Interaction effects and group comparisons
Contents
1. black Interaction effects and group comparisons Page 5 Again you see two parallel lines with the black line 2 55 points below the white line Note that the Y axis is different in the two graphs because education has a stronger effect than job experience it produces a wider range of predicted values but the distance between the parallel lines is the same in both graphs Model 2 Intercepts and one or more but not all slope coefficients differ across groups We will now regress Y on the IVs black and one interaction term For reasons we will explain later when using interaction terms you should generally include the variables that were used to compute the interaction even if their effects are not statistically significant In this case this would mean including black and the IV that was used in computing the interaction term Here is the Stata output for our current example where we test to see if the effect of Job Experience is different for blacks and whites reg income educ jobexp i black i black c jobexp Source Ss df MS Number of obs 500 F 4 495 604 39 Model 33352 2559 4 8338 06397 Prob gt F 0 0000 Residual 6828 99339 495 13 7959462 R squared 0 8300 Adj R squared 0 8287 Total 40181 2493 499 80 5235456 Root MSE 3 7143 income Coef Std Err t P gt t 95 Conf Interval educ 1 834776 0463385 39 60 0 000 1 743732 1 925821 jobexp 7128145 0395293 182. ELSE 0 INTO DUMMY RECODE X 3 1 ELSE 0 INTO DUMMY3 Note that group 4 is coded 0 on all three dummy variables Category 4 is sometimes referred to as the excluded category or reference category One of several shortcuts for doing this in Stata is tab x gen dummy If x had 4 categories this would create dummy1 dummy2 dummy3 and dummy4 If you then regress Y on dummy1 dummy2 dummy3 e The intercept is the mean for group 4 i e the reference group e The intercept by is the mean for group k e The T values for the betas tell you whether that group s mean significantly differs from the mean of the excluded category Note that this is equivalent to a one way ANOVA where the dependent variable is Y and the independent variable is X Or if X only has 2 values it is the same as a t test Example Suppose Religion is coded 1 Catholic 2 Protestant 3 Jewish 4 Other If a 10 b1 3 b2 2 and b3 7 the other mean is 10 the Catholic mean is 13 the Protestant mean is 8 and the Jewish mean is 17 The T values for each dummy variable indicate whether the mean for that group significantly differs from the Other mean For our current example the average white income is 30 04 the average black income is 18 79 i e 11 25 less than the average white income Running a regression we get reg income black Source SS df MS Number of obs 500 F 1 498 167 76 Model 1012
3. quietly margins black at jobexp 1 1 21 atmeans Marginsplot noci scheme sj name intedjob_jobexp Variables that uniquely identify margins jobexp black Interaction effects and group comparisons Page 10 Adjusted Predictions of black 30 35 25 Linear Prediction 20 T T T T T T T T T T T T T T T T T T T T 123 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 jobexp white black quietly margins black at educ 2 1 21 atmeans marginsplot noci ylabel 10 scheme sj name intedjob_educ Variables that uniquely identify margins educ black Adjusted Predictions of black Linear Prediction 30 35 L L 25 l 20 T T T T T T T T T T T T T T T T T T T T 123 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 jobexp white black Comparing the two approaches Now let s compare these with our earlier results from when we ran separate models by race 1 893338 722255 Constant 6 461190 Interaction effects and group comparisons Page 11 Notice that the coefficients in the interactions model for the intercept Educ and Jobexp are the same as the coefficients we got in the earlier whites only equation Further if you add the interactions model coefficients for Intercept Black Educ Blacked and Jobexp Blackjob you get the coefficients from the earlier blacks only equation Why this works read on your own if we don
4. 0 0000 The variable names created by xi are fairly logical but you might still prefer just to compute variables on you own so you can easily get the names you want Also computing them on your own will get rid of all the annoying dropped terms in the printout on the other hand Stata may be less likely to screw up the computations of the dummy variables and interaction terms than you are Also note that xi includes the lower order terms i e even though you didn t explicitly tell it to include the non interaction terms for educ jobexp and black it did for the SPSS commands that allow similar shortcuts you have to explicitly specify both the main and interaction effect If you want a little more control over how terms appear in the printout you can explicitly specify the main effects e g Interaction effects and group comparisons Page 29 xi reg income educ jobexp i black i black educ i black jobexp i black _Tblack_0 1 naturally coded _Iblack_0 omitted i black educ _IblaXeduc_ coded as above i black jobexp _IblaxXjobex_ coded as above Source SS df MS Number of obs 500 F 5 494 487 60 Model 33411 2623 5 6682 25246 Prob gt F 0 0000 Residual 6769 98696 494 13 7044271 R squared 0 8315 Adj R squared 0 8298 Total 40181 2493 499 80 5235456 Root MSE 3 7019 income Coef Std Err t P gt t 95 Conf Interval educ 1 893338 054125 34 98 0 000 1 786994 1 999681 jobexp 2122255 03
