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.

Blocking

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.

Treatments

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.

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.

Exercise 1: Randomizing Treatments For a RCBD

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.

Steps:

1. Enter the formula =RAND() in cell D2.
2. 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.
3. 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!
4. Select Sort from the Data menu at the top of the main window.
5. In the Sort by box in the Sort dialog box, select Block.
6. If not already present, add another sort level by clicking on Add Level.
7. 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

• matrix
• for
• gl
• runif
• cbind
• _[order]

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 field into the Console.

field

    [,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)

pop

[1] 1 2 3

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)

rand

[1] 0.1165839 0.5730972

0.3469669

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)

block

    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.

block<-cbind(pop,rand)

block

   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,]

field

    [,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).

field<-matrix(0,3,3)

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)

rand<-runif(3)

block<-cbind(cultivar,rand)

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

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

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

field

   [,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)

rand<-runif(3)

block<-cbind(pop,rand)

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

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.

for (i in 1:3)

{pop<-gl(3,1)

rand<-runif(3)

block<-cbind(pop,rand)

randblock <-

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

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

Look at the field you’ve just created.

field

    [,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.

field<-matrix(0,3,3)

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

for (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)

rand<-runif(3)

block<-cbind(cultivar,rand)

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

field[,i]<-randblock[,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.

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.

R Code Functions

• read.csv
• as.factor
• attach
• summary
• aov
• sqrt

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.

Summary

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.

Analysis of Variance for RCBD

• 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.