a randomized complete block experimental design

What is a Randomized Complete Block Design (RCBD)?

A Randomized Complete Block Design (RCBD) is defined by an experiment whose treatment combinations are assigned randomly to the experimental units within a block. Generally, blocks cannot be randomized as the blocks represent factors with restrictions in randomizations such as location, place, time, gender, ethnicity, breeds, etc. It is not simply possible to randomly assign a particular gender to a person. It is not possible to pick a country and call X country. However, the presence of these factors (also known as nuisance factors ) will introduce systematic variation in the study. For example, the crops produced in the northern vs the southern part will get exposed to different climate conditions. Therefore, they should be controlled whenever possible. Controlling these nuisance factors by blocking will reduce the experimental error, thereby increasing the precision of the experiment and many other benefits. In the completely randomized design (CRD) , the experiments can only control the random unknown and uncontrolled factors (also known as lucking nuisance factors). However, the RCBD is used to control/handle some systematic and known sources ( nuisance factors ) of variations if they exist.

Randomized Complete Block Design (RCBD) is arguably the most common design of experiments in many disciplines, including agriculture, engineering, medical, etc. In addition to the experimental error reducing ability, the design widens the generalization of the study findings. For example, if the study contains the place as a blocking factor, the results could be generalized for the places. A fertilizer producer can only claim that it is effective regardless of the climate conditions when it is tested in various climate conditions.

The “ complete block ” part of the name indicates that each treatment combination is applied in all blocks. If a block misses one or more treatment combinations, the experiment would be called Randomized Incomplete Block Design. The design would still be called randomized because the treatment combinations are randomly assigned to the experimental units within the blocks. If a block is only missing data points from a couple of observation units , the experiment will still be called randomized complete block design (RCBD) with missing data, but not “incomplete block design.”

Randomized Complete Block Design Analysis Model

The effects model for the RCBD is provided in Equation 1.

The primary interest is the treatment effect in any RCBD, therefore the hypothesis for the design is statistically written as.

Test Your Knowledge

Example problem on randomized complete block design.

Chapter 11: Randomized Complete Block Design

M. L. Harbur; Ken Moore; Ron Mowers; Laura Merrick; and Anthony Assibi Mahama

It is important when conducting an experiment that the experimental units be as homogenous as possible. This ideal may be met without much difficulty in a lab or within a field with particularly uniform soils. In many field locations, however, the landscape can vary greatly over a short distance. How can we assure, then, that an observed agronomic difference is the result of a specific treatment, rather than the result of the experimental units to which it was allocated? In other words, how do we prevent our treatment results from being confounded with our experimental units? The Randomized Complete Block Design (RCBD) offers one solution.

• How heterogeneity of experimental units can reduce the sensitivity of an experiment
• How the Randomized Complete Block Design (RCBD) can be used to reduce the heterogeneity of experimental units
• How to conduct the analysis of variance (ANOVA) for an experiment that employs the RCBD
• How to test for the efficiency of the RCBD versus that of the Completely Randomized Design

The rationale for blocking is to achieve homogeneous experimental units within blocks. In Chapter 1 on Basic Principles, you learned about the importance of replication in designing a valid experiment. We applied those concepts in Chapter 8 on The Analysis of Variance (ANOVA) when we introduced the analysis of variance. Treatments and replications were assigned to experimental units through the process of randomization . The result of this effort is referred to as a Completely Random Design (CRD) .

The CRD is an appropriate experimental design when all experimental units are assumed to be similar or homogeneous (as statisticians like to say). If this is the case, then any observed differences among treatments will cause us to conclude that there was a treatment effect. As mentioned in the introduction, however, homogeneity of experimental units can be difficult to achieve in field plot experiments.

Heterogeneity

Heterogeneity of experimental units presents two problems. First, the failure to recognize differences between experimental units may lead us to conclude that differences in our variates are the result of the treatments applied, when they were actually caused by the pre-existing condition of the experimental units.

Variance of the Error

Differences between plots (experimental units) not related to the treatments applied to them can inflate the variance of the error associated with the experiment. Recall from Chapter 8 on The Analysis of Variance (ANOVA) that we tested the significance of our treatment using an F-test (Equation 1 below).

If the residual mean square error is increased, then the treatment mean square must also increase in order to maintain the same F-value. In other words, the greater the pre-existing differences between our plots (experimental units), the greater and more profound the difference between treatments must be just to be recognized as statistically significant. Heterogeneity of experimental units thus reduces the sensitivity of our experiment.

[latex]\small F = \dfrac{TMS}{RMS}[/latex]

[latex]\textrm{Equation 1}[/latex] Formula for calculating F value.

where: [latex]\small TMS[/latex]= treatment mean square, [latex]\small RMS[/latex]= residual (error) mean square.

How to Block

The first step in using the RCBD is to recognize the source(s) of potential heterogeneity among plots (experimental units). In field research, this potential most often exists between plots situated on different soil map units, for example, as the slope changes. We viewed some of the potential differences in soils related to landscape in Study Question 1. There are many other possible sources of heterogeneity among plots, though. These map units often differ in their yield potential, with the result that some of the variation in the yield measurement is due to the plot location. One example of site heterogeneity is found in Fig. 2.

In the map in Figure 1, we have four map units (A-D) labeled with the difference in yield between each map unit and the mean for the entire field. As we move from map unit A across to map unit D, the yield potential decreases. In such a case, we say that we have a “production gradient” across the map units. Now let’s suppose that we are comparing four different levels of fertilizer (0, 50, 100, 150 kg/ha). If we used a completely randomized design (CRD) across these map units, we risk the possibility of placing all of the high-fertility treatments on extreme map units. This would lead to an unfair comparison of the treatments.

Answer the following question using the treatment map above.

In blocking, we generally place an equal-sized block on every map unit. Each block, in this case, contains four experimental units (plots). Each treatment is applied to one experimental unit within the block (Fig. 3).

Each block can be thought of as a replication. Every treatment is forced to occur within each block. Treatments are randomly allocated within each block so that separate randomization is made for each block. In this way, the treatments are prevented from being affected by a second, unrecognized source of heterogeneity that could exist between experimental units within the blocks.

Design Control

Forcing each treatment to occur once in every block is sometimes referred to as a restriction on randomization. The restriction in the case of a RCBD is that every treatment must occur in every block. In a CRD, every plot would have the same chance of receiving any treatment, so there is no restriction on randomization; hence the name.

Blocking is a form of design control that was discussed in Chapter 1 on Basic Principles. It is one of the three characteristics of designed experiments (do you remember the other two?). Blocking in a field experiment amounts to grouping plots into more similar sets such that the variation associated with the blocks (whatever is causing them to differ) can be estimated and associated with the block. In a CRD, this variation would be associated with and show up in the Error MS, thus inflating the estimate and reducing the precision of the experiment. When blocking is effective (i.e. there is some variation in the measured response associated with the blocking criterion), this variation is removed from the Error MS and the ability to detect true treatment differences (i.e. precision of the experiment) is improved.

Further Thought : Discussion

What other possible influences could affect each block beyond the known yield potential gradient?

Randomization

Fig. 4 below illustrates a RCBD randomization scheme. Notice that the Blocks are numbered 100, 200, 300, and 400 and that each of the 5 treatments is on one plot in each block.

Randomization should be done using some sort of randomization method, not just arbitrarily. This precludes any bias which may be unintentionally introduced due to the assignment of treatment. Check out the exercise in the next screens to learn how to randomize treatments for a RCBD using Excel.

Ex. 1: Randomizing Treatments for a RCBD

As we said before, experimental design is really about how treatments are assigned to experimental units (plots). In the case of the randomized complete block design (RCBD), treatments are blocked into groups of experimental units that are similar for some characteristic. In field experiments, treatments are usually blocked perpendicular to some perceived gradient present in the field. The gradient may be related to such characteristics as soil properties, previous crop history, or any number of other factors that can occur naturally or unnaturally in a field. In a RCBD, treatments are allotted to plots at random within each block of experimental units (plots) such that within any given block, each plot has the same probability of receiving a particular treatment as any other. The only restriction is that every treatment has to occur in every block.

For the purpose of this exercise, we will develop a plot plan for an oat variety trial experiment. The treatments consist of five cultivars replicated four times for a total of twenty plots. The treatments and plots are listed in the Treatments worksheet of the Excel file QM-mod11-ex1data.xls . A map of the field layout is presented in the Plot Plan worksheet. Our goal is to randomize the plot order within blocks and permanently associate it with the treatments.

Ex. 1: Create a Random Assignment

Randomly assign the five oat cultivars to each of the four blocks identified in the Treatments worksheet. The idea now is to create a random order of the treatments under the restriction that each Cultivar must occur once in each Block .

• Enter the formula =RAND() in cell D2.
• Copy the formula in cell D2 to cells D3:D21. A fast way to do this is to double-click the square that appears in the lower right-hand corner of the cell when you select cell D2. The RAND() function will return a random number between 0 and 1 to each cell the formula is copied to.
• Using your mouse, select all the cells in the Cultivar, Block, and Rand columns, including the column headings (B1:D21). Do not select the Plot column!
• Select Sort from the Data menu at the top of the main window.
• In the Sort by box in the Sort dialog box, select Block.
• If not already present, add another sort level by clicking on Add Level.
• In the second Sort by box, select Rand and then click OK.

Ex. 1: Finished Random Assignment

You should now have a random assignment of Cultivars in column B, and each cultivar should occur once in every block. Your worksheet should look something like Fig. 5.

Ex. 1: Plot Plan

The treatments are now sorted according to plot order. Look at the Plot Plan worksheet to see the field layout of the plots according to the new randomization (Fig. 6). Your plan should look something like the one below. Note that every cultivar occurs once in every block. However, the order of treatments within each block will vary with each randomization.

Ex. 1: RCBD vs. CRD Randomization

You may have noticed that randomizing treatments for the CRD and RCBD follow a similar process in Excel. The main difference is in the number of sort levels you use. In the CRD, only one sort level was used which corresponded to the random number generated by the RAND() function. In the case of the RCBD, you used two sort levels: 1) the first one assigned to Blocks, and 2) the second assigned to the random number. You can compare and contrast the plot map from the previous page for an RCBD with one generated for a CRD by removing the first sort level so that the only one that remains is the random number.