5. 0 8254 Total 40181 2493 499 80 5235456 Root MSE 3 7499 income Coef Std Err te P gt t 95 Conf Interval 1 black 2 55136 4736266 5 39 0 000 3 481921 1 620798 educ 1 840407 0467507 39 37 0 000 1 748553 1 932261 jobexp 6514259 0350604 18 58 0 000 5825406 7203111 _cons 4 72676 9236842 5 12 0 000 6 541576 2 4911943 The i black notation tells Stata that black is a categorical variable rather than continuous As the Stata 11 User Manual explains section 11 4 3 1 i group is called a factor variable although more correctly we should say that group is a categorical variable to which factor variable operators have been applied When you type i group it forms the indicators for the unique values of group In other words Stata in effect creates dummy variables coded 0 1 from the categorical variable In this case of course black is already coded 0 1 but margins and other post estimation commands still like you to use the i notation so they know the variable is categorical rather than say being a continuous variable that just happens to only have the values of 0 1 in this sample But if say we had the variable race coded white 2 black the new variable would be coded 0 white 1 black Or if the variable religion was coded 1 Catholic 2 Protestant 3 Jewish 4 Other saying i religion would cause Stata to create three 0 1 dummies By default the first category in this case Catholic i
6. 650 B B B B As the above make clear Interactions model Q Bi Bo Baummy Baummyx1 B dummyX2 e The interaction terms indicate the difference in effects between group and group 0 If the intercept is larger in group than in group 0 the coefficient for the dummy variable will be positive If the effect of a variable is larger i e more positive or less negative in group 1 than in group 0 then the interaction term will have a positive value e Ifthe intercept and regression coefficients are the same in both populations then the expected values of the interaction terms are all zero Hence a test of whether the interaction and dummy terms zero which is what the incremental F test is testing is equivalent to a test of whether there are any group differences Other comments on interaction effects and group comparisons Interpretation of the main effects i e the non interaction terms can be a little confusing when interaction terms are in the model We ll discuss these interpretation issues more and ways to make the interpretation clearer in a subsequent handout People often get confused by the following If lines are not parallel at some point the group that seems to be behind has to have a predicted edge over the other group although that point may never actually occur within the observed or even any possible data Consider the following hypothetical example where Education X is regre
7. F 2 497 1103 96 Model 32798 4018 2 16399 2009 Prob gt F 0 0000 Residual 7382 84742 497 14 8548238 R squared 0 8163 Adj R squared 0 8155 Total 40181 2493 499 80 5235456 Root MSE 3 8542 income Coef Std Ere P gt t 95 Conf Interval educ 1 94512 0436998 44 51 0 000 1 859261 2 03098 jobexp 7082212 0343672 20 61 0 000 6406983 775744 _cons 7 382935 8027781 9 20 0 000 8 960192 5 805678 Block 2 black Source Ss df MS Number of obs 500 EFG 3 496 787 14 Model 33206 4588 3 11068 8196 Prob gt F 0 0000 Residual 6974 79047 496 14 0620776 R squared 0 8264 Adj R squared 0 8254 Total 40181 2493 499 80 5235456 Root MSE 3 7499 income Coef Std Err t P gt t 95 Conf Interval educ 1 840407 0467507 39 2 3 0 000 1 748553 1 932261 jobexp 6514259 0350604 18 58 0 000 5825406 7203111 black 2 55136 4736266 5 39 0 000 3 481921 1 620798 _cons 4 72676 9236842 5 12 0 000 6 541576 2 911943 Interaction effects and group comparisons Block 3 blacked blackjob Source SS af MS Number of obs 500 F 5 494 487 60 Model 33411 2623 5 6682 25246 Prob gt F 0 0000 Residual 6769 98696 494 13 7044271 R squared 0 8315 Adj R squared 0 8298 Total 40181 2493 499 80 5235456 Root MSE 3 7019 income Coef Std Err t P gt t 95 Conf Interval educ 1 893338 054125 34 98 0 000 1 786994 1 999681 jobexp 722255 0396598 18 21 0 000 6443322 8001777
8. SSE SSE N K 1 _ Ri R N K 1 ee SSE J 1 R J _ 7383 6975 496 82642 81626 496 _ 29 01 6975 1 82642 l Confirming with the ftest command ftest intonly baseline Assumption baseline nested in intonly F 1 496 29 02 prob gt F 0 0000 You can also do a likelihood ratio test lrtest intonly baseline Likelihood ratio test LR chi2 1 28 43 Assumption baseline nested in intonly Prob gt chi2 0 0000 Interpretation of a Model that allows only the Intercepts to Differ We ll simplify things a bit and consider the case where there is only one X variable Suppose Y is regressed on X1 and Dummy1 where X1 is a continuous variable and Dummy is coded 1 if respondent is a member of group 1 O otherwise Note that there are no interaction terms in the model In this case the model assumes that X1 has the same effect i e slope for both groups However the intercept is Interaction effects and group comparisons Page 3 different for group 1 than for others The coefficient for Dummy tells you how much higher or lower the intercept is for group 1 Put another way the reported intercept is the intercept for those not in Group 1 the intercept baummy1 is the intercept for group 1 For example suppose that a 0 b 3 Daummy1 2 Graphically this looks something like Y Grp 1 Grp2 That is you get two parallel lines but for each value of X the predicted value of Y is 2 units high
9. t have time in class The model with interaction terms represents an alternative way of expressing the unconstrained model instead of running separate regressions for each group we run a single regression with additional variables The coefficients for the dummy variable and the interaction terms indicate whether the groups differ or not With the interactions approach the unconstrained model can be written as Y at BX T BX Bummy Dummy Btummyex1 Dummy X1 Bummy x2 Dummy X2 E But for Group 0 Dummy and the interaction terms computed from it all equal 0 hence for group 0 this simplifies too Y a pX p X E a POX X 6 That is both the model using interaction terms and the separate model estimated only for group 0 will yield identical estimates of the intercept and the non interaction terms also known as the main effects For group 1 where Dummy 1 DUMMYX1 X1 and DUMMYX2 X2 the model simplifies to Y d BX X Pisces Baummy X1 1 Bronmy X2 2 E a Pri B E Bimma X 2 Pisces X E a BOX BPX 6 That is adding the main effect to the corresponding interaction term gives you the parameters for when a regression is run on Group 1 separately The following tables illustrate how to go from parameters estimated using one approach to parameters estimated using the other Interaction effects and group comparisons Page 12 Separate regressions a 8 B
