1. 1 Psych 5510/6510 Chapter 13: ANCOVA: Models with Continuous and Categorical Predictors Part 3: Within a Correlational Design Spring, 2009

2. 2 Contexts We will be looking at three contexts in which this will be useful: 1.Within a ‘true experimental’ design, where we can use this approach to increase the power of the design and to add sophistication to our model. 2.Within a ‘quasi-experimental’ or ‘static group’ design, where we can use this approach to control a confounding variable. 3.Within a correlational design, where we can introduce a categorical variable to better understand a continuous variable.

3. 3 Context 3: Using Categorical Variables to Better Understand Continuous Variables So far we have used a continuous variable to help our analysis of categorical variables. Now we switch to focusing on using a categorical variable to help our understanding of the relationship between a continuous variable and the dependent variable.

4. 4 The Variables 1.Dependent Variable: support for a woman’s right to choose abortion (1-10 scale). 2.Continuous Independent Variable: political ideology (1-10 scale of conservative to liberal) 3.Categorical Independent Variable: Religious preference (Protestant, Catholic, or Jewish).

5. 5 Right to Choose (RTC) & Political Ideology The focus of our research is on the relationship between right to choose (RTC) and political ideology (conservative to liberal...a continuous variable). Dependent variable: Y RTC = view on right to choose Independent variable: X Pol = political ideology Model C: Ŷ RTC = β 0 Model A: Ŷ RTC = β 0 + β 1 X Pol Ŷ RTC = 3.60 +.48X Pol PRE=.117 F*=52.53 p<.0001 It is worthwhile to add ideology to a model of RTC.

6. 6 Our attention turns to the value of adding religious preference (a categorical variable) to the model. We are particularly interested in what religious preference might add to our understanding of the relationship between the variables of primary interest (RTC and ideology). We will begin by examining the relationship between religious preference and RTC. Note this example came from the text, I apologize if anyone is offended by anything about this example.

7. 7 Religion and RTC Dependent variable: Y RTC = view on right to choose Independent variable: X Rel = religion (Catholic, Protestant, Jewish). This is a categorical variable with three values so it will take two contrast codes to code it. Comparison 1: Catholic vs. (Protestant & Jewish) Comparison 2: Protestant vs. Jewish Model C: Ŷ RTC = β 0 Model A: Ŷ RTC = β 0 + β 1 X 1 + β 2 X 2

8. 8 Ŷ RTC = 5.97 -.84X 1 +.13X 2 PRE=.17 F*=41.02 p<.0001 It is worthwhile to use religion (coded by X1 and X2) in a model of RTC. To answer more specific questions: For b1=-.84, p<.0001 Can conclude Catholics differ from (Jews and Protestants combined) in RTC. For b2=.23, p=.51 Unable to conclude Jews differ from Protestants in RTC. Even though this comparison is not significant, we will continue to include it in later analyses to see what crops up as we work with the model (e.g. to see if it interacts with our other variable).

9. 9 What We Know So Far We are modeling people’s attitudes on ‘right to choose’, we have established that: 1. Ideology predicts RTC. 2. Religion predicts RTC. We know that Catholics differ from the group of (Protestants and Jews combined) in RTC We were unable to show that Protestants and Jews differ in RTC.

10. 10 Ideology & Religion Both political ideology (a continuous variable ) and religion (a categorical variable) predict RTC, next let’s examine the relationship between our two predictor variables. To accomplish that we can see to what degree religion predicts ideology. By the way, why not the other way around? (think about it) Model C: Ŷ Pol = β 0 Model A: Ŷ Pol = β 0 + β 1 X 1 + β 2 X 2

11. 11 Ideology & Religion Ŷ Pol = 5.18 -.21X 1 + 1.06X 2 PRE=.126 F*=28.72 p<.0001. Yes, there is a relationship between ideology and religion. To answer more specific questions: For b1=-.21, p=.0036 Can conclude Catholics differ from the group of (Jews and Protestants combined) in their political ideology. For b2=1.06, p=.001 Can conclude Jews differ from Protestants in their political ideology.

12. 12 What This Tells Us We have found that our two predictor variables are related, this is interesting for at least two reasons. 1.It adds to our knowledge about the predictor variables in our model, we now know that religion and ideology are related. We also have more specific information about this relationship thanks to our contrasts. 2.It tells us that our predictor variables (religion and ideology) are somewhat redundant, which is important to know as it might influence our interpretation of significance when all the variables are in the model at once.