The restriction on randomization has been removed so that any cultivar can be assigned to any plot (Fig. 7).

Ex. 1: R Code Functions

Experimental design is really about how treatments are assigned to experimental units (or plots). In the case of the randomized complete block design (RCBD), treatments are blocked into groups of experimental units that are similar for some characteristic. In field experiments, treatments are usually blocked perpendicular to some perceived gradient present in the field. The gradient may be related to such characteristics as soil properties, previous crop history, or any number of other factors that can occur naturally or unnaturally in a field. In a RCBD, treatments are allotted to plots at random within each block of experimental units (plots) such that within any given block, each plot has the same probability of receiving a particular treatment (location) as any other. The only restriction is that every treatment has to occur in every block.

Ex. 1: Maize Yield Test

You are a maize breeder in charge of designing a field for a yield test of 3 synthetic maize populations. The field has never been used before by your company, and the area surrounding the field is known to have very localized deposits of clay which inhibit root growth and thus negatively affect the yield of plants planted directly on top of the deposits. You want to account for the heterogeneity of the field by assigning each of the 3 maize populations to one of the 3 positions in each of 3 blocks.

In other words, you want to create a random order of the three populations (or treatments) within each block under the restriction that each population must occur once in each block. To make the coding a bit easier for this exercise, we will rename Population 7.5 to Population 1, Population 10.0 to Population 2, and Population 12.5 to Population 3.

Exercise 1:

We will learn two ways to do this. The first way will take a little longer than the second but involves less coding. The second way will be much faster than the first. However, it will involve at least a basic understanding of some coding tools, such as loops.

Ex. 1: Creating a Field

Let’s start by creating a matrix of all zeros, where each entry represents a plot that will be assigned to a cultivar, and each column represents a block. We have 3 cultivars in each of the 3 blocks, thus the dimensions of this ‘field’ matrix should be 3×3. We will use the matrix command to create the field in R. In the Console window, enter in the parenthesis after matrix, the 0 indicates the type of element we want the matrix to be composed of (i.e., in this case, zeros because we are going to fill in the cultivar numbers, which will be randomly assigned to each plot in each block). The first 3 indicates the number of rows we want in the matrix; since we have 3 populations in each block, we need our matrix to have 3 rows. Finally, the second 3 indicates that we want 3 columns (blocks) in our matrix.

field<-matrix(0,3,3)

Have a look at the field you’ve created. Enter f ield  into the Console.

    [,1][,2][,3]

[1,]   0   0   0

[2,]   0   0   0

[3,]   0   0   0

Ex. 1: Creating a Vector

Now, let’s create a vector representing the cultivars we have in our breeding program. We’ll accomplish this by using the gl command. This command allows us to create a vector (or column) of factor variables. In the parenthesis after the gl  command, the first entry ( 3 ) indicates the number of factor levels, and the second number ( 1 ) indicates the number of replications of each factor variable. You may be asking yourself why we are not entering a 3 for the number of replications, as we have 3 randomized complete blocks. The reason we do not enter a 3 for replications is due to the fact that we will create randomized complete blocks individually and enter them into each block (column) in the field matrix. This will become apparent in the following steps.

Enter into the Console:

pop<-gl(3,1)

Levels: 1 2 3

Ex. 1: Vector with 3 Entries

Now let’s create a vector with 3 entries, where each entry is a random number between 0 and 1. We’ll use the runif command to do this. This command is used by entering the number of entries of numbers between 0 and 1 you want your vector to be composed of in parenthesis after the runif command (i.e. for this example, we will enter 3, since we have 3 maize populations). Let us call this vector with 3 entries of random numbers between 0 and 1 rand . Create the vector rand , then look at it by entering the vector name ( rand ).

rand<-runif(3)

[1] 0.1165839 0.5730972

Ex. 1: New Matrix with Block

Now, let us create a new matrix called block by putting the cultivar and rand vectors together, so each random number in the rand vector corresponds to one of the 3 maize populations. The cbind command can be used to put two vectors together. The command is used by entering the cbind   command followed by the two vectors you want to be put together (or concatenated) in parenthesis separated by a comma. Create the matrix called block with the cbind  command, then look at the matrix by entering the name of the matrix (block).

block<-cbind(pop,rand)

    pop     rand

[1,]      1   0.1165839

[2,]      2      0.5730972

[3,]      3      0.3469669

Ex. 1: Ordering the Population in Block

We can now order the maize populations in the first column of the block matrix by each of the 3 population’s corresponding random number in the rand vector. We will use the order  command to sort the populations based on their corresponding number in the rand column, by the population with the smallest random number first to the population with the largest random number last. The population with the smallest random number is Population 1 ( 0.1165839 ), and the population with the largest random number is Population 2 ( 0.5730972 ). Thus, the order for this randomized complete block should be 1,3,2. We will now go through the order  command.

Let us call this randomized complete block randblock . Create matrix randblock and take a look at it by entering the following in the Console.

   pop     rand

[1,]     1   0.1165839

[2,]     2      0.5730972

[3,]     3      0.3469669

Ex. 1: Filling the Block

To use the order command, we first write the matrix or vector that we’d like ordered (i.e., in this case, block), followed by brackets. Then, we write the command order followed by the column that we would like the ordering of the rows based off of (i.e., in this case the second column of the block matrix, which contains the random numbers between 0 and 1 for each population. Note: the row number is left blank in the brackets before the comma block[order(block[,2]),] to specify that we want all of the rows sorted based on the values of column 2 in the block matrix. Since the second column in the block matrix is the random number column, this is the column that we want to order the populations by; thus, we enter block[,2] to specify we want the rows ordered based on the values in the second column in the block matrix. The default of the order function is to sort in a ascending order, with the lowest value at the top of the column and the highest value at the bottom. We must now enter this randomized block into the first block (column) of the field matrix we created previously. We’ll do this by setting the first column of the field matrix (specified by field[,1] ) to equal the first column of the randblock matrix (specified by randblock[,1] ). Carry out this operation, then look at the field matrix to make sure you’ve entered the first column from the randblock matrix into the field matrix.

field[1,]<-randblock[1,]

    [,1] [,2] [,3]

[1,]     1   0    0

[2,]     2      0     0

[3,]     3      0     0

Again, we specify the first column of the field matrix with field[,1], and set it equal to the first column of the randblock matrix, specified by randblock[,1] . Note: The order of the cultivars in the first column of your field matrix may not be the same as in this lesson due to the randomization process.

To fill in the rest of the blocks, carry out all of the same steps you just did, but when entering the next block into the field matrix, change the field[,1]<-randblock[,1] to field[,2]<-randblock[,1] to specify that you want to enter the randomized order for the second block (or column) in the field matrix.

Ex. 1: Review RCBD Method 1

To summarize what we’ve just learned, the first method for creating 3 randomized complete blocks with 3 populations is reviewed succinctly below.

First, we create a field matrix of all zeros, with the dimensions of the number of entries in each block, and the number of blocks desired (i.e., in this case we have 3 populations or entries, and 3 blocks).

Then, go through steps 1-5 (presented below) 3 times. After each time through steps 1-5, add 1 to the column indicated in the field matrix in step 5. For example, the second time through steps 1-5, step 5 will be field[,2]<-randblock[,1 ], and the third time through step five will be field[,3]<- randblock[,1] , etc.

cultivar<-gl(3,1)

block<-cbind(cultivar,rand)

randblock <- block[order(block[,2]),]

field[,1]<-randblock[,1]

Have a look at the field you’ve just created:

   [,1] [,2] [,3]

[1,]    1     1     2

[2,]    3     3     1

[3,]    2     2     3

Ex. 1: RCBD Method 2

This method can save time in comparison with the first method, especially if you have many treatments that you want randomized in many blocks. The two methods are computationally equivalent, however the second method utilizes a loop command to repeat the operations that we previously did for each block in Method 1.

We can use a for loop to go through a set of operations a specified number of times. Using the for loop, we must first assign an iteration variable, which corresponds to the number of times the set of operations has been completed. For example, if we assign i=1:3 as the iteration variable, the first time through the set of commands i=1, the second time i=2, the third time i=3. In this example, we’ll use the letter i to indicate the iteration variable in our loop.

Let’s clear the entire data frame before starting Method 2. Use the rm(list=ls() ) command to clear the entire data frame/environment. The upper-right window should now be clear of all variables and data.

rm(list=1s())

Ex. 1: Creating a Field Matrix

Great! Now create the field matrix in the same way as in Method 1.

rm(-matrix(0,3,3)

Now we need to enter the loop with the number of cycles, or iterations we want carried out. In this case, we want to create 3 randomized complete blocks, so we the total number of iterations is 3. Enter the for command into the Console with the iteration variable i  ndicating that we want 3 iterations carried out

for (i in 1:3)

Good, we have indicated that we want i to be our iteration variable ranging from 1 to 3. Now, we need to enter the bracket { , then enter lines 1-5 from the method 1 code with line 5 ending in a } bracket.

{pop<-gl(3,1)

field[,i]<-randblock[,1]}

Ex. 1: Finished Field Matrix

Look at line 5 for Method 2 directly above. Do you notice anything different than line 5 in Method 1? Instead of manually entering the block number in the field[,1]<-randblock[,1] command, we simply enter i , so that line 5 in Method 2 becomes field[,i]<-randblock[,1] . The value of i is 1 for the first iteration, 2 for the second iteration, and 3 for the third iteration. This can save us a lot of time if we are trying to create many randomized complete blocks for many treatments.

Let’s now go through method 2 in its entirety.

randblock <-

block[order(block[,2]),]

Look at the field you’ve just created.

    [,1] [,2] [,3]

[1,]     1    1    2

[2,]     3    3    1

[3,]     2    2    3

Ex. 1: Review RCBD Method 2

To reiterate, Method 2 is accomplished by first creating the field matrix.

Then entering the for command, specifying the iteration variable (i.e., i in 1:3)

and finally entering a { bracket, lines 1 to 5 from Method 1, with a }  after the final line (line 5).