10. 1 1 21 atmeans Marginsplot noci scheme sj name intonly_jobexp Interaction effects and group comparisons Page 4 Variables that uniquely identify margins jobexp black Adjusted Predictions of black Linear Prediction 25 30 35 L L 20 1 15 T T T T T T T T T T T T T T T T T T T T 123 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 jobexp white black This graph plots the relationship between job experience and income for values of job experience that range between 1 year and 21 years the observed range in the data More specifically because education is also in the model and I specified the atmeans option it plots the relationship between job experience and income for individuals who have average values of education 13 16 years I could have used some other value for education but doing so would have simply shifted both lines up or down by the same amount As you see we get two parallel lines with the black line always 2 55 points below the white line Doing the same thing for education quietly margins black at educ 2 1 21 atmeans Marginsplot noci ylabel 10 scheme sj name intonly_educ Variables that uniquely identify margins educ black Adjusted Predictions of black Linear Prediction 15 20 25 30 35 40 45 E L 10 5 f T T j T T T T T T T T T T T T T T T T T 23 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 educ white
11. Coef Std Err t P gt t 95 Conf Interval jobexp 3262549 0691292 4 72 0 000 1904335 4620764 black 10 30386 8739031 11 79 0 000 12 02086 8 586861 _cons 25 43981 1 04632 24 31 0 000 23 38405 27 49556 predict whiteline if black option xb assumed fitted values 100 missing values generated predict blackline if black option xb assumed fitted values 400 missing values generated label variable whiteline Line for whites label variable blackline Line for blacks twoway connected whiteline blackline jobexp 9 4 34 9 4 g wo 0 5 10 15 20 jobexp Interaction effects and group comparisons Page 20 Model 2 Intercepts and one or more but not all slope coefficients differ across groups reg income educ jobexp black blackjob Source Ss df MS Number of obs 500 F 4 495 604 39 Model 33352 2559 4 8338 06397 Prob gt F 0 0000 Residual 6828 99339 495 13 7959462 R squared 0 8300 Adj R squared 0 8287 Total 40181 2493 499 80 5235456 Root MSE 3 7143 income Coef Std Err P gt t 95 Conf Interval educ 1 834776 0463385 39 60 0 000 1 743732 1 925821 jobexp 7128145 0395293 18 03 0 000 6351486 7904805 black 4686862 1 040728 0 45 0 653 1 576102 2 513475 blackjob 2556117 0786289 3 25 0 001 4100993 1011242 _cons 5 514076 9464143 5783 0 000 T3 T3561 3 654592 est store intjob The significant negative coefficient for BLACKJOB indicates that blacks benef
12. black 3 409988 1 756477 1 94 0 053 0410984 6 861075 blacked 2153886 1038015 2 08 0 039 4193354 0114418 blackjob 3002799 0812705 3 69 0 000 4599584 1406015 _cons 6 461189 1 0479 6 17 0 000 8 520079 4 402298 Block Residual Change Block F df df Pr gt F R2 in R2 1 1103 96 2 497 0 0000 0 8163 2 29 02 1 496 0 0000 0 8264 0 0102 3 7 47 2 494 0 0006 0 8315 0 0051 In the table at the end Block 1 gives us the statistics for the baseline model in which there are no differences across groups In effect you are contrasting a model with no variables with the model that includes educ and jobexp The F of 1103 96 is therefore the global F statistic for the baseline model In Block 2 the baseline model is contrasted with the model that allows the intercepts to differ The F of 29 02 is the F from the Wald test of black which is the same as the incremental F test In Block 3 the model that allows only the intercepts to differ is contrasted with the model that also allows the two slope coefficients to differ The significant F value of 7 47 tells us that at least one of the slope coefficients significantly differs from 0 Interaction effects and group comparisons Page 26 Appendix Y regressed on dummy variables only Suppose X is a K category variable with nominal level measurement From X we construct K 1 Dummy variables e g in SPSS RECODE X 1 1 ELSE 0 INTO DUMMY1 RECODE X 2 1
13. dummy2 protestant rename dummy3 jewish rename dummy4 other e Compute interaction terms for the dummy variable and each of the IVs whose effects you think may differ across groups In Stata do something like gen dummyx1 dummy x1 gen dummyx2 dummy x2 NOTE If you want you can think of DUMMY as being an interaction term too DUMMY DUMMY X0 where X0 1 for all cases Baseline Model No differences across groups As before we can begin with a model that does not allow for any differences in model parameters across groups We will also compute the interaction terms that we will need later the dummy variable black is already in the data set use http www3 nd edu rwilliam stats2 statafiles blwh dta clear gen blacked black educ gen blackjob black Jjobexp Interaction effects and group comparisons Page 18 reg income educ jobexp Source ss df MS Number of obs 500 4 F 2 497 1103 96 Model 32798 4018 2 16399 2009 Prob gt F 0 0000 Residual 7382 84742 497 14 8548238 R squared 0 8163 4 Adj R squared 0 8155 Total 40181 2493 499 80 5235456 Root MSE 3 8542 income Coef Std Err t P gt t 95 Conf Interval educ 1 94512 0436998 44 51 0 000 1 859261 2 03098 jobexp 7082212 0343672 20 61 0 000 6406983 775744 _cons 7 2382935 8027781 9 20 0 000 8 960192 5 805678 est store baseline Model 1 Only the intercepts differ across groups To allow the i
