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Comparing Regression Lines

Comparing Regression Lines. From Independent Samples. The Design. You have two or more groups. One or more continuous predictors (C). And one continuous outcome variable (Y). You want to know if Y = a + b 1 C 1 + … + b p C p + error Is the same across groups. Poteat, Wuensch, & Gregg.

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Comparing Regression Lines

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  1. Comparing Regression Lines From Independent Samples

  2. The Design • You have two or more groups. • One or more continuous predictors (C). • And one continuous outcome variable (Y). • You want to know if • Y = a + b1C1 + … + bpCp + error • Is the same across groups.

  3. Poteat, Wuensch, & Gregg • Predictive validity study • Children referred for school psychology services • Does Grades = a + bIQ + error • Differ across races? • Called a “Potthoff analysis” by school psychologists • Differences fell short of significance.

  4. Two Groups, One X • Y = a + b1C + b2G + b3CG • If there are more than two groups, groups is represented by k-1 dummy variables • Wuensch, Jenkins, & Poteat • Y = Attitude to animals • C = Misanthropy • G = High idealism or not

  5. SAS • Potthoff.sas -- download • Potthoff.dat – download • Point program file to data file • Run the program • Data step: MxI = MisanthIdealism • Page 1: Ignoring idealism, is a .2 corr between misanthropy and attitude to animals.

  6. Zero-Order Correlations

  7. Interpretation • Misanthropy is significant related to support for animal rights. • The two idealism groups do not differ significantly on support for animal rights – rpb = .092. • The two idealism groups do not differ significantly on misanthropy – rpb = -.099.

  8. Four Regression Models • ProcReg; • CGI: model ar = misanth idealism MxI; • C: model ar=misanth; • CG: model ar = misanth idealism; • CI: model ar = misanth MxI;

  9. Model CGI

  10. Model C

  11. Test of Coincidence • Compare model CGI with model C • Use a partial F test p = .029

  12. Conclusion • This was a simultaneous test of intercept and slopes. • We conclude that the two groups differ with respect to • The intercepts, or • The slopes, • Or both.

  13. An Easier Way procreg; model ar = misanth idealism MxI; TEST idealism=0, MxI=0; run;

  14. Model CI

  15. Test of Intercepts • Compare model CGI with model CI. • The intercepts differ significantly. p = .011

  16. F(1, 150) = 6.632, p = .011 • As you know, on one df, t = SQRT(F) • Look back at Model CGI • For the test of main effect of idealism, t= SQRT(6.632) = 2.58, p= .011. • If we had more than two groups we could not take this shortcut.

  17. Model CGI

  18. Test of Parallelism • Do the slopes differ significantly? • Compare model CGI with model CG • Is model fit significantly reduced when we remove the interaction term? p = .026

  19. Model CG

  20. F(1, 150) = 5.073, p = .026 • As you know, on one df, t = SQRT(F) • Look back at Model CGI • For the test of the interaction, t = 2.25,p = .026. • If we had more than two groups we could not take this shortcut.

  21. Get the Separate Regression Lines • Sort by groups. • Run the bivariate regressions • For nonidealists, • For idealists,

  22. Prepare Plots Procsgplot; scatter x = misanth y = ar; reg x = misanth y = ar; yaxis label='Attitude to Animals‘ grid values=(1 to 5 by 1); xaxis label='Misanthropy‘ grid values=(1 to 5 by 1); by idealism; run;

  23. Another Plot procsgplot; reg x = misanth y = ar / group = idealism nomarkers; yaxis label='Attitude to Animals'; xaxis label='Misanthropy'; run;

  24. Full Model Slope 1 • AR = 1.626 + (.300)Misanthropy + (.779)Idealism + (- .285)Interaction. • This is a conditional slope. • predicted increase in AR accompanying a one-point increase in misanthropy is .3 given that idealism has value zero (the idealists).

  25. the conditional effect of X on Y given a particular value of the moderator is the conditional slope for predictor X + the interaction slope times the value of the moderator. Idealism as moderator, simple effect of misanthropy

  26. Simple Slopes for Misanthropy • Idealism = 0 (nonidealists) • Each one point increase in Misanthropy lead to a .3 point increase in AR. • Idealism = 1 (idealists) • Each one point increase in Misanthropy leads to a .015 point increase in AR.