{cultivar&lt;-gl(3,1)

Linear Additive Model

The Linear Additive Model also applies for the RCBD. In Chapter 8 on The Analysis of Variance (ANOVA) we introduced the linear additive model in describing the analysis of variance. The model showed how the measured or dependent variable is affected by different factors in the model (independent or classification variables). According to the linear model, the observed result can be estimated by adding (summing) the effects of the model terms. By introducing blocking into the experimental design, another possible source of variation is included in the linear additive mode that is associated with blocks. Thus, the model for a RCBD includes an additional term for blocks. The error term can now be thought of as that variability among plots (experimental units) that cannot be accounted for by blocks or treatments.

The linear additive model for the RCBD must include the effects of blocking, treatment(s), and error (Equation 2). The model for a single-factor RCBD is:

[latex]Y_{ij}=\mu+\beta_i+T_j+\beta{T}_{ij}[/latex]

[latex]\textrm{Equation 2}[/latex] Linear Additive Model.

where: [latex]Y_{ij}[/latex] = response observed for the [latex]ij^{th}[/latex] experimental unit, [latex]\mu[/latex] = grand mean, [latex]B[/latex] = block effect, [latex]T[/latex] = treatment effect, [latex]BT[/latex] = block error x treatment interaction.

Differences in Models

This model differs from the linear additive model for the CRD in two ways. First, it differs by including blocks as an effect. Blocks capture the effect related to the blocking criterion, which is often soil heterogeneity in field experiments. The RCBD model, by having a block effect, inherently includes a restriction on randomization . The RCBD is not completely randomized — instead, each level of the treatment is “forced” to occur in each block.

The second difference is that the block x treatment interaction is the error term. Why should we use this interaction to test the effect of treatment? The answer is that we test treatment differences to see if they remain relatively large compared with their random changes for different blocks. These random changes in treatments over blocks comprise the BT interaction.

Suppose that soybean yield means for three herbicide treatments are 2.7 t/ha for herbicide A, 3.0 for B, and 3.3 for C. If these differences remain fairly consistent over each block, the block x treatment interaction will be small relative to the mean yield differences, and the F-ratio of treatment MS to error (BT interaction) will be large. In effect, we are testing whether the yield differences remain relatively large in comparison with their (random) changes over blocks, i.e., block x treatment interaction. The block x treatment is the proper error term for testing for treatment differences and it is used for the F-Test in the ANOVA. A large F-value implies treatment differences are large relative to the error.

Treatment Differences

In both graphs in Fig. 8, the average soybean yields are the same (2.7, 3.0, and 3.3 t/ha). We trust the results more if these differences are consistent across blocks (left-hand side). When the B x T interaction is large, (right-hand side), the random error obscures the treatment differences.

Estimate Effects Using ANOVA

Perhaps the linear model will be clearer if we use an example. Recall the corn population experiment from Chapter 5 on Categorical Data—Multivariate. In that experiment, corn was planted at three populations in order to determine the effect on grain yield. If we treat the repetitions as being blocks, the linear model for that specific experiment is (Equation 3):

[latex]\small Y_{ij}=\mu+Blk_i+POP_j+BlkxPOP_{ij}[/latex]

[latex]\textrm{Equation 3}[/latex] Linear Additive Model.

where: [latex]\mu[/latex] = grand mean, [latex]\small Blk[/latex] = block effect, [latex]\small POP[/latex] = population effect.

The linear model provides a convenient method of listing the effects which are to be estimated using an ANOVA. As you will see, every effect from the linear model (with the exception of the mean) will be included in the ANOVA table. The arguments you list in the aov() function in R correspond directly to the terms that comprise the linear additive model.

R Code Functions

• as.factor()

In this example, you will learn how to analyze data from an experiment that has a restriction on randomization. This exercise will give us an opportunity to evaluate how blocking affects the sum of squares for our model. More specifically, it will allow us to discern how Total SS are partitioned between the two designs (blocking vs. not blocking).

Exercise 2: Analyze an RCBD experiment

We return to the synthetic maize population scenario which we used in the previous exercise, where we created randomized complete blocks. You are again a maize breeder and have now been asked by your supervisor to analyze the yield data for the 3 synthetic maize populations that were planted in a yield trial in 3 randomized complete blocks. Conduct an ANOVA on this data with cultivar and block as factors.

Ex. 2: Beginning Analysis

Reading the Data To begin the analysis, first set the working directory and read the data into R …steps presented in the CRD activity.

Analyzing The Data

The code required to analyze the experiment is similar to what we used to analyze the Maize Population Example in the CRD activity. The linear additive model for an RCBD includes an additional term to account for the linear effect of blocks.

Let’s first do an analysis without incorporating block into the model. This is exactly what we did in the analysis of the CRD experiment, where the only factor in the model was population.

data<-read.csv(“exercise.11.2.data.csv”, header =T)

head(data, n=3)

    Pop    Block    Yield

1   7.5       1         8.50

2   7.5       2         7.71

3   7.5       3         8.50

attach(data)

Ex. 2: Running ANOVA

Set population as a factor, equal to variable Pop.

Pop<- as.factor(data$Pop)

out<- summary(aov(Yield ~ Pop))

             Df  Sum Sq  Mean Sq  F value  Pr(>F)

Pop           2  24.809   12.405    19.15  0.00249  **

Residuals     6   3.887    0.648

Signif. codes:  0  ‘***’  0.001  ‘**’  0.01  ‘*’  0.05  ‘.’  ‘ ‘  1

Run the ANOVA as we did in the CRD activity. The one-factor model for the ANOVA is: Yield = Population. After you run the ANOVA, look at the ANOVA table.

Let’s run the ANOVA incorporating block as a second factor. Now we can run the ANOVA. The model we will use is: Yield = Population + Block.

Block<- as.factor(data$Block)

out<- summary(aov(data$Yield ~ Pop+Block))

             Df   Sum Sq   Mean Sq   F value   Pr(>F)

Pop           2   24.809    12.405     25.65   0.00523  **

Block         2    1.953     0.977      2.02   0.24756

Residuals     4    1.934     0.484

Signif. codes:  0  ‘***’  0.001  ‘**’  0.01  ‘*’  0.05  ‘.’  ‘ ‘  1

Ex. 2: Interpreting Results, Comparing The CRD Vs. RCBD ANOVA

The information that most interests us is the last 2 columns of the first row in the ANOVA table; the F-value and P-value for factor variable pop. The F-value is 25.65, which corresponds to a P-value of 0.00523. We would therefore conclude that population had a significant effect at 0.1% on yield in this experiment.

Our next step in the analysis would normally be to evaluate the mean differences among the three populations. However, at this point, we will take the opportunity to compare and contrast the two designs.

The first question we want to answer is, “How did including blocks as a factor in the analysis affect the partitioning of df and SS”? You will notice right away that the total df and SS are exactly the same for the two analyses. What has changed is how these are partitioned between the Model and Residuals. The Model df in the RCBD analysis increased by 2, reflecting the two additional df from adding the block term. Note that these were partitioned out of the Error df, which was reduced by 2, from 6 to 4. Something changed with the SS from the one-factor ANOVA to the two-factor ANOVA. Some of the variation that was unexplained and associated with error has been partitioned into the model SS; i.e. it went from being unexplained to being explained by the model. By viewing the ANOVA at the bottom of the figure, you can see that this variation was attributed to blocks in the RCBD.

Ex. 2: Conclusions

This improves the precision of the F-test for population because the Error MS decreased as a result of partitioning some of the variation into blocks. Since the Error MS is smaller for the RCBD, there will also be improved precision for comparing means because the standard errors used to calculate these tests are calculated from the Error MS.

You may have noticed that R computed an F-test for Blocks and indicated that it was nonsignificant (with a P-value of 0.24756). Normally we are not interested in this test because our primary goal with blocking is to reduce the Error MS. In this case, blocking resulted in a modest reduction in the Error MS, so the test turned out to be nonsignificant. Regardless of the magnitude of the Block effect, we would use this analysis since our experimental treatments were arranged in an RCBD.

There is a cost to blocking that affects the precision of statistical tests in cases like this where df for error is marginal. If you have a look at a distribution of F-values, you can see that the critical F-value (the one that you have to exceed to demonstrate significance) decreases rapidly as you add error df up to about 10, after which it decreases much more slowly. The reason for this is that we are much less confident in variances estimated from a small number of samples than those estimated from larger numbers. In our example, even though the F-value was larger for the RCBD, the probability of it occurring at random was actually higher than the CRD (P > F = 0.0052 vs. 0.0025). This is a direct consequence of the reduction in df for Error.

Analysis of Variance for RCBD

The analysis of variance used with RCBD is similar to that used with the Completely Randomized Design for a factorial experiment. The differences are the inclusion of the block effect and the replacement of the error term with the block x treatment effect. The ANOVA table is structured to account for these effects (Table 1).

Compare these to the linear model from the previous section.

Example Using RCBD

For our example, we will use the same dataset as in Chapter 8 on The Analysis of Variance (ANOVA), corn planted at three populations. In this example, however, the three replicates are arranged in blocks, in contrast to the Completely Randomized Design in Chapter 9 on Two Factor ANOVAS. Three replications were included in the experiment for a total of 9 (3 pop x 3 rep) experimental units. The data are listed in Table 2.

If the replications were blocked, then the ANOVA table for the corn population experiment example would include the following sources of variation: blocks, population (the treatment), the interaction (block x population), and total.

Degrees of Freedom

The degrees of freedom are also calculated in a manner similar to the factorial experiment. The block and treatment degrees of freedom are simply the number of blocks or treatment levels minus one. The error (interaction) df is the product of the block and treatment degrees of freedom. These are summarized below:

For the sample experiment, there are 9 total experimental units in the experiment, leaving eight total df. The df associated with blocks and treatment are, in each case, one less than the number of levels for each factor, or 2 = (3 – 1) df. The interaction df is the product of the block and treatment df, or 4 = (2 x 2).

Sum of Squares

The sums of squares for the Randomized Complete Block Design are similar to those calculated in Chapter 8 on The Analysis of Variance (ANOVA). Recall we first calculate the correction factor (CF) (Equation 4):

[latex]\textrm{CF} =\frac{(\sum{x})^2}{n}[/latex].

[latex]\textrm{Equation 4}[/latex] Formula for calculating correction factor.

where: [latex]x[/latex]= each observation, [latex]n[/latex]= number of observations.