14. not shown testparm i black c jobexp 1 1 black c jobexp 0 F 1 495 Prob gt F 10 57 0 0012 Or doing a likelihood ratio test lrtest intonly intjob 10 56 0 0012 Likelihood ratio test LR chi2 1 Assumption intonly nested in intjob Prob gt chi2 Interpreting a Model in which the slopes are allowed to differ across groups Suppose Y is regressed on X1 Dummy1 and Dummy X1 The coefficient for Dummy X1 will indicate how the effect of X1 differs across groups For example if the coefficient is positive this means that X1 has a larger effect i e more positive or less negative in group 1 than it does in the other group For example we might think that whites gain more from each year of education than do blacks Or we might even think that the effect of a variable is positive in one group and zero or negative in another The coefficient for X1 is the effect i e slope of X1 for those not in group 1 by Daummyx1 iS the effect slope of X1 on those in group 1 When interaction terms are added lines are no longer parallel and you get something like Interaction effects and group comparisons Page 7 Y Grp 1 Grp 2 For both groups as X increases Y increases However the increase slope is much greater for group than it is for group 2 The T value for the interaction term tells you whether the slope for that group differs significantly from the slope for the reference group To generate
15. such a graph in Stata quietly margins black at jobexp 1 1 21 atmeans Marginsplot noci scheme sj name int job_jobexp Variables that uniquely identify margins jobexp black Adjusted Predictions of black 30 1 25 Linear Prediction 20 T T T T T T T T T T T T T T T T T T T 1 3 4 5 6 7 8 Q9 10 11 12 13 14 15 16 17 18 19 20 21 jobexp white black At low levels of job experience there is virtually no difference between blacks and whites people with little experience don t make much money no matter what their race is As job experience goes up the gap between blacks and whites gets bigger and bigger because whites benefit more from job experience than blacks do Model 3 All coefficients freely differ across groups Before we estimated separate models for blacks and whites We can achieve the same thing by estimating a model that includes a Interaction effects and group comparisons Page 8 dummy variable for race and interaction terms for race with each independent variable Remember that this is called a Chow test reg income educ jobexp i black i black c educ i black c jobexp Source SS df MS Number of obs 500 4 F 5 494 487 60 Model 33411 2623 5 6682 25246 Prob gt F 0 0000 Residual 6769 98696 494 13 7044271 R squared 0 8315 4 Adj R squared 0 8298 Total 40181 2493 499 80 5235456 Root MSE 3 7019 income Coef Std Err
16. t P gt t 95 Conf Interval educ 1 893338 054125 34 98 0 000 1 786994 1 999681 jobexp 722255 0396598 18 21 0 000 6443322 8001777 1 black 3 409988 1 756477 1 94 00 53 0410984 6 861075 black c educ t 2 153886 1038015 2 08 0 039 4193354 0114418 black c jobexp i 3002799 0812705 3 69 0 000 4599584 1406015 _cons 6 461189 1 0479 6 17 0 000 8 520079 4 402298 est store intedjob Note that Nu 500 SSE 6770 DFE 494 These are the exact same numbers we got using the earlier procedure where we estimated separate models for each race and if we want to test the hypothesis that there are no differences across groups the calculation of the incremental F is identical Or if you prefer to do the calculation using the constrained and unconstrained R values you get R R2 N 2K 2 _ 83151 81626 494 14 90 1 R K 1 1 83151 3 Fk n 2K 2 To confirm ftest baseline intedjob Assumption baseline nested in intedjob F 3 494 14 91 prob gt F 0 0000 Also in Stata you can easily use the test or testparm command test 1 black 1 black c educ 1 black c jobexp output not shown testparm i black i black c educ i black c jobexp a 1 black 0 2 1 black c educ 0 3 1 black c jobexp 0 F 3 494 14 91 Prob gt F 0 0000 Interaction effects and group comparisons Page 9 For good measure we can add a likelihood ratio test as
17. 03 0 000 6351486 7904805 1 black 4686862 1 040728 0 45 0 653 1 576103 2 513475 black c jobexp 1 2556117 0786289 3 20 0 001 4100993 1011242 _cons 5 514076 9464143 5 83 0 000 75373561 3 654592 est store intjob The significant negative coefficient for black c jobexp indicates that blacks benefit less from job experience than do whites Specifically each year of job experience is worth about 256 less for a black than it is for a white Doing an incremental F test we contrast the unconstrained model immediately above with the constrained model in which blackjob is excluded Note that Model 1 is now the constrained model it is constrained in that the effect of jobexp is constrained to be the same across groups Remember the terms constrained and unconstrained are always relative and that the unconstrained model in one contrast may the constrained model in another SSE 6829 R 83005 K 4 SSE 6975 R 82642 J 1 Interaction effects and group comparisons Page 6 _ SSE SSE N K 1 _ RU R N K 1 e SSE J 1 R2 J _ 6975 6829 495 _ 83005 82642 495 6829 1 83005 1058 To confirm ftest intonly intjob Assumption intonly nested in intjob F 1 495 prob gt F 10 57 0 0012 The incremental F the squared T value for blackjob Or doing a Wald test with the test or testparm command test 1 black c jobexp Output