13. 13 Expanding our Understanding Again our primary interest was in the relationship between attitudes on RTC and political ideology. We now know that ideology and religion are somewhat redundant. This raises the possibility that religion is a confounding variable in the relationship between ideology and RTC. If religion is influencing both RTC and ideology then perhaps that is the only reason why we are getting a relationship between RTC and ideology. At this point what we want to know is what is the relationship between RTC and ideology when religion is held constant?

14. 14 Answering that Question We can answer that question quite easily: Model C: Ŷ RTC = β 0 + β 1 X 1 + β 2 X 2 Model A: Ŷ RTC = β 0 + β 1 X 1 + β 2 X 2 + β 3 X Pol Ŷ RTC = 3.28 -.73X 1 -.42X 2 +.52X Pol The partial correlation coefficient for X Pol =.382, this is the correlation between ideology and RTC when religion is held constant (i.e. the correlation between ideology and RTC for people who are of the same religion). PRE=.382²=.146 F*=67.85 p<.0001. It is worthwhile to add political ideology to a model that already contains the person’s religion.

15. 15 Looking at the b’s Ŷ RTC = 3.28 -.73X 1 -.42X 2 +.52X Pol b1=-.73, p<.001 b2=-.42, p=.025 b3=.52, p<.001 Catholics vs. (Jews and Protestants combined), Jews vs. Protestants, and political ideology, are all worthwhile to add last to a model that contains the other variables.

16. 16 Continuing to Expand Understanding Because religion is a categorical variable, it is easy to examine the relationship between ideology and RTC within each religion. This will shed quite a bit more light on the relationship of primary interest (ideology and RTC).

17. 17 Protestant Respondents Ŷ RTC = 3.28 -.73X 1 -.42X 2 +.52X Pol Being Protestant is coded by a score of -1 on X1, and -1 on X2 (see slide 7). Plugging in those values: Ŷ RTC = 3.28 -.73(-1) -.42(-1) +.52X Pol Ŷ RTC = 4.43 +.52X Pol This is the relationship between RTC and ideology for Protestant respondents.

18. 18 Catholic Respondents Catholic respondents have a score of 2 on X1, and 0 on X2. Plugging in those values: Ŷ RTC = 3.28 -.73X 1 -.42X 2 +.52X Pol Ŷ RTC = 3.28 -.73(2) -.42(0) +.52X Pol Ŷ RTC = 1.82 +.52X Pol This is the relationship between right to choose and political ideology for Catholic respondents.

19. 19 Jewish Respondents Jewish respondents have a score of -1 on X1, and 1 on X2. Plugging in those values: Ŷ RTC = 3.28 -.73X 1 -.42X 2 +.52X Pol Ŷ RTC = 3.28 -.73(-1) -.42(1) +.52X Pol Ŷ RTC = 3.59 +.52X Pol This is the relationship between right to choose and political ideology for Jewish respondents.

20. 20 The Relationship Within Each Religion Putting these slides together, the relationships between political ideology and right to choose within each of the three religions are: –Jews: Ŷ RTC = 3.59 +.52X Pol –Catholic: Ŷ RTC = 1.82 +.52X Pol –Protestant: Ŷ RTC = 4.43 +.52X Pol

21. 21 Political Ideology RTC

22. 22 Additive Model The regression equations and the previous graph show the same slope for all three religions, this is because our model so far is an additive model (it has no interaction terms): Ŷ RTC = β 0 + β 1 X 1 + β 2 X 2 + β 3 X Pol

23. 23 Interactions By not having any interaction terms in our model we have precluded having the slopes change as we move from one religious preference to another. Let’s add some interaction terms to see if it is worthwhile to include them in the model.

24. 24 Testing for Interactions We will introduce two new variables to represent the interaction between political ideology and the two contrasts that code religion. Interaction 1: β 4 X 1 X Pol Interaction 2: β 5 X 2 X Pol Now we regress Y on all six predictors. Ŷ RTC = β 0 + β 1 X 1 + β 2 X 2 + β 3 X Pol + β 4 X 1 X Pol + β 5 X 2 X Pol.

25. 25 Testing for an Interaction First, we will test to see if a non-additive model is worthwhile: Model C: Ŷ RTC = β 0 + β 1 X 1 + β 2 X 2 + β 3 X Pol Model A: Ŷ RTC = β 0 + β 1 X 1 + β 2 X 2 + β 3 X Pol + β 4 X 1 X Pol + β 5 X 2 X Pol Ŷ RTC = 3.28 +.07X 1 -.28 X 2 +.51X Pol + -.16X 1 X Pol + -.06X 2 X Pol PRE=.042, F*=8.53, p=.0002 OK, a non-additive model is worthwhile.

