called statistics as principled argument ( Abelson 1995 ) . Employing useless statistical analysis without reference to the actual scientific question is surrogate science ( Gigerenzer and Marewski 2014 ) and adaptive thinking is integral to meaningful statistical analysis ( Gigerenzer 2002 ) .

In an experiment, the investigator has full control over the experimental conditions applied to the experiment material. The experimental design gives the logical structure of an experiment: the units describing the organization of the experimental material, the treatments and their allocation to units, and the response. Statistical design of experiments includes techniques to ensure internal validity of an experiment, and methods to make inference from experimental data efficient.

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• Explanatory vs Response Variables | Definitions & Examples

Explanatory vs Response Variables | Definitions & Examples

Published on 4 May 2022 by Pritha Bhandari .

In research, you often investigate causal relationships between variables using experiments or observations. For example, you might test whether caffeine improves speed by providing participants with different doses of caffeine and then comparing their reaction times.

• An explanatory variable is what you manipulate or observe changes in (e.g., caffeine dose).
• A response variable is what changes as a result (e.g., reaction times).

The words ‘explanatory variable’ and ‘response variable’ are often interchangeable with other terms used in research.

Table of contents

Explanatory vs response variables, explanatory vs independent variables, visualising explanatory and response variables, frequently asked questions about explanatory and response variables.

The difference between explanatory and response variables is simple:

• An explanatory variable is the expected cause, and it explains the results.
• A response variable is the expected effect, and it responds to explanatory variables.

You expect changes in the response variable to happen only after changes in an explanatory variable.

There’s a causal relationship between the variables that may be indirect or direct. In an indirect relationship, an explanatory variable may act on a response variable through a mediator .

If you’re dealing with a purely correlational relationship, there are no explanatory and response variables. Even if changes in one variable are associated with changes in another, both might be caused by a confounding variable .

Examples of explanatory and response variables

In some studies, you’ll have only one explanatory variable and one response variable, but in more complicated research, you may predict one or more response variable(s) using several explanatory variables in a model.

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Explanatory variables and independent variables are very similar, but there are subtle differences between them.

In research contexts, independent variables supposedly aren’t affected by or dependent on any other variable – they’re manipulated or altered only by researchers. For example, if you run a controlled experiment where you can control exactly how much caffeine each participant receives, then caffeine dose is an independent variable.

But sometimes, the term ‘explanatory variable’ is preferred over ‘independent variable’, because in real-world contexts, independent variables are often influenced by other variables. That means they’re not truly independent.

You gather a sample of young adults and ask them to complete a survey in the lab. They report their risk perceptions of different threatening scenarios while you record their stress reactions physiologically.

In your analyses, you find that gender and risk perception are highly correlated with each other. Women are likely to rate situations as riskier than men.

You’ll often see the terms ‘explanatory variable’ and ‘response variable’ used in regression analyses , which focus on predicting or accounting for changes in response variables as a result of explanatory variables.

The easiest way to visualise the relationship between an explanatory variable and a response variable is with a graph.

On graphs, the explanatory variable is conventionally placed on the x -axis, while the response variable is placed on the y -axis.

• If you have quantitative variables , use a scatterplot or a line graph.
• If your response variable is categorical, use a scatterplot or a line graph.
• If your explanatory variable is categorical, use a bar graph.

When you have only one explanatory variable and one response variable, you’ll collect paired data . This means every response variable measurement is linked to an explanatory variable value for each unit or participant.

• Your explanatory variable is academic motivation at the start of the academic year.
• Your response variable is the grade point average (GPA) at the end of the academic year.

Academic motivation is assessed using an 8-point scale, while GPA can range from 0 to 4. To visualise your data, you plot academic motivation at the start of the year on the x -axis and GPA at the end of the year on the y -axis. Each data point reflects the paired data of one participant.

From the scatterplot, you can see a clear explanatory relationship between academic motivation at the start of the year and GPA at the end of the year.

• A response variable is the expected effect, and it responds to other variables.

The term ‘ explanatory variable ‘ is sometimes preferred over ‘ independent variable ‘ because, in real-world contexts, independent variables are often influenced by other variables. This means they aren’t totally independent.

Multiple independent variables may also be correlated with each other, so ‘explanatory variables’ is a more appropriate term.

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• Published: 29 July 2024

Early-stage recovery of lithium from spent batteries via CO 2 -assisted leaching optimized by response surface methodology

• Ksenija Milicevic Neumann   ORCID: orcid.org/0000-0001-8645-0915 1 ,
• Muhammad Ans 1 &
• Bernd Friedrich 1  

Scientific Reports volume  14 , Article number:  17369 ( 2024 ) Cite this article

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Recycling lithium (Li) from spent lithium-ion batteries (LIBs) due to the depletion of natural resources and potential toxicity is becoming a progressively favourable measure to realize green sustainability. Presently, the prevalent recycling technique relying on pyrometallurgy lacks the capability to extract lithium. Meanwhile, conventional hydrometallurgical processes frequently employ robust acidic solutions like sulfuric acid and precipitation agents such as sodium carbonate. Unfortunately, this approach tends to result in the extraction of lithium at the end of a lengthy process chain, leading to associated losses and creating challenges in managing complex waste. This study addresses a cost-effective and environmentally friendly early-stage lithium recovery from the thermally conditioned black mass. In this sense, a thermally conditioned black mass is subjected to the carbonization process in a water solution to transform the water-insoluble Li phase into soluble lithium bicarbonate (LiHCO 3 ) and carbonate (Li 2 CO 3 ) facilitating its selective separation from other elements. Response surface methodology (RSM)—a statistical tool integrated with central composite design (CCD) is employed to optimize the parameters for Li recovery. Temperature, solid–liquid (S/L) ratio, leaching time and CO 2 flow rate are considered as variable factors in modelling the optimum recycling process. A quadratic regression model is developed for Li recovery and based on ANOVA analysis, (S/L) ratio, temperature and time are identified as statistically significant factors. Experimental results demonstrate a maximum leaching efficiency of lithium with optimized parameter set, achieving a recovery rate of 97.18% with a fit response of 93.54%.

