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 =

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

9. Model CGI Analysis of Variance Source DF Sum of Squares Mean Square
F Value
Pr > F
Model 3
5.10
0.0022
Error
150
Corrected Total
153

10. Model C Analysis of Variance Source DF Sum of Squares Mean Square
F Value
Pr > F
Model 1
7.78
0.0060
Error
152
Corrected Total
153

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. Test 1 Results for Dependent Variable ar
An Easier Way
proc reg; model ar = misanth idealism MxI;
TEST idealism=0, MxI=0; run;
Test 1 Results for Dependent Variable ar
Source DF
Mean Square
F Value
Pr > F
Numerator 2
3.62
0.0291
Denominator
150

14. Model CI Analysis of Variance Source DF Sum of Squares Mean Square
F Value
Pr > F
Model 2
4.18
0.0172
Error
151
Corrected Total
153

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 Parameter Estimates Variable DF Standard Error t Value
Pr > |t|
Intercept 1
8.17
<.0001
misanth
3.72
0.0003
idealism
2.58
0.0110
MxI
-2.25
0.0258

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 Analysis of Variance Source DF Sum of Squares Mean Square
F Value
Pr > F
Model 2
4.98
0.0081
Error
151
Corrected Total
153

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
Proc sgplot; 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‘ by idealism; run;

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

27. Full Model Slope 1
AR = (.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).

28. 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

29. 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.

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

31. Full Model Slope 2
AR = (.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.

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

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

34. 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.

35. 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.

36. 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.

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

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

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

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

41. 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.

42. The SAS Macro
%process (data=kevin,vars=ar misanth idealism,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

43. The SAS Macro
%process (data=kevin,vars=ar misanth idealism,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.

44. 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.

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

46. 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).

48. 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.

49. Different Slopes, Similar Correlations

50. Identical Slopes, Different Correlation Coefficients

51. Comparing the Groups on r
At SPSS and SAS programs for comparing Pearson correlations and OLS regression coefficients is the code for this analysis.
r is significant for nonidealists, not for idealists.
For more than two groups, use this chisquare (also available at link above).

52. Analysis of Covariance
You already know how to do this.
Just drop the interaction term from the model.
Here that would not be appropriate, as well have heterogeneity of regression.

53. A Couple of t Tests
You may also want to compare the groups on the Y (ignoring C) and/or C.
I have include those tests in the program.
This is redundant with the initial Proc Corr output (point biserial correlations).

54. SPSS This analysis is easy to do with SPSS too. See my handout.
You can do the analysis in a sequential fashion.
And get the partial F tests from SPSS, even with df > 1: Leave your calculator in the desk drawer.

55. Three Groups Two dummy variables, G1 and G2
Two interaction terms, G1C and G2C
To test the slopes you would see if the model fit were significantly reduced by simultaneously removing G1C and G2C .
That would be an F with two df in its numerator.
To test the intercepts, remove both G1 and G2

56. Let’s Go Fishing Length = a + bWeight for flounder
Does the relationship differ across regions?
Pamlico Sound
Pamlico River
Tar River
Potthoff3.sas and Potthoff3.dat
Download and run.

57. Proc GLM Proc GLM; class Location;
Model Length = WeightSR|Location;
GLM creates the (2) dummy variables for you
And the (2) interaction terms.

58. Full Model Source DF Sum of Squares Mean Square F Value Pr > F5
<.0001
Error
745
Corrected Total
750

59. Covariate Only Model
Proc GLM; class Location; model Length = WeightSR / solution;
Weights are significantly correlated with lengths, r2 = .95, F(1, 749) = 14,544, p < .001.

60. Test of Coincidence On 4, 745 df, p < .001
The lines are not coincident.

61. Look at Full Model
The slopes do not differ significantly, F(2, 745) = 1.63, p = .20.
The intercepts do differ significantly, F(2, 745) = 4.90, p = .008.
Since the slopes do not differ significantly
But the intercepts do,
The group means must differ.

62. Source DF
Type III SS
Mean Square
F Value
Pr > F
WeightSR 1
<.0001
Location 2
4.90
0.0077
WeightSR*Location
1.63
0.1961

63. Analysis of Covariance
Location significantly affects mean length of flounder, after adjusting for the effect of weight.

64. Unadjusted Means (notice the different pattern)