Our correction factor is the same as in Chapter 8 on The Analysis of Variance (ANOVA):

[latex]\textrm{CF}= \frac{8.50 + 7.71 + 9.05 + 10.30 + 9.14 + 8.85 + 6.53 +5.36 + 4.65)^2}{9} =\frac{4912.61}{9}=\small 545.85[/latex].

We also calculate the Treatment sum of squares as in Chapter 8 on The Analysis of Variance (ANOVA) (Equation 5):

[latex]\small \textrm{Treatment SS} = \sum (\frac{T^2{}}{r})-\text{CF}[/latex].

[latex]\textrm{Equation 5}[/latex] Formula for calculating treatment sum of squares.

where: [latex]\small T[/latex]= each treatment level, [latex]r[/latex]= number of replications, [latex]\small CF[/latex] = correction factor.

Sum of Squares Example

In our example, the Treat SS is:

[latex]\small \textrm{Treatment SS} = \normalsize (\frac{25.26^2}{3} + \frac{28.29^2}{3}+\frac{16.54^2}{3}) - \small \textrm{CF} =(212-69 + 266.77 + 91.19) - 545.85 = 24.81[/latex].

The total sum of squares is (Equation 6):

[latex]\small \textrm{Total SS}=\sum {x}^{2}-\text{CF}[/latex].

[latex]\textrm{Equation 6}[/latex] Formula for calculating total SS

where: [latex]x[/latex]= each observation, [latex]\small CF[/latex] = correction factor.

Thus, the total SS in our example is:

[latex]\small \textrm{Total SS}=(8.5^2+7.71^2+9.05^2+10.3^2+9.14^2+8.85^2+6.53^2+5.36^2+4.65^2) - 545.85=28.70[/latex].

Difference in RCBD and CRD

So far, every source of variation in the Randomized Complete Block Design is exactly the same as the Completely Randomized Design. The RCBD differs from the CRD in that it includes Blocks as a source of variation (Equation 7).

[latex]\small \textrm{Block SS}=\sum (\frac{B^2{}}{t})-\text{CF}[/latex].

[latex]\textrm{Equation 7}[/latex] Formula for calculating block SS.

where: [latex]B[/latex]= each block total, [latex]t[/latex]= number of treatments, [latex]CF[/latex] = correction factor.

For example, the Block SS is:

[latex]\small \textrm{Block SS}=\textrm{Block SS} = (\frac{25.33^2}{3} + \frac{22.21^2}{3}+\frac{22.55^2}{3}) - \textrm{CF} =(213.87 + 164.43 + 169.50) - 545.85 = 547.80-545.85=1.95[/latex].

We calculate the residual (error) sum of squares for the RCBD similar to how we did for the CRD, only now we subtract both the Treat SS and Block SS from the Total SS (Equation 8).

[latex]\small \textrm{Residual SS} = \textrm{Total SS - Block SS - Treatment SS = 28.70 - 1.95 - 24.80 = 1.95}[/latex].

[latex]\textrm{Equation 8}[/latex] Formula for calculating residual SS.

You may see the Residual SS listed in some tables as the Block*Treatment interaction.

Sum of Squares Table

The sums of squares for each source of variation in the experiment are shown below.

Mean Squares

Mean squares are calculated in the same manner regardless of the design used: in each case, the mean square is equal to the sum of squares divided by the degrees of freedom for each source of variation. The mean squares for the sample experiment are shown below.

F-Values and F-Test

The observed F-value for the treatment is calculated for the RCBD experiment by dividing the treatment mean square (TMS) by the residual mean square (RMS). In other words, F = TMS / RMS (Equation 1).

The observed F-value for treatment must be compared with a critical F-value in order to test the significance of the treatment effect. This critical F-value is determined using the same procedure as for the CRD: the value is selected from Appendix 4a, using the treatment df to select the column and the error df to select the row. The desired significance level (P=0.05, 0.025, 0.01, or 0.001) determines which of the 4 numbers is chosen.

The observed and critical F-values for the sample experiment are shown below:

RCBD Analysis Exercises using R

Since the calculated F (25.72) exceeds the critical F (6.94), we reject the null hypothesis and conclude that there is a significant difference due to treatments.

We see next that R can be used to do an RCBD analysis.

Exercise: Analyzing Another RCBD

You are a forage breeder and have been asked by your supervisor to analyze data from a variety trial in which 10 cultivars of red clover were evaluated for dry yield. The experimental design was an RCBD with four replications (4 randomized complete blocks, each consisting of all 10 cultivars in a randomized order). The yield data represent seasonal totals in tons/acre. Carry out an ANOVA on the yield data, and determine the effect of blocking on the partitioning of residual SS.

The data are in a .xls file found here . Download it and name it exercise.11.3.data.csv .

Ex. 3: Two-Factor ANOVA

Run the analysis of variance using a two-factor model (with Cultivar and Block as factors).

cult<-as.factor(data$Cultivar)

block<-as.factor(data$Block)

out <- summary(aov(Yield ~ cult + block))

              Df    Sum Sq    Mean Sq   F value   Pr(>F)

cult           9   0.04257   0.004730     5.326  0.00033  ***

block          3   0.07997   0.026657    30.013  9.5e-90  ***

Residuals     27   0.02398   0.000888

Signif. codes:  0  ‘***’  0.001  ‘**’  0.01  ‘*’  0.05  ‘.’  0.1  ‘ ‘  1

Look at the ANOVA table.

Ex. 3: Interpreting Results

Looking first at the F-test for Cultivar, we see that the calculated F-value is 5.326, which is significant at P = 0.00033. The chance of making a Type I Error in declaring that there is a difference in yield among the ten cultivars is very small, so we conclude that such a difference exists. You can also see from the ANOVA that blocking was much more effective in this example than in the previous one. In fact, Block explained more variation among our plots than did Cultivar. If you were to analyze this data as a CRD, you would find that Cultivar did not affect red clover yield in this experiment. The SS associated with Cultivar would not differ between the two analyses. What differs is that some of the variation associated with plots in the CRD analysis has been partitioned into a Block effect in the RCBD. Therefore, the error MS is smaller for the RCBD giving you more precision for the F-test and subsequent mean comparisons.

Ex. 3: Standard Error of the Mean (SEM)

The Standard Error of the Mean (SEM) is often reported in association with the treatment means of an experiment. The SEM is the square root of the variance of the mean. It is a very useful statistic for comparing means, as the SEM can be used to calculate significant ranges in a number of multiple comparison procedures, such as the Fisher’s LSD mean comparison. The SEM is calculated as (Equation 9):

[latex]SEM =\sqrt{\frac{2RMS}{r}}[/latex].

[latex]\textrm{Equation 9}[/latex] Formula for calculating standard error of the mean.

where: [latex]\small RMS[/latex]=residual (or error) mean square, [latex]r[/latex]= is the number of observations used to calculate the mean. Usually, r is equal to the number of replications or blocks.

Ex. 3: RCBD – Red Clover Variety Trial

Use the ANOVA table that you just made for the RCBD Red Clover variety trial to calculate the standard error of the mean for the experiment.

• Look up the error mean square in the analysis of variance table from Exercise 3.
• Compute the SEM from the above formula and the error MS from the ANOVA table.

The mean square of the residuals is 0.0888. The number of blocks per treatment is 4. We can use the sqrt command in R to calculate the square root of RMS/r.

sqrt(0.000888/4)

[1] 0.01489966

The SEM is, therefore, 0.01489966.

Ex. 4: Mean Comparisons with RCBD

• install.packages(””)

The mean comparison procedure we’ll use for the Red Clover variety trial is the least significant difference (LSD) comparison. This is because we are comparing a large number of qualitative treatments for which there are no obvious preplanned comparisons. The LSD tells us the minimum mean difference that we should consider between individuals in the sample population that we are analyzing.

Ex. 4: Calculating LSD

A useful formula for calculating an LSD is (Equation 10)

[latex]\textrm{LSD} = t\times\sqrt{\frac{2*MSE}{n}}[/latex].

[latex]\textrm{Equation 10}[/latex] Formula for calculating studentized range statistic.

where: [latex]\small MSE[/latex]= error mean square, [latex]n[/latex]= the number of observations used to calculate each mean.

 Remember that we calculated the Standard Error of the Mean (SEM) in the last exercise with the equation [latex]\textrm{SEM} = \sqrt{\frac{MSE}{n}}[/latex].

LSD can also be calculated as (Equation 11)

[latex]\textrm{LSD} = \textrm{t}\times\sqrt{{2*MSE}}[/latex].

[latex]\textrm{Equation 11}[/latex] Formula for calculating least significant difference, LSD.

 Ex. 4: LSD Calculation Exercise

In this exercise, we will calculate the LSD for comparing means of the Red Clover data using the following steps:

• Obtain the residual (or error) MS from the ANOVA table for the Red Clover variety trial. We did this in the last exercise. The ANOVA table is presented below.

out<- summary(aov(Yield ~ cult + block))

cult          9  0.04257  0.004730     5.326  0.00033  ***

block         3  0.07997  0.026657    30.013  9.5e-09  ***

Residuals    27  0.02398  0.000888

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ‘ 1

• Compute the LSD using the appropriate t-value.
• We can calculate the two-sided t-value at the 0.05 P level with 27 degrees of freedom and set it equal to variable t with the following command:

t<-abs(qt(0.5/2,27)) # Calculate the two-sided P-level of 0.05 with 27df

[1] 2.051831

Let us go through the command above: the P-level (or significance value) is specified after the second parenthesis (i.e. 0.05/2), and the residual degrees of freedom are entered to the right of the P-level after a comma (i.e. 27 in this example). The two-sided t value at the 0.05 P level with 27 degrees of freedom is thus 2.051831.

• Calculate the LSD from the ANOVA using Formula 1.

LSD<-t*sqrt((2*0.000888)/4)

[1] 0.04323475

Ex. 4: Second LSD Calculation

Good, now let’s go through the second LSD calculation using the SEM, which we calculated in the previous exercise. First, let us set the variable SEM equal to the SEM, then look at the answer R returns.

SEM <-sqrt(0.000888/4)

Great, now we can carry out the second LSD calculation by entering

LSD2<- t * SEM * sqrt(2)

You can easily see that the same result is obtained using both equations.

