Supplemental Digital Content 2 for Psychosomatic Medicine
Myers et al., Multilevel Modeling in Psychosomatic Medicine Research
Supplemental Digital Content 2.
Supplemental path diagrams (44), equations, and text
Curran and Bauer’s system of path diagrams for MLRMs contains a number of
components. A box represents an observed variable. Font within a box communicates the
centering decision: no centering (plain font), group-mean centered (italicized font), grand-mean
centered (bold italic font). A triangle labeled with the number “1level” defines an intercept term
where the subscript denotes the level at which the intercept is specified. A circle represents a
random coefficient, where the particular coefficient is declared within the circle. A straight
single-headed arrow represents a regression parameter which is taken as fixed unless
superimposed within a circle. A multiheaded arrow indicates a covariance that is estimated as a
model parameter.
For indexing purposes, let Yij  RSBij, X 1ij  PAij, X 2ij  SFij, X 3ij  GAPij, and X 4ij 
FAMij. Adolescent-level independent variables were aggregated to the school-level to create
Level-2 predictors that adopted the relevant acronym while altering subscript notation (e.g., PAij
changed to PA. j from Level-1 to Level-2). For indexing purposes, let X 1. j  PA. j , X 2 . j  SF. j ,
X 3 . j  GAP. j , and X 4 . j  FAM. j .
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12
γ00
11
Yij
0j
rij
u0j
Figure 1. One-way ANOVA with random effects: Yij   00  u0 j  rij .
The reader is referred to Equation 1 in the main text for a description of this model.
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Model 2: Means as Outcomes Regression Model
Suppose that the interest is in modeling unadjusted mean risky sexual behavior at the
school-level. This model will be unconditional at Level-1 and conditional at Level-2. If each
Level-2 predictor is GMC then the interpretation of 00is the same as in the previous model
(where there also was no adjustment to the school-level means,  0 j ). This model can be written:
RSBij   0 j  rij
 

GAP.  GAP..    FAM .  FAM ..  u

 0 j   00   01 PA. j  PA..   02 SF . j  SF ..
  03
j
04
j
(1)
0j
 01 = change in mean risky sexual behavior given a one-unit increase in mean perceived peer
abstinence after controlling for the effect of mean adolescent functioning in school, mean
gap between parental and adolescents Americanism, and mean family functioning. The text
following “after controlling” for is denoted … for the remaining terms. Interpretation of
 02 through  04 follows the same form as the interpretation of  01 .
u0j = residual mean risky sexual behavior for the jth school after controlling for …
τ00 = Var(u0j) = residual school-level variance in mean risky sexual behavior after controlling
for…
From this point forward all Level-2 independent variables are GMC.
As can be viewed in Table 1, mean gap between parental and adolescents Americanism at
the school-level was the only statistically significant predictor of the intercepts (i.e., school-mean
risky sexual behavior), ˆ03  0.036, p  .011. The set of predictors combined to explain 15.3% of
the variance in the intercepts  i.e., .072  .061 .072 . The italicized text emphasizes that in a
MLM the notion of variance accounted for can be complex due to the fact that the variance in the
outcome(s) can be conceptualized as being partitioned by level. Because the independent
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variables were specified as predictors of only the intercepts, variance explained can be focused at
Level-2 (e.g., the proportion of between-school variance accounted for by the four school-level
predictors). That ˆ00 = 0.061, p <.001 suggested there was substantial between-school variance
unaccounted for by the set of predictors.
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X1.j
12
X2.j
γ01
γ00
γ02
X3.j
11
γ03
X4.j
β0j
γ04
u0j
Yij
rij
Figure 2. Means as outcomes regression model:








Yij   00   01 X 1. j  X 1..   02 X 2 . j  X 2 ..   03 X 3 . j  X 3 ..   04 X 4 . j  X 4 ..  u0 j  rij .
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Model 3: One-Way ANCOVA with Random Effects
Suppose there are theoretical reasons to believe that school means need to be adjusted for
some adolescent-level variables because students are not randomly assigned to schools; thereby
making comparisons of unadjusted school means misleading. The goal then is to model adjusted
mean risky sexual behavior at the school-level. This model will be conditional at both Level-1
and Level-2. Suppose further that there is reason to believe that each Level-1 slope coefficient
should be treated as fixed (i.e., the relationship between each Level-1 predictor and the outcome
is believed to be homogenous across schools). In this case it is appropriate to GMC (or RAS)
each Level-1 predictor because CWC results in unadjusted school means. This model can be
written:

