1 Balanced 3-Way Factorial Design

Use the following commands to initialize R before answering the questions in this section.

options(contrasts=c("contr.sum","contr.poly") )  # set definition of contrasts
load(url("http://pnb.mcmaster.ca/bennett/psy710/datasets/3-way-data-6.rda") )

An experiment examined how performance in three tasks was affected by divided attention in three age groups of subjects using a 3 (group) x 3 (task) x 2 (attention) crossed-factorial design. The data are stored in the data frame df1. The independent variables are group, task, and attention, and the dependent variable is Y

  1. Confirm that the data are balanced.

  2. Use ANOVA to evaluate the all of the main effects and interactions. List the ANOVA table.

  3. The 3-way interaction is significant. What null hypothesis is evaluated by this \(F\) test?

  4. Evaluate the simple task x attention interaction in each group. If a simple interaction is significant, the analyze the simple simple main effect of task at each level of attention. Finally, if the simple simple main effect is significant, evaluate the pairwise differences among tasks with Tukey HSD.

  5. Analyze the data with a linear contrast that tests the null hypothesis that the difference between tasks 1 and 2 (t1 & t2) is the same in groups 1 and 3 (g1 & g3).

2 Unbalanced Factorial Design

Use the following commands to initialize R before answering the questions in this section.

options(contrasts=c("contr.sum","contr.poly") )  # set definition of contrasts
load(url("http://pnb.mcmaster.ca/bennett/psy710/datasets/DfCD-2.rda") )

An experiment was done to measure the effects of treatment C and treatment D on a dependent variable, y, using a crossed-factorial design. Six subjects were assigned randomly to each condition; however, the data from two subjects in one of the conditions were lost. The data are stored in the data frame Df.CD.

  1. Confirm that the data are unbalanced.

  2. Verify that the results of the two-way ANOVA depend on the order of the terms in the full linear model. Calculate the Type I and Type II sum of squares for C and D.

  3. Evaluate the main effects of C and D and the C x D interaction using Type III sums of squares. What null hypotheses are evaluated by these \(F\) tests?