Ideal Answers to Chapter 10 Questions (Interactions in Multiple Regression)
QUESTION 10.1a. A simultaneous multiple regression analysis revealed that both Homer’s presence and
Marge’s presence were uniquely associated with the disappearance of cookies. The association for
Homer was β = .577, t (9) = 3.00, p = .015. All else being equal, cookies were more likely than usual to
disappear on days on which Homer visited the kitchen. The association for Marge was also significant, β
= -.577, t (9) = -3.00, p = .015. However, as indicated by the negative Beta weight for Marge, cookies
were less likely than usual to disappear on days when she visited the kitchen. So we observed main
effects for both Homer and Marge.
The adjusted R2 in this regression analysis was .593, which indicates a very strong connection between
these two predictors and our criterion (missing cookies). Knowing about both Homer and Marge’s
presence allows us to predict very well on which days cookies will be missing.
QUESTION 10.1b. Adding the interaction term to this regression analysis revealed that we are now
better able to predict the disappearance of the cookies. The adjusted R2 has increased from a highly
respectable R2 = .593 to a perfect R2 = 1.00! The equivalent of a main effect for Homer is now β = 1.16, t
(8) = 1.38 x 108, p < .001. There is no main effect for Marge, was β = .00, t (8) = 0.00, p = 1.00. However,
there is an extraordinarily large Marge x Homer interaction, β = -1.00, t (8) = -9.77 x 108, p < .001. So
knowing about Homer and Marge’s combined presence seems to tell us more than knowing about
either one of them alone.
QUESTION 10.1c. To follow up on this significant interaction, we simply correlated Homer’s presence
and the disappearance of cookies on days when Marge (a) was and (b) was not present. These results
revealed that on days on which Marge was not in the kitchen, cookies always disappeared when Homer
visited and never disappeared when he did not, r (4) = 1.0, p < .001. In contrast, on days, when Marge
visited the kitchen, cookies simply never disappeared, whether Homer was there or not. In fact,
because there was no variance on “missing” on days on which Marge was present, it was not possible to
calculate a correlation between Homer’s presence and the disappearance of cookies. Collectively, these
findings point the finger of blame at Homer while strongly supporting Marge’s innocence. In fact, they
strongly suggest that Marge was consistently playing the role of cookie police officer. The simple effects
tests also show why the adjusted multiple R2 was perfect. We can now predict perfectly when cookies
will disappear. They will only and always disappear when Homer is in the kitchen without Marge!
QUESTION 10.2. In this simultaneous multiple regression analysis, we entered three predictors: (a) the
centered measure of negative interpersonal life events, (b) the centered measure of implicit sociotropic
vulnerability to depression, and (c) an interaction term that was the cross-product of these two centered
predictors.
The results of this multiple regression analysis appear below.
Predictor
β
equivalent t
p value
negative life events
.450
4.18
< .001
implicit sociotropy
.126
1.27
.209
interaction term
.226
2.07
.041
Note. There were n = 92 participants with complete data on all of these measures.
Not surprisingly, this analysis showed that the more negative life events people reported, the more
depressive symptoms they reported, p < .001. There was no unique association between the measure
of implicit sociotropy and depressive symptoms, p > .20. However, as expected, there was a significant
implicit sociotropy x negative life events interaction, p = .041.
In a series of simple slopes tests, we estimated the association between negative life events and
depressive symptoms for participants who scored one standard deviation above and below the mean on
the measure of implicit sociotropy. For participants who were 1 SD above the mean in implicit
vulnerability, there was a very strong association between negative life events and self-reported
depressive symptoms, β = .637, t (88) = 3.83, p < .001. In contrast, for participants 1 SD below the mean
in implicit vulnerability, there was a much more modest, albeit still significant, association between
negative life events and self-reported depressive symptoms, β = .263, t (88) = 2.41, p = .018. Thus, those
whose implicit sociotropy scores suggested they were less vulnerable than most were not completely
invulnerable to the impact of negative life events. Instead, it appears that they were simply less
vulnerable.
Of course, because these data were cross-sectional, we must be cautious about making any strong
causal leaps from these findings. It is possible that higher levels of depressive symptoms contributed to
a greater number of negative life events in these data (rather than the reverse). This being said, though,
it must still be the case that this reverse association is stronger for people who score higher on the
implicit sociotropic vulnerability measure. It would also be useful to see whether these implicit
diathesis-stress effects would remain significant if we were to control for any diathesis-stress effects
observed for a traditional (explicit) measure of sociotropic vulnerability.
QUESTION 10.3. In this third data set, we tested the hypothesis that the relation between implicit and
explicit self-esteem would be stronger among people who score higher on a new measure of
intuitiveness. To test this hypothesis, we conducted a simultaneous multiple regression analysis in which
our criterion variable was explicit self-esteem. In this analysis, there were three predictors: (a) a
measure of implicit self-esteem (people’s nonconscious thoughts and feelings about their value or
worth), (b) a measure of intuitiveness, and (c) an implicit self-esteem x intuitiveness interaction term. In
keeping with standard practice, we centered both implicit self-esteem and intuitiveness prior to analysis
(and prior to computing the interaction term).