 
GAP  GAP..    FAM
RSBij   0 j  1 j PAij  PA..   2 j SFij  SF ..
 3 j
ij
4j
ij


 FAM ..  rij
 

GAP.  GAP..    FAM .  FAM ..  u

 0 j   00   01 PA. j  PA..   02 SF . j  SF ..
  03
j
04
j
(3)
0j
1 j   10
 2 j   20
3 j   30
 4 j   40
 01 = change in adjusted mean risky sexual behavior given a one-unit increase in mean perceived
peer abstinence after controlling for the effect of mean adolescent functioning in school,
mean gap between parental and adolescents Americanism, and mean family functioning.
Interpretation of  02 through  04 follows the same form as the interpretation of  01 .
u0j = residual adjusted mean risky sexual behavior for the jth school after controlling for …
τ00 = Var(u0j) = residual school-level variance in adjusted mean risky sexual behavior after
controlling for …
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 20 = change in expected risky sexual behavior across schools given a one-unit increase in
Adolescent functioning in school after controlling for the effect of perceived peer
abstinence, gap between parental and adolescents Americanism, and family functioning.
Interpretation of  10 ,  30 , and  40 follow the same form as the interpretation of  20 .
rij = residual risky sexual behavior of the ith adolescent in the jth school after controlling for …
σ2 = Var(rij) = residual adolescent-level variance in risky sexual behavior after controlling for …
Each Level-1 predictor had a statistically significant direct effect on adolescent-level (or
“within school”) risky sexual behavior (see Table 1). The set of Level-1 predictors combined to
explain 4.3% of the adolescent-level variance. Note that the set of Level-2 predictors explained
less of the variance in the intercepts than in the previous model (i.e., from 15.3% to 13.8%). This
can largely be explained by the differing definition of the intercepts between the two models
(i.e., unadjusted versus adjusted means), which confounds a direct comparison of the estimate of
variance accounted for in the intercepts across these two models. If the Level-1 predictors had
been CWC the explained variance in the intercepts would have been nearly identical (i.e., from
15.3% to 15.0%). Finally, the statistical significance of both residual variances, ˆ 2 = 1.012, p
<.001 and ˆ00 ,= 0.065, p <.001, suggested that there was substantial unexplained variance at both
levels.
The centering decision at Level-1 (GMC) along with including the means for each Level1 predictor at Level-2, allowed ˆ10 , ˆ20 , ˆ30 , and ˆ40 to each be interpreted as the relevant
estimated within effect (e.g., the effect of school functioning at the adolescent-level), while
ˆ01 , ˆ02 , ˆ03 , and ˆ04 are interpreted as the relevant estimated contextual effect. The contextual
effect is defined as the difference between the within effect and the between effect (19; see p.
140 for a visual display). It should be noted that this same distinction (i.e., the possibility for
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three different “types” of effects of an observed Level-1 predictor: within, between, contextual)
could also be viewed from a latent variable perspective (49,50).
The estimated effect of each Level-1 predictor ˆ10 , ˆ20 , ˆ30 , ˆ40  was listed under the
“Within Level” results of the output (see Appendix C). Treating each Level-1 slope as
homogenous across schools (i.e., fixed) was akin to ignoring the nesting of the data for these
effects; thereby relegating each effect to Level-1. For example, ˆ20  0.009, p  .009, within the
context of the fuller model implied that the effect of school functioning on risky sexual behavior
was negative and constant across schools (because u2j was omitted in the model specification).
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X1.j
12
X2.j
γ01
X3.j
γ02
γ00
γ03
11
X4.j
γ04
u0j
β0j
β1
X1ij
Yij
β2
X2ij
β3
X3ij
rij
β4
X4ij
Because each Level-1 slope coefficient was treated as fixed:
1   10 , 2   20 , 3   30 , and 4   40 .
Figure 3. One-way ANCOVA with random effects:







Yij   00   01 X 1. j  X 1..   02 X 2 . j  X 2 ..   03 X 3 . j  X 3 ..   04 X 4 . j  X 4 ..

