1. The following data consists of a 3 (age) x 2 (sex) natural design in which the proportion of pretend play between parents and infants changes as a function of age.  The DV (Y) is the observed proportion of parent-infant play that consists of pretend play.  The first independent variable (X1) is age, where 1 = 7-month old infants; 2 = 10-month old infants; and 3 = 13-month-old infants.   The second independent variable (X2) is sex, where 1=girls and 2=boys.   There are 8 infants in each cell of the design. 

The data are listed below in triplets to be read as (X1, X2, Y)  (e.g. input x1 x2 y @@;)

1 1 0.02  1 1 0.01 1 1 0.07 1 1 0.04 1 1 0.01 1 1 0.09 1 1 0.05 1 1 0.06

1 2 0.05 1 2 0.01 1 2 0.04 1 2 0.03  1 2 0.02 1 2 0.02 1 2 0.13 1 2 0.06

2 1 0.15 2 1 0.11 2 1 0.22 2 1 0.05  2 1 0.09 2 1 0.05 2 1 0.15 2 1 0.11

2 2 0.14 2 2 0.21 2 2 0.06 2 2 0.12 2 2 0.11 2 2 0.19 2 2 0.12 2 2 0.04

3 1 0.09 3 1 0.03 3 1 0.18 3 1 0.12 3 1 0.18 3 1 0.43 3 1 0.24 3 1 0.4

3 2 0.02 3 2 0.19 3 2 0.15 3 2 0.07 3 2 0.45 3 2 0.2 3 2 0.49 3 2 0.19

The questions to be answered for this dataset are as follows:

a.  Reproduce a basic  ANOVA Table (main effects + interaction) for this analysis using SAS. 

b.  What is the value of RMSE for this dataset?

c.  What is the F and p-value for the age main effect? What are the models being compared?

d. What is the F and p-value for the sex main effect? 

e.  What models are being compared for the test of the interaction effect?  What is the F and p-value for this test?

f.  Based on your analysis, what do you conclude is the final model for this dataset?

g.   Your instructor is on an island, and can only be rescued if you solve the following problem.  Test the significance of the linear and quadratic trends in age using  a) regression modeling, and b) ANOVA contrasts (the contrasts are linear  = -1 0 1  and quadratic = 1 -2  1).  What, if any, differences do you see and why?

h. Now, test the age linear trend by sex interaction in the data above, using ANY METHOD you are comfortable with. Report the results and the hypotheses being tested in this model.