Ex. 4: Interpretation of LSD

Again, the LSD tells us the minimum mean difference that we should consider between individuals in the sample population that we are analyzing. In this example, the minimum mean difference that we would consider among the red clover cultivars is 0.432 tons per hectare.

Comparing RCBD Means

To perform an LSD comparison in R with the red clover data that we’ve been working with, we first need to install the ‘agricolae’ package, if you have not done so before, use the use the library command to access the functions in the package.

Now we can use the LSD.test command. Let’s set the output of the command equal to variable out

out<-LSD.test(data$Yield, data$Cultivar,27,0.000888, p.adj=”bonferroni”,group=TRUE)

The inputs in the parenthesis after the LDS.test function are as follows : data$Yield specifies the experimental unit (Yield), data$Cultivar indicates the treatment (Cultivar), 27 is the residual (or error) degrees of freedom from the ANOVA table, 0.000888 is the MSerror (also from the ANOVA table), p.adj=”bonferroni” indicates that we are using the Bonferroni p-value correction method, and group=TRUE indicates that each treatment (cultivar) should be treated as a separate group (mean calculations should be done for each cultivar).

Ex. 4: R Output

Let’s look at the output.

$statistics

   Mean        CV    MSerror         LSD

0.64345   4.63118   0.000888  0.07689099

$parameters

   Df  ntr  bonferroni

   27   10    3.649085

    data$Yield         std  r        LCL        UCL    Min    Max

1      0.66175  0.05882956  4  0.6311784  0.6923216  0.607  0.737

2      0.61075  0.03508442  4  0.5801784  0.6413216  0.577  0.643

3      0.63425  0.01590335  4  0.6063784  0.6648216  0.620  0.657

4      0.57675  0.03398406  4  0.5461784  0.6073216  0.541  0.611

5      0.64775  0.04358421  4  0.6171784  0.6783216  0.601  0.705

6      0.63300  0.09809519  4  0.6024284  0.6635716  0.526  0.734

7      0.69925  0.05518076  4  0.6686784  0.7298216  0.634  0.766

8      0.67950  0.09113177  4  0.6489284  0.7100716  0.559  0.763

9      0.63700  0.06243397  4  0.6064284  0.6675716  0.580  0.725

10     0.65450  0.04219400  4  0.6239284  0.6850716  0.628  0.717

$comparison

   trt    means    m

1   7   0.69925    a

2   8   0.67950   ab

3   1   0.66175   ab

4  10   0.65450   ab

5   5   0.64775  abc

6   9   0.63700  abc

7   3   0.63425  abc

8   6   0.63300  abc

9   2   0.61075   bc

10  4   0.57675    c

Ex. 4: Interpret the Results/Make a Decision

At the bottom of the results, you’ll find the mean data for each cultivar in the $groups table. In the column labeled M at the far right, we are given a new piece of information; the means of the 10 cultivars fall into 3 distinct groups based on the LSD. Means that have the same letters are not statistically different from one another; i.e. the difference between them is less than the LSD (0.0432 tons/ha). Cultivars 7, 8, and 1 clearly outyielded the others and should be the ones selected for advancement in the breeding program.

Blocking Efficiency

Blocking efficiency can be tested vs. CRD. Blocking is not always beneficial. When it is not necessary or is done inappropriately, blocking can actually reduce the precision of an experiment. Let us compare the results of using an RCBD to those which we would have obtained using a CRD for our sample experiment. First, we have the results using RCBD:

Blocking is a tradeoff; We reduce the error variance by blocking, but we also reduce the degrees of freedom that we use to determine our critical F-value.

Calculating Blocking Efficiency

Whenever we block, therefore, we must ask ourselves the following question: “Will the increase in our F-value for treatment be large enough to offset the increase in the critical F-value?” The relative efficiency of blocking for an RCBD experiment can be calculated as (Equation 12):

[latex]\small \textrm{Block Efficiency}= \normalsize \frac{(n_{B}+1)(n_{C}+3)MS_{eC}}{(n_{C}+a)(n_{B}+3)MS_{eB}} \times 100[/latex].

[latex]\textrm{Equation 12}[/latex] Formula for calculating blocking efficiency.

where: [latex]\small MS_eC[/latex]= error mean square for CRD, [latex]\small MS_eB[/latex]= error mean square for RCBD, [latex]n_C[/latex]= error df for CRD, [latex]n_B[/latex]= error df for RCBD.

Now you might expect that we can just estimate MS eC by running the analysis without blocks in the model as we did in Chapter 5 on Categorical Data—Multivariate. However, this does not give a proper estimate of the CRD error mean square over all possible randomizations, but rather just for the one having each treatment in each block. Cochran and Cox (1957 Experimental Designs, 2nd Edition, p.112) prove that MS eC should be estimated as (Equation 13):

[latex]\textrm{}MS_{eC}= \frac{df_{B}(MS_{B})+(df_{T}+df_{E})MS_{E}}{df_{B}+df_{T}+df_{E}}[/latex].

[latex]\textrm{Equation 13}[/latex] Formula for calculating error mean square for CRD.

where: [latex]MS_eC[/latex]= error mean square for CRD, [latex]df_B, df_T, df_E[/latex] = degree of freedom for blocks, treatments, and error in the RCBD ANOVA, [latex]MS_B, MS_E[/latex] = mean squares for blocks and for error in the RCBD ANOVA.

Calculating Error Mean Square for CRD

For our experiment, this is (using Equation 13):

[latex]\textrm{}MS_{eC}= \frac{2(0.997)+(2+4)0.484}{2+2+4}=0.607[/latex].

This error mean square estimate for a CRD is somewhat less than the error mean square computed by just re-running the model without blocks, which is 168.6.

A value for Blocking Efficiency greater than 1.00 suggests that we gained efficiency in our experiment by blocking. The blocking efficiency for the sample experiment is (using Equation 12):

[latex]\textrm{Block Efficiency}= \frac{(4+1)(6+3)0.607}{(6+1)(4+3)0.484} \times 100 =115.2[/latex].

Thus, our sample experiment was improved by blocking. The efficiency is about 15.2% greater than what it would have been in a CRD.

It has been said that one can “never lose by blocking.” While this is not always the case, it is true that blocking will generally improve an experiment whenever a production gradient is recognized and blocks are appropriately arranged across that gradient.

Reason for Blocking

• To achieve more homogeneous conditions for experimental units.
• Allows better separation of treatment effects and error.

RCBD Linear Model

• Includes term for Blocks.
• Error term is the Block * Treatment interaction.
• Has sources for Treatment, Block, and Error.
• Degrees of freedom are (t-1), (b-1) and (t-1)(b-1), respectively.
• Sums of squares are computed in the same manner (as in CRD).
• Mean Squares are SS/df.
• F = MST/MSE

Relative Efficiency of Blocking

• Can be compared with no blocking as in CRD.

How to cite this chapter : Harbur, M.L., K. Moore, R. Mowers, L. Merrick, and A. A. Mahama. 2023. Randomized Complete Block Design. In W. P. Suza, & K. R. Lamkey (Eds.), Quantitative Methods. Iowa State University Digital Press.

Process of assigning experimental units without any bias. Usually executed by random number generation.

A design where every plot has an equal chance of any treatment.

Chapter 11: Randomized Complete Block Design Copyright © 2023 by M. L. Harbur; Ken Moore; Ron Mowers; Laura Merrick; and Anthony Assibi Mahama is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License , except where otherwise noted.

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• Published: 16 October 2020

The “completely randomised” and the “randomised block” are the only experimental designs suitable for widespread use in pre-clinical research

• Michael F. W. Festing   ORCID: orcid.org/0000-0001-9092-4562 1  

Scientific Reports volume  10 , Article number:  17577 ( 2020 ) Cite this article

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• Experimental models of disease
• Medical research

Too many pre-clinical experiments are giving results which cannot be reproduced. This may be because the experiments are incorrectly designed. In “Completely randomized” (CR) and “Randomised block” (RB) experimental designs, both the assignment of treatments to experimental subjects and the order in which the experiment is done, are randomly determined. These designs have been used successfully in agricultural and industrial research and in clinical trials for nearly a century without excessive levels of irreproducibility. They must also be used in pre-clinical research if the excessive level of irreproducibility is to be eliminated. A survey of 100 papers involving mice and rats was used to determine whether scientists had used the CR or RB designs. The papers were assigned to three categories “Design acceptable”, “Randomised to treatment groups”, so of doubtful validity, or “Room for improvement”. Only 32 ± 4.7% of the papers fell into the first group, although none of them actually named either the CR or RB design. If the current high level of irreproducibility is to be eliminated, it is essential that scientists engaged in pre-clinical research use “Completely randomised” (CR), “Randomised block” (RB), or one of the more specialised named experimental designs described in textbooks on the subject.

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Introduction.

Excessive numbers of randomised, controlled, pre-clinical experiments give results which can’t be reproduced 1 , 2 . This leads to a waste of scientific resources with excessive numbers of laboratory animals being subjected to pain and distress 3 . There is a considerable body of literature on its possible causes 4 , 5 , 6 , 7 , but failure by scientists to use named experimental designs described in textbooks needs further discussion.

Only two designs are suitable for widespread use in pre-clinical research: “Completely randomised” (CR) shown in Fig.  1 A, and “Randomised block” (RB) shown in Fig.  1 B. In the CR design, each subject (experimental unit) has one of the treatments randomly assigned to it, so that subjects receiving different treatments are randomly intermingled within the research environment. Results can be statistically analysed using a one-way analysis of variance, with the usual assumptions of homogeneity of variances and normality of the residuals.