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Leaching of NMC industrial black mass in the presence of LFP

A closer look at lithium-ion batteries in E-waste and the potential for a universal hydrometallurgical recycling process

Sustainable regeneration of spent cathodes for lithium-ion and post-lithium-ion batteries

Introduction.

Presently, the demand for lithium-ion batteries (LIBs) as electrochemical power sources, driven by their widespread use in electric vehicles, mobile and smartphones, and other applications due to long-life cycles, high energy density, and low self-discharge is higher than ever before. Considering the global market growth of LIB products, it is expected that a large number of spent LIBs will be increased 1 , 2 , 3 , 4 , 5 . For instance, cylindrical (NMC-18650) and prismatic cells are the most popular types for electronic and automotive applications 6 . However, spent LIBs pose severely human health and environmental risks due to the presence of various organic chemicals and heavy metals 7 , 8 , 9 , 10 . Despite, spent LIBs contain valuable metals, such as nickel (Ni), cobalt (Co), and lithium (Li), underscoring the high economic value 11 , 12 . Therefore, recycling and treatment of spent LIBs have become dominant and imperative from the viewpoint of ecological protection and resource preservation.

The recycling of Li garners significant attention due to the substantial environmental impact associated with primary production from natural resources 13 and potential supply risk 14 . To date, the exploration of metallurgical Li recovery methods from spent LIBs have covered both pyrometallurgical and hydrometallurgical techniques and their combination 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 . However, the pyrometallurgical routes exhibits notable drawbacks, such as the emission of toxic gasses that contributes to air pollution, high energy consumption, and significant Li losses as it becomes part of the slag system and flue dust 23 .

The conventional pyrometallurgical recycling process for lithium-ion batteries entails subjecting them to high-temperature smelting, resulting in the recovery of nickel, cobalt, and copper in the form of an alloy. However, crucial battery materials such as lithium, aluminum, and iron become constituents of a generated slag, rendering their further extraction economically unviable. Notably, the combustion of graphite from the anode during this process contributes to carbon dioxide emissions. Subsequent to the smelting phase, the alloy derived necessitates additional hydrometallurgical processing involving multiple steps to recuperate salts suitable for reuse in battery production. Therefore, hydrometallurgy emerges as a viable alternative, gaining extensive traction in both industrial applications and academic research due to its high metal recovery efficiency and cost-effectiveness. Before hydrometallurgical treatment, lithium-ion batteries are mechanically shredded, electrolyte evaporated, plastic and metallic housing material separated by diverse screening methods. The most critical elements (lithium, nickel, cobalt and graphite) coming from cathode and anode are obtained in form of black powder called black mass. Traditional hydrometallurgical methods for processing of black mass and recovery of these critical materials often rely on strong acids, posing severe environmental and human health issues. The most common processing route of black mass to recover lithium goes through multiple stages starting from dissolution of all metals with sulphuric acid and removal of graphite. The obtained solution is further undergoing the processing via precipitation, cementation and other methods to remove the impurities—aluminium, iron and copper. Such a purified sulphuric solution containing nickel, cobalt, manganese and lithium is entering the solvent extraction method, in which after multiple stage extraction, nickel, cobalt and manganese are recovered. Lithium is recovered at the end from the leftover solution by precipitation adding sodium carbonate to form lithium carbonate and waste stream of sodium sulphate. Owing to the extensive processing pathway, lithium is undergoing substantial losses, resulting in the generation of multiple waste streams 24 , 25

As an alternative, methods like the carbonation process have been explored to convert insoluble solid Li-compounds into H 2 O-soluble compounds, mitigating these environmental and economic concerns 26 , 27 , 28 . This process was established based on the kinetic reactions of spent electrodes with CO 2 solution by means of ion exchange extraction under different conditions. The dissolution rate of lithium carbonate (Li 2 CO 3 ) exponentially increased with increasing CO 2 flow rate, with key reactions during the process are expressed in Eqs. ( 1 – 4 ), followed by the precipitation of Li 2 CO 3 through solution heating 28 , 29 .

Schwich et al. 18 investigated an effective eco-friendly “Early-Stage Lithium Recovery” (ESLR) method involving Li leaching through carbonation with supercritical CO 2 in a cost-intensive autoclave process, achieving an efficiency of 79%. In contrast, developing a cost-efficient carbonation process under atmospheric pressure possess a high potential in tackling the drawbacks of this approach. So far, no systematic research has explored the optimal parameters, such as, leaching time, temperature and S/L ratio in CO 2 -assisted hydrometallurgy under atmospheric pressure, particularly using the statistical design of experiments for selectively Li recovery.

Response surface methodology (RSM) is a statistical technique used for collecting functional relationship between influential factors and adequate response. It establishes predict response values of multivariant experimental design to determine process optimization. The RSM evaluates an appropriate operating condition which is significantly reported in the literature 30 , 31 . Furthermore, due to diagnostic or screening studies to locate ideal settings in the experimental design, this modelling approach is suitable for implementing the quadratic polynomial model 32 . Therefore, RSM's analytical and experimental processes are typically more advanced and modern than any other modelling technique 33 , 34 .

