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Using Multilevel Modeling to Analyze Longitudinal Data Mark A. Ferro, PhD Offord Centre for Child Studies Lunch & Learn Seminar Series January 22, 2013

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Recommended Readings 1.Singer JD, Willett JB. Applied longitudinal data analysis. Modeling change and event occurrence. New York: Oxford University Press; 2003. 2.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. 3.Singer JD. Using SAS PROC MIXED to fit multilevel models, hierarchical models, and individual growth models. J Educ Behav Stat 1998;24: 323-55.

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Objectives 1.Explore longitudinal data a)Wrong approaches 2.Understand multilevel model for change a)Specify the level-1 and level-2 models b)Interpret estimated fixed effects and variance components 3.Data analysis with the multilevel model a)Adding level-2 predictors b)Comparing models

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Research Questions Broadly speaking, we are interested in two types of questions: 1.Start by asking about systematic change over time for each individual 2.Next ask questions about variability in patterns of change over time (what factors may help us explain different patterns of growth?)

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Wrong Approaches 1.Estimated correlation coefficients: Problem: only measures status, not change (tells whether rank order is similar at both time-points) 2.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-linear

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Less-than-Ideal Approaches 1.Aggregate data Reduced power No intra-individual variation 2.Repeated Measures ANOVA Reduced power Equal linear change Compound symmetry Class 1 Patient 1 Time 1 Time 2 Time 3 Patient 2 Time 1 Time 2 Time 3 Class 2 Patient 3 Time 1 Time 2 Time 3 Patient 4 Time 1 Time 2 Time 3 Level 2 Level 1 Class 2 Patient 3 Time 1 Time 2 Patient 4 Time 1 Time 2 Time 3 Level 2 Level 1 Class 1 Patient 1 Time 1 Time 2 Time 3 Patient 2 Time 1 Time 2 028028 012012

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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 time

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Example Dataset Longitudinal Study of American Youth (LSAY) N=1322 Caucasian and African-American students Change in mathematics achievement between grades 7-11 1.At what rate does mathematics achievement increase over time? 2.Is the rate of increase related to student race, controlling for the effects of SES and gender?

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How to Answer the Questions? 1.Exploratory analysis 2.Fit taxonomy of progressively more complex models a)Unconditional means model (not shown) b)Unconditional linear growth model c)Add race as level-2 predictor of initial status and rate of change in match achievement d)Add SES as level-2 control variable, testing impact on initial status and rate (does effect of race change?) e)Add gender as level-2 control variable,… 3.Select final model and plot prototypical trajectories 4.Residual analysis to evaluate tenability of assumptions

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Multilevel Model for Change Level-1 model: Level-2 model: Composite model: structural stochastic

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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 )

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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 race

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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 race

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Exploratory Analysis - OLS

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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;

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Unconditional Linear Growth – Fixed Effects Solution for Fixed Effects EffectEstimateStandard Error DFt ValuePr > |t| Intercept52.36600.25411321206.10<.0001 grade_c2.81580.0732510238.46<.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 population

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Unconditional Linear Growth – Random Effects Covariance Parameter Estimates Cov ParmSubjectEstimateStandard Error Z ValuePr Z UN(1,1)LSAYID62.49443.363818.58<.0001 UN(2,1)LSAYID6.45500.70119.21<.0001 UN(2,2)LSAYID3.21640.290611.07<.0001 Residual 37.16450.855243.46<.0001 Estimated variance in intercept Estimated variance in slope Estimated variance in level-1 residuals Estimated covariance between intercept and slope

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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;

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Adding the Effect of Race – Fixed Effects Solution for Fixed Effects EffectEstimateSEDFt ValuePr > |t| Intercept53.01700.26381320201.00<.0001 grade_c2.86880.0775510137.03<.0001 aa-5.93360.79691320-7.45<.0001 grade_c*aa-0.48220.23415101-2.060.0395 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 AA

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Adding the Effects of Race – Random Effects Covariance Parameter Estimates Cov ParmSubjectEstimateSEZ ValuePr Z UN(1,1)LSAYID59.04503.231318.27<.0001 UN(2,1)LSAYID6.17650.68688.99<.0001 UN(2,2)LSAYID3.19300.289911.01<.0001 Residual 37.16710.855343.46<.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 race

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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;

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Adding the Effects of SES – Fixed Effects EffectEstimateSEDFt ValuePr > |t| Intercept52.80640.25371319208.13<.0001 grade_c2.84620.0774510036.79<.0001 aa-4.66200.77341319-6.03<.0001 ses3.62100.3379131910.72<.0001 grade_c*aa-0.34910.23585100-1.480.1389 grade_c*ses0.37180.102951003.610.0003 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 race

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Adding the Effects of SES – Random Effects Cov ParmSubjectEstimateStandard Error Z ValuePr Z UN(1,1)LSAYID52.46352.979417.61<.0001 UN(2,1)LSAYID5.50220.65878.35<.0001 UN(2,2)LSAYID3.12600.287410.88<.0001 Residual 37.16840.855343.46<.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 SES

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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;

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Removing the Effect of Race on Rate of Change Solution for Fixed Effects EffectEstimateStandard Error DFt ValuePr > |t| Intercept52.81830.25361319208.28<.0001 grade_c2.80740.0729510138.53<.0001 aa-4.76980.77001319-6.19<.0001 ses3.61390.3379131910.70<.0001 grade_c*ses0.39540.101851013.890.0001

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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;

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Final Model with Gender Solution for Fixed Effects EffectEstimateStandard Error DFt ValuePr > |t| Intercept52.40130.35041318149.55<.0001 grade_c2.80770.0729510138.53<.0001 aa-4.79820.76931318-6.24<.0001 ses3.61590.3375131810.71<.0001 female0.81830.475113181.720.0852 grade_c*ses0.39530.101751013.890.0001

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Goodness-of-Fit Model AModel BModel CModel DModel EModel F Deviance45443.445383.045253.245255.445252.245252.4 AIC45455.445399.045723.245273.245274.245272.4 BIC45486.545440.545325.145320.145331.245324.3 Deviance -2LL statistic Worse fit = larger -2LL Can be compared in nested models χ 2 distribution, df = difference in number of parameters AIC & 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 samples

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Presenting Results Ferro & Boyle. Journal of Pediatric Psychology 2013;38(4):425-37

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Plotting Trajectories for Prototypical Individuals RaceSESInitial StatusRate of Change CaucasianLow52.401-4.798(0)+3.616(-0.693)+0.818(1)=50.7132.808+0.395(-0.693)=2.534 CaucasianHigh52.401-4.798(0)+3.616(0.735)+0.818(1)=55.8772.808+0.395(0.735)=3.098 AALow52.401-4.798(1)+3.616(-0.693)+0.818(1)=45.9152.808+0.395(-0.693)=2.534 AAHigh52.401-4.798(1)+3.616(0.735)+0.818(1)=51.0792.808+0.395(0.735)=3.098 Estimates of initial status and rate of change for Caucasian and African- American girls of high and low SES

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Prototypical Trajectories

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Assumptions & Evaluation Assumption 1.Level-1 growth model is linear 2.Level-2, relationship between predictors and intercept and slope is linear 3.Level-1 and level-2 residuals are normal and homoscedastic Evaluation 1.Examine empirical growth plots for evidence of linearity 2.Plot OLS estimates of growth parameters vs. each predictor 3.Standard diagnostics for level-1 and level-2

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