Representation of three experimental designs, each with three treatments (colours) with a sample size of four (for illustration). Each small rectangle represents an experimental unit (for example, a single animal in a cage). Designs A and B can have any number of treatments and sample sizes as, well as additional factors such as both sexes, or more than one strain. Design C is not statistically valid. ( A ) The “Completely randomised” (CR) design. Both assignment of treatments to subjects, and the order in which the experiment is done are randomly determined. This design can accommodate unequal sample sizes. Randomisation was done using EXCEL: Four “A”s, four “B”s and four “C”s were entered into column one and 12 random numbers were put in column two using the command “ = rand()”, and pulling down on the small box on the lower right of the cell. Columns one and two were then marked and sorted on column two using “data, sort”. The row numbers represent individual identification numbers. Different results will be obtained each time. ( B ) The “Randomised block” (RB) design. In this example the experiment has four blocks (outer rectangles) each having a single individual receiving each of the three treatments, in random order. The blocks can be separated in time and/or location. Randomisation was done as follows: Four “A”s, “B”s and “C”s, were put in column 1 and the numbers 1–4 repeated three times were put in column 2. Twelve random numbers were then put in column three, as above. All three columns were then marked and sorted first on column two and then on column three. Row numbers are the individual identity numbers. ( C ) The “Randomisation to treatment group” (RTTG) “design”. This is not a valid design because treatment and environmental effects are confounded. Any environmental effect that differs between groups may be mistaken for the effects of the treatment, leading to bias and irreproducible results.

In the RB design, the experiment is split up into a number of independent “blocks” each of which has a single subject assigned at random to each treatment. When there are only two treatments, this is known as a “matched pairs” design. The whole experiment consists of “N” such blocks where N is sample size. A two-way analysis of variance without interaction is used to analyse the results. The matched pairs design can also be analysed using a one-sample t-test.

Unfortunately, most authors appear to use the in-valid “Randomisation to treatment group” (RTTG) design, shown in Fig.  1 C. In this design, subjects are randomly assigned to physical treatment groups but the order in which the experiment is done is not randomised. This is not valid because each treatment group will occupy a different micro-environment, the effects of which may be mistaken for treatment effects, leading to bias and irreproduciblity.

The origin of randomized controlled experiments

Randomized controlled experiments have a long history of successful use in agricultural research. They were developed largely by R. A. Fisher in the 1920s as a way of detecting small but important differences in yield of agricultural crop varieties or following different fertilizer treatments 8 . Each variety was sown in several adjacent field plots, chosen at random, so that variation among plots growing the same and different crop varieties could be estimated. He used the analysis of variance, which he had invented in previous genetic studies, to statistically evaluate the results.

Fisher noted that in any experiment there are two sources of variation which need to be taken into account if true treatment differences are to be reliably detected. First, is the variation among the experimental subjects, due for example, to the number of grains in a given weight of seed, or to individual variation in a group of mice. Second is the variation caused during the course of the experiment by the research environment and in the assessment of the results. B o th types of variation must be controlled if bias and irreproducibility are to be avoided.

In most pre-clinical research the inter-individual variation can be minimised by careful selection of experimental subjects. But variation associated with the environment caused, for example, by cage location, lighting levels, noise, time of day and changes in the skill of investigators must also be considered. Fisher’s designs minimised bias by using uniform material and by replication and randomisation so that plots receiving different varieties were randomly “intermingled” in the research environment.

According to Montgomery 9 “By randomization we mean that both the allocation of the experimental material, and the order in which the individual runs or trials of the experiment are to be performed, are randomly determined”.

The RB design often provides better control of both inter-individual and environmental variation. Subjects within a block can be matched and each block has a small environmental footprint, compared with the CR design. In one example this resulted in extra power equivalent to using about 40% more animals 10 . The RB design is also convenient because individual blocks can be set up over a period of time to suit the investigator. Positive results will only be detected if the blocks give similar results, as assessed by statistical analysis 11 , 12 . Montgomery 9 ,p 12 even suggests that blocking is one of the three basic principles of experimental design, along with “replication” and “randomisation”.

Fisher and others invented a few other named designs including the “Split plot”, the “Latin square” and the “Cross-over” designs. These can also be used in pre-clinical research in appropriate situations 13 , although they are not discussed here.

The research environment is an important source of variation in pre-clinical research

In most pre-clinical experiments inter-individual variation can be minimised by choosing animals which are similar in age and/or weight. They will have been maintained in the same animal house and should be free of infectious disease. They may also be genetically identical if an inbred strain is used. So the research environment may be the main source of inter-individual variation.

Temporal variation due to circadian and other rhythms such as cage cleaning and feeding routines can affect the physiology and behaviour of the animals over short periods, as do physical factors such as cage location, lighting and noise 14 . If two or more animals are housed in the same cage they will interact, this can increase physiological variation. Even external factors such as barometric pressure can affect the activity of mice 15 . Staff may also become more proficient at handling animals, applying treatments, doing autopsies and measuring results during the course of an experiment, leading to changes in the quality of data.

To avoid bias, cages receiving different treatments must be intermingled (see Fig.  1 A,B), and results should be assessed “blind” and in random order. This happens automatically if subjects are only identified by their identification number once the treatments have been given.

The RB design, is already widely used in studies involving pre-weaned mice and rats 11 . No litter is large enough to make up a whole experiment. So each is regarded as a “block” and one of the treatments, chosen at random, is assigned to each pup within the litter. Results from several litters are then combined in the analysis 16 .

Possible confusion associated with the meaning of the word “group”

Research scientists are sometimes urged to “randomise their subjects to treatment groups”. Such advice is ambiguous. According to Chambers Twentieth Century Dictionary (1972) the word “ group ” can mean “a number of persons or things together” or “a number of individual things related in some definite way differentiating them from others” .

Statisticians involved in clinical trials sometimes write about “randomising patients to treatment groups”. Clearly, they are using the second definition as there are no physical groups in a clinical trial. But if scientists assign their animals to physical groups (“….things together”), they will be using the invalid “Randomisation to treatment group” (RTTG) design shown in Fig.  1 C, possibly leading irreproducibility.

A sample survey of experimental design in published pre-clinical papers

A survey of published papers using mice or rats was used to assess the use of CR, RB, or other named experimental designs. PubMed Central is a collection of several million full-text pre-clinical scientific papers that can be searched for specific English words. A search for “Mouse” and “Experiment” retrieved 682,264 papers. The first fifty of these had been published between 2014 and 2020. They were not in any obvious identification number or date order. For example, the first ten papers had been published in 2017, 17, 19, 19, 19, 18, 15, 16, 19, and 18. And the first two digits of their identification numbers were 55, 55, 66, 65, 66, 59, 71, 61, 46 and 48. In order to introduce a random element to the selection, only papers with an even identification number were used.

Each paper was searched for the words “random”, “experiment”, “statistical”, “matched” and other words necessary to understand how the experiments had been designed. Tables and figures were also inspected. The discipline and type of animals which had been used (wild-type, mutant, or genetically modified) was also noted. The aim was to assess the design of the experiments, not the quality of research.

Most papers involved several experiments, but the designs were usually similar. All were assessed and re-assessed, blind to the previous scores, after an interval of approximately 2 weeks. The results in seventeen of the papers were discordant so they were reassessed.

Papers which used laboratory mice

The results for mice and rats are summarised in Table 1 . Thirty six (72 ± 3.2%) of the “mouse” papers involved genetically modified or mutant mice. Each was assigned to one of three categories:

“Apparently well designed” (13 papers, 26 ± 1.6%). None of these papers mentioned either the CR or RB design by name, although a few of them appeared to have used one of these designs. For example, one stated: "All three genotypes were tested on the same day in randomized order by two investigators who were blind to the genotypes." This was scored as a CR design.

“Room for improvement” (22 papers, 44 ± 3.5%). None of these papers used the word “random” with respect to the assignment of treatments to the animals, or the order in which the experiment was done, although it was sometimes used in other contexts. So these papers had not, apparently, used any named experimental design, so were susceptible to bias.

“Randomised to group” (15, papers, 30 ± 2.1%). These papers stated that the subjects had been “Randomised to the treatment groups”. The most likely interpretation is that these were physical groups, so the experiments had used the statistically invalid RTTG design as shown in Fig.  1 C. However, as noted above, the word group is ambiguous. if it meant that one of the treatments, chosen at random, had been assigned to each animal, then this would have constituted a “Completely randomised” (CR) design. As the first interpretation seems to be most likely, these experiments were classified as being of doubtful validity.

Papers which used laboratory rats

A similar search in Pubmed on “rat” and “experiment” found 483,490 papers. The first 50 of these with even identification numbers were published between 2015 and 2020. Four of them used mutant or genetically modified, the rest used wild-type rats. Twenty two of them involved experimental pathology, nineteen behaviour, seven physiology, one immunology and one pharmacology. Again, it was only the quality of the experimental design which was assessed, not the biological validity of results.

Nineteen (38 ± 0.3%) of the rat papers were placed in the “Design acceptable” category. Those involving behaviour were of notably high statistical quality (and complexity). Three stated that they had used the “Matched pairs” design and one had used a RB design without naming it. None of them mentioned either the CR or RB designs.

In eleven (22 ± 2.9%) papers, the rats were assigned to treatment groups. So it is unclear whether these papers had used the valid CR design or the in-valid RTTG design as discussed above for mice, although the latter seems to be more likely.

The “Room for improvement” group consisted of 20 (40 ± 4.0%) of the papers. These had not used the word “random” with respect to the assignment of treatments to subjects or vice versa and there was no evidence that they had used the RB, CR or other recognised experimental designs.

Conclusions from the sample survey

Results for both mice and rats are summarised in Table 1 . The quality of the experimental design in papers involving rats was slightly higher than that involving mice (Chi-sq. = 1.84, p  = 0.04). This was largely due to the high quality of the behaviour (psychological) studies in the rat.

Combining the two species, 32 ± 4.7% of the papers were judged to have been designed and randomised to an acceptable standard, although none of them stated that they had used either the CR or RB design. One mouse paper had used a “Latin square” design. Another had used a “Completely randomised” design without naming it, and a mouse paper noted that “All experiments were performed independently at least three times.” Such repetition can lead to tricky statistical problems if results are to be combined 17 . Scientists wishing to build repeatability into their experiments could use the RB design, spreading the blocks over a period of time.

Discussion and conclusions

Names matter. People, places, species and scientific procedures have names which can be used to identify and describe a subject or a procedure. Experimental designs also have names; “Completely randomised”(CR), “Randomised block”(RB), “Latin square”, “Matched pairs” etc. These can be found in textbooks which describe the characteristics and uses of each design 13 . However, none of the papers in the above survey mentioned either the CR or the RB design by name, although these are the only designs suitable for general use.

The widespread use of the statistically in-valid RTTG design, which is not found in any reputable textbooks, may account for a substantial fraction of the observed irreproducibility. Organisations which support pre-clinical research and training should ensure that their literature and web sites have been peer reviewed by qualified statisticians and that they refer to named, statistically valid experimental designs.

