Transcript Other Topics - of David A. Kenny

Multilevel Modeling:
Other Topics
David A. Kenny
January 25, 2014
Presumed Background
• Multilevel Modeling
• Nested Design
• Growth Curve Models
Outline
•
•
•
•
Centering and the Three Effects
Multiple Correlation
Tau Matrix
Modeling Nonindependence with Overtime
Data
• Significance Testing
• Non-normal Outcomes
• GEE
3
Centering and
the Three Effects
• The Three Effects of X (a level 1
variable) on Y
–Within: effect of X on Y estimated for
each level 2 unit and then averaged
–Between: effect of mean X on Y
–Pooled: an “average” of the two
4
Example: Effect of Daily
Stress on Mood
• Within: the effect of daily stress on
mood computed for each person and
then averaged
• Between: Stress is averaged for each
person and then average stress is used
to predict mood.
• Pooled: Stress is used to predict mood
using all people and all days.
5
Types of Centering and the Effect
• Grand mean centering
– X effect: Pooled estimate
• Grand mean centering with mean X as a
predictor
– X effect: Within estimate
– Mean X effect: Between minus within
estimate
• Group (or person) mean centering
– X effect: Within estimate
6
Multiple Correlation
• Not outputted by any MLM program.
• Estimate a second model without any fixed
effects besides the intercept, the empty
model.
• Measure the relative changes in variances
with predictors in and out of the model.
– sE2 from the empty model; sM2 from the model
– (sE2 - sM2)/sE2
– If negative, report as zero.
– Sometimes called pseudo R2.
7
Illustration: Doctor-Patient
Variances
>
Terma Empty Model
Model
R2
DD
0.105
0.102
.040
DP
0.131
0.131
.004
PD
0.009
0.008
.054
PP
0.184
0.184
.000
aD implies that the respondent is the doctor
P
and that level is that of the patient.
8
Tau Matrix
• Whenever there is more than one
random effect at level 2, there is a
variance-covariance matrix of
random effects.
• That matrix is called the “tau matrix”
in the program HLM.
• Different programs make different
restriction on this matrix.
9
Programs
• HLM: Unstructured only
• SPSS and R’s nlme: Allows
various possibilities but not any
matrix.
• SAS and MLwiN: User can enter
own matrix which gives maximal
flexibility.
10
Example: Growth Curve Model with
Indistinguishable Dyad Members
Slope P1 (1)
Int. P1 (2)
Slope P2 (3)
Int. P2 (4)
a
c
b
d
e
a
e
f
c
b
(1)
(2)
(3)
(4)
Letters symbolize different elements of
the tau matrix, some of which are set
equal.
11
Modeling Nonindependence
with Overtime Data
• k overtime measurements
• There are k(k + 1)/2 variances and
covariances.
• That makes k(k + 1)/2 potential
parameters that could be estimated.
• Note that with a growth curve model
only 4 parameters are estimated or 5
with autoregressive errors.
12
We Should Test
• Fit of a growth-curve model can be
compared to the fit of a repeated
measures model that exactly fits all
variances and covariances (saturated
error model).
• If the growth-curve model fits as well as
the saturated model, the simpler model
(growth-curve model) is preferred).
13
Significance Testing
• SPSS uses the Wald test for
variances.
• The likelihood ratio test involving
deviance differences is used by
other programs and provides a
more powerful and accurate test of
significance.
14
Non-normal Outcomes
• Types
–Dichotomous or binary outcomes
–Counts
• For these cases, the error variance is
not an additional parameter.
• Basic model is often multiplicative.
• Can access in SPSS: Mixed Models,
Generalized Linear
15
GEE: Generalized
Estimating Equations
• An alternative to MLM
• Does not test variance components, but
rather using a “working model.”
• Weaker assumptions about the
distribution of random variables.
• Used often in medical research.
• Used also with non-normal outcomes.
16
References
References (pdf)
17