Considering process-related challenges posed by conventional recovery methods, specifically extraction of Li, this study proposed an early-stage Li recovery process from spent LIBs using environmentally friendly green hydrometallurgy i.e., CO 2 -H 2 O leaching under atmospheric pressure. It comprises low-cost and eco-friendly carbonation processes at the early stages of process chain, diverging from traditional acidic leaching or smelting 15 , 24 , 25 . Using RSM, a statistical modelling technique is employed to assess the efficiency of lithium leaching and validate operational parameters, aiming to minimize losses typically encountered during the multi-step precipitation chain for separating Cu, Fe/Al, Mn and Ni/Co. Quadratic regression models are employed for each variable involved in the leaching process, enabling the exploration of interaction effects among several factors and the determination of significance/insignificance of various terms. Laboratory trials are conducted to verify these findings. The proposed Li recovery process outlined in this study is based on a fundamental structure, incorporating well-defined particle size distribution and scientifically optimized solid/liquid ratios. It serves as a benchmark in the field and has the potential to catalyze advancements in industry by reducing toxic waste and increasing Li recovery rates.

Experimental

Material preparation & leaching method.

Spent NMC cells have been pyrolyzed at 600 ℃ under vacuum. Subsequently, the material underwent shredding in a cutting mill and sorted to obtain black mass sieved to particles < 1 mm in size. These initial steps were conducted by an external company. The obtained black mass was further milled to reduce the particle grain size less than 63 µm, enhancing the interaction of liquid solution with solid during leaching and thus improving the Li recovery yield. A planetary ball mill ( PKM—Pulverisette 6, FRITSCH GmbH, Germany ) with a stainless steel vail (400 mL in volume) was used for the milling process. The stainless steel vail was filled with 2/3rd of powder and 14 balls of 20 mm diameter were introduced for the milling process. The rotational speed was set at 450 rpm and the black mass milled for 5 min, as reported in the previous literature 35 . Subsequently, dry sieve analysis was conducted using a sieving tower ( AS200, RETSCH GmbH, Germany ) to collect particles sized < 63 μm. During the sieve analysis, a frequency of 2.0 mm/g with an interval of 30 s was maintained for 2 min.

All leaching processes were conducted in deionized (DI) water with a continuous flow of CO 2 . The leaching setup consisted of a four-neck round-bottomed reactor of double-wall, connected to a heating bath circulation thermostat ( Huber CC-304B, Kältemaschinenbau AG, Germany ). The desired amount of black mass dissolved into a 1.5 L water solution and stirred at a constant rate of 350 rpm with a mechanical stirrer throughout all experiments. A CO 2 glass lance (Ø10 mm outer diameter) was immersed into the solution through the lid and a thermometer was used to measure the reactor's temperature. The entire setup is depicted in Fig.  1 . Following leaching, solid residue filtration was performed on a suction funnel using Macherey–Nagel MN-619 ¼ filter paper. The solid residue was collected and dried in a heating furnace at 80 ℃ for 12 h. Finally, the filtered Li solution was boiled to precipitate the Li-carbonate in a solid state. Before precipitation, samples of the leached solution were taken for chemical analyses to determine Li concentration, i.e., recovery rate.

Hydrometallurgical setup for carbonation process and lithium recovery from the black mass.

The concentration of Li in the solution sample has been determined by ion-selective electrode ( Mettler Toledo, DX207-Li) and further confirmed through the inductively coupled plasma optical emission spectroscopy (ICP-OES) method ( Ciros Vision, Spectro Analytical Instruments GmbH, Germany ). The lithium leaching efficiency (η Li ) has been calculated using the Eq. ( 5 ):

where c Li is the measured concentration of Li with ion selective electrode in obtained solution after leaching, V is volume of the reaction mixture, m BM is the mass of the input material (black mass), ν Li is the percentage of Li in input material (black mass) measured by ICP-OES method.

Optimization of leaching parameters by response surface methodology (RSM) and central composite design (CCD)

The aim of optimization is to maximize Li yield from the spent black mass while minimizing the experimental trials. To achieve this, we employed a statistical modelling tool (known as RSM) with a central composite design (CCD). This approach allows us to evaluate the uniform precision design within various parameters and reduce prediction errors. CCD is widely used to optimize variables based on multivariant nonlinear regression models derived from the appropriate experimental parameters. It enables the assessment of adequate operating conditions and facilitates the interactions of various parameters influencing the process 32 . The CCD technique acquires experimental values for fitting the model (a second-order model also known as the rotatable variance model). In our study, the set of four variable parameters are temperature (10–77 ℃), time (10–180 min), solid–liquid (S-L) ratio (10:1—70:1 g/L), and CO 2 flow (3–6 L/min) were considered in this analysis. The stirring rate and particle size remained constant, as these parameters were deemed to have a lower impact compared to others and were not varied significantly by other researchers 18 in the field, based on prior experience. The parameter ranges were selected based on consideration such as solubility of CO 2 and lithium carbonate, equipment capabilities and existing literature. Additionally, the Li-yield (η Li [%]) was chosen as the response variable. A two-level factorial design was employed to determine appropriate parametric conditions corresponding to maximum predicted values. The two-level factorial in the statistical modelling was achieved as 31 (= 2  k  + 2 k + 7), where k is the number of factors = 4, to ensure randomness and avoid biases. Table 1 shows the parametric levels with coded and uncoded values. To verify the reproducibility and reliability of the optimum parameters, an additional experiment was conducted using defined parameters to demonstrate Li yield via leaching. The coefficient correlation (R 2 ) values indicated the polynomial fit. The RMS and CCD technique were implemented using MINITAB 19.0 software, facilitating graphical analysis, desirability functions and optimizer plots.

