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 =  (s_y / s_x1)( r_x1,y - r_x1,x2 r_x2,y)/(1  -  r_x1,x2² ).

b) Show that the variance of this estimate equals:

s_b1² =  [ MSE/(N-1) ] (1/s_x1²)[ 1/(1 - r_x1,x2²) ] .

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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 = M_y

b) Show that the least squares estimate for the predictor variable is equal to:

b₁ = (s_y / s_x1) r_x1,y

c) Show that the variance of the least squares estimate for the predictor variable is equal to:

s_b1²  = [ MSE / (N-1) ] (1/s_x1²)

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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