  27. Misanthropy as moderator, simple effects of idealism (group differences)

  28. Full Model Slope 2 • AR = 1.626 + (.300)Misanthropy + (.779)Idealism + (- .285)Interaction. • This is a conditional slope. • The predicted increase in AR accompanying a one-point increase in idealism (idealism groups were coded 0,1) is .779 given that misanthropy has value zero.

  29. Treating Idealism as the moderator, the simple slope for the effect of misanthropy on AT is

  30. Simple Slopes for Idealism • predict the difference between the two idealism groups (idealist minus nonidealist) when misanthropy = 1) • .779 -.285(1) = .505. • If misanthropy = 4, the predicted difference in means is .779 - .285(4) = -.361

  31. Probing the Interaction • Same as simple effects analysis in ANOVA • We have already shown that the relationship between misanthropy and support for animal rights is significant for nonidealists but not for idealists. • Change perspectives -- how does misanthropy moderate the relationship between idealism (group) and support of animal rights.

  32. Analysis of Simple Slopes • Arbitrarily pick two or more values of misanthropy and compare the groups at those points. • The points are often 1 SD below the mean, the mean, and 1 SD above the mean. • Here, that would be misanthropy = 1.65, 2.32, and 2.99.

  33. Testing the Simple Slopes • To test the null that mean AR does not differ between groups when misanthropy = 1.65, we center the misanthropy scores around 1.65, recomputed the interaction term, and run the full model again. • We repeat this action with the scores centered around 2.32 and then again centered around 2.99. • See the code in the program.

  34. The Code Data Centered; set kevin; MisanthLow = misanth - 1.65; InteractLow= MisanthLow * Idealism; MisanthMean = misanth - 2.32; InteractMean= MisanthMean * Idealism; MisanthHigh = misanth - 2.99; InteractHigh= MisanthHigh * Idealism; procreg; Low: model ar = MisanthLow idealism InteractLow; Mean: model ar = MisanthMean idealism InteractMean; High: model ar = MisanthHigh idealism InteractHigh; run; Quit;

  35. Low Misanthropy When MIS is low, AR is significantly higher (by .309) in the idealistic group than it is in the nonidealistic group.

  36. Average Misanthropy • The groups do not differ significantly when MIS is average.

  37. High Misanthropy • The groups do not differ significantly when MIS is High.

  38. Process Hayes • Makes it way easier to do this analysis. • Bring the process.sas program into SAS and run it. • You have already read the data into the work file “kevin.” • Hayes also provides a script to do the same in SPSS.

  39. The SAS Macro %process (data=kevin,vars=armisanthidealism,y=ar,x=idealism,m=misanth, model=1,jn=1,plot=1); • Data= points to the SAS data file • Vars= identifies the variables • Y= identifies the outcome variable • X= identifies the focal predictor variable • M= identifies the moderator variable • Model=1 identifies the simple moderation model – see the templates document

  40. The SAS Macro %process (data=kevin,vars=armisanthidealism,y=ar,x=idealism,m=misanth, model=1,jn=1,plot=1); • jn=1 invokes the Johnson-Neyman technique • Plot=1 requests the values for making a plot to visualize the interaction. • Notice that the output includes all of the tests we did earlier, the hard way.

  41. Johnson-Neyman Technique • Maps out the values of the moderator for which the effect of the focal predictor is significant versus those values for which it is not significant. • I’ll use idealism groups as the focal predictor and misanthropy as the moderator.

  42. The Boundary • When misanthropy = 2.1286 or less, the difference between the groups is statistically significant (higher for the idealists), otherwise it is not.

  43. If we were to extrapolate beyond misanthropy = 4, we would find a second region where the difference between the groups would be significant (with the mean higher for the nonidealists).

  44. Don’t Confuse Test of Slopes with Test of Correlation Coefficients • If the slopes are the same across groups, the correlation coefficients (standardized slopes) may or may not. • If the correlation coefficients are the same across groups, the slopes may or may not.

  45. Different Slopes, Similar Correlations

  46. Identical Slopes, Different Correlation Coefficients

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