Question

2) Suppose the regression model y = B0 + B1x1 + B2x2 + B3x3 + B4x1x2 + B5x1x3 + B6x2x3 was fit to n = 27 data points with SSE = 2000.0.

a) Set up the null and alternative hypotheses for testing whether the interaction terms are significant.

b) Give the reduced model necessary to test the significance of the interaction terms.

c) The reduced model resulted in SSE = 2800. Calculate the value of the test statistic appropriate for testing the significance of the interaction terms.

d) Establish the decision rule for the test using a = 0.05 and calculate the test statistic.

e) Do the interaction terms contribute anything to the model's ability to predict y

Answer 1

a)

The coefficients of the interaction terms are B4, B5 and B6 for x1x2, x1x3 and x2x3 respectively.

Null Hypothesis H0: B4 = B5 = B6 = 0

Alternative hypothesis H1: At least one the coefficients B4, B5 or B6 is not 0.

b)

The reduced model will be a model without the interaction terms.

Reduced model is,

y = B0 + B1x1 + B2x2 + B3x3

c)

Test Statistic will follow F distribution with numerator and denominator degree of freedom as q, n - k - 1

where q is number of predictors removed to get the reduced model. q = 3

k is number of predictors in the full mode. k = 6

n-k-1 = 27-6-1 = 20

Test Statistic will follow F distribution with df= 3, 20

Test statistic, F = [(SSE Reduced - SSE Full)/q ] /[SSE Full) / (n - k - 1)]

= [(2800 - 2000)/3 ] /[2000) / (20)]

= 2.67

(d)

Critical value of F at a = 0.05 and df = 3,20 is 3.10

We reject H0 if F > 3.10

(e)

Since the observed F (2.67) is less than the critical value, we fail to reject H0 and conclude that there is no significant evidence that the at least one the coefficients B4, B5 or B6 is not 0 and thus here is no significant evidence that the interaction terms contribute anything to the model's ability to predict y.