The Randomized Complete Block Design Lesson ANOVA - C The Randomized Complete Block Design

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

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

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

Randomized Block Designs The Randomized Complete Block Design is 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

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

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

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

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

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

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

Tukey Test The Tukey Test can be interpreted using either confidence intervals or hypothesis tests For confidence intervals If the confidence interval does not include 0, then we reject the null hypothesis that the two means are equal For hypothesis tests If the P-value is less than 0.05, then we reject the null hypothesis that the parameter is 0

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

Tukey Test: Confidence Intervals Significantly different (Diets 1 vs 4) Significantly different (Diets 2 vs 4) Significantly different (Diets 3 vs 4)

Tukey Test: Hypothesis Tests Significantly different (Diets 1 vs 4) Significantly different (Diets 2 vs 4) Significantly different (Diets 3 vs 4)

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 x2 vs x4: q = 5.043 reject H0 x3 vs x4: q = 4.454 reject H0

Summary and Homework Summary Homework 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