This analysis revealed, first of all, that there was no unique association between implicit and explicit selfesteem, β = .051, t (114) = 0.55, p = .585. Although this might seem surprising, research on the
association between implicit and explicit self-esteem has often shown either a dissociation (i.e., a lack of
association) or only a modest association between measures of implicit and explicit self-esteem (e.g.,
see Bosson, Swann, & Pennebaker, 1999). The same analysis also revealed that there was a significant
association between intuitiveness and explicit self-esteem, β = .290, t (114) = 3.20, p = .002. More
important, this analysis revealed that these effects were qualified by a nearly significant intuitiveness x
implicit self-esteem interaction, β = .181, t (114) = 1.97, p = .052.
Instructor’s Note: These values and those that follow may vary slightly depending on the number of
decimal places to which one carries the calculations when centering and creating interaction terms
based on this centering. After selecting the 118 participants who answered all of the questions in this
case, we carried the centering calculations out to 4 decimal places. Thus, our three compute
statements read:
COMPUTE cent_ise=IMPLSE - 1.9369.
COMPUTE cent_intuit = INTUIT - 6.6525.
COMPUTE cent_inter = cent_ise*cent_intuit.
EXECUTE.
Simple slopes tests revealed a pattern that was loosely consistent with predictions. Among participants
who scored 1 SD below the mean on the measures of intuitiveness, there was a non-significant negative
association between implicit and explicit self-esteem, β = -.133, t (114) = -1.16, p = .25. In contrast, for
participants who scored 1 SD above the mean on the measures of intuitiveness, there was a nonsignificant positive association between implicit and explicit self-esteem, β = .235, t (114) = 1.59, p =
.114. It is not so clear why the association between implicit and explicit self-esteem was so weak in this
particular study. However, we can say, if we are generous enough to round the p value in the main
regression analysis to p = .05, that the association between implicit and explicit self-esteem is, in fact,
stronger among people who seem to be more inclined to trust their intuitions. Needless to say, it would
be nice to see a replication of this effect before we place too much confidence in it. It would also be
useful to see what role, if any, gender played in this analysis. Do women score higher on this new
measure of intuitiveness than men do? If so, is the intuitiveness effect really just an effect of gender in
disguise?
QUESTION 10.4a. Simple slopes tests focusing on people one standard deviation above the mean on
the measure of specific self-views (people who generally think they have a lot of skills and talents)
revealed that there was no association between differential importance and global self-esteem. That is,
the degree to which these participants valued their strengths more than their weaknesses did not
predict their global self-esteem scores, β = .082, t (46) = 0.53, p = .596. In stark contrast, the simple
slopes tests focusing on people one standard deviation below the mean in specific self-views revealed a
strong tendency for those who valued their strengths more than their weaknesses to be higher in global
self-esteem β = .605, t (46) = 5.00, p < .001.
QUESTION 10.4b. These findings show that it would be misleading indeed to argue that this interaction
is unimportant. Despite the fact that the interaction term only allows one to account for an additional
4.9 percent of the total variance in self-esteem, it turns out to be extremely important to know about
differential importance for some participants, namely people who do not evaluate themselves very
favorably on the overall measure of specific self-views. For this subset of people, differential importance
uniquely accounts for more than 36% of the total variance (.6052) in their global self-esteem. In
contrast, knowing about differential importance tells us little or nothing about the global self-esteem of
people who generally think they have a lot of talents. Rather than being problematic this difference is
both interesting and potentially useful to know about. For example, it’s likely to be very hard to get
people with negative self-views to view themselves favorably, but it might not necessarily be all that
difficult to get them to value or appreciate their self-perceived strengths relative to their self-perceived
weaknesses.
QUESTION 10.5a. In these data, the hypothesis under evaluation is that the relation between BMI (body
mass index) and self-reported physical health is stronger for women than for men. To test this
hypothesis, we followed exactly the same procedures we would have if gender had been a continuous
variable, with the assumption that the simple slopes analysis would allow us to focus separately on the
association between BMI and health for women and men.
To test the hypothesis that gender would moderate the association between BMI and self-reported
health, we conducted a simultaneous multiple regression analysis in which our three predictors were:
(a) the centered BMI measure, (b) the centered gender measure, and (c) the centered BMI x centered
gender cross-product. Our criterion variable was the self-report measure of health, reported on a 5point scale. It is important to note that on this measure, higher scores indicate poorer health (1 =
excellent, 5 = poor). In addition, gender was coded such that being female was given a higher score (1 =
male, 2 = female).
