Overview of Talk. Motivating Example: Antisocial Behavior (AB) Raw Data

Transcription

1 An Evaluative Comparison of Random Coefficient Growth Models for Individual Development May 6, 7 Association for Psychological Science Overview of Talk Motivating example Definition of Random Coefficient Growth Model (RCGM) Three basic types of RCGMs Controversies in model selection and interpretation Daniel J. Bauer L.L. Thurstone Psychometric Laboratory Department of Psychology The University of North Carolina at Chapel Hill H. Luz McNaughton Reyes Department of Health Behavior and Health Education The University of North Carolina at Chapel Hill Motivating Example: Antisocial Behavior (AB) Raw Data Concerns: What do trajectories of change in AB look like from age 6 to age 15? Are there sex-differences in these developmental trends? Do supportive home environments protect against increases in AB? Can AB be predicted by poor academic performance? Sample: 894 children assessed biennially from 1986 to 199 as part of the NLSY-Child Sample. Between 6 and 8 years old in Measures: Antisocial Behavior: Sum of 6 items from BPI Early Home Environment: HOME-SF cognitive and emotional support scores from first assessment Academic Performance: PIAT Math scores 5 Random Cases

2 The Idea Behind RCGMs The Idea Behind RCGMs 5 Random Cases AB = β + β Age + r ti i 1i ti ti Random Coefficients 5 Random Cases AB = β + β Age + r ti i 1i ti ti How do individuals differ from one another? The Idea Behind RCGMs Three Types of RCGMs 5 Random Cases AB = β + β Age + r ti i 1i ti ti How are random coefficients distributed? 1. RCN: Models that assume the random coefficients are normally distributed (conditional on predictors) Latent curve/trajectory/growth models, HLM, multilevel growth models, mixed-effects models

3 Trends in Use of RCN Models (HLMs) Trends in Use of RCN Models (LCMs) Bryk & Raudenbush (1987) McArdle & Epstein (1987) Meredith & Tisak (199) Citation Count Citation Year Citation Count Citation Year RCN RCN Fit to Antisocial Data Within-Person Model: = β + β1 ( 6) + rti ~ N(, σ ) AB Age r ti i i ti ti Between-Person Model: βi = γ + ui β = γ + u 1i 1 1i u τ ~ N, i u 1i τ1 τ 11 Assumes continuous individual differences in change over time. Parameter Estimate (SE) Fixed effects Intercept ( γ ) 1.87 (.7)** Age ( γ 1 ).5 (.1)** Variance / Covariance Parameters Intercept ( τ ) 1.43 (.5)** Age ( τ 11 ). (.1)* Covariance ( τ 1 ).5 (.4) Residual ( σ ).9 (.14)** Note. Robust standard errors reported. * p <.5 ; ** p <.1

4 of RCN of RCN Mean ± SD Adding Predictors to RCN Parameter Prediction of Intercepts ( β ) i Estimate (SE) Intercept 1.18 (.14)** Sex.85 (.13)** Home -. (.4)** Prediction of Slopes ( β 1i ) Intercept.1 (.3)** Sex.1 (.3) Home -.1 (.1) Time Varying Covariates Math -.17 (.4)** Variance / Covariance Parameters Intercept 1.5 (.16)** Age. (.1)* Covariance.6 (.3)* Residual.11 (.6)** * p<.5; ** p<.1 Three Types of RCGMs 1. RCN: Models that assume the random coefficients are normally distributed (conditional on predictors) Latent curve/trajectory/growth models, HLM, multilevel growth models, mixed-effects models. RCD: Models that assume the random coefficients are discretely distributed across K groups Semi-parametric group-based trajectory method, nonparametric random coefficient model, latent class growth analysis, latent class regression

5 Trends in Use of RCD Models (SPGBM) RCD Nagin (1999) 5 Within-Person Model: = β + β1 ( 6) + rti ~ N(, σ ) AB Age r ti i i ti ti 4 Between-Person Model: Citation Count Citation Year βi = γ ( k) if Ci = k β = γ if C = k 1i 1( k) i = 1,,, K Assumes individual differences in change over time are discretely distributed K types of trajectories. Ci PC ( = k) = π i ( k) RCD Fit to Antisocial Data Latent Class of RCD 4% Parameter High Increasing Low Increasing High Declining Abstaining Class Size Class probability ( π ( k )) Sample N Fixed effects Intercept ( γ ( k ) ) 3.7 (.91)**.35 (.)** 6.9 (1.95)** 1.17 (.7)** Age ( γ 1( k ) ).53 (.)**.17 (.4)** -.34 (.37) -.3 (.1)* Variance / Covariance Parameters Residual ( σ ). (.1)**. (.1)**. (.1)**. (.1)** Note. Robust standard errors reported. * p <.5 ; ** p <.1 8% 5% 63%

