Lesson 13 - 3

1. Lesson 13 - 3 The Randomized Complete Block Design

2. Objectives • Conduct analysis of variance on the randomized complete block design • Perform the Tukey test

3. Vocabulary • factor – a level manipulated or set at different levels for a designed experiment

4. Experimental Design Three ways researchers deal with factors • Control their levels so they remain fixed throughout the experiment • Manipulate or set them at fixed levels • Randomize so that any effects not identified or uncontrollable are minimized

5. Randomized Block Designs • The RandomizedCompleteBlockDesignis a method for assigning subjects to treatments that can further reduce experimental error • The subjects should be assigned in blocks • A block could be a car (assigning 4 tires) • A block could be twins (assigning 2 treatments) • A block could be schools (assigning 3 math tests) • Blocks are homogeneous

6. Randomized Block (cont) • Every treatment is applied to every block • For each treatment, at least one subject in each block receives that treatment • Within each block, treatments are assigned to the subjects in a random way • There are designs that are even fancier, that further balances the design, such as Latin Square designs

7. Requirements • Response variable for each of the k populations • is normally distributed • has the same variance • Note: design your experiment so that the sample size for each factor is the same, because analysis of variance is not affected too much when we have equal sample sizes even if there is inequality in the variances

8. Example • Effect of diet on rats • Each block is a set of 4 newborn rats from the same mother • 5 sets of rats (5 blocks) are chosen for the experiment • The diets are randomly assigned -- the 4 diets to the 4 rats • Average weight gain (in ounces) is recorded for each block and is shown in the table below

9. Excel Example Results (Without Replication - what does this mean?) Rows Columns p < α, so diets make a difference

10. Which Diets are Different? • Now that we know that some of the treatments are significantly different, we would like to find which ones are the ones that are different • For one-way ANOVA, we used the Tukey Test • For this two-way ANOVA, we also will use the Tukey Test

11. Critical Value for Tukey qα,v,k • The parameter α is the significance level, as usual • The parameter ν is equal to (r – 1)(c – 1) where r is the number of rows (blocks) and c is the number of columns (treatments) • The parameter k is the number of populations

12. Example (cont) x2 – x1 q = -------------------------- s2 1 1 --- · --- + --- 2 n1 n2 • Tukey Test:MSE = 3.052x1 = 16.42x2 = 16.68x3 = 16.22x4 = 12.74ni = 5

13. Example (cont) • Qc = 4.199 • x1 vs x2: q = 0.3328 • x1 vs x3: q = 0.2560 • x1 vs x4: q = 4.7101 reject H0 • x2 vs x3: q = 0.5888 reject H0 • x2 vs x4: q = 5.043 • x3 vs x4: q = 4.454 reject H0

14. Summary and Homework • Summary • The randomized complete block design provides a method for balancing treatments across blocks, thus reducing the impact of other (unmeasured) factors • The Tukey Test provides a method for determining which treatments are significant, i.e. which population means are significantly different • Homework • pg 707 - 710: 1 – 7, 10, 15