13.3.1 The CRD Model

Let denote the experimental unit for the treatment, with , and . We assume that the number of observations for each treatment is the same as for any other treatment. Suppose that the experimenter believes that the average yield due to treatment is . The CRD model is then expressed by

13.1

where . This model 13.1 is known as the means model. The more general and useful mathematical model for the CRD is

13.2

The CRD model 13.2 in this form is known as the effects model. The effects model is more feasible from a practical point of view. In this form the mean is thought of as some guaranteed yield in the absence of any kind of treatment. This parameter is also known as the baseline or control treatment. The parameter values are a reflection of the effect due to the treatment . We will consider the second format of the model throughout the rest of this section. The error component, or the noise factor, are noise factors associated with the experimental unit. We will assume that the errors are iid as , where the variance of the normal distribution is not known. Thus, the CRD model says that the probability distribution of the experimental units is . The means and effects model are also related in the sense of defining .

In both models 13.1 and 13.2, each treatment receives an equal number of experimental units, that is, each treatment receives number of units. These kinds of models are called balanced designs. In practical setups, it may not be feasible to allocate equal numbers of units, and we allow the -th treatment of , number of units. In this case, the model is called the unbalanced design. The inferential aspects of balanced or unbalanced models do not vary drastically from each other, at least for the CRD model, and the coverage for the balanced model is provided in the rest of this section.

The movement to Design Matrix from Covariate Matrix. The covariates 's are missing in the effects model 13.2! In fact, the 's will not appear in the rest of this chapter either. Recollect that the median polish model 4.14 also did not have the covariates 's. However, the covariates are very much present in these models and they have a well-defined format. Indeed, the covariates are designed to appear in a specific way and this is the reason why we call the covariate matrix the Design Matrix. The models in this broad area completely determine the exact structure of the covariate matrix. In R, the function model.matrix will generate the exact design matrix, and we will come to this function later.

Consider the effects in model 13.2. Suppose we have treatments and observations for each of the treatments. Let the treatment effect be denoted by , and let take the value of 1 if treatment is assigned to the observation, and 0 otherwise, . The design matrix is then defined as in Table 13.1. Note that it is important to drop one of the !

Observation	Intercept
1	1	1	0
2	1	1	0
3	1	1	0
4	1	1	0
5	1	0	1
6	1	0	1
7	1	0	1
8	1	0	1
9	1	0	0
10	1	0	0
11	1	0	0
12	1	0	0

The next small sub-section will help in random allocation of the treatments to the experimental units.

13.3.2 Randomization in CRD

Suppose that we have number of treatments and that the treatment, , is allocated number of experimental units. Let be the total number of available experimental units.

The statistical inference for the CRD model will now be discussed.

13.3.3 Inference for the CRD Models

A few standard notations are in order. The treatment sample sum (mean), denoted by () and total sample sum (mean), (), are defined by

13.3

13.4

Define the total (corrected) sum of squares, denoted by , as

13.5

The ANOVA technique partitions the as the sum of two components: (i) the sum of squares due to treatments , and (ii) the sum of squares due to error . Here, and are defined by

13.6

13.7

Note that accounts for the between treatments effect, and accounts for the within treatments difference. Here, has degrees of freedom, whereas and have respectively and degrees of freedom. It is easy to verify that

Define

That is, is the sampling variance of the -th treatment. We can pool these sampling variances and obtain the following:

where denotes the mean error sum of squares. Note that is an estimator of the variance for the -th treatment, and it may be seen that is an estimator of the common variance within each of the treatments. Similarly, we can also use the variation of the treatment averages from the grand average, under the assumption that there is no difference among the treatment means, for estimation of . That is, the mean treatment sum of squares is given by

An interesting hypothesis testing problem is about the equality of effect of the treatment means: against the alternative . The details can then be presented in the ANOVA Table 13.2.

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	-Statistic
Between Treatments
Error within Treatments
Total

In light of Theorem 6.6.2, the sampling distribution of and may be seen as an -distribution with and degrees of freedom respectively. Finally, the sampling distribution of is seen to be an -distribution, see Theorem 6.3.4, with degrees of freedom.

Before a formal illustration of the CRD model, let us take an EDA route for understanding ANOVA. The package granova has a very interesting graphical tool granova.1w, where 1w stands for “one-way layout”. The illustration through example(granova.1w) is first considered. The granova may be useful for outlier identification, skewness, etc.

In the next example, we will find out how R handles the covariates for the CRD model!

We will consider one more example for the CRD model 13.2 from Dean and Voss (1999).

Validation of the model assumptions is considered next.

13.3.4 Validation of Model Assumptions

The CRD model is a linear regression model, as in 12.1. The assumptions for the model are the same as detailed in Section 12.2, that is, linearity, independent, and normality assumptions. For the model under consideration, the plots, as in Section 12.2.6, give us all that is required. The demonstration is carried out for Example 13.3.3.

13.3.5 Contrasts and Multiple Testing for the CRD Model

Let us begin with a definition.

