Exercise 1

Suppose we wish to compare two methods of teaching (traditional [T] and modern [M]) and two types of media (classroom lecture [CL] and programmed instruction [PI]).

We assign 12 students randomly to each of the four possible cells in the design and obtain a score for each subject on the same 50 point test at the end of the semester. The number of correct answers for each subject is as follows:

T&CL T&PI M&CL M&PI

2 9 10 21

5 12 13 25

6 14 14 31

7 15 16 33

4 10 10 22

6 13 13 26

7 14 14 32

8 16 17 34

4 10 11 22

6 13 13 30

7 14 15 32

10 17 17 35

(a) By hand, estimate the following effects for the two-way analysis of variance model:

• Effects of teaching method (Factor A)

• Effects of media (Factor B)

• Interaction effects

(b) Construct the appropriate ANOVA table by hand.

(c) Use R to verify your results for parts (a) and (b).

(d) Present the interaction effects graphically (Hint: Try the function interaction.plot()). Interpret the graphs in terms of the

main effects and interaction effect.

(e) In R, carry out a test of all pairwise comparisons among the interaction terms from the two-way ANOVA model using Tukey's

procedure. Let the overall Type I error rate be bounded above by .05.

(f) Represent the two-way interaction ANOVA model as a regression model by hand.

(g) Extra Credit: The analysis of the two-way ANOVA performed in (a)-(b) can be rephrased in terms

of orthogonal comparisons if we treat the factorial design as a one-way ANOVA with four cells. By

hand, show this to be true numerically by finding the orthogonal comparisons and testing each of them against zero separately.