Material characterization

The black mass used in this investigation originates from thermally and mechanically pre-treated NMC cells, which is further milled to particle size below 63 µm. Dynamic image analysis has been performed to determine the average particle size of the prepared input material ( QuickPick Oasis, Sympatec GmbH, Clausthal-Zellerfeld, Germany ). As shown in Fig.  2 , over 90.3% of particles were found to be smaller than 45.27 µm, with 50.3% measuring below 23.55 µm. This indicates that after ball milling and sieving processes, the desired particle fractions < 63 µm were successfully obtained. Also, the decision to use lower particle fraction was influenced by the noticeable presence of current collectors (Cu and Al foils) in coarser size, as shown in Fig.  3 . These foils pose a hinderance to Li extraction, underscoring the importance of selecting finer particle sizes for improved processing efficiency.

Density and distribution of particles size of the input material.

Black mass fractions: ( a ) > 500 µm, ( b ) 500 µm < x > 125 µm ( c ) < 63 µm.

The ICP-OES method determined the chemical composition and the percentage of elements in the input mixture (see Table 3 ). The fluorine contents are analysed by combustion-ion chromatography (CIC)—A1 combustion-IC, while the carbon contents are assessed by total carbon analysis (TC) via Analytic Jena multi N/C 2001 S instrument.

The ICP-OES analysis of the input material was performed twice on the residual Li samples, and the medium value of 2.67 wt.-% of Li was taken for yield calculations (Eq.  5 ). X-ray diffraction (XRD) was carried out to evaluate the crystalline phases of the powder samples. The crystalline phases (i.e., C, LiF, Ni , Mn 0.95 O, CoO) were identified when powder diffraction treated by HighScore Plus, Malvern Panalytical B.V

Results and discussion

Model development by central composite design (ccd) for the leaching process.

A total of 31 experimental trials of Li leaching (Table 2 ) were carried out using various regression models via CCD (MINITAB® software). Table 3 illustrates the analysis of Li (Y Li )—carried out with ICP-OES. The design of experiments (DOE) is essential for completing statistical analysis and verifying the correctness of variables (coded/uncoded). Therefore, each response was fitted by the second-order multivariable polynomial (as provided by the regression, Eq.  6 ) to showcase the accuracy and reliability of the results.

To further evaluate the development of the model for several indicator performances Eqs. ( 7 – 10 ) are employed 30 , 31

where, Y Li is the experimental response, N is the total number of trails. The low values of root mean sq. root (RMSE) (< 3.8) and mean absolute error (MAE) (< 3.0) indicate that the model’s predictions closely align with the experimental values.

The orthogonal design yields approximation model values, resulting in a less complex and reliable analysis. Figure  4 shows the regression plot of experimental data (vs. predicted response). The R 2 value of the experimental response (91.30%) exceeds that of the predicted value (75.47%) underscoring the importance of optimization. ANOVA analysis (at a 5% significance level) confirms the quadratic model's importance and identifies the effect of each parameter. The p- and F-values assess the significance of each coefficient in the parameter set for the correlations between the numerical values (see Table 4 ). These values assess the decision of the statistical model, providing crucial evidence against the null hypothesis. Notably, the p-values < 0.05 (except for CO 2 flow) indicate satisfactory model performance, as deviations from the null hypothesis are negligible in standard distribution data 36 .

Fits regression plot of experimental and predicted response (S = standard deviation, R-sq (adj) = adjusted R 2 , R-sq (pred) = predicted R 2 ).

Main effect plots containing experimentally fitted data are used to assess the yield of Li during the leaching process—see Fig.  5 . In Fig.  5 a, it is observed that leaching efficiency remains relatively unchanged at lower temperature ranges (10 to 45 ℃). However, a significance increase in Li efficiency is noted at higher temperatures (77 ℃). In the case of time, a slight increase of Li efficiency is observed (up to 95 min), followed by a plateau reached at 180 min, which persists until 265 min. Although CO 2 flow has a minimal impact on the leaching rate, a slight increase is observed at 6 L min -1 with time dependency. Notably, varying S-L ratios drastically enhance the leaching behaviour, with higher efficiency observed at a ratio of 10:1 g/L, which decreases as the ratio increases. Figure  5 b.shows the interaction plots depicting individual leaching efficiency as a function of responses. All interactions involving CO 2 flow are parallel with the x-axis confirming this parameter's lack of interaction effect with Li recovery efficiency. Other plots show varying degrees of interaction between the effects, with some showing nonlinear trends. The interaction effect of leaching temperature vs. time shows excessive tendency values at 77 °C and 180 min, respectively. A significantly higher response is observed for a lower S-L ratio in the interaction plot of S-L ratio vs. temperature and time. Based on the interaction graphs, maximum leaching rates are determined at a low S-L ratio, coupled with high temperature and time.

( a ) Main effect and ( b ) interaction plots for responses of optimum parameters.

Pareto and normal diagrams were designed for leaching efficiencies with all responses to support the validity and satisfactory approximation of the created model (see Fig.  6 .) A reference line (red line in Fig.  6 a) passes through the main effects with the highest magnitude starting with S-L ratio (C), then time (B) and temperature (C). However, this reference line does not intersect with the CO 2 flow (D) main effect, indicating that this variable may not significantly influence the leaching processes.

( a ) The Pareto and ( b ) Normal plot of the standardized effect of variables (temperature, time, S-L ratio, and CO 2 flow), as a function of efficiency response, α = 0.1.

The residuals of all variables are positioned near the diagonal line (Fig.  6 b), suggesting the normal data distribution function. The red squares represent the significant effects contributing to maximum efficiency, whereas the blue points express a non-significant effect. As can be seen, the S-L ratio exhibits a significant impact, positioned furthest from the 0 on the x-axis, confirming its prominence in the Pareto chart (Fig.  6 a). Its location on the negative side of the x-axis suggests that the response decreases with a shift from low to high values of this factor. On the other hand, time and temperature demonstrate a positive effect, indicating an increase in the response as a factor value transitions from low to high. Also, significant interactions between temperature and time (AB) and squared terms for S-L ratio (CC) and temperature (AA) are evident.