The RB and CR designs are quite versatile. They can be used for any number of treatments and sample sizes as well as for additional factors such as both sexes or several strains of animals, often without increasing the total numbers.

The first clinical trials were supervised by statisticians who adapted the CR design for such work. But scientists doing pre-clinical research have received little statistical support, so it is not surprising that so many of their experiments are incorrectly designed. High levels of irreproducibility are unlikely to be found in pre-clinical research in the pharmaceutical industry because the “PSI”, (the Association of Statisticians in the UK pharmaceutical industry), has about 800 members employed in the U.K.

Irreproducibility is wasteful and expensive. The employment of more applied statisticians in Academia to assist the scientists doing pre-clinical research would be an excellent investment.

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The author wishes to thank Laboratory Animals Limited for financial support in the publication of this paper.

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Festing, M.F.W. The “completely randomised” and the “randomised block” are the only experimental designs suitable for widespread use in pre-clinical research. Sci Rep 10 , 17577 (2020). https://doi.org/10.1038/s41598-020-74538-3

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Randomized Complete Block Designs

Last updated on 2024-05-14 | Edit this page

• What is randomized complete block design?
• A randomized complete block design randomizes treatments to experimental units within the block.
• Blocking increases the precision of treatment comparisons.

Design issues

Imagine that you want to evaluate the effect of different doses of a new drug on the proliferation of cancer cell lines in vitro. You use four different cancer cell lines because you would like the results to generalize to many types of cell lines. Divide each of the cell lines into four treatment groups, each with the same number of cells. Each treatment group receives a different dose of the drug for five consecutive days.

Group 1: Control (no drug) Group 2: Low dose (10 μM) Group 3: Medium dose (50 μM) Group 4: High dose (100 μM)

When analyzing a random complete block design, the effect of the block is included in the equation along with the effect of the treatment.

Randomized block design with a single replication

Sizing a randomized block experiment, true replication, balanced incomplete block designs.

• Replication, randomization and blocking determine the validity and usefulness of an experiment.

Plant Breeding and Genomics

Randomized Complete Block Design

Trudi Grant, The Ohio State University

Introduction

The randomized complete block design (RCBD) is a standard design for agricultural experiments in which similar experimental units are grouped into blocks or replicates. It is used to control variation in an experiment by, for example, accounting for spatial effects in field or greenhouse. The defining feature of the RCBD is that each block sees each treatment exactly once.

In the event that the slideshare tutorial does not work, the tutorial is also attached as a pdf at the bottom of the page.

External Links

• SAS [Online]. SAS Institute. Available at: http://www.sas.com/ (verified 30 Aug 2012).

Additional Resources

• Clewer, A. G., and D. H. Scarisbrick. 2001. Practical statistics and experimental design for plant and crop science. John Wiley & Sons Ltd., New York.

Funding Statement

Development of this lesson was supported in part by the National Institute of Food and Agriculture (NIFA) Solanaceae Coordinated Agricultural Project, agreement 2009-85606-05673, administered by Michigan State University. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the view of the United States Department of Agriculture.

Attachments:

Randomized Complete Block Design Tutorial.pdf (1.18 MB)

PBGworks 924

• 1. Introduction
• 2. Concepts
• 3. Experimental Design I
• 4. Data Exploration
• 5.1 Statistical Inference 1
• 5.2 Statistical Inference 2
• 6. Linear Regression
• 7. Analysis of Variance
• 8. Multiple Regression
• 8.2 Multiple Regression: Factorial Designs
• 8.3. Experimental Design II
• 8.4. Experimental Design II
• 9.1. Nonparametric Statistics 1
• 9.2. Nonparametric Statistics 2
• 10. Categorical Data Analysis
• 5. Statistical inference
• 6. Simple linear regression
• 8. Multiple linear regression
• 9. Non-parametric statistics
• Show All Code
• Hide All Code
• Download Rmd

8.4 Experimental Design III: Randomized Complete Block Designs and Pseudo-replication

Lieven clement, statomics, ghent university ( https://statomics.github.io ).

1 Randomized complete block designs

\[\sigma^2= \sigma^2_{bio}+\sigma^2_\text{lab} +\sigma^2_\text{extraction} + \sigma^2_\text{run} + \ldots\]

• Biological: fluctuations in protein level between mice, fluctations in protein level between cells, …
• Technical: cage effect, lab effect, week effect, plasma extraction, MS-run, …

2 Nature methods: Points of significance - Blocking

https://www.nature.com/articles/nmeth.3005.pdf

Oneway anova is a special case of a completely randomized design:

• the experimental units are sampled at random from the population
• the treatments are randomized to the experimental units
• Every experimental unit receives one treatment and its response variable is measured once.

In a block design the experimental units are blocks which are sampled at random of the population of all possible blocks.

• The RCB restricts randomization: the treatments are randomized within the blocks.
• it cannot be analysed using oneway anova.
• The paired design is a special case of an RCB: block size equals 2.
• Within block effects can be assessed using the lm function in R.

3 Mouse example

3.1 background.

Duguet et al. (2017) MCP 16(8):1416-1432. doi: 10.1074/mcp.m116.062745

• All treatments of interest are present within block!
• We can estimate the effect of the treatment within block!

We focus on one protein

• The measured intensities are already on the log2-scale. Differences between the intensities can thus be interpreted as log2 FC.
• P16045 or Galectin-1.
• Function: “Lectin that binds beta-galactoside and a wide array of complex carbohydrates. Plays a role in regulating apoptosis, cell proliferation and cell differentiation. Inhibits CD45 protein phosphatase activity and therefore the dephosphorylation of Lyn kinase. Strong inducer of T-cell apoptosis.” (source: uniprot )

3.2 Data Exploration

The plots show evidence for - an upregulation of the protein expression in regulatory T-cells and - a considerable variability in expression from animal to animal!

3.3 Paired Analysis

This is a paired design, which is the most simple form of randomized complete block design.

In the introduction to statistical inference we would have analyzed the data by differencing.

3.3.1 Data exploration

• Boxplot of difference

• Summary statistics

Note, that the intensity data are not independent because we measured the expression in regulatory and ordinary T-cells for every animal

• Covariance and correlation between expression in both celltypes

There is indeed a large correlation between the expression of the protein in conventional and regulatory T-cells.

Standard deviation of difference?

\[ \begin{array}{lcl} \text{sd}_{x_r - x_c} &=& \sqrt{1^2\hat \sigma_r^2 + (-1)^2 \hat \sigma_c^2 + 2 \times 1 \times -1 \times \hat\sigma_{r,c}}\\ &=&\sqrt{\hat \sigma_r^2 + \hat \sigma_c^2 - 2 \times \hat\sigma_{r,c}} \end{array} \]

• The standard deviation on the difference is much lower because of the strong correlation!
• Note, that the paired design enabled us to estimate the difference in log2-expression between the two cell types in every animal (log2 FC).

3.4 Randomized complete block analysis

We can also analyse the design using a linear model, i.e. with

• a main effect for cell type and
• a main effect for the block factor mouse

Because we have measured the two cell types in every mouse, we can thus estimate the average log2-intensity of the protein in the T-cells for each mouse.

If you have doubts on that your data violates the assumptions you can always simulate data from a model with similar effects as yours but where are distributional assumptions hold and compare the residual plots.

The deviations seen in our plot are in line with what those observed in simulations under the model assumptions. Hence, we can use the model for statistical inference.

Notice that

the estimate, se, t-test statistic and p-value for the celltype effect of interest is the same as in the paired t-test!

the anova analysis shows that we model the total variability in the protein expression in T-cells using variability due to the cell type (CT), the variability from mouse to mouse (M) and residual variability (R) within mouse that we cannot explain. Indeed,

\[ \begin{array}{lcl} \text{SSTot} &=& \text{SSCT} + \text{SSM} + \text{SSR}\\ 14.6 &=& 5.1 + 9.1 + 0.4 \end{array} \]

So the celltype and the mouse effect explain respectively \[ \begin{array}{ll} \frac{\text{SSCT}}{\text{SSTot}}\times 100&\frac{\text{SSM}}{\text{SSTot}}\times 100\\\\ 34.7& 62.4\\ \end{array} \]

percent of the variability in the protein expression values and \[ \frac{\text{RSS}}{\text{SSTot}} \times 100 = 2.8 \] percent cannot be explained.

• the variability from mouse to mouse is the largest source of variability in the model,
• This variability can be estimated with the RCB design and
• can thus be isolated from the residual variability
• leading to a much higher precision on the estimate of the average log2 FC between regulatory and conventional T-cells that what would be obtained with a CRD!

Note, that the RCB also has to sacrifice a number of degrees of freedom to estimate the mouse effect, here 6 DF.

Hence, the power gain of the RCB is a trade-off between the variability that can be explained by the block effect and the loss in DF.

If you remember the equation of variance covariance matrix of the parameter estimators \[ \hat{\boldsymbol{\Sigma}}^2_{\hat{\boldsymbol{\beta}}} =\left(\mathbf{X}^T\mathbf{X}\right)^{-1}\hat\sigma^2 \]

you can see that the randomized complete block will have an impact on

• \(\left(\mathbf{X}^T\mathbf{X}\right)^{-1}\) as well as
• on \(\sigma^2\) of the residuals!

\(\rightarrow\) We were able to isolate the variance in expression between animals/blocks from our analysis!

\(\rightarrow\) This reduces the variance of the residuals and leads to a power gain if the variability between mice/blocks is large.

Also note that,

\[ \hat\sigma^2 = \frac{\text{SSR}}{n-p} = \frac{SSTot - SSM - SSCT}{n-p} \]

• Hence, blocking is beneficial if the reduction in sum of squares of the residuals is large compared to the degrees of freedom that are sacrificed.
• Thus if SSM can explain a large part of the total variability.

Further, the degrees of freedom affect the t-distribution that is used for inference, which has broader tails if the residual degrees of freedom \(n-p\) are getting smaller.

4 Power gain of an RCB vs CRD

In this section we will subset the original data in two experiments:

• A randomized complete block design with three mice
• A completely randomized design with six mice but where we only measure one cell type in each mouse.

So in both experiments we need to do six mass spectrometry runs.