18. 5 1 10125 Prob gt F 0 0000 Residual 30056 2493 498 60 3539142 R squared 0 2520 Adj R squared 0 2505 Total 40181 2493 499 80 5235456 Root MSE 7 7688 income Coef Std Err t P gt t 95 Conf Interval black 11 25 8685758 12 95 0 000 12 95652 9 543475 _cons 30 04 3884389 77 34 0 000 29 27682 30 80318 Interaction effects and group comparisons Page 27 Commands that give equivalent results oneway income black tabulate Summary of income black Mean Std Dev Freq white 30 04 7 7943748 400 black 18 79 7 6647494 100 Total 2719 8 9734913 500 Analysis of Variance Source SS dE MS F Prob gt F Between groups 10125 iL 10125 167 76 0 0000 Within groups 30056 2493 498 60 3539142 Total 40181 2493 499 80 5235456 Bartlett s test for equal variances chi2 1 0 0442 Prob gt chi2 0 834 ttest income by black Two sample t test with equal variances Group Obs Mean Std Err Std Dev 95 Conf Interval white 400 30 04 3897187 7 794375 29 27384 30 80616 black 100 T879 7664749 7 664749 17 26915 20 31085 combined 500 21509 4013067 8 973491 27 00154 28 57846 diff 11 25 8685758 9 543475 12495632 diff mean white mean black t 12 9522 Ho diff 0 degrees of freedom 498 Ha diff lt 0 Ha diff 0 Ha diff gt 0 Pr T lt t 1 0000 Pr T gt t 0 0000 Pr T gt t 0 0000 Interaction effects and group comparisons Page 28 Appendix The Stata x
19. 96598 Les 2L 0 000 6443323 8001777 _Iblack_1 3 409988 1 756477 1 94 0 053 0410983 6 861074 _Iblack_1 dropped educ dropped _IblaXxXeduc_1 2153886 1038015 2 08 0 039 4193354 0114418 _Tblack_1 dropped jobexp dropped _IblaxXjobe l 3002799 0812705 3 69 0 000 4599584 1406015 _cons 6 461189 1 0479 6 17 0 000 8 520079 4 402298 test _Iblack_1 _IblaXeduc_1 _IblaXjobex_1 1 _Iblack 1 0 2 _IblaXeduc_1 0 3 _IblaXjobex_1 0 F 3 494 14 91 Prob gt F 0 0000 Note Again remember that starting with Stata 11 the xi command still works but it is often preferable to use factor variables instead Interaction effects and group comparisons Page 30
20. Interaction effects and group comparisons Richard Williams University of Notre Dame http www3 nd edu rwilliam Last revised February 20 2015 Note This handout assumes you understand factor variables which were introduced in Stata 11 If not see the first appendix on factor variables The other appendices are optional If you are using an older version of Stata or are using a Stata program that does not support factor variables see the appendix on Interaction effects the old fashioned way also the appendices on the nest reg command which does not support factor variables and the xi prefix an older alternative to the use of factor variables may also be useful Finally there is an appendix that shows the equivalences between t tests and one way ANOVA with a regression model that only has dummy variables Also there are a lot of equations in the text e g for calculations of incremental F tests You can just skip over most of these if you are content to trust Stata to do the calculations for you Alternative strategy for testing whether parameters differ across groups Dummy variables and interaction terms We have previously shown how to do a global test of whether any coefficients differ across groups This can be a good starting point in that it tells us whether any differences exist across groups It may also be useful when we have good reason for believing that the models for two or more groups are substantially diffe
21. across groups use http www3 nd edu rwilliam statafiles blwh dta clear reg income educ jobexp Source ss df MS Number of obs 500 4 F 2 497 1103 96 Model 32798 4018 2 16399 2009 Prob gt F 0 0000 Residual 7382 84742 497 14 8548238 R squared 0 8163 4 Adj R squared 0 8155 Total 40181 2493 499 80 5235456 Root MSE 3 8542 income Coef Std Err t P gt t 95 Conf Interval educ 1 94512 0436998 44 51 0 000 1 859261 2 03098 jobexp 7082212 0343672 20 61 0 000 6406983 775744 _cons 7 382935 8027781 9 20 0 000 8 960192 5 805678 est store baseline Model 1 Only the intercepts differ across groups To allow the intercepts to differ by race we add the dummy variable black to the model reg income educ jobexp i black Source Ss df MS Number of obs 500 d F 3 496 787 14 Model 33206 4588 3 11068 8196 Prob gt F 0 0000 Residual 6974 79047 496 14 0620776 R squared 0 8264 4 Adj R squared 0 8254 Total 40181 2493 499 80 5235456 Root MSE 3 7499 income Coef Std Err t P gt t 95 Conf Interval educ 1 840407 0467507 B96 33 0 000 1 748553 1 932261 jobexp 6514259 0350604 18 58 0 000 5825406 7203111 1 black 2 55136 4736266 5 39 0 000 3 481921 1 620798 _cons 4 72676 9236842 5 12 0 000 6 541576 2 911943 est store intonly There are several ways to test whether the intercepts differ by race a Since there are only two grou
22. ckline2 Interaction Line for blacks twoway connected whiteline2 blackline2 jobexp wo oO oJ oO 1O N oy o_O OOP to T T T T T 0 5 10 15 20 jobexp Interaction effects and group comparisons Page 22 Model 3 All coefficients freely differ across groups reg income educ jobexp black blacked blackjob Source SS df MS Number of obs 500 F 5 494 487 60 Model 33411 2623 5 6682 25246 Prob gt F 0 0000 Residual 6769 98696 494 13 7044271 R squared 0 8315 Adj R squared 0 8298 Total 40181 2493 499 80 5235456 Root MSE 3 7019 income Coef Std Err t P gt t 95 Conf Interval educ 1 893338 054125 34 98 0 000 1 786994 1 999681 jobexp 122255 0396598 18 21 0 000 6443323 8001777 black 3 409988 1 756477 1 94 0 053 0410983 6 861074 blacked 2153886 1038015 2 08 0 039 4193354 0114418 blackjob 3002799 0812705 3 69 0 000 4599584 1406015 _cons 6 461189 1 0479 6 17 0 000 8 520079 4 402298 est store intedjob To test whether there are any racial differences in effects ftest baseline intedjob Assumption baseline nested in intedjob F 3 494 14 91 prob gt F 0 0000 Also in Stata you can easily use the test command test black blacked blackjob 1 black 0 2 blacked 0 3 blackjob 0 F 3 494 14 91 Prob gt F 0 0000 Here we are using the baseline model that did not allow for any differences across group