  10  X 1ij  X 1..   20  X 2ij  X 2 ..   30  X 3ij  X 3 ..   40  X 4ij  X 4 ..  u0 j  rij .
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Model 4: Non-Randomly Varying Slopes
Suppose that the previous model is altered to reflect the belief that the within-school
effect (i.e., slope) of risky sexual behavior regressed on the adolescent functioning in school
,  2 j , should be changed from fixed ,  2 j   20 , to non-randomly varying based on mean family


functioning:  2 j   20   24 FAM . j  FAM .. . Conceptually, the within-school effect of
adolescent functioning in school on risky sexual behavior can vary from school to school,
depending on each school's mean family functioning ˆ24  0). Note the continued absence of a
random component for this within-school slope (i.e., the absence of a u2 j term). This model can
be written:

 
 GAP  GAP..    FAM
RSBij   0 j  1 j PAij  PA..   2 j SFij  SF ..
 3 j
ij
4j
ij


 FAM ..  rij
 

GAP.  GAP..    FAM .  FAM ..  u

 0 j   00   01 PA. j  PA..   02 SF . j  SF ..
  03
1 j   10
j
04

 2 j   20   24 FAM . j  FAM ..
j
0j
(4)

3 j   30
 4 j   40
 20 = change in expected risky sexual behavior given a one-unit increase in adolescent
functioning in school (after controlling for the effect of peer abstinence, gap between
parental and adolescents Americanism, and family functioning) for schools that have a
mean family functioning value equal to the grand mean family functioning value.
 24 = change in the change in expected risky sexual behavior given a one-unit increase in school
functioning (conditional on all other Level-1 predictors) given a one-unit increase in mean
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family functioning.
It should be noted that in cases where a Level-1 effect, qj, is specified such that the relevant
random effect is omitted at Level-2, uqj, statistical tests are available to test the veracity of the
assumption - regardless of the degree to which an a priori argument exists for omitting uqj.
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12
γ00
X1.j
γ20
X2.j
γ01
X3.j
γ02
γ03
11
γ04
X4.j
β0j
u0j
γ24
β1
X1ij
Yij
β2j
X2ij
β3
rij
β4
X3ij
X4ij
Because certain Level-1 slope coefficients were treated as fixed:
1   10 , 3   30 , and 4   40 .
Figure 4. Non-randomly varying slopes:







Yij   00   01 X 1. j  X 1..   02 X 2 . j  X 2 ..   03 X 3 . j  X 3 ..   04 X 4 . j  X 4 ..
  10  X 1ij  X 1..   20  X 2ij  X 2 ..   24
 X
2 ij
  30  X 3ij  X 3 ..   40  X 4ij  X 4 ..  u0 j  rij .
12

 X 2 .. * X 4 . j  X 4 ..


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Model 5: Random Coefficients Regression
Suppose that the interest is in modeling adolescent-level effects on risky sexual behavior
while specifying random intercepts, a mix of both random and fixed slopes, and no school-level
predictors. This model will be conditional at Level-1 and unconditional at Level-2. The effect of
family functioning on risky sexual behavior is believed to be heterogeneous across schools,
justifying the inclusion of the u 4 j term, while each of the other within-school slopes is believed to
be homogenous across schools. CWC each Level-1 predictor is appropriate because the interest
is estimating pure adolescent-level effects  i.e.,  10 ,  20 ,  30 ,  40  . Failing to CWC would yield
coefficients  i.e.,  10 ,  20 ,  30 ,  40  that are a blend of the relevant adolescent-level (or within)
effect and school-level (or between) effect (19). This model can be written:

 
GAP  GAP.     FAM
RSBij   0 j  1 j PAij  PA. j   2 j SFij  SF . j
 3 j
ij
j
4j
ij