The reduction of the nominal terms in the regression Eq. ( 6 ) and the confidence level increase was tested to improve the model’s reliability. Despite these adjustments, no significant changes were revealed, confirming the high conformity of the applied model and obtained results.

Contour plots of lithium leaching

2D contour plots provide a comprehensive depiction of the relationship between central composite values' responses and the variables of the resulting models, as shown in Fig.  7 . The curved contours signify the inclusion of statistically significant quadratic terms in the model, as observed in previous sections. The contour plots are obtained only for central values (hold values in Fig.  7 ) and represent how the efficiency rate changes with variations in parameters. The bell shape (the core of the counterplots), enable the direct establishment of maximum efficiency between the time*temperature, S/L ratio*temperature, and S/L ratio*time can be predicted. According to Fig.  7 , temperature and S-L ratio are supreme factors in Li extraction. The Li recovery rates experience significant enhancement (from 50 to 80%) with prolonged time (> 180 min) and elevated temperature (> 60 ℃) at a minimum S-L (10:1). However, the influence of CO 2 flow*S-L ratio, CO 2 flow*time and CO 2 flow*temperature remain limited. These findings align with previous explanations, indicating that CO 2 flow and interactions of CO 2 flow with other variables do not exert a substantial influence the Li recovery rates.

Contour plots as a function of response.

Desirability function of lithium recovery

The contour plots could not determine the desired optimum conditions for the exact parameters. To address this, the desirability function plots were generated to fine-tune the appropriate model with optimal conditions. This approach effectively validated the model using various experimental variables to achieve the key trade-offs. Based on the multiple response and desirability function, the optimum parameters of maximum Li efficiency are defined as 77 ℃, 180 min, S-L ratio (10:1) g/L, and CO 2 flow rate of 6.0 L/min. The fitted result confirms that these values yield an approximate response of ~ 93.54%, as detailed in Table 5 .

The program combines individual desirability data into a single composite number, further maximizing the function. The composite desirability for Li leaching was defined as 0.92980, aiming for an efficiency of ~ 97% (Table 5 ), indicating the attainment of maximum yield. In Fig.  8 , the blue dotted lines represent the maximum efficiency obtained based on the optimum conditions (highlighted in red), which were used in the experimental trials.

Response optimization diagrams for leaching parameters.

The acceptable desirability reported by Resentera et al. 37 was ~ 0.95 for optimizing Li extraction at low temperatures. In our study, the achieved composite desirability values are ~ 0.93, validating our analytical approach and confirming the actual response. The critical phase reaction requires sufficient dispersion of S-L mass transfer to dissolve solid particles in a CO 2 -assisted water solution. Consequently, adequate time is necessary for achieving uniform dispersion and S-L equilibrium of the particles during leaching. This observation is further supported by the favourable effect of immersion duration on Li dissolution in H 2 O.

Effect of operating conditions as a function response

The optimum efficiency of Li recovery from NMC powder was determined at a S-L ratio of 10:1 (g L -1 ) by introducing carbonated water. As depicted in Fig.  5 , the behaviour of Li leaching interactions and main effect plots with optimized variables predicted by RSM and verified by lab-based experimental trials. The results show that the Li 2 CO 3 conversion improved with decreasing solid concentration from 70:1 (~ 75% leaching efficiency) to 10:1 (~ 97% leaching efficiency), taking the same temperature, retention time and gas flow parameters. The carbonation process, involving the conversion of Li + into the CO 2 -water, follows a typical noncatalytic three-phase reaction (gas–liquid-solid) 28 . Increasing solid concentration in the mixture leads to higher bulk diffusion resistance at the S-L boundary and liquid phase, hindering complete dissolution in the reaction 38 . Besides, higher solid concentration reduces the interfacial reaction area of the particles due to increased powder density and friction between solid phases 39 . Consequently, decreased mixing efficiency results in difficulty in gas–liquid mass transfer difficulty, slowing down the reaction and resulting to lower Li + conversion. This suggests that an excess of H 2 O is required for high Li + dissolution in the system.

The investigation into the effect of temperature and time on Li recycling revealed a notable enhancement in efficiency with varying temperature ranges, as shown in Fig.  5 . Specifically, the result indicated that the Li recovery rate increased from 69 to 97% by increasing the temperature from 10 ℃ to 77 ℃ for 180 min. This observation was made with a S/L ratio of 10:1 and a CO 2 flow of 6 l/min. Clearly, the higher temperature ranges are more favourable for Li + dissolution. The literature shows diverse findings on this subject, Yi et. al. 24 reported better efficiencies at lower temperatures, Zhang et. al. 27 observed a slight increase in efficiency with rising temperatures, while Makuza et. al. 40 noted an increase in efficiency rates with higher leaching temperatures. Although the solubility of Li 2 CO 3 and CO 2 generally decreases with temperature 41 , 41 . Makuza’s 40 hypothesis suggests that higher temperatures lead to enhancement of reaction kinetics. Also, higher CO 2 concentration at lower temperatures enforces reaction by increasing the pH values of the solution. However, its important to note that a lower pH value of the solution can result in the simultaneous dissolution of other heavy metals, consequently leading to lower Li recovery efficiency. Indeed, our experiments analyzed higher concentrations of other metals (such as Mn, Ni, Co etc.,) in leached solution at lower temperatures, thus confirming these findings.

Furthermore, achieving optimal S-L mass transfer and phase dispersion during the carbonation reaction necessitates sufficient time. Specifically, the system must reach an equilibrium where particles are uniformly distributed, and S-L equilibrium is established before carbonation. In our research, it was observed that efficient carbonation was achieved at 180 min, indicating a positive impact on the dissolution of Li 2 CO 3 .

