Mark A Ferro PhD Offord Centre for Child Studies Lunch amp Learn Seminar Series January 22 2013 Recommended Readings Singer JD Willett JB Applied longitudinal data analysis Modeling change and event occurrence New York Oxford University Press 2003 ID: 714289
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Slide1
Using Multilevel Modeling to Analyze Longitudinal Data
Mark A. Ferro, PhD
Offord Centre for Child Studies Lunch & Learn Seminar Series
January 22, 2013Slide2
Recommended Readings
Singer JD, Willett JB.
Applied longitudinal data analysis. Modeling change and event occurrence. New York: Oxford University Press; 2003
.
Singer JD.
Fitting individual growth models using SAS PROC MIXED. In:
Moskowitz
DS, Hershberger SL, editors. Modeling
intraindividual
variability with repeated measures data. Methods and applications. Mahwah: Lawrence Erlbaum Associates; 2002
.
Singer JD.
Using SAS PROC MIXED to fit multilevel models, hierarchical models, and individual growth models. J
Educ
Behav
Stat 1998;24: 323-55
.Slide3
Objectives
Explore longitudinal data
Wrong approaches
Understand multilevel model for change
Specify the level-1 and level-2 models
Interpret estimated fixed effects and variance components
Data analysis with the multilevel model
Adding level-2 predictors
Comparing modelsSlide4
Research Questions
Broadly speaking, we are interested in two types of questions:
Start by asking about
systematic change over time
for each individual
Next ask questions about
variability in patterns of change over time
(what factors may help us explain different patterns of growth?)Slide5
Wrong Approaches
Estimated correlation coefficients:
Problem: only measures status, not change (tells whether rank order is similar at both time-points)
Use difference score to measure change and use this as an estimate of rate of change
Problem: assumes linear growth over time, but change may be non-linearSlide6
Less-than-Ideal Approaches
Aggregate data
Reduced power
No intra-individual variation
Repeated Measures ANOVA
Reduced power
Equal linear change
Compound symmetry
Level 2
Level 1
Level 2
Level 1
0
2
8
0
1
2Slide7
Advantages of MLM
Flexibility in research design
Different data collection schedules
Varying number of waves
Identify temporal patterns in the data
Inclusion of time-varying predictors
Interactions with time
Effects that get smaller or larger over timeSlide8
Example Dataset
Longitudinal Study of American Youth (LSAY)
N=1322 Caucasian and African-American students
Change in mathematics achievement between grades 7-11
At what rate does mathematics achievement increase over time?
Is the rate of increase related to student race, controlling for the effects of SES and gender?Slide9
How to Answer the Questions?
Exploratory analysis
Fit taxonomy of progressively more complex models
Unconditional means model (not shown)
Unconditional linear growth model
Add race as level-2 predictor of initial status and rate of change in match achievement
Add SES as level-2 control variable, testing impact on initial status and rate (does effect of race change?)
Add gender as level-2 control variable,…
Select final model and plot prototypical trajectories
Residual analysis to evaluate tenability of assumptionsSlide10
Multilevel Model for Change
Level-1 model:
Level-2 model:
Composite model:
structural stochasticSlide11
Level-1 Model
Within-individual
Intercept of individual i’s trajectory (initial status)
Centred at a time 0
Math achievement at time 0
Slope of individual i’s trajectory (rate of change)
Change in math achievement between each time point
Deviations of individual i’s trajectory from linearity on occasion j (error term)
~N(0,
σ
2)Slide12
Level-2 Model
Between-individual