• we are able to pick up the upregulation between regulatory T-cells and ordinary T-cells with the RCB but not with the CRD.
• the standard error of the \(\log_2\text{FC}_\text{Treg-Tcon}\) estimate is a factor 4.4 smaller for the RCB design!

A poor data analysis who forgets about the blocking will be back at square one:

So, the block to block variability is then absorbed in the variance estimator of the residual.

Of course, we are never allowed to analyse an RCB with a model for a CRD (without block factor) because blocking imposes a randomization restriction: we randomize the treatment within block.

4.1 Powergain of blocking?

4.1.1 power for completely randomized design.

Because we have a simple 2 group comparison we can also calculate the power using the power.t.test function.

4.2 Power for randomized complete block design

Note, that the power is indeed much larger for the randomized complete block design. Both for the designs with 6 and 14 mass spectrometer runs.

Because we have an RCB with a block size of 2 (paired design) we can also calculate the power using the power.t.test function with type = "one.sample" and sd equal to the standard deviation of the difference.

Note, that the power is slightly different because for the power.t.test function we conditioned on the mice from our study. While in the simulation study we generated data for new mice by simulating the mouse effect from a normal distribution.

4.3 Impact of the amount of variability that the blocking factor explains on the power?

We will vary the block effect explains \[ \frac{\sigma^2_\text{between}}{\sigma^2_\text{between}+\sigma^2_\text{within}}=1-\frac{\sigma^2_\text{within}}{\sigma^2_\text{between}+\sigma^2_\text{within}} \] So in our example that is the ratio between the variability between the mice and the sum of the variability between and within the mice. Note, that the within mouse variability was the variance of the errors of the RCB. The ratio for our experiment equals

So if the variance that is explained by the block effect is small you will loose power as compared to the analysis with a CRD design. Indeed, then

• SSR does not reduce much and
• n \(_\text{blocks}\) -1 degrees of freedom have been sacrificed.

As soon as the block effect explains a large part of the variability it is very beneficial to use a randomized complete block design!

Note, that the same number of mass spectrometry runs have to be done for both the RCB and CRD design. However, for the RCB we only need half of the mice.

5 Penilin example

The production of penicillin corn steep liquor is used. Corn steep liquor is produced in blends and there is a considerable variability between the blends. Suppose that

• four competing methods have to be evaluated to produce penicillin (A-D),
• one blends is sufficient for four runs of a penicillin batch reactor and
• the 20 runs can be scheduled for the experiment.

How would you assign the methods to the blends.

5.2 Analysis

We analyse the yield using

• a factor for blend and
• a factor for treatment.

We conclude that the effect of the treatment on the penicillin yield is not significant at the 5% level of significance (p = 0.34.

We also observe that there is a large effect of the blend on the yield. Blend explains about 47.1% of the variability in the penicillin yield.

6 Pseudo-replication

A study on the facultative pathogen Francisella tularensis was conceived by Ramond et al. (2015) [12].

F. tularensis enters the cells of its host by phagocytosis.

The authors showed that F. tularensis is arginine deficient and imports arginine from the host cell via an arginine transporter, ArgP, in order to efficiently escape from the phagosome and reach the cytosolic compartment, where it can actively multiply.

In their study, they compared the proteome of wild type F. tularensis (WT) to ArgP-gene deleted F. tularensis (knock-out, D8).

The sample for each bio-rep was run in technical triplicate on the mass-spectrometer.

We will use data for the protein 50S ribosomal protein L5 A0Q4J5

6.1 Data exploration

• Experimental unit?
• Observational unit?

\(\rightarrow\) Pseudo-replication, randomisation to bio-repeat and each bio-repeat measured in technical triplicate. \(\rightarrow\) If we would analyse the data using a linear model based on each measured intensity, we would act as if we had sampled 18 bio-repeats. \(\rightarrow\) Effect of interest has to be assessed between bio-repeats. So block analysis not possible!

If the same number of pseudo-replicates/technical replicates are available for each bio-repeat:

• average first over bio-repeats to obtain independent measurements
• averages will then have the same precision
• assess effect of treatment using averages
• Caution: never summarize over bio-repeats/experimental units

6.2 Wrong analysis

Note, that the analysis where we ignore that we have multiple technical repeats for each bio-repeat returns results that are much more significant because we act as if we have much more independent observations.

6.2.1 No Type I error control!

So we observe that the analysis that does not account for pseudo-replication is too liberal!

The analysis that first summarizes over the technical repeats leads to correct p-values and correct type I error control!

What to do when you have an unequal number of technical repeats: more advanced methods are required

• e.g. mixed models
• The mixed model framework can model the correlation structure in the data

Mixed models can also be used when inference between and within blocks is needed, e.g. split-plot designs.

But, mixed models are beyond the scope of the lecture series.

7 Nature methods: Split-plot designs

Nature Methods - Point of Significance - Split-plot Designs

Randomized Block Design

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A randomized block design is an experimental design where the experimental units are in groups called blocks. The treatments are randomly allocated to the experimental units inside each block. When all treatments appear at least once in each block, we have a completely randomized block design. Otherwise, we have an incomplete randomized block design.

This kind of design is used to minimize the effects of systematic error. If the experimenter focuses exclusively on the differences between treatments, the effects due to variations between the different blocks should be eliminated.

See experimental design .

A farmer possesses five plots of land where he wishes to cultivate corn. He wants to run an experiment since he has two kinds of corn and two types of fertilizer. Moreover, he knows that his plots are quite heterogeneous regarding sunshine, and therefore a systematic error could arise if sunshine does indeed facilitate corn cultivation.

The farmer divides the land into...

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© 2008 Springer-Verlag

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(2008). Randomized Block Design. In: The Concise Encyclopedia of Statistics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-32833-1_344

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7.3 - randomized complete block design (rcbd).

A CRD for the greenhouse experiment is reasonable, provided the positions on the bench are equivalent. In reality, this is rarely the case. For example, some micro-environmental variation can be expected due to the glass wall on one end, and the open walkway at the other end of the bench.

A powerful alternative to the CRD is to restrict the randomization process to form blocks . In a block design, general blocks are formed such that the experimental units are expected to be homogeneous within a block and heterogeneous between blocks. The number of experimental units within a block is called its block size.

In a randomized complete block design (RCBD) , each block size is the same and is equal to the number of treatments (i.e. factor levels or factor level combinations). Furthermore, each treatment will be randomly assigned to exactly one experimental unit within every block. So if we think of the greenhouse example, in a RCBD we will have 6 blocks, each with block size of 4 (the number of fertilizer levels).

To establish an RCBD for this data, the assignments of fertilizer levels to the experimental units (the potted plants) have to be done within each block separately. Examples of how to do this using statistical software will be detailed in the section below. 

It is important to mention that blocks are usually (but not always) treated as random effects as they typically represent the population of all possible blocks. In other words, the mean comparison among specific blocks is not of interest. However, the variation between blocks must be incorporated into the model. In a RCBD, the variation between blocks is partitioned out of the MSE of the CRD, resulting in a smaller MSE for testing hypotheses about the treatments. 

The statistical model corresponding to the RCBD is similar to the two-factor studies with one observation per cell (i.e. we assume the two factors do not interact). 

Here is Dr. Shumway stepping through this experimental design in the greenhouse.

We will consider the greenhouse experiment with one factor of interest (Fertilizer). We also have the identifications for the blocks. In this example, we consider  Fertilizer  as a  fixed  effect (as we are only interested in comparing the 4 fertilizers we chose for the study) and  Block  as a  random  effect.

Therefore the statistical model would be

\(Y_{ij} = \mu + \rho_i + \tau_j + \epsilon_{ij}\)

with \(i=1,2,\dots,6\) and \(j=1,2,3,4\). \(\rho_i\)s and \(\epsilon_{ij}\) are independent random variables such that \(\rho_i \sim \mathcal{N}\left(0, \sigma^2_{\rho}\right)\) and  \(\epsilon_{ij}\sim \mathcal{N}\left(0, \sigma^2\right)\).

Using Technology Section  

•   Example

To obtain the block design in SAS, we can use the following code:

The output we obtain would be as follows:

Once we collect the data for this experiment, we can use SAS to analyze the data and obtain the results.

Let us read the data into SAS and obtain the proc summary  output.

The proc summary  output would be as follows. We see that the first line in the table with _TYPE_=0 identification is the estimated overall mean (i.e. \(\bar{y}_{\cdot\cdot}\)). The estimated treatment means (i.e. \(\bar{y}_{\cdot j}\)) are displayed with _TYPE_=1  identification and the estimated block means are displayed with _TYPE_=2  identification. Since we only have one observation per treatment within each block, we cannot estimate the standard error using the data.

To run the model in SAS we can use the following code

We obtain the ANOVA table below for the RCBD.

For comparison, let us obtain the ANOVA table for the CRD for the same data. We use the following SAS commands:

The CRD ANOVA table for our data would be as follows:

Comparing the two ANOVA tables, we see that the MSE in RCBD has decreased considerably in comparison to the CRD. This reduction in MSE can be viewed as the partition in SSE for the CRD (61.033) into SSBlock (53.32) + SSE (7.715). The potential reduction in SSE by blocking is offset to some degree by losing degrees of freedom for the blocks. But more often than not, is worth it in terms of the improvement in the calculated F -statistic. In our example, we observe that the F -statistic for the treatment has increased considerably for RCBD in comparison to CRD. It is reasonable to assume that the result from the RCBD is more valid than that from the CRD as the MSE value obtained after accounting for the block to block variability is a more accurate representation of the random error variance.

To obtain the design in Minitab, we do the following.

For Block 1, manually create two columns: one with each treatment level and the other with a position number 1 to n, where n is the block size (i.e. n = 4 in this example). The third column will store the assignment of fertilizer levels to the experimental units.

Next, select Calc >  Sample from Columns > fill in the dialog box as seen below, and click OK .

Here, the number of rows to be specified is our block size (and number of treatment levels) which yields a random assignment from Block 1.

The same process should be repeated for the remaining blocks. The key element is that each treatment level or treatment combination appears in each block (forming complete blocks), and is assigned at random within each block.

To obtain the RCBD in R, we can use the blocks function from the blocksdesign package.

We can then construct a data frame to view the design.