The effect of the CO 2 flow rate is shown in Fig.  5 a. It is evident that increasing the CO 2 flow does not exhibit a clear trend or significant impact on the leaching rate. However, according to the optimisation method, the higher CO 2 flow rates resulted in increased Li concentration in the water solution. This phenomenon aligns with the findings of Yi et. al. 43 and is contributed to the augmentation of the mass transfer volumetric coefficient in the gas–liquid phases. Despite the statistical evaluation not showing a straightforward tendency, the higher CO 2 flow rate is beneficial in increasing the stirring effect and, herewith better interaction between gas–liquid-solid phases.

Confirmation of reproducibility of experiments

To validate the obtained experimental results, the trial with the highest Li recovery efficiency (77 ℃, 180 min. 10:1 g/L, 6 l/min) was repeated 5 times, yielding an average efficiency value of 94.73%. This value closely aligns with both our experimental and statistically optimized efficiency rates. Also, reference tests were conducted using the same optimal parameters (77 ℃, 180 min. 10:1 g/L) but without CO 2 injection, repeated multiple times, with an average value of 59.08%. This further supports the effectiveness of the carbonation process, which improves the Li leaching efficiency by 35.65%. Furthermore, the Li concentration in solution after the carbonation process was analyzed by two analytical methods, (1) ion selective electrode (ISE) and (2) ICP-OES. The values obtained from both analyses are plotted against each other in Fig.  9 . The Pearson coefficient R is 0.93875, indicating a strong linear correlation between the values obtained from both analyses.

Lithium concentration measured by ISE and ICP-OES methods.

This study provides a novel approach to achieve high-efficiency recovery of spent LIBs using environmentally friendly and acid-free hydrometallurgical methods. We investigated the effect of temperature, time, S-L ratio, and CO 2 flow on effective lithium recovery from spent batteries, using modelling techniques such as RSM and CCD. Experimental findings aligned closely with the predicted values obtained by quadratic and statistical models. Temperature, time and S-L ratio emerged as the significant factors in ANOVA analysis, contributing prominently to achieving the highest Li leaching efficiency, whereas the impact of CO 2 flow rate on Li recovery was comparatively less pronounced. The optimized conditions yielding the highest Li yield of 97.18% and a response fit of 93.54% were determined at 77 ℃, 180 min, S-L ratio (10:1) g/L, and CO 2 6.0 L min -1 . The significant value of composite desirability (0.92980) of the predicted model suggests the reliability and precision of the optimum combination for all responses. Further investigation will explore the feasibility of translating these laboratory-scale results into the semi-pilot scale operations, including considerations for a reactor volume of 100 L.

Data availability

Data generated or analysed during this study are mainly included in this published article. Any additional data required will be provided upon request by corresponding author.

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The project on which this report/ publication is based was funded by the German Federal Ministry of Education and Research within the Competence Cluster Recycling & Green Battery (greenBatt) under the grant number 03XP0332C. The authors are responsible for the contents of this publication.

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Milicevic Neumann, K., Ans, M. & Friedrich, B. Early-stage recovery of lithium from spent batteries via CO 2 -assisted leaching optimized by response surface methodology. Sci Rep 14 , 17369 (2024). https://doi.org/10.1038/s41598-024-67761-9

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Multi-condition adaptive detail characterization model of magnetorheological dampers and experimental verification

• Lei, Bingyue
• Fu, Benyuan
• Liao, Changrong

The theoretical model for predicting the damping characteristics of magnetorheological dampers (MRDs) is significant for enhancing the design efficiency of the control algorithm. However, some existing theoretical models face limitations in characterizing MRD damping characteristics simultaneously in terms of nonlinear detail characterization and adaptability to variable working conditions. Therefore, this paper proposed the Composite Double-Boltzmann (CDB) model combining the Double-Boltzmann (DB) function widely used in the field of biology and chemistry for its strong nonlinear characterization capability. Utilizing this model to fit the sinusoidal vibration testing data of the MRD prototype under variable combination working conditions, obtaining quantitative relationships between the undetermined parameters in the CDB model and the excitation current, vibration frequency, and amplitude to enable the model to address both the nonlinear details characterization of MRDs and adaptability to variable working conditions. Subsequently, the validity of the quantitative relationships were verified by comparing the calculated parameter values using the quantitative relationships with the original accurate parameter values. In order to verify the validity of the CDB model, extensive unknown working condition vibration tests were conducted on the MRD prototype under variable excitation currents, vibration frequencies, amplitudes and random excitation working conditions, employing the CDB and Tanh models to predict the damping characteristics, to compare to demonstrate the CDB model's capability of adapting to variable working conditions while accurately characterizing the nonlinear details of MRD damping characteristics.

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1.4 Experimental Design and Ethics

Does aspirin reduce the risk of heart attacks? Is one brand of fertilizer more effective at growing roses than another? Is fatigue as dangerous to a driver as speeding? Questions like these are answered using randomized experiments. In this module, you will learn important aspects of experimental design. Proper study design ensures the production of reliable, accurate data.

The purpose of an experiment is to investigate the relationship between two variables. In an experiment, there is the explanatory variable which affects the response variable . In a randomized experiment, the researcher manipulates the explanatory variable and then observes the response variable. Each value of the explanatory variable used in an experiment is called a treatment .

You want to investigate the effectiveness of vitamin E in preventing disease. You recruit a group of subjects and ask them if they regularly take vitamin E. You notice that the subjects who take vitamin E exhibit better health on average than those who do not. Does this prove that vitamin E is effective in disease prevention? It does not. There are many differences between the two groups compared in addition to vitamin E consumption. People who take vitamin E regularly often take other steps to improve their health: exercise, diet, other vitamin supplements. Any one of these factors could be influencing health. As described, this study does not prove that vitamin E is the key to disease prevention.

