I just noticed that the Gregory Chow citation below was missing the year. The full citation is: Chow, G.C. (1960). Tests of equality between sets of coefficients in two linear regressions, Econometrica, 28(3), 591-605. I'll also take this opportunity to stress that this approach is testing the equality of slopes AND intercepts across groups. The coefficient for the dummy variable for group is the estimated increase in the intercept for the group coded 1.

David Matheson SPSS Technical Support

-----Original Message----- From: Matheson, David Sent: Thursday, November 09, 2000 11:38 AM To: SPSSX-L@LISTSERV.UGA.EDU Subject: Re: comparing b's or beta's

The Chow test provides a test of the equality of a set of coefficients across groups. You set up a dummy variable for gender and interaction variables involving gender and the other predictors. Here is a technical note on running Chow's test in SPSS. It is available as solution 100000298 on the AnswerNet at http://www.spss.com/tech/answer/index.cfm . I've actually modified the text slightly below. When the solution was written, the change in R^2 could not be requested in the menus. That statistic is now available in the Statistics dialog of the Regression procedure, as of SPSS 7.5. If you're running an earlier version of SPSS than that, you'll need to run the procedure as a syntax command. You can build most of the command in the dialog boxes, paste it to a syntax window and add the keyword CHA to the /STATISTICS subcommand. The example in the solution uses only 1 predictor (other than those computed for the purpose of the Chow) test, but the approach is generalizable to multiple predictors. You would then have the 1 dummy variable for sex and a sex*predictor interaction term for each of the predictors in the original model. The unstandardized coefficient for the edlevel*sex interaction in this example is the estimated increase in the slope for edlevel for those cases where sex = 1.

David Matheson SPSS Technical Support

Chow test for equal sets of regression coefficients across groups

Q. What is the formula for the Chow test for equal regression parameters across groups? Will SPSS perform this test?

A. There is no SPSS procedure or keyword which requests the Chow test by name, but the test is easy to obtain from the REGRESSION procedure. The Chow test provides a test of whether the set of linear regression parameters, i.e. the intercepts and slopes, is equal across groups. For example, suppose we use the variable SALNOW from SPSS for Windows' bank.sav sample data set as our dependent variable and EDLEVEL as our predictor. Also suppose that we want to know whether the intercept and slope for this regression are equal for men and women. The algorithm for the Chow test is as follows:

1. Run the regression on men and women together and note the residual sum of squares and degrees of freedom. Call this RSS1 and DF1. 2. Run the regression separately for men and women and total the residual sums of squares and degrees of freedom from the two regressions. Call these RSS2 and DF2. 3. Find (RSS1 - RSS2)/(DF1 - DF2) . 4. Divide the result of step3 by RSS2/DF2 and compare this result to the F distribution with (DF1-DF2) and DF2 degrees of freedom. The null hypothesis for this test is that the regression intercept and slope are both independent of gender.

You can perform this test in SPSS REGRESSION and also obtain separate tests for the equality of intercept and slope across genders. The group variable should be a dummy variable which equals 0 for 1 group; 1 for the other. The bank.sav variable SEX is already in this form, with a value of 1 representing female. An interaction term is computed as the product of the predictor of interest (EDLEVEL) and SEX. REGRESSION is run first with only EDLEVEL as a predictor. SEX and the interaction term, called EDSEX in this example, are then entered in a second step with a second /METHOD = ENTER subcommand. The change in R square is requested with the keyword CHA in the /STATISTICS subcommand. This keyword also requests a test of whether the change in R square is greater than 0. This test is equivalent to the CHOW test as calculated from steps 1 to 4 above. The standard statistical output from REGRESSION also provides tests and confidence intervals for the SEX and EDSEX coefficients, which are effectively adjustments to the intercept and slope parameters, respectively, for female respondents. The REGRESSION command to perform this analysis is presented below.

COMPUTE edsex = edlevel * sex . REGRESSION /DESCRIPTIVES MEAN STDDEV CORR SIG N /MISSING LISTWISE /STATISTICS COEFF OUTS CI R ANOVA END CHA /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT salnow /METHOD=ENTER edlevel /METHOD=ENTER sex edsex .

If there were K groups whose regression coefficients were to be compared, you would compute K-1 dummy variables and multiply each of these by the independent variable(s) to produce K-1 (sets of) interaction variables.

The Chow Test is introduced in Gregory Chow's paper, 'Tests of Equality Between Sets of Coefficients in Two Linear Regressions', Econometrica, 28(3), 591-605.

For discussions of the dummy variable approach to the Chow test, see a pair of papers by Damodar Gujarati in The American Statistician, 1970, 24(1), 50-52; and 1970, 24(5), 18-22.

-----Original Message----- From: William B. Ware [mailto:wbware@EMAIL.UNC.EDU] Sent: Thursday, November 09, 2000 10:53 AM To: SPSSX-L@LISTSERV.UGA.EDU Subject: Re: comparing b's or beta's

Would it be appropriate to "dummy" the gender variable and include it along with its interactions... the tests on the interactions would be tests for the equality of the regression coefficients, would they not?

WBW

__________________________________________________________________________ William B. Ware, Professor and Chair Educational Psychology, CB# 3500 Measurement, and Evaluation University of North Carolina PHONE (919)-962-7848 Chapel Hill, NC 27599-3500 FAX: (919)-962-1533 http://www.unc.edu/~wbware/ EMAIL: wbware@unc.edu __________________________________________________________________________

On Thu, 9 Nov 2000, Dale Glaser wrote:

> Wim.......in the past I have used the formula from Cohen and Cohen (1983) on > page 56 when comparing the significance of the difference between > independent Betas....Zar (1999) in chapter 18 (p. 360) in his > Biostatistical Analysis text also has a section on comparing two slopes; > however, and unfortunately I did not save it, I know some one developed a > macro for the test of two independent betas..............hope this > helps..........dale glaser > > -----Original Message----- > From: SPSSX(r) Discussion [mailto:SPSSX-L@LISTSERV.UGA.EDU] On Behalf Of Wim > Beyers > Sent: Wednesday, November 08, 2000 11:22 PM > To: SPSSX-L@LISTSERV.UGA.EDU > Subject: comparing b's or beta's > > Dear all, > > After having applied a hierarchical regression procedure in two > groups separately (boys and girls) I want to compare the estimated > parameters between the two groups, e.g. the b's (regression weights) or > beta's (standardized). Is there any way to do it, either with computer > power, either by hand (e.g. a kind of z-test like we have for comparing > correlations) ? > Someone with experience in this matter ? > -- > Wim Beyers >