Homework For GLM
1. Define X as a (N x 2) deviation score matrix, containing deviation scores for N subjects on 2 predictor variables. Define Y as a (N x 1) matrix of deviation scores for the same N subjects on the criterion variable. Consider the deviation score regression model: Y* = Xb.
a) Show that the least squares estimate for the first predictor variable is equal to:
b1 = (sy / sx1)( rx1,y - rx1,x2 rx2,y)/(1 - rx1,x22 ).
b) Show that the variance of this estimate equals:
sb12 = [ MSE/(N-1) ] (1/sx12)[ 1/(1 - rx1,x22) ] .
2. Define X as a (N x 2) score matrix, containing a column of ones for the intercept and a column of deviation scores for N subjects on one predictor variable. Define Y as a (N x 1) matrix of raw scores for the same N subjects on the criterion variable. Consider the simple linear regression model: Y* = Xb.
a) Show that the intercept is equal to:
b0 = My
b) Show that the least squares estimate for the predictor variable is equal to:
b1 = (sy / sx1) rx1,y
c) Show that the variance of the least squares estimate for the predictor variable is equal to:
sb12 = [ MSE / (N-1) ] (1/sx12)
3. Enter the following data into matlab and spss or sas. Compute the parameter estimates, standard errors, and 95% confidence intervals for the regression model. Plot the predicted and observed values.
P = b0 + b1A + b2E+ b3A×E
Ability |
Effort |
Perf |
2 |
5 |
22.0993 |
1 |
9 |
2.16695 |
4 |
8 |
22.73277 |
7 |
4 |
29.82942 |
2 |
1 |
38.59505 |
5 |
6 |
31.1994 |
9 |
2 |
11.44924 |
6 |
3 |
24.29746 |
8 |
5 |
32.28507 |
3 |
7 |
16.7399 |