Stat 705: Completely randomized and complete block designs, Exams of Design

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Stat 705: Completely randomized and complete

block designs

Timothy Hanson

Department of Statistics, University of South Carolina

Stat 705: Data Analysis II

Experimental design

Our department offers an entire course, STAT 706, on experimental design. In Stat 705 we will focus mainly on the analysis of common models: completely randomized designs, randomized complete block designs, ANCOVA, multifactor studies, hierarchical models (mixed-effects models), split-plots (e.g. longitudinal data analysis), Latin squares, and nested models. Some of the material in these notes is lifted from Ron Christensen’s book Analysis of Variance, Design and Regression (Chapman and Hall, 1996). The rest of it is paraphrased from your textbook.

Basic object of experimental design Obtain a valid estimate of variability σ^2 ; make the treatment inferences as sharp as feasibly possible by making the error variability σ^2 small.

Components of an experimental design

An experimental design includes: (^1) Treatments (^2) Subjects (^3) A subject-specific response to be recorded (^4) A rule to assign treatments to subjects or vise-versa

Completely randomized designs

In a completely randomized design, the experimenter randomly assigns treatments to experimental units in pre-specified numbers (often the same number of units receives each treatment yielding a balanced design). Every experimental unit initially has an equal chance of receiving a particular treatment. The data collected is typically analyzed via a one-way (or multi-way) ANOVA model.

Denise, continued

Denise uses the one-way ANOVA model

Yij = μi + ij

where i = 1, 2 for the treatment effects (dogs barking and silence, respectively) and j = 1,... , 5 for the replications, to model subject mood after being exposed to the treaments. She, not surprisingly, finds a 95% confidence interval for μ 2 − μ 1 to be (2. 2 , 4 .5). On average, those psychology students subjected to silence were 2.2 to 4.5 “mood points” higher than those subjected to dogs barking.

Randomized complete block designs

Say there are b treatments to be considered. In a randomized complete block design, the experimenter constructs a blocks of b homogeneous subjects and (uniformly) randomly allocates the b treatments to everyone in each block. The treatments are assumed to act independent of the blocks, and the overall error variability σ^2 is reduced as some variability will be explained by the block differences. Initially we consider fixed block effects, but will explore random block effects shortly. A simple randomized complete block design is analyzed as a two-way ANOVA without replication. A valid estimate of σ^2 is obtained through blocking and assuming an additive model.

Noise pollution, continued

Denise refines her experiment by considering blocks defined by the number of dogs owned by students: (i = 1, 2 , 3 , 4 for no dogs, 1 dog, 2 dogs, 3 or more dogs); among her nT = 8 participants she now requires two from each of the a = 4 blocking categories. For each of the a = 4 blocks of b = 2 subjects she makes one endure the barking-tape and the other one gets silence. She proposes the following simple model for the mood scores:

Yij = μ + αi + βj + ij ,

where j = 1, 2 denotes treatment.

Denise, continued

Denise finds the confidence interval for ¯μ• 2 − μ¯• 1 = β 2 − β 1 now to be (2. 8 , 3 .4). Again, she concludes there is a significant mean mood difference in subjects exposed to dogs barking versus not, and this difference is more precise than before, as the difference is examined within blocks. The p-value associated with the blocking variable “number of dogs” is 0.03 indicating that blocking made a significant difference in the analysis (i.e. we would reject that the blocking effect is null). Notice that we are assuming that there is no interaction between being exposed to dogs barking or not and number of dogs owned. Is this assumption reasonable? Whether you think so or not, it can be tested using Tukey’s test for additivity.

Another model...

What does this analysis have in common with an ANCOVA (coming up in Chapter 22) model

Yij = μ + γxij + τj + ij ,

where xij is the actual number of dogs (owned by the ith subject receiving treatment j) as a concomitant variable? Which model is simpler (requires fewer parameters?) Which is to be preferred?

Diagnostics for R.C.B. designs

(^1) Residuals eij versus the block index i may be used to check for unequal error variance within blocks. Residuals eij vs. the treatment index j may be used to check for unequal error variance by treatment. Also look at eij vs. ˆμij (automatic in SAS), normal probability plot of {eij }, and the deleted residuals tij to check for outliers. (^2) An interaction plot Yij versus j (or i) may be used to check for a possible interaction between the blocking variate and the treatments. NOTE: No replication of treatments within blocks means there is only one observation to estimate ˆμij , μˆij = Yij. Therefore, when there truly is no interaction present, we expect to see the deviation from parallel curves to be much greater than when we have replication, unless σ^2 is very small relative to treatment/blocking effects. Tukey’s test for additivity is a formal way to check the appropriateness of model IV with a p-value.

Noise pollution, finished

Denise decides that she would like to test the effect of leaf-blowers as well. Now she selects nT = 16 students to “participate” and subjects each block of a = 4 students sorted according to how many dogs they own (i = 1, 2 , 3 , 4 as before) to one of: j = 1, k = 1 nothing, j = 2, k = 1 dogs barking, j = 1, k = 2 a running leaf-blower nearby, or j = 2, k = 2 both a tape of dogs barking and the leaf-blower. She used the following model:

Yijk = μ + ρi + αj + βk + (αβ)jk + ijk.

The model is fit as a three-way ANOVA and interpreted as usual. If there is no interaction between treatments and blocks (again, testable via Tukey), the model is valid and contrasts in the effects of interest, for example ¯μ• 2 k − μ¯• 1 k for k = 1, 2, are examined. If (αβ)jk = 0 is accepted, simply ¯μ• 2 • − μ¯• 1 • = α 2 − α 1 , may be examined. This is a R.C.B. design with factorial treatments.