Sometimes your research may predict that the size of a
regression coefficient should be bigger for one group than for another. For example, you
might believe that the regression coefficient of height predicting
weight
would be higher for men than for women. Below, we have a data file with 10 fictional
females and 10 fictional males, along with their height in inches and
their weight in pounds.
DATA htwt; INPUT id Gender $ height weight ; CARDS; 1 F 56 117 2 F 60 125 3 F 64 133 4 F 68 141 5 F 72 149 6 F 54 109 7 F 62 128 8 F 65 131 9 F 65 131 10 F 70 145 11 M 64 211 12 M 68 223 13 M 72 235 14 M 76 247 15 M 80 259 16 M 62 201 17 M 69 228 18 M 74 245 19 M 75 241 20 M 82 269 ; RUN;
We analyzed their data separately using the proc reg below.
PROC REG DATA=htwt; BY gender; MODEL weight = height ; RUN;
The parameter estimates (coefficients) for females and
males are shown below, and the results do seem to suggest that height is a
stronger predictor of weight for males (3.18) than for females (2.09).
GENDER=F
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
INTERCEPT -2.397470040 -0.34 0.7427 7.05327189
HEIGHT 2.095872170 18.97 0.0001 0.11049098
GENDER=M
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
INTERCEPT 5.601677149 0.63 0.5480 8.93019669
HEIGHT 3.189727463 25.88 0.0001 0.12323669
We can compare the regression coefficients of males with
females to test the null hypothesis Ho: Bf =
Bm,
where Bf is the regression coefficient for females, and
Bm
is the regression coefficient for males. To do this analysis, we first make a dummy
variable called female that is coded 1 for female and 0 for male,
and a variable femht
that is the product of female and height. We then use
female height and femht as predictors in the regression
equation.
data htwt2; set htwt; female = . ; IF gender = "F" then female = 1; IF gender = "M" then female = 0; femht = female*height ; RUN; PROC REG DATA=htwt2 ; MODEL weight = female height femht ; RUN;
The output is shown below.
Model: MODEL1
Dependent Variable: WEIGHT
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 3 60327.09739 20109.03246 4250.111 0.0001
Error 16 75.70261 4.73141
C Total 19 60402.80000
Root MSE 2.17518 R-square 0.9987
Dep Mean 183.40000 Adj R-sq 0.9985
C.V. 1.18603
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
INTERCEP 1 5.601677 8.06886167 0.694 0.4975
FEMALE 1 -7.999147 11.37054598 -0.703 0.4919
HEIGHT 1 3.189727 0.11135027 28.646 0.0001
FEMHT 1 -1.093855 0.16777741 -6.520 0.0001
The term femht tests the null
hypothesis Ho: Bf = Bm.
The T value is -6.52 and is significant, indicating that the regression coefficient
Bf
is significantly different from Bm.
Let’s look at the parameter estimates to get a better understanding of what they mean and
how they are interpreted.
First, recall that our dummy variable
female is 1 if female and 0 if
male; therefore, males are the omitted group. This is needed for proper interpretation
of the estimates.
Parameter
Variable Estimate
INTERCEP 5.601677 : This is the intercept for the males (omitted group)
This corresponds to the intercept for males in
the separate groups analysis.
FEMALE -7.999147 : Intercept Females - Intercept males
This corresponds to differences of the
intercepts from the separate groups analysis.
and is indeed -2.397470040 - 5.601677149
HEIGHT 3.189727 : Slope for males (omitted group), i.e., Bm.
FEMHT -1.093855 : Slope for females - Slope for males
(i.e. Bf - Bm).
From the separate groups, this is indeed
2.095872170 - 3.189727463 .
It is also possible to run such an analysis in proc glm, using syntax like that below.
PROC GLM DATA=htwt2 ; CLASS gender ; MODEL weight = gender height gender*height / SOLUTION ; RUN;
As you see, the proc glm output
corresponds to the output obtained by proc reg.
General Linear Models Procedure
Class Level Information
Class Levels Values
GENDER 2 F M
Number of observations in data set = 20
General Linear Models Procedure
Dependent Variable: WEIGHT
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 60327.097387 20109.032462 4250.11 0.0001
Error 16 75.702613 4.731413
Corrected Total 19 60402.800000
R-Square C.V. Root MSE WEIGHT Mean
0.998747 1.186031 2.1751812 183.40000
Source DF Type I SS Mean Square F Value Pr > F
GENDER 1 55125.000000 55125.000000 11650.85 0.0001
HEIGHT 1 5000.982757 5000.982757 1056.97 0.0001
HEIGHT*GENDER 1 201.114630 201.114630 42.51 0.0001
Source DF Type III SS Mean Square F Value Pr > F
GENDER 1 2.3416157 2.3416157 0.49 0.4919
HEIGHT 1 4695.8308766 4695.8308766 992.48 0.0001
HEIGHT*GENDER 1 201.1146303 201.1146303 42.51 0.0001
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
INTERCEPT 5.601677149 B 0.69 0.4975 8.06886167
GENDER F -7.999147189 B -0.70 0.4919 11.37054598
M 0.000000000 B . . .
HEIGHT 3.189727463 B 28.65 0.0001 0.11135027
HEIGHT*GENDER F -1.093855293 B -6.52 0.0001 0.16777741
M 0.000000000 B . . .
NOTE: The X'X matrix has been found to be singular and a generalized inverse
was used to solve the normal equations. Estimates followed by the
letter 'B' are biased, and are not unique estimators of the parameters.
The parameter estimates appear at the end of the proc glm output. They correspond to the output from
proc reg and from
the separate analyses, that is:
INTERCEPT 5.601677149 : This is the intercept for the males GENDER F -7.999147189 : Intercept Females - Intercept males HEIGHT 3.189727463 : Slope for males HEIGHT*GENDER F -1.093855293 : Slope for females - Slope for males