Additional variables that can cloud a study are called lurking variables . In order to prove that the explanatory variable is causing a change in the response variable, it is necessary to isolate the explanatory variable. The researcher must design her experiment in such a way that there is only one difference between groups being compared: the planned treatments. This is accomplished by the random assignment of experimental units to treatment groups. When subjects are assigned treatments randomly, all of the potential lurking variables are spread equally among the groups. At this point the only difference between groups is the one imposed by the researcher. Different outcomes measured in the response variable, therefore, must be a direct result of the different treatments. In this way, an experiment can prove a cause-and-effect connection between the explanatory and response variables.

Confounding occurs when the effects of multiple factors on a response cannot be separated, for instance, if a student guesses on the even-numbered questions on an exam and sits in a favorite spot on exam day. Why does the student get a high test scores on the exam? It could be the increased study time or sitting in the favorite spot or both. Confounding makes it difficult to draw valid conclusions about the effect of each factor on the outcome. The way around this is to test several outcomes with one method (treatment). This way, we know which treatment really works.

The power of suggestion can have an important influence on the outcome of an experiment. Studies have shown that the expectation of the study participant can be as important as the actual medication. In one study of performance-enhancing substances, researchers noted the following:

Results showed that believing one had taken the substance resulted in [ performance ] times almost as fast as those associated with consuming the substance itself. In contrast, taking the substance without knowledge yielded no significant performance increment. 1

When participation in a study prompts a physical response from a participant, it is difficult to isolate the effects of the explanatory variable. To counter the power of suggestion, researchers set aside one treatment group as a control group . This group is given a placebo treatment, a treatment that cannot influence the response variable. The control group helps researchers balance the effects of being in an experiment with the effects of the active treatments. Of course, if you are participating in a study and you know that you are receiving a pill that contains no actual medication, then the power of suggestion is no longer a factor. Blinding in a randomized experiment designed to reduce bias by hiding information. When a person involved in a research study is blinded, he does not know who is receiving the active treatment(s) and who is receiving the placebo treatment. A double-blind experiment is one in which both the subjects and the researchers involved with the subjects are blinded.

Sometimes, it is neither possible nor ethical for researchers to conduct experimental studies. For example, if you want to investigate whether malnutrition affects elementary school performance in children, it would not be appropriate to assign an experimental group to be malnourished. In these cases, observational studies or surveys may be used. In an observational study, the researcher does not directly manipulate the independent variable. Instead, he or she takes recordings and measurements of naturally occurring phenomena. By sorting these data into control and experimental conditions, the relationship between the dependent and independent variables can be drawn. In a survey, a researcher’s measurements consist of questionnaires that are answered by the research participants.

Example 1.20

Researchers want to investigate whether taking aspirin regularly reduces the risk of a heart attack. 400 men between the ages of 50 and 84 are recruited as participants. The men are divided randomly into two groups: one group will take aspirin, and the other group will take a placebo. Each man takes one pill each day for three years, but he does not know whether he is taking aspirin or the placebo. At the end of the study, researchers count the number of men in each group who have had heart attacks.

Identify the following values for this study: population, sample, experimental units, explanatory variable, response variable, treatments.

The population is men aged 50 to 84. The sample is the 400 men who participated. The experimental units are the individual men in the study. The explanatory variable is oral medication. The treatments are aspirin and a placebo. The response variable is whether a subject had a heart attack.

Example 1.21

The Smell & Taste Treatment and Research Foundation conducted a study to investigate whether smell can affect learning. Subjects completed mazes multiple times while wearing masks. They completed the pencil and paper mazes three times wearing floral-scented masks, and three times with unscented masks. Participants were assigned at random to wear the floral mask during the first three trials or during the last three trials. For each trial, researchers recorded the time it took to complete the maze and the subject’s impression of the mask’s scent: positive, negative, or neutral.

• Describe the explanatory and response variables in this study.
• What are the treatments?
• Identify any lurking variables that could interfere with this study.
• Is it possible to use blinding in this study?
• The explanatory variable is scent, and the response variable is the time it takes to complete the maze.
• There are two treatments: a floral-scented mask and an unscented mask.
• All subjects experienced both treatments. The order of treatments was randomly assigned so there were no differences between the treatment groups. Random assignment eliminates the problem of lurking variables.
• Subjects will clearly know whether they can smell flowers or not, so subjects cannot be blinded in this study. Researchers timing the mazes can be blinded, though. The researcher who is observing a subject will not know which mask is being worn.

Example 1.22

A researcher wants to study the effects of birth order on personality. Explain why this study could not be conducted as a randomized experiment. What is the main problem in a study that cannot be designed as a randomized experiment?

The explanatory variable is birth order. You cannot randomly assign a person’s birth order. Random assignment eliminates the impact of lurking variables. When you cannot assign subjects to treatment groups at random, there will be differences between the groups other than the explanatory variable.

Try It 1.22

You are concerned about the effects of texting on driving performance. Design a study to test the response time of drivers while texting and while driving only. How many seconds does it take for a driver to respond when a leading car hits the brakes?

• Describe the explanatory and response variables in the study.
• What should you consider when selecting participants?
• Your research partner wants to divide participants randomly into two groups: one to drive without distraction and one to text and drive simultaneously. Is this a good idea? Why or why not?
• How can blinding be used in this study?

The widespread misuse and misrepresentation of statistical information often gives the field a bad name. Some say that “numbers don’t lie,” but the people who use numbers to support their claims often do.

A recent investigation of famous social psychologist, Diederik Stapel, has led to the retraction of his articles from some of the world’s top journals including, Journal of Experimental Social Psychology, Social Psychology, Basic and Applied Social Psychology, British Journal of Social Psychology, and the magazine Science . Diederik Stapel is a former professor at Tilburg University in the Netherlands. Over the past two years, an extensive investigation involving three universities where Stapel has worked concluded that the psychologist is guilty of fraud on a colossal scale. Falsified data taints over 55 papers he authored and 10 Ph.D. dissertations that he supervised.