26. 26 Simple Relationships Ŷ RTC = 3.28 +.07X 1 -.28 X 2 +.51X Pol + -.16X 1 X Pol + -.06X 2 X Pol If we plug in the values for X1 and X2 that code the various religions then we arrive at the following formulas: Protestants: Ŷ RTC =3.49+.73X Pol Catholics: Ŷ RTC =3.42 +.19X Pol Jews: Ŷ RTC =2.93 +.61X Pol

27. 27 Political Ideology RTC

28. 28 Individual Interactions Ŷ RTC = 3.28 +.07X 1 -.28 X 2 +.51X Pol -.16X 1 X Pol -.06X 2 X Pol X 1 X Pol This is the interaction between the first contrast (Protestant and Jewish combined) vs. (Catholic) and political ideology. Go back to the previous slide. It looks like if we combined the Protestant and Jewish slopes and compared that to the slope for the Catholics we might find a difference in the slopes. β 4 = -.16, p<.0001. That difference in the slopes is indeed statistically significant.

29. 29 Individual Interactions Ŷ RTC = 3.28 +.07X 1 -.28 X 2 +.51X Pol + -.16X 1 X Pol + -.06X 2 X Pol X 2 X Pol This is the interaction between the first contrast (Protestant vs. Jewish) and political ideology. Go back to the graph. It looks like there is not much of a difference between the slopes for Protestant and Jewish groups. β 5 = -.06, p=.68. This interaction is indeed not statistically significant

30. 30 Interpreting the b’s Ŷ RTC = bo + b1X 1 +b2 X 2 + b3X Pol + b4X 1 X Pol + b5X 2 X Pol Ŷ RTC = 3.28 +.07X 1 -.28 X 2 +.51X Pol + -.16X 1 X Pol + -.06X 2 X Pol b1=.07 Variable X1 codes the contrast Catholic vs. (Jewish and Protestant). In an additive model the p value for b1 tells us whether we can conclude that contrast is significant. When we add the interaction of X1 and X Pol to the model then the p value for b1 tells us whether we can conclude the contrast is significant for the special case where X Pol = 0.

31. 31 When X Pol =0 there is little difference between Catholics and (Jews and Protestants combined). Not only that, it is of little theoretical interest, for our scale for X Pol runs from 1 to 10, it can’t even be zero. RTC Xpol

32. 32 Centering X Pol Let’s solve that problem by centering X Pol by subtracting the mean of X Pol (4.8) form each X Pol score. X Pol’ = X Pol - 4.8 X1 and X2 are contrasts, they are automatically centered (they have a mean of zero).

33. 33 Interpreting the b’s Ŷ RTC = 3.28 +.07X 1 -.28 X 2 +.51X Pol + -.16X 1 X Pol + -.06X 2 X Pol Ŷ RTC = 5.75 -.72X 1 -.56 X 2 +.51X Pol’ + -.16X 1 X Pol’ + -.06X 2 X Pol’ Using X Pol’ rather than X Pol changed the values of b1 and b2 but not the values of b3, b4, b5. Remember, b1 is the slope of the relationship between X1 and Y when all the other variables equal 0, and we have changed X Pol to X Pol’, so that would effect X1. The same goes for b2. b3 didn’t change because X1 and X2 didn’t change. And, the interaction terms (b4 and b5) are not affected by centering data.

34. 34 The effect of centering X Pol is just to move the Y axis to the mean of X Pol (4.8). b1 reflects Catholic vs. (Protestant and Jewish) at the new axis, b2 reflects Protestant vs. Jewish at the new axis. Note that the contrast reflected by b1 is now much larger and the contrast reflected by b2 is slightly larger. Both are more meaningful at the new axis (reflecting the contrast when X Pol equals its mean, rather than when it equals zero).

35. 35 Our Model Ŷ RTC = b0 +b1X 1 +b2 X 2 + b3X Pol’ + b4X 1 X Pol’ + b5X 2 X Pol’ Ŷ RTC = 5.75 -.72X 1 -.56 X 2 +.51X Pol’ + -.16X 1 X Pol’ + -.06X 2 X Pol’ All the b’s are statistically significant except for b5. We can see from the graph that the slopes for Protestant and Jewish don’t differ much, it’s not worth having that difference in our model. Final model (to date): Ŷ RTC = 0 + β1X 1 + β2 X 2 + β3X Pol’ + β4X 1 X Pol’