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How can I compare regression coefficients between two groups?

Sometimes your research hypothesis 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 list free / id * gender (A8) height * weight. begin data. 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 end data. execute.

We analyzed their data separately using the **regression** commands below. Note that we have to do two regressions, one
with the data for females only and one with the data for males only. We
can use the **split file** command to split the data file by **gender**
and then run the regression. 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).

sort cases by gender. split file by gender. regression /dep weight /method = enter height. split file off.

We can compare the regression coefficients of males with
females to test the null hypothesis Ho: **B _{f}** =

split file off. compute female = 0. if gender = "F" female = 1. compute femht = female*height. execute. regression /dep weight /method = enter female height femht.

The output is shown below.

The term **femht** tests the null
hypothesis Ho: **B _{f}** =

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. B_{m}. FEMHT -1.093855 : Slope for females - Slope for males (i.e. B_{f}- B_{m}). From the separate groups, this is indeed 2.095872170 - 3.189727463 .

It is also possible to run such an analysis
using** glm**, using syntax like that below. Note
that other statistical packages, such as SAS and Stata, omit the group of the dummy variable
that is coded as zero. However, SPSS omits the group coded as one. Therefore, when you compare
the output from the different packages, the results seem to be different. To make the SPSS results
match those from other packages, you need to create a new variable that has the opposite coding (i.e.,
switching the zeros and ones). We do this with the **male** variable. We do not know of an option in SPSS
**glm** to easily change which group is the omitted group. (Please
note that you can use the **contrast** subcommand to get the contrast
coefficient for **female** using 0 as the reference group; however, the
coding of **female** in the interaction is such that 1 is used as the
reference group, so the use of the **contrast** subcommand is not very
helpful in this situation.)

compute male = not female. glm weight by male with height /design = male height male by height /print = parameter.

As you see, the **glm** output
corresponds to the output obtained by **regression**. The parameter estimates appear at the end of the **glm** output. They also correspond to the output from
**regression**.

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