Population average intercept and slope for math achievement for reference group (Caucasian)
Difference in population average intercept and slope for math achievement between African-American and Caucasian
Difference between population average and individual i’s intercept and slope for math achievement, controlling for raceSlide13
Level-2 Model Residuals
Variance-covariance matrix
Population variance in intercept, controlling for race
Population variance in slope, controlling for race
Population covariance between intercept and slope, controlling for raceSlide14
Exploratory Analysis - OLSSlide15
SAS Syntax
proc
mixed
data
=
lsay
noclprint
noinfo
covtest
method
=ml;
title 'Model A: Unconditional Linear Growth Model';class
lsayid;model math =
grade_c / solution
ddfm=bw notest;
random intercept
grade_c /subject=lsayid
type=un;run;Slide16
Unconditional Linear Growth – Fixed Effects
Solution for Fixed Effects
Effect
Estimate
Standard Error
DF
t Value
Pr
> |t|
Intercept
52.3660
0.2541
1321
206.10
<.0001
grade_c
2.8158
0.0732
5102
38.46
<.0001
Estimated math achievement in 7
th
grade
Estimated yearly rate of change in math achievement
t-test for null H0 of no average change in achievement in the populationSlide17
Unconditional Linear Growth – Random Effects
Covariance Parameter Estimates
Cov Parm
Subject
Estimate
Standard Error
Z Value
Pr
Z
UN(1,1)
LSAYID
62.4944
3.3638
18.58
<.0001
UN(2,1)
LSAYID
6.4550
0.7011
9.21
<.0001
UN(2,2)
LSAYID
3.2164
0.2906
11.07
<.0001
Residual
37.1645
0.8552
43.46
<.0001
Estimated variance in intercept
Estimated variance in slope
Estimated variance in level-1 residuals
Estimated covariance between intercept and slopeSlide18
SAS Syntax
proc
mixed
data
=
lsay
noclprint
noinfo
covtest
method
=ml;
title 'Model B: Adding the Effect of Race';
class lsayid;
model math = grade_c
aa aa*grade_c
/ solution ddfm
=bw notest;
random intercept grade_c
/
subject
=
lsayid
type
=un;
run
;Slide19
Adding the Effect of Race – Fixed Effects
Solution for Fixed Effects
Effect
Estimate
SE
DF
t Value
Pr > |t|
Intercept
53.0170
0.2638
1320
201.00
<.0001
grade_c
2.8688
0.0775
5101
37.03
<.0001
aa
-5.9336
0.7969
1320
-7.45
<.0001
grade_c
*
aa
-0.4822
0.2341
5101
-2.06
0.0395
Estimated math achievement in 7
th
grade for Caucasians
Estimated yearly rate of change in math achievement for Caucasians
Estimated difference in yearly rate of change in math achievement between Caucasian and AA
Estimated difference in math achievement in 7
th
grade between Caucasians and AASlide20
Adding the Effects of Race – Random Effects
Covariance Parameter Estimates
Cov Parm
Subject
Estimate
SE
Z Value
Pr Z
UN(1,1)
LSAYID
59.0450
3.2313
18.27
<.0001
UN(2,1)
LSAYID
6.1765
0.6868
8.99
<.0001
UN(2,2)
LSAYID
3.1930
0.2899
11.01
<.0001
Residual
37.1671
0.8553
43.46
<.0001
Estimated variance in intercept, controlling for race
Estimated variance in slope, controlling for race
Estimated variance in level-1 residuals
Estimated covariance between intercept and slope, controlling for raceSlide21
SAS Syntax
proc
mixed
data
=
lsay
noclprint
noinfo
covtest
method
=ml;
title 'Model B: Adding the Effect of Race';
class lsayid;
model math = grade_c
aa aa*grade_c
ses ses*
grade_c / solution
ddfm=bw notest
;
random
intercept
grade_c
/
subject
=
lsayid
type
=un;
run
;Slide22
Adding the Effects of SES – Fixed Effects
Effect
Estimate
SE
DF
t Value
Pr > |t|
Intercept
52.8064
0.2537
1319
208.13
<.0001
grade_c
2.8462
0.0774
5100
36.79
<.0001
aa
-4.6620
0.7734
1319
-6.03
<.0001
ses
3.6210
0.3379
1319
10.72
<.0001
grade_c
*
aa
-0.3491
0.2358
5100
-1.48
0.1389
grade_c
*
ses
0.3718
0.1029
5100
3.61
0.0003
Estimated math achievement in 7
th
grade for Caucasians of average SES
Estimated yearly rate of change in math achievement for Caucasians of average SES