Stapel did not deny that his deceit was driven by ambition. But it was more complicated than that, he told me. He insisted that he loved social psychology but had been frustrated by the messiness of experimental data, which rarely led to clear conclusions. His lifelong obsession with elegance and order, he said, led him to concoct results that journals found attractive. “It was a quest for aesthetics, for beauty—instead of the truth,” he said. He described his behavior as an addiction that drove him to carry out acts of increasingly daring fraud . 2

The committee investigating Stapel concluded that he is guilty of several practices including

• creating datasets, which largely confirmed the prior expectations,
• altering data in existing datasets,
• changing measuring instruments without reporting the change, and
• misrepresenting the number of experimental subjects.

Clearly, it is never acceptable to falsify data the way this researcher did. Sometimes, however, violations of ethics are not as easy to spot.

Researchers have a responsibility to verify that proper methods are being followed. The report describing the investigation of Stapel’s fraud states that, “statistical flaws frequently revealed a lack of familiarity with elementary statistics.” 3 Many of Stapel’s co-authors should have spotted irregularities in his data. Unfortunately, they did not know very much about statistical analysis, and they simply trusted that he was collecting and reporting data properly.

Many types of statistical fraud are difficult to spot. Some researchers simply stop collecting data once they have just enough to prove what they had hoped to prove. They don’t want to take the chance that a more extensive study would complicate their lives by producing data contradicting their hypothesis.

Professional organizations, like the American Statistical Association, clearly define expectations for researchers. There are even laws in the federal code about the use of research data.

When a statistical study uses human participants, as in medical studies, both ethics and the law dictate that researchers should be mindful of the safety of their research subjects. The U.S. Department of Health and Human Services oversees federal regulations of research studies with the aim of protecting participants. When a university or other research institution engages in research, it must ensure the safety of all human subjects. For this reason, research institutions establish oversight committees known as Institutional Review Boards (IRB) . All planned studies must be approved in advance by the IRB. Key protections that are mandated by law include the following:

• Risks to participants must be minimized and reasonable with respect to projected benefits.
• Participants must give informed consent . This means that the risks of participation must be clearly explained to the subjects of the study. Subjects must consent in writing, and researchers are required to keep documentation of their consent.
• Data collected from individuals must be guarded carefully to protect their privacy.

These ideas may seem fundamental, but they can be very difficult to verify in practice. Is removing a participant’s name from the data record sufficient to protect privacy? Perhaps the person’s identity could be discovered from the data that remains. What happens if the study does not proceed as planned and risks arise that were not anticipated? When is informed consent really necessary? Suppose your doctor wants a blood sample to check your cholesterol level. Once the sample has been tested, you expect the lab to dispose of the remaining blood. At that point the blood becomes biological waste. Does a researcher have the right to take it for use in a study?

It is important that students of statistics take time to consider the ethical questions that arise in statistical studies. How prevalent is fraud in statistical studies? You might be surprised—and disappointed. There is a website dedicated to cataloging retractions of study articles that have been proven fraudulent. A quick glance will show that the misuse of statistics is a bigger problem than most people realize.

Vigilance against fraud requires knowledge. Learning the basic theory of statistics will empower you to analyze statistical studies critically.

Example 1.23

Describe the unethical behavior in each example and describe how it could impact the reliability of the resulting data. Explain how the problem should be corrected.

A researcher is collecting data in a community.

• She selects a block where she is comfortable walking because she knows many of the people living on the street.
• No one seems to be home at four houses on her route. She does not record the addresses and does not return at a later time to try to find residents at home.
• She skips four houses on her route because she is running late for an appointment. When she gets home, she fills in the forms by selecting random answers from other residents in the neighborhood.
• By selecting a convenient sample, the researcher is intentionally selecting a sample that could be biased. Claiming that this sample represents the community is misleading. The researcher needs to select areas in the community at random.
• Intentionally omitting relevant data will create bias in the sample. Suppose the researcher is gathering information about jobs and child care. By ignoring people who are not home, she may be missing data from working families that are relevant to her study. She needs to make every effort to interview all members of the target sample.
• It is never acceptable to fake data. Even though the responses she uses are real responses provided by other participants, the duplication is fraudulent and can create bias in the data. She needs to work diligently to interview everyone on her route.

Try It 1.23

Describe the unethical behavior, if any, in each example and describe how it could impact the reliability of the resulting data. Explain how the problem should be corrected.

A study is commissioned to determine the favorite brand of fruit juice among teens in California.

• The survey is commissioned by the seller of a popular brand of apple juice.
• There are only two types of juice included in the study: apple juice and cranberry juice.
• Researchers allow participants to see the brand of juice as samples are poured for a taste test.
• Twenty-five percent of participants prefer Brand X, 33 percent prefer Brand Y and 42 percent have no preference between the two brands. Brand X references the study in a commercial saying “Most teens like Brand X as much as or more than Brand Y.”
• 1 McClung, M. and Collins, D. (2007 June). "Because I know it will!" Placebo effects of an ergogenic aid on athletic performance. Journal of Sport & Exercise Psychology, 29(3), 382-94 .
• 2 Bhattacharjee, Y. (2013, April 26). The mind of a con man. The New York Times . Retrieved from http://www.nytimes.com/2013/04/28/magazine/diederik-stapels-audacious-academic-fraud.html?_r=3&src=dayp&.
• 3 Tillburg University. (2012, Nov. 28). Flawed science: the fraudulent research practices of social psychologist Diederik Stapel. Retrieved from https://www.tilburguniversity.edu/upload/3ff904d7-547b-40ae-85fe-bea38e05a34a_Final%20report%20Flawed%20Science.pdf.

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