 FAM . j  rij
0 j   00  u0 j
1 j   10
(5a)
 2 j   20
3 j   30
 4 j   40  u4 j
u0j = unique effect of the jth school on risky sexual behavior
u4j = unique effect of the jth school on the change in expected risky sexual behavior given a oneunit increase in family functioning (after controlling for the effect of peer abstinence,
adolescent functioning in school, and gap between parental and adolescents Americanism at
Level-1).
τ44 = unconditional variance of the u4j (or equivalently, the 4j
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τ04 = unconditional school-level covariance between the u0j and the u4j (or equivalently, the 0j
and the  4j, respectively
From a practical perspective, the degree to which estimating a blended coefficient as opposed to
a pure within-level effect, represents a noteworthy problem depends, in part, on empirical
characteristics of the data such as the magnitude of difference between the within-effect versus
the between-effect and the ICC for the relevant predictor (19, 32).
As can be viewed in Table 1, while the average change in expected risky sexual behavior
across schools given a one-unit increase in family functioning was statistically non-significant,
ˆ40  0.026, p  .068, the variance around this average was statistically significant, ˆ44  0.018,
p  .001. The statistical significance of ˆ44 suggested that there was substantial variance in this
set of within-school slopes that may be explained by the Level-2 predictors (i.e., treating this
slope as an outcome – see Model 6). The covariance between the set of unadjusted risky sexual
behavior school means and the set of within-school risky sexual behavior on family functioning
slopes was not statistically significant, ˆ04  0.001, p  .900.
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12
γ00
u0j
11
β0j
X1ij
X2ij
γ40
β1
β2
Yij
β3
X3ij
β4j
X4j
u4j
Because certain Level-1 slope coefficients were treated as fixed:
1   10 , 2   20 , and 3   30 .
Figure 5. Random coefficients regression:
Yij   00   10  X 1ij  X 1. j    20  X 2ij  X 2 . j    30  X 3ij  X 3 . j    40  X 4 ij  X 4 . j 
 u0 j  u4 j  X 4ij  X 4 . j   rij .
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12
X1.j
γ00
X2.j
X3.j
γ02
γ01
γ03
11
γ43
X4.j
γ04
β0j
γ20
γ24
u0j
γ40
Yij
β1
X1ij
β2j
rij
X2ij
β3
β4j
X3ij
u4j
X4ij
Because certain Level-1 slope coefficients were treated as fixed:
1   10 , 3   30 , and 4   40 .
Figure 6. Intercepts and slopes as outcomes:







Yij   00   01 X 1. j  X 1..   02 X 2 . j  X 2 ..   03 X 3 . j  X 3 ..   04 X 4 . j  X 4 ..
  10  X 1ij  X 1. j    20  X 2ij  X 2 . j    24
  30  X 3ij  X 3 . j    40  X 4ij  X 4 . j    43
 u0 j  u4 j  X 4ij  X 4 . j   rij .
 X
 X
16
2 ij
4 ij


 X .  *  X .  X .. 
 X 2 . j  * X 4 . j  X 4 ..
4 j
3 j
3

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12
13
γ000
γ100
u00j
β00j
γ001
Wj
γ101
r0ij
β10j
u10j
11
π0ij
lifemij
π1ij
Amij
r1ij
Figure 7. Stress Management Example:
Ymij   000   001W j   100 Amij   101  Amij *W j 
u00 j  u10 j Amij  r0 ij  r1ij Amij  emij .
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emij
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Ymij  observed quality of life at time m for the ith participant in the jth therapist
emij = residual quality of life at time m for the ith participant in the jth therapist
r0ij = unique effect of the ith participant on expected quality of life at baseline in the jth
therapist
r1ij = unique effect of the ith participant on expected rate of weekly linear change in
quality
of life in the jth therapist
 000 = expected quality of life at baseline for participants in the control condition
 001 = difference in expected quality of life at baseline for participants in the experimental
condition as compared to the control condition
u00 j = residual quality of life at baseline in the jth therapist after controlling for the
treatment
effect
 100 = expected rate of weekly linear change in quality of life for participants in the
control
condition
 101 = difference in expected rate of weekly linear change in quality of life for participants
in the
experimental condition as compared to the control condition
u10 j = residual rate of weekly linear change in quality of life in the jth therapist after
controlling for the treatment effect
18