Estimated difference in yearly rate of change in math achievement between Caucasian and AA, controlling for SES
Estimated difference in math achievement in 7
th
grade between Caucasians and AA, controlling for SES
Estimated effect of SES on average 7
th
grade achievement, controlling for race
Estimated effect of SES on rate of change of achievement, controlling for raceSlide23
Adding the Effects of SES – Random Effects
Cov
Parm
Subject
Estimate
Standard Error
Z Value
Pr Z
UN(1,1)
LSAYID
52.4635
2.9794
17.61
<.0001
UN(2,1)
LSAYID
5.5022
0.6587
8.35
<.0001
UN(2,2)
LSAYID
3.1260
0.2874
10.88
<.0001
Residual
37.1684
0.8553
43.46
<.0001
Estimated variance in intercept, controlling for race and SES
Estimated variance in slope, controlling for race and SES
Estimated variance in level-1 residuals
Estimated covariance between intercept and slope, controlling for race and SESSlide24
SAS Syntax
proc
mixed
data
=
lsay
noclprint
noinfo
covtest
method
=ml;
title 'Model B: Adding the Effect of Race';
class lsayid;
model math = grade_c
aa aa*grade_c
ses ses*
grade_c / solution
ddfm=bw notest
;
random
intercept
grade_c
/
subject
=
lsayid
type
=un;
run
;Slide25
Removing the Effect of Race on Rate of Change
Solution for Fixed Effects
Effect
Estimate
Standard Error
DF
t Value
Pr > |t|
Intercept
52.8183
0.2536
1319
208.28
<.0001
grade_c
2.8074
0.0729
5101
38.53
<.0001
aa
-4.7698
0.7700
1319
-6.19
<.0001
ses
3.6139
0.3379
1319
10.70
<.0001
grade_c*ses
0.3954
0.1018
5101
3.89
0.0001Slide26
SAS Syntax
proc
mixed
data
=
lsay
noclprint
noinfo
covtest
method
=ml;
title 'Model B: Adding the Effect of Race';
class lsayid;
model math = grade_c
aa ses ses*
grade_c female / solution
ddfm=bw notest
;random intercept
grade_c
/
subject
=
lsayid
type
=un;
run
;Slide27
Final Model with Gender
Solution for Fixed Effects
Effect
Estimate
Standard Error
DF
t Value
Pr > |t|
Intercept
52.4013
0.3504
1318
149.55
<.0001
grade_c
2.8077
0.0729
5101
38.53
<.0001
aa
-4.7982
0.7693
1318
-6.24
<.0001
ses
3.6159
0.3375
1318
10.71
<.0001
female
0.8183
0.4751
1318
1.72
0.0852
grade_c*ses
0.3953
0.1017
5101
3.89
0.0001Slide28
Goodness-of-Fit
Model
A
Model
B
Model
C
Model
D
Model
E
Model
F
Deviance
45443.4
45383.0
45253.2
45255.445252.245252.4
AIC45455.445399.045723.2
45273.245274.245272.4
BIC45486.545440.545325.145320.1
45331.245324.3
Deviance-2LL statisticWorse fit = larger -2LL
Can be compared in nested modelsχ2 distribution, df = difference in number of parametersAIC & BIC
Can be used for non-nested models
AIC corrects for number of parameters estimated
BIC corrects for sample size and number of parameters, so larger improvement needed for larger samplesSlide29
Presenting Results
Ferro & Boyle.
Journal of Pediatric Psychology
2013;38(4):425-37Slide30
Plotting Trajectories for Prototypical Individuals
Race
SES
Initial Status
Rate of Change
Caucasian
Low
52.401-4.798(0)+3.616(-0.693)+0.818(1)=50.713
2.808+0.395(-0.693)=2.534
Caucasian
High
52.401-4.798(0)+3.616(0.735)+0.818(1)=55.877
2.808+0.395(0.735)=3.098
AA
Low
52.401-4.798(1)+3.616(-0.693)+0.818(1)=45.915
2.808+0.395(-0.693)=2.534
AA
High
52.401-4.798(1)+3.616(0.735)+0.818(1)=51.079
2.808+0.395(0.735)=3.098
Estimates of initial status and rate of change for Caucasian and African-American girls of high and low SESSlide31
Prototypical TrajectoriesSlide32
Assumptions & Evaluation
Assumption
Level-1 growth model is linear
Level-2, relationship between predictors and intercept and slope is linear
Level-1 and level-2 residuals are normal and homoscedastic
Evaluation
Examine empirical growth plots for evidence of linearity
Plot OLS estimates of growth parameters vs. each predictor
Standard diagnostics for level-1 and level-2