The statistical interest is in the problem of testing the hypothesis against the alternative . An estimate of the contrast is given by

The test procedure is to reject if , where

The test statistic is rewritten as

where and is defined by

Some contrasts are considered in the next example.

The problem of multiple testing was dealt with in Section 17.15. Here, the results are specialized to the CRD model.

If the hypothesis is rejected, we know that at least two treatment levels have significantly different effects. This calls for techniques which will help the reader to identify which treatments are significantly different. The details of the multiple comparison problem may be found in Miller (1981), Hsu (1996), and Bretz, et al. (2011).

For the multiple testing problem we are familiar with the Bonferroni's method and the Holm's method. Tukey's honest significant differences, HSD, and Dunnett's procedures are detailed next.

In a CRD model with treatment levels, the interest is in comparison of the equality for each possible combination of the levels, that is, there are possible hypotheses. Tukey's procedure uses the distribution of the Studentized range statistic:

13.9

where and are the largest and smallest sample means out of the possible means. Tukey's HSD procedure declares two means to be significantly different if the absolute value of their differences exceeds

where is the upper percentage point of and is the number of degrees of freedom associated with the . Thus, a % confidence interval for all possible pairs of means is given by

Suppose that one of the treatment levels is a control level and comparisons are required with respect to this control level. The Dunnett's procedure offers a solution in this case. The hypotheses are then , which then need to be tested against the set of hypotheses . The Dunnett's procedure begins with computation of the differences:

The test procedure is to reject the hypothesis at size if

where corresponds to Dunnett's distribution.

The four methods of multiple testing are next illustrated for the tensile strength experiment.

The tensile strength experiment is an example of a balanced design. Fortunately, there is a slight change of , instead of the , but all the results continue to hold. In R there are no changes in the structure of the commands and functionality. Thus, the user can easily solve the multiple testing problem for the Olson dataset, which is an example of an unbalanced design.

13.4.1 Randomization and Analysis of Balanced Block Designs

In a block model, a comparison needs to be carried out for the different treatment levels and blocks. In a balanced block design, each block will have one observation per treatment level. It is to be noted that randomization is only carried out on the treatments within a block. The effects model for a balanced block design model is specified by

13.10

where is the overall mean, is the effect of the -th treatment, and is the effect of the -th block. Also, define . As previously we assume that the error term follows a Gaussian distribution .

The randomization for the above design 13.10 is illustrated next.

Towards inference for the randomized block design, define the following quantities:

The decomposition of the total sum of squares is given below:

13.11

where

Observe that has degrees of freedom, has df, has df, and finally, has df. Thus, the ANOVA table is then set up as given in Table 13.3. The balanced block design will be illustrated now.

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	-Statistic
Between Treatments
Between Blocks
Error
Total

As seen in Models 13.2, 13.10, etc., these linear models need to be investigated for model assumptions and other regression diagnostics too. Can it be said without loss of generality that leverages are not a concern in DOE since all the covariate values are fixed? Now, we consider an example from Montgomery (2005), wherein we will investigate issues related to model adequacy.

13.4.2 Incomplete Block Designs

The constraints of the real world may not allow each block to have an experimental unit for every treatment level. The restriction may force allocation of only units within each block. Such models are called incomplete block designs. If in an incomplete block design, any two treatments pair appearing together, across the blocks, occurring an equal number of times, the designs are then called balanced incomplete block designs, abbreviated as BIBD. It may be seen that a BIBD may be constructed in different ways.

The setup is now recollected again. We have treatments, distinct blocks, is the number of units appearing in a block, and is the total number of observations, where is the number of times a treatment appears in the design. Then, the number of times each pair of treatments appears in the same block is given by

13.12

The statistical model for BIBD is given by

13.13

where is the observation in the block, is the overall mean, is the effect of the treatment, and is the effect of the block. The error term is assumed to follow . The total variability needs to be handled in a slightly different manner. Define

13.14

where

Using the 's, the adjusted treatment sum of squares is defined by

Thus, the total variability is partitioned by

13.15

where

The ANOVA table for BIBD model is given in Table 13.4.

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	-Statistic
Between Treatments
Between Blocks
Error
Total

The function BIB.test from the agricolae package is useful to fit a BIBD model.

Two very important variations of the BIBD will be considered in the rest of the section.

13.4.3 Latin Square Design

In each of the Examples 13.4.1 to 13.4.4, and the blocking models 13.2 and 13.10, we had a single source of known and controllable source of variation and we use the blocking principle to overcome this. Suppose now that there are two ways or sources of the variation which may be controlled through blocking! Consider the following example from Montgomery (2001), page 144.

In the above example, we need to ensure that each type of formulation occurs exactly once for the type of raw material and also that each formulation is prepared exactly once by each of the five operators. This type of problem is handled by the Latin Square Design, abbreviated as LSD, and we use the Latin letters to represent the type of formulations. A simple technique is to set up an LSD for say formulations, set up in the first row of the design as 1, 2, …, , the second row as 2, 3, …, ,1, the third row as 3, 4, …, , 2, 1, and so forth until the last row as , , …, 3, 2, 1. There are many other techniques available for creating an LSD, such as Wichmann-Hill, Marsaglia-Multicarry, Mersenne-Twister, Super-Duper, etc. A couple of illustrations are given next using the design.lsd function from the agricolae package.

