Consider the same study by Heffner et al. (1974) and the data set described in Problems 4 and 5, where the response measured was the lever press rate (LPR) taken by a thirsty rat to press a lever and receive water. The three factors identified as possible predictors were Drug (three doses plus a placebo), IPR (initial press rate, categorized as slow, medium, or fast), and PRS (number of presses required to obtain water: two or five). Another factor of importance is the random factor Rat (with 24 levels). In this problem, we consider the effects of IPR and PRS individually, as well as the effects of Drug and Rat and various interactions among these factors, in a repeated measures analysis.
The ANOVA model for this situation is a modification of the model used in Problem 5, except that the effect of Factor A is split into individual components. The structure for the subject-specific scalar version of this model involves the following fixed-effect parameters and random effects:
μ = overall mean
αj; = jth fixed effect of IPR, j = 1, 2, 3
βk = kth fixed effect of PRS, k = 1,2
γl = lth fixed effect of Drug, l = 1, 2, 3, 4
(aβ)jk = fixed interaction effect of the jth level of IPR with the kth level of PRS
{aγ)jl = fixed interaction effect of the j'th level of IPR with the lth level of Drug
(βγ)kl = fixed interaction effect of the kth level of PRS and the lth level of Drug
{aβγ)Jkl = three-way fixed interaction effect for the (j, k, I) combination of IPR, PRS and Drug
Si(jk) = random effect of rat i within levels j and k of IPR and PRS, respectively
El(ijk) = random error of the lth level of Drug within levels j and k of IPR and PRS, respectively, for rat i
a. Using the computer output based on fitting the above model, carry out tests for main effects and interactions of each predictor in the model. (Assume that all such tests are orthogonal.) What do you conclude about whether or not the Drug factor has a significant effect on the response?
b. Use the output to test whether there is a significant random effect of the factor Rat.