23. d 0 7576 Total 40181 2493 499 80 5235456 Root MSE 4 4179 income Coef Std Err t P gt t 95 Conf Interval 1 black 6 298638 5424112 11 61 0 000 7 364345 5 232931 educ 5775958 2176483 2 65 0 008 1 005222 1499695 c educ c educ 0859208 0081894 10 49 0 000 0698305 1020111 _cons 20 41186 1 470897 13 88 0 000 17 5219 23 30181 The pronounced cross operator is used for interactions and product terms The use of implies the i prefix i e unless you indicate otherwise Stata will assume that the variables on both sides of the operator are categorical and will compute interaction terms accordingly Hence we use the c notation to override the default and tell Stata that educ is a continuous variable So c educ c educ tells Stata to include educ 2 in the model we do not want or need to compute the variable separately Similarly i race c educ produces the race educ interaction term Stata also offers a notation called factorial cross It can save some typing and or provide an alternative parameterization of the results Interaction effects and group comparisons Page 16 At first glance the use of factor variables might seem like a minor convenience at best They save you the trouble of computing dummy variables and interaction terms beforehand Further factor variables have some disadvantages e g as of this writing they cannot be used with nestreg or stepwise The advantages of factor variables become
24. er for group than it is for group 2 Such a model implies some sort of flat advantage or disadvantage for members of group 1 For example if Y was income and X was education this kind of model would suggest that for blacks and whites with equal levels of education whites will average 2 000 a year more For both blacks and whites however each year of education is worth an additional 3 000 on average Hence whites with 10 years of education will average 2 000 more a year than blacks with 10 years of education whites with 12 years of education will average 2 000 more a year than blacks with 12 years of education etc If there are more than two groups you can just include additional dummy terms and add additional parallel lines to the above graph The T value for the dummy variable tells you whether the intercept for that group differs significantly from the intercept for the reference group Here is how we could generate such a graph for our race data using Stata There are different ways of doing this e g see the graphics in the Appendix on Interaction terms the old fashioned way Iam going to use the margins command whose output can be hard to read so I won t show it but try it on your own and the marginsplot command which as you might guess is graphically displaying all the numbers that were generated by margins est restore intonly results intonly are active now quietly margins black at jobexp
25. g to be careful of When comparing groups by estimating separate models it is entirely possible that a variable will have a significant effect in one group and an insignificant effect in the other Yet the difference in effects between the groups may not be statistically significant This might occur if say the sample size for one group is larger than the sample size for the other It would therefore be very misleading to say that a variable was important for one group but not the other Likewise apparently large differences in effects may not be statistically significant When comparing groups you should do formal statistical tests such as those described here if you want to claim there are group differences don t rely on just eyeballing Interaction effects and group comparisons Page 14 Appendix Factor Variables Stata 11 and higher Factor variables not to be confused with factor analysis were introduced in Stata 11 Factor variables provide a convenient means of computing and including dummy variables interaction terms and squared terms in models They can be used with regress and several other albeit not all commands For example use http www3 nd edu rwilliam statafiles blwh dta clear reg income i black educ jobexp Source Ss df MS Number of obs 500 F 3 496 787 14 Model 33206 4588 3 11068 8196 Prob gt F 0 0000 Residual 6974 79047 496 14 0620776 R squared 0 8264 Adj R squared
26. i command Stata has some shortcuts for computing dummy variables and interaction terms In particular there is the xi interaction expansion command If you have Stata 11 or higher you will probably want to use factor variables instead although xi can still be helpful for commands that do not support factor variables although even in those cases I usually prefer to compute the interactions myself A typical syntax is xi reg income i black educ i black jobexp i black _Iblack_0O 1 naturally coded _Iblack_0 omitted i black educ _IblaXeduc_ coded as above i black jobexp _IblaxXjobex_ coded as above Source SS df MS Number of obs 500 F 5 494 487 60 Model 33411 2623 5 6682 25246 Prob gt F 0 0000 Residual 6769 98696 494 13 7044271 R squared 0 8315 Adj R squared 0 8298 Total 40181 2493 499 80 5235456 Root MSE 3 7019 income Coef Std Err t P gt jt 95 Conf Interval _Iblack_1 3 409988 1 756477 1 94 0 053 0410983 6 861074 educ 1 893338 054125 34 98 0 000 1 786994 1 999681 _IblaXeduc_1l 2153886 1038015 2 08 0 039 4193354 0114418 _Tblack_1 dropped jobexp lt 122255 0396598 18 21 0 000 6443323 8001777 _IblaXjobe 1 3002799 0812705 3 69 0 000 4599584 1406015 _cons 6 461189 1 0479 6 17 0 000 8 520079 4 402298 test _Iblack_1 _IblaXeduc_1 _IblaXjobex_1 1 _Iblack_1 0 2 _IblaXeduc_1 0 3 _IblaXjobex_1l 0 F 3 494 14 91 Prob gt F