This analysis revealed a significant effect of all three predictors. First, participants with higher BMI
reported poorer health, β = .150, t (1711) = 6.13, p < .001. Second, women reported slightly poorer
health than men, β = .051, t (1711) = 2.12, p = .034. Finally, there was a significant gender x BMI
interaction, β = .073, t (1711) = 3.01, p = .003. Simple slopes tests clarified that the nature of this
interaction was consistent with predictions. In the analysis focusing on men, there was only a weak, and
marginally significant, association between BMI and self-reported health, β = .069, t (1711) = 1.73, p =
.084. In contrast, in the analysis focusing on women, the association between BMI and self-reported
health was clearly significant, β = .219, t (1711) = 7.30, p < .001. In short, these analyses supported the
hypothesis. To a greater degree than was the case for men, the heavier women were the more likely
they were to report poorer health.
QUESTION 10.5b. For the sake of comparison with the simple slopes tests, we also correlated BMI and
self-reported health for women and men separately. This much simpler approach focusing separately
on women and men yielded results very similar to the simple slopes reported here. The zero-order
correlation between BMI and self-reported health for men was r (779) = .061, p = .089. For women, the
correlation was r (932) = .236, p < .001. Presumably difference in the variances for BMI and health for
the total sample versus for each gender contributed to these very small differences in results.
There are several potential explanations for this finding. First, compared with men, women may be
socialized to be more attentive to their physical well-being. It has been shown, for example, that
women are somewhat more attentive than are men to their emotional states. On average, women also
make use of health care services and adhere to the instructions of health care practitioners at a greater
rate than do men. In other words, heavier women may correctly report not feeling healthy whereas
heavy men may not be so attuned to their health.
It is also plausible that these findings are rooted in men and women’s endorsements of gender
stereotypes about what kind of body type should be considered healthy. It is possible, for example, that
the weak correlation between BMI and health we observed for men simply reflects the possibility that
very thin men, unlike very thin women, may often consider themselves unhealthy because they do not
meet the cultural ideal of being “manly.”). In other words, both men and women may consider
themselves healthy to the degree that they think they look the way their culture tells them they should
look!
In more statistical terms, because linear regression is sensitive to linear rather than curvilinear
associations, this kind of curvilinear association among men would convert to a very weak, or nonexistent linear association. Either curvilinear regression analyses or follow-up comparisons between
women and men in different BMI categories (e.g., “normal weight” versus “underweight”) could yield
further insights into this possibility.
Whatever the preferred interpretation, it seems clear that gender does moderate the association
between BMI and self-reported health. Thin versus heavy men differ very little in their self-reported
health whereas heavy women report being less healthy than thin women.
QUESTION 10.6. When I focused on the contrast that made Republicans the focal group, the most
important statistics for the two way Democrat vs. Republican x percentage of Black neighbors
interaction were conceptually identical to the ones reported in the text (when Democrats were the focal
group). I say “most important” and “conceptually identical” because the sign of the t value is reversed
(+2.67 versus the original -2.67). This makes sense because now we changed which of the two groups is
focal. However, the p value is identical (p = .011) to the conceptually identical p value presented in the
text, which also makes sense. To my surprise, the absolute value of the beta weight for this two-way
interaction term changed quite a bit (β = .757 versus the original β = -.463).
The simple slopes test for the association between the percentage of Black neighbors that White
Republicans have and the favorability of their attitudes toward Obama yielded a marginally significant
negative association, β = -.712, t (39) = -1.83, p = .076. So depending on how much stock one places in a
marginally significant finding, either nothing is happening for the Republicans or there is a negative
association, such that the more Black neighbors White Republicans have, the less likely they are to
approve of Obama. It would be nice to see if this replicates in a larger sample. At any rate, it is clear
that what is happening for Republicans is very different than what is happening for Democrats.
Traditional contact effects only occur for the Democrats, perhaps because Republicans are less likely to
have the kind of positive equal status contact with Blacks that are thought to facilitate positive contact
effects.
Here is a copy of the syntax commands I used to conduct this analysis:
IF (pol_party eq 1) rep_v_dem = 1.
IF (pol_party eq 2) rep_v_dem = 0.
IF (pol_party eq 3) rep_v_dem = 0.
IF (pol_party eq 1) rep_v_ind = 0.
IF (pol_party eq 2) rep_v_ind = 1.
IF (pol_party eq 3) rep_v_ind = 0.
COMPUTE interrep_v_dem = cent_perc_Black*rep_v_dem.
COMPUTE interrep_v_ind = cent_perc_Black*rep_v_ind.
EXE.
QUESTION 10.7. The table below contains dummy codes that would allow us to represent contrasts
between four different groups, based on their status in prison environments. With four levels of the
categorical variable we would need three variables to cover each contrast.
Dummy-Coding Options for Representing Contrasts between Four Different Groups of Prisoners
-----------------------------------------------------------------------------------------------------------------------------------------------------------
Wardens
Guards
Medical Staff
Prisoners
Wardens as Focal Group
D1w D2w D3w
0
0
0
1
0
0
0
1
0
0
0
1
Guards as Focal Group
D1G D2G D3G
1
0
0
0
0
0
0
1
0
0
0
1
Medical Staff as Focal Group Prisoners as Focal Group
D1M
D2M D3M
D1P
D2P D3P
1
0
0
1
0
0
0
1
0
0
1
0
0
0
0
0
0
1
0
0
1
0
0
0