Kristen Jones
Faye Huie
Project Random Coefficient Modeling
You hypothesize that Mood and Emotion Regulation are positively related in children.
That is, when a child is in a good mood, s/he engages in more emotion regulation than
when in a bad mood. You hypothesize further that this is more true for some children
than for others. Specifically, whereas mood and regulation are strongly positively related
in children with calm temperaments (i.e., kids who score high on temperament), mood
and regulation are only weakly related in children who are total spazzes (technical term).
Mood and regulation, therefore, vary within child, while temperament is a stable
individual difference variable.
The rcmproj file has data on these variables for each of 40 children. Mood and
Regulation data were collected at each of five time points for each child. Temperament
was measured at the outset of the study. Your task is to use rcm to test your hypotheses.
1. What does one of the Level 1 equations look like in its general form?
yij = B1jxij + B0j + rij where j symbolizes any single group (i.e., child) and i
represents any single occasion within child j
We can assume that each of the 40 children have their own separate regression
equation. Since the 40 children are randomly sampled from the population, this
means our slope and intercept values for each child’s regression equation are now
also random effects. The above equation represents the regression of emotional
regulation onto mood for child j, case i.
2. What do the two Level 2 equations look like, again in general form?
Boj = ϒ00 + u0j
The above equation is a representation of a particular intercept value of any given
child’s individual regression equation (generated from Level 1) and how it
deviates from the population regression intercept. This equation shows how the
estimation of any child’s given intercept value is function of the overall
population intercept value (fixed effect) and the random deviation of that
particular child’s intercept value from the population intercept value. Insofar as
clustering influences emotional regulation (i.e., to the degree that group
membership explains variance in emotional regulation), this random deviation
between a child’s intercept and the population intercept will be greater.
B1j = ϒ10 + u1j
The above equation is a representation of a particular slope value of any given
child’s individual regression equation (generated from Level 1) and how it
deviates from the population regression slope. This equation shows how the
estimation of any child’s given slope value is function of the overall population
slope value (fixed effect) and the random deviation of that particular child’s slope
value from the population slope value. Insofar as clustering influences emotional
regulation (i.e., to the degree that group membership explains variance in
emotional regulation), this random deviation between a child’s slope and the
population slope will be greater.
3. Do a simple Variance Components analysis on centered variables to answer the
question, What proportion of the variance in regulation can be attributed to
between-child differences? What proportion is within-child? Show your work.
ICC = (.5812)/(.5812 + 3.0175) = .16
Thus, 16% of the total variance in emotion regulation is attributable to betweenchild differences.
(3.0175)/(.5812 + 3.0175) = .84
The remaining percent of the variance (84%) can then be attributable to withinchild differences.
4.
Now do another analysis with centered mood as a covariate. What proportion of
between subject variance is explained by differences in average mood? What
proportion of within subject variance is explained by within person variability in
mood? Do significant amounts of each remain? Show your work.
(.5812 -.2967)/.5812 = .49
49% of the between-child variance is explained by differences in average mood
which leaves an nonsignificant amount of variance in emotion regulation between
children left to be explained.
(3.0175-2.652)/3.0175 =
12% of the within-child variance in emotion regulation is explained by within
person variability in mood leaving a significant amount of within-child variance
in emotion regulation left unexplained.
5. Now do a final analysis that also includes Temperament and the Mood*
Temperament interaction. What proportion of between subject variance is
explained by differences in temperament? What proportion of within subject
variance is explained by within person variability in temperament? Do significant
amounts of each remain? Show your work.
Estimates of Fixed Effectsa
95% Confidence Interval
Parameter
Estimate
Std. Error
df
t
Sig.
Lower Bound
Upper Bound
Intercept
-.327691
.406888
31.176
-.805
.427
-1.157354
.501972
Cenmood
-.466966
.216433
41.453
-2.158
.037
-.903916
-.030016
Temperament
.049706
.092522
32.479
.537
.595
-.138647
.238059
Cenmood * Temperament
.101174
.046032
34.525
2.198
.035
.007678
.194670
a. Dependent Variable: CenReg.
Estimates of Covariance Parametersa
95% Confidence Interval
Parameter
Estimate
Residual
Wald Z
Sig.