27. it less from job experience than do whites Specifically each year of job experience is worth about 256 less for a black than it is for a white Doing an incremental F test ftest intonly intjob Assumption intonly nested in intjob Be Dy 495 LO 57 prob gt F 0 0012 Or doing a Wald test with the test command test black job 1 blackjob 0 F 1 495 Prob gt F 10 57 0 0012 To generate a graph of an interaction in Stata again using jobexp only note that the effect of job experience for blacks is almost zero here Interaction effects and group comparisons Page 21 reg income jobexp black blackjob Source Ss df MS Number of obs 500 F 3 496 68 11 Model 11723 42 3 3907 80666 Prob gt F 0 0000 Residual 28457 8293 496 57 3746558 R squared 0 2918 F Adj R squared 0 2875 Total 40181 2493 499 80 5235456 Root MSE 7 5746 income Coef Std Err t P gt t 95 Conf Interval jobexp 417038 0791602 5427 0 000 2615073 5725687 black 5 874446 2 097077 2 80 0 005 9 994694 1 754198 blackjob 3719771 160237 2 32 0 021 6868041 0571501 _cons 24 15976 1 178664 20 50 0 000 21 84397 26 47555 predict whiteline2 if black option xb assumed fitted values 100 missing values generated predict blackline2 if black option xb assumed fitted values 400 missing values generated label variable whiteline2 Interaction Line for whites label variable bla
28. much more apparent when used in conjunction with post estimation commands such as margins Note Not all commands support factor variables In particular user written commands often will not support factor variables sometimes because the commands were written before Stata 11 came out Chapters 11 and 25 of the Stata Users Guide provide more information Or from within Stata type help fvvarlist Interaction effects and group comparisons Page 17 Appendix Interaction Effects the Old Fashioned Way Older versions of Stata do not support factor variables and even some programs you can use in Stata 12 especially older user written programs do not support factor variables Therefore you may need to compute the interaction terms yourself Preliminary Steps If the dummy variables and interaction terms are not already in our data set we need to compute them e Compute a DUMMY variable for group membership Code it 1 for all members of one of the groups 0 for all members of the others For example you could do something like gen dummy group 1 amp missing group Here dummy will equal if group equals 1 It will equal 0 if group has any other nonmissing value dummy will be missing if group is missing Another possible approach tab x gen dummy If x had 4 categories this would create dummy dummy2 dummy3 and dummy4 You could use the rename command to create clearer names e g rename dummyl catholic rename
29. ntercepts to differ by race we add the dummy variable black to the model reg income educ jobexp black Source SS df MS Number of obs 500 4 F 3 496 787 14 Model 33206 4588 3 11068 8196 Prob gt F 0 0000 Residual 6974 79047 496 14 0620776 R squared 0 8264 4 Adj R squared 0 8254 Total 40181 2493 499 80 5235456 Root MSE 3 7499 income Coef Std Err a P gt t 95 Conf Interval educ 1 840407 0467507 396 3 0 000 1 748553 1 932261 jobexp 6514259 0350604 18 58 0 000 5825406 7203111 black 2 55136 4736266 5 39 0 000 3 481921 1 620798 _cons 4 72676 9236842 5 12 0 000 6 541576 2 911943 est store intonly To do Wald and F tests of the effect of black test black 1 black 0 F 1 496 29 02 Prob gt F 0 0000 ftest intonly baseline Assumption baseline nested in intonly F 1 496 29 02 prob gt F 0 0000 Here is how we could generate such a graph for our race data using Stata note that I am only using jobexp and not educ on average blacks earn 10 300 less than whites with comparable levels of job experience Interaction effects and group comparisons Page 19 reg income jobexp black Source SS af MS Number of obs 500 F 2 497 98 60 Model 11414 229 2 5707 11449 Prob gt F 0 0000 Residual 28767 0203 497 57 8813285 R squared 0 2841 Adj R squared 0 2812 Total 40181 2493 499 80 5235456 Root MSE 7 608 income
30. ps we can look at the t value for black It is highly significant implying that the intercepts do differ Note however that if there were more than 2 groups a t test would not be sufficient b We can also do a Wald test Since only one parameter is being tested the F value will as usual be the square of the corresponding T value Since we are using factor variables you refer to 1 black rather than black Interaction effects and group comparisons Page 2 test 1 black 1 l black 0 F 1 496 Prob gt F 29 02 0 0000 However I find that testparm is often a little easier to use especially if the categorical variables have more than 2 categories This is because I can just copy part of the syntax that was used in the estimation command without having to get the numbers correct for coefficients e g 1 black like I did above From here on out I will show the commands for both test and testparm but I will only show the output from testparm testparm i black 1 1 black 0 F 1 496 Prob gt F 29 02 0 0000 c If we aren t using software that makes life so simple for us we can compute an incremental F test In this case the constrained model is the baseline model which forced all parameters to be the same for blacks and whites SSE 7383 DFE 497 N 500 The unconstrained model is Model 1 which allows the intercepts to differ SSE 6975 DFE 496 N 500 The incremental F is then f _
31. rent This approach however has some major limitations First it does not tell you which coefficients differ across groups Possibilities include a only the intercepts differ across groups b the intercepts and some subset of the slope coefficients differ across groups or c all of the coefficients both intercepts and slope coefficients differ across groups A related problem is that running separate models for each group can be quite unwieldy estimating many more coefficients than may be necessary It becomes even more unwieldy if there are multiple group characteristics you are interested in e g race gender and religion Recall that when extraneous parameters are estimated it becomes more difficult to detect those effects that really do differ from zero Further theory may give you good reason for believing that the effects of only a few variables may differ across groups rather than all of them In this handout we consider an alternative strategy for examining group differences that is generally easier and more flexible Specifically by incorporating dummy variables for group membership and interaction terms for group membership with other independent variables we can better identify what effects if any differ across groups Interaction effects and group comparisons Page 1 Model 0 Baseline Model No differences across groups As before we can begin with a model that does not allow for any differences in model parameters