6 of RCD Adding Predictors to RCD Latent Class Parameter High Increasing Low Increasing High Declining Abstaining Class Size a Class probability Sample N Between-Class Effects b Sex 4.5 (.15,8.43) 3.7 (.3,5.94) 9.7 (3.15,9.88) N/A Home.59 (.47,.7).65 (.56,.75).68 (.48,.96) N/A Within-Class Effects Intercept 3.8 (.)**.3 (.14)** 6.77 (.33)** -.91 (.11)** Age.56 (.5)**.17 (.3)** -.3 (.7)*.3 (.3) Math -.11 (.4)** -.11 (.4)** -.11 (.4)** -.11 (.4)** Variance / Covariance Parameters Residual 1.96 (.4)** 1.96 (.4)** 1.96 (.4)** 1.96 (.4)** * p <.5 ; ** p <.1 a Based on estimated posterior probabilities b Odds Ratio (95% CI); Reference class is Abstaining Three Types of RCGMs Trends in Use of RCNM Models (GMMs) 1. RCN: Models that assume the random coefficients are (conditionally) normally distributed Latent curve/trajectory/growth models, HLM, multilevel growth models, mixed-effects models 5 4 Muthen & Shedden (1999) Muthen & Muthen (). RCD: Models that assume the random coefficients are discretely distributed across K latent groups Semi-parametric group-based trajectory method, nonparametric random coefficient model, latent class growth analysis, latent class regression 3. RCNM: Models that assume the random coefficients are from a (conditional) normal mixture distribution with K latent groups General growth mixture models Citation Count Citation Year

7 RCNM Within-Person Model: = β + β1 ( 6) + rti ~ N(, σ ( k )) AB Age r ti i i ti ti Between-Person Model: βi = γ ( k) + u i( k) if Ci = k β = γ + u if C = k 1i 1( k) 1 i( k) i u τ ( i k) ( k ) ~ N, u 1( i k) τ1( k) τ 11( k) Ci = 1,,, K PC ( = k) = π Assumes K discrete trajectory groups, within which individual differences in change over time are continuously distributed. i ( k) RCNM Fit to Antisocial Data Latent Class Parameter Increasing Decreasing Class Size Class probability ( π ( k )).5.5 Sample N Fixed effects Intercept ( γ ( k ) ).78 (.13)**.98 (.9)** Age ( γ 1( k ) ).13 (.)** -.4 (.)** Variance / Covariance Parameters Intercept ( τ ( k ) ).83 (.4)*.43 (.1)** Age ( τ 11( k ) ). (.).1 (.4)* Covariance ( τ 1( k ) ).1 (.7) -.6 (.)** Residual ( σ ) 3.55 (.3)**.6 (.7)** ( k ) Note. Robust standard errors reported. * p <.5 ; ** p <.1 of RCNM of RCNM Mean ± SD 5% 5%

8 Adding Predictors to RCNM Latent Class Parameter Increasing Decreasing Class Size a Class probability Sample N Between-Class Effects b Sex.8 (1.93,4.11) N/A Home.66 (.58,.75) N/A Within-Class Prediction of Intercepts ( β i ) Intercept 1.89 (.4)**.76 (.1) Sex.6 (.4)**.31 (.1)** Home -.6 (.4) -.6 (.4) Within-Class Prediction of Slopes ( β 1i ) Intercept.6 (.5)**.1 (.) Sex -.1 (.5) -.5 (.3)* Home.1 (.1).1 (.1) Time Varying Covariates Math -.3 (.7)** -.6 (.3) Variance / Covariance Parameters Intercept.45 (.34).35 (.11)** Age.1 (.).1 (.4)** Covariance.5 (.6) -.5 (.)** Residual 3.6 (.17)**.61 (.5)** * p <.5 ; ** p <.1 a Based on estimated posterior probabilities b Odds Ratio (95% CI); Reference class is Decreasing Comparison of Three Models Overall trends are similar Most children had low levels of AB at age 6. Some children showed increases in AB from age 6 to 15, others showed stable or decreasing AB. Those with highest initial levels of AB tended to show greatest increases over time. Male children, and children from less supportive home environments displayed higher levels of AB. Children scoring poorly on the PIAT-M displayed higher levels of AB. Despite general consistency, much controversy over appropriate model Controversy in the Selection of RCGMs Key Issue: The Interpretation of Groups 5 recent papers on RCD and RCNM models have been followed by commentaries and rejoinders Bauer & Curran (3, Psychological Methods), commentaries by Cudeck & Henly, Muthen, & Rindskopf. Eggleston, Laub & Sampson (4, J. Quant. Criminology), commentary by Nagin. Nagin & Tremblay (5, Annals AAPSS), commentaries by Maughan and Raudenbush. Nagin & Tremblay (5, Criminology), commentary by Sampson & Laub. Connell & Frye (6, Infant & Child Dev), commentaries by Hoeksma & Kelderman, Muthen, and Stanger. Reflects disagreements, misunderstandings about relative merits of different RCGMs. Often, the latent classes estimated from RCD or RCNM models are interpreted as true taxa (i.e., real groups). This can be problematic Are there 4 groups (RCD), groups (RCNM), or no groups (RCN)? Should groups be strictly homogeneous (RCD) or do we permit within-group variability (RCNM)? Spurious groups can compensate for errors in model specification, e.g., lack of normality of residuals (Bauer & Curran, 3, 4). Number/nature of groups can change with minor alterations in model specification, covariates, measurement, or design (Eggleston et al, 4; Jackson & Sher, 5, 6). RCN does permit groups as a function of observed covariates (when strