In Gruber's (1970) study of n = 104 individuals (discussed in Problem 10), the relationship between blood pressure change (SBPSL) and relative weight (RW), controlling for initial blood pressure (SBP1), was compared for three different geographical backgrounds and for three different psychosocial orientations, using the following 15 variables:
Y = SBPSL
X1 = SBP1 (initial blood pressure)
X2 = R (1 if rural background, 0 if town, -1 if urban)
X3 = T (1 if town background, 0 if rural, -1 if urban)
X4 = TD (1 if traditional orientation, 0 if transitional, -1 if modern)
X5 = TN (1 if transitional orientation, 0 if traditional, -1 if modern)
X6 = RW (relative weight)
X7 = T × TD
X8 = T × TN
X9 = R × TD
X10 = R × TN
X11 = R × TD × RW
X12 = R × TN × RW
X13 = T × TD × RW
X14 = T × TN × RW
A standard stepwise regression program was run using these data, yielding the following ANOVA table (variables were forced to enter in the order presented) based on the model
Y = β0 + β1X1 + β2X2 + … + β14X14 + E
a. Using this regression model, determine the form of the nine fitted regression equations corresponding to the nine possible combinations of background with orientation (i.e., R = 1 and TD = 1, R = 0 and TD = 1, R = -1 and TD = l, etc). [Each of the nine equations will be of the form Ŷ = β*0 + β*1 (SBP1) + β*2 (RW).]
b. Test the null hypothesis that the nine regression equations determined in part (a) are parallel. State the null hypothesis in terms of the regression coefficients of the original 14-variable regression model.
c. Test the hypothesis H0: "The three regression equations corresponding to the three backgrounds (rural, town, and urban) are parallel (but not necessarily coincident)" against the alternative HA: "They are not parallel."
d. Set up the multiple partial F-test formula for testing H0: "The nine regression equations dealt with in part (a) are coincident" against HA: "They are not coincident." State the null hypothesis in terms of the coefficients in the regression equation.