32. s as our constrained model 7t is also quite common indeed perhaps more common to treat Model 1 the model that allows the intercepts to differ as the constrained model Hence if we want to test whether either or both of the slope coefficients differ across groups we can give the command test blackjob blacked 1 blackjob 0 2 blacked 0 F 2 494 Prob gt F 7 47 0 0006 Interaction effects and group comparisons Page 23 Or using incremental F tests ftest intonly intedjob Assumption intonly nested in intedjob 7 47 0 0006 F 2 494 prob gt F This tells us that at least one slope coefficient differs across groups Further the T values for blackjob and blacked indicate that both significantly differ from 0 Interaction effects and group comparisons Page 24 Appendix The nestreg command Warning As of this writing the nest reg command does not work with factor variables The nest reg command provides a convenient means for estimating and contrasting nested models By default variables are added one at a time If you put parentheses around a set of variables the entire set will be entered in the same step use http www3 nd edu rwilliam statafiles blwh dta clear gen blacked black educ gen blackjob black job nestreg reg income educ jobexp black blacked blackjob Block 1 educ jobexp Source SS df MS Number of obs 500
33. s the reference category but we can easily change that e g ib2 religion would make Protestant the reference category or ib Jast religion would make the last category Other the reference Factor variables can also be used to include squared terms and interaction terms in models For example to add interaction terms Interaction effects and group comparisons Page 15 reg income i black educ jobexp black c educ black c jobexp Source Ss df MS Number of obs 500 F 5 494 487 60 Model 33411 2623 5 6682 25246 Prob gt F 0 0000 Residual 6769 98696 494 13 7044271 R squared 0 8315 Adj R squared 0 8298 Total 40181 2493 499 80 5235456 Root MSE 3 7019 income Coef Std Ere t P gt t 95 Conf Interval 1 black 3 409988 1 756477 1 94 0 053 0410984 6 861075 educ 1 893338 054125 34 98 0 000 1 786994 1 999681 jobexp 722259 0396598 Les20 0 000 6443322 8001777 black c educ 1 2153886 1038015 2 08 0 039 4193354 0114418 black c jobexp 1 3002799 0812705 3 69 0 000 4599584 1406015 _cons 6 461189 1 0479 6 17 0 000 8 520079 4 402298 If you wanted to add a squared term to the model you could do something like reg income i black educ c educ c educ Source SS df MS Number of obs 500 F 3 496 520 90 Model 30500 3792 3 10166 7931 Prob gt F 0 0000 Residual 9680 87009 496 19 5178833 R squared 0 7591 Adj R square
34. ssed on Income Y with separate lines for men and women Interaction effects and group comparisons Page 13 Male Line Female Line In the present example women happen to have a predicted edge over men when education equals 0 They d have an even bigger edge if you extended the lines to include negative values of job education But since you don t observe such negative and zero values in reality the predicted lead for women at these values doesn t mean much e Estimating separate models for each group can result in loss of statistical power i e you can be less likely to reject the null when it is false Similarly including too many interaction terms can lead to the same problem As we have seen many times before inclusion of extraneous variables in this case extraneous interaction terms should be avoided if possible e The same model can include interactions involving more than one categorical variable For example it might be felt that the effect of education is different for whites than for nonwhites and the effect of income is different for women than for men Hence the model could include the variables EDUC WHITE and INCOME FEMALE If you have a lot of categorical variables you should think carefully about what interaction terms to include if any e As noted earlier in the course a failure to include interactions in models can lead to problems like heteroscedasticity omitted variable bias etc e One thin
35. usual note that chi square divided by DF is very close to the value of the corresponding F test lrtest baseline intedjob Likelihood ratio test LR chi2 3 43 33 Assumption baseline nested in intedjob Prob gt chi2 0 0000 In the above tests we are using the baseline model that did not allow for any differences across groups as our constrained model It is also quite common indeed probably more common to treat Model 1 the model that allows the intercepts to differ as the constrained model Hence if we want to test whether either or both of the slope coefficients differ across groups we can give the command test 1 black c educ 1 black c jobexp Output not shown testparm i black c educ i blacki c jobexp 1 1 black c educ 0 2 1 black c jobexp 0 EC 2 494 7 47 Prob gt F 0 0006 Or using incremental F tests ftest intonly intedjob Assumption intonly nested in intedjob F 2 494 7 47 prob gt F 0 0006 The likelihood ratio test is as usual note that chi square divided by DF is very close to the value of the corresponding F test lrtest intonly intedjob 14 90 0 0006 Likelihood ratio test LR chi2 2 Assumption intonly nested in intedjob Prob gt chi2 These tests tell us that at least one slope coefficient differs across groups Further the T values for blackjob and blacked indicate that both significantly differ from 0 You can plot these results using the same commands as before
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