Lower Bound
Upper Bound
2.647070
.319384
8.288
.000
2.089600
3.353265
.335828
.254901
1.317
.188
.075864
1.486611
UN (2,1)
-.083624
.073742
-1.134
.257
-.228156
.060907
UN (2,2)
.103287
.057407
1.799
.072
.034749
.307006
Intercept + Cenmood [subject UN (1,1)
= Subjnum]
Std. Error
a. Dependent Variable: CenReg.
(.2967 - .3358)/.2967 = -.1318
After the inclusion of temperament, our unexplained between-child variance
increases by 13%. We can interpret this as ZERO. Thus, none of the betweenchild variance in emotional regulation can be explained by the addition of
temperament. If we look back at the unexplained variance BEFORE the inclusion
of temperament, we can see that we no longer had a significant amount of
variance in between-child differences in emotion regulation left to be explained.
Thus, after the addition of temperament, there is not a significant (p = .24) amount
of variance in between-child emotion regulation to explain.
(2.652 - 2.647)/2.652 = .0019
After the inclusion of temperament, we not surprisingly don’t explain additional
within-child variance in emotion regulation. This is because each child has the
same constant temperament value; thus, temperament does not vary across
observations within a child. Therefore, a group level predictor will never be able
to account for within-group variance, which is why we observe no change in
unexplained within-child variance after the inclusion of temperament.
6. Now explain which of your hypotheses were supported and which weren’t.
H1: You hypothesize that Mood and Emotion Regulation are positively related in
children. That is, when a child is in a good mood, s/he engages in more emotion
regulation than when in a bad mood.
When we look at the fixed effects after the inclusion of mood, we can observe the
weight for mood (-.028) is not statistically significant (p = .73). This tells us that
the addition of mood to the equation did not explain a significant amount of
variance in emotion regulation. Furthermore, for every single point increase in
mood, a slight (albeit) insignificant decrease in emotion regulation was predicted.
Therefore, we cannot reject the null hypothesis since our results failed to provide
support for a positive relationship between mood and emotion regulation.
H2: You hypothesize further that this is more true for some children than for
others. Specifically, whereas mood and regulation are strongly positively related
in children with calm temperaments (i.e., kids who score high on temperament),
mood and regulation are only weakly related in children who are total spazzes
(technical term).
When we look at the second output (after the inclusion of mood) we get a
variance component (tau value) for the variance in slopes across the 40 children
(.134) that is significant (p = .038). This tells us that there is a significant amount
of variance in the relationship between mood and emotion regulation across
children. This indicates that there is some group level (or in this case, child-level)
variable accounting for this between-child variance in the mood-emotion
regulation slope.
In the third set of output (after the inclusion of temperament and the interaction
between temperament and mood), we see that our weight for the interaction term
is significant (p=.035), meaning the inclusion of the interaction term explains
significant variance in emotion regulation. Thus, the weight for the product term
(.1012) represents the rate of change in the mood-emotion regulation slope per
single point increase in temperament. That is, as children increase in
temperament, the relationship between mood and emotion regulation strengthens.
This is in line with H2; therefore, we can reject the null hypothesis.
7. Now do an ordinary moderated regression analysis (i.e., a disaggregated analysis
ignoring group membership) and explain how your results differ from the RCM
analysis.
After running this example as an ordinary moderated regression, we can see that
our standard errors for the regression coefficients decrease when we ignore group
membership. For the RCM analysis, the SE of the intercept = .16, SE of cenmood
= .078, SE of centemp = .092, and SE of the interaction = .046 whereas in the
disaggregated analysis, the SE of the intercept = .134, SE of cenmood = .06, SE
of centemp = .077, and SE of the interaction = .036.
This is because in the disaggregated approach, the only error that is accounted for
is random variability in Y. Thus, random variability in intercepts and random
variability in slopes are ignored, resulting in smaller but less accurate SEs. This
means that in OLS, to the degree that clustering matters, we higher alpha
inflation, a higher Type I error rate, and an easier time achieving statistical
significance.
The error term in RCM accounts for random variability in Y, random variability
in intercepts, and random variability in slopes which SHOULD be accounted for
resulting in larger (but more accurate) SEs of the weights.