The LSD model and its statistical analyses is now detailed. The LSD statistical model is specified by

13.16

Here, will continue to represent the effect of the -th treatment (or formulation), for the row (block) effect (raw material), and for the column (block) effect (operator). The errors are assumed to follow normal distribution . The ANOVA decomposition of the sum of squares for the LSD model 13.16 will be as follows:

13.17

where

The ANOVA table for the LSD is given in Table 13.5. For Example 13.4.5, the R method is illustrated next.

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	-Statistic
Between Treatments
Between Rows
Between Columns
Error
Total

There are many other useful variants of LSD and the reader may refer to Chapter 10 of Hinkelmann and Kempthorne (2008) for the same. An extension of the LSD is considered in the next topic.

13.4.4 Graeco Latin Square Design

Suppose that there is an extra source of randomness for the Rocket Propellant problem in Example 13.4.5 in test assemblies, which forms an additional type of treatment. Now, it is obvious that this is a second treatment which may be again addressed by another LSD. Let us denote the second treatment by Greek letters, say , , , , and . There is symbolic confusion in the sense that the Greek letters of , , and may be confused with the notations used in Model 13.16. However, we will use the notations as elements of the LSD matrix. It should be clear from the context whether the notations are as in the Greek letters required or in the statistical model. Now, the LSD matrices for the two treatments are superimposed on each other to obtain the Graeco-Latin Square Design model. Table 13.6 shows how superimposing is done for two simple LSDs to obtain the GLSD.

LSD			GSD			GLSD
A	B	C				A	B	C
B	C	A				B	C	A
C	A	B				C	A	B

In more formal terms, the Graeco-Latin square design model, abbreviated as GLSD, is specified by

13.18

where now denotes the effect of the additional treatment. A couple of GLSDs will be first set up.

The ANOVA decomposition of the sum of squares for the GLSD model 13.18 will be as follows:

13.19

where

The ANOVA table for the GLSD is given in Table 13.7. For Example 13.4.5, the R method is illustrated next.

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	-Statistic
Between Latin Treatments
Between Greek Treatments
Between Rows
Between Columns
Error
Total

We now move to more important and complex experimental designs.

13.5.1 Two Factorial Experiment

Consider the case of two factor variables. Suppose that factor is at levels, and at levels. The two factorial design model is given by

13.20

Here, is the overall mean effect, is the effect of the -th factor level of , is the -th effect of the factor , and is the interaction effect between the variables. The ANOVA decomposition of the sum of squares for the two factorial experiments is given by

13.21

where

The computation of involves two stages. Since such issues do not exist when we use R, we skip this detail. The ANOVA table for the two-factorial model is given in Table 13.8. The hypotheses problems of interest here are the following:

1. 1. Testing for Factor :
2. 2. Testing for Factor :
3. 3. Testing for the Interaction of factors:

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	-Statistic
Treatment A
Treatment B
Interaction
Error
Total

The two-factorial experiment is now illustrated with an example from Montgomery (2005).

The extension of the two-factorial experiment is discussed in the next subsection.

13.5.2 Three-Factorial Experiment

A natural extension of the two-factorial experiment will be the three-factorial experiment. Let the three factors be denoted by , , and , each respectively at , , and factor levels. Consequently, there are now three more interaction terms to be modeled for in , , and . Towards a complete crossed model, we need each replicate (of index ) to cover all possible factor combinations. Thus, the three-way factorial model is then modeled by

13.22

The total sum of squares decomposition equation is then

13.23

where

The ANOVA table for the three-factorial design model is given in Table 13.9.

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	-Statistic
Treatment A
Treatment B
Treatment C
Interaction (AB)
Interaction (AC)
Interaction (BC)
Interaction (ABC)
Error
Total

Since the details of the two-factorial experiment extend in a fairly straightforward manner to the current model, the model will be illustrated with two examples from Montgomery (2005).

The principle of blocking in the context of factorial experiments is taken up in the next subsection.

13.5.3 Blocking in Factorial Experiments

The two- and three-factorial experiments discussed above are extensions of the basic CRD model. In Section 13.4, we saw that replication does not alone eliminate the source of error and that the principle of blocking helps improve the efficiency of the experiment. Similarly, in the case of factorial experiments, practical considerations sometimes dictate that the replication principle cannot be implemented uniformly and that the factorial experiments may be improved through the introduction of a blocking route. The blocking model for a two-factorial experiment is stated next:

13.24

The total sum of squares decomposition equation for the model 13.24 is then

13.25

where

The corresponding ANOVA table for the blocking factorial experiment 13.24 is given in Table 13.10.

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	-Statistic
Blocks
Treatment A
Treatment B
Interaction (AB)
Error
Total

The theory and applications of Experimental Designs extend far beyond the models discussed here and thankfully the purpose of the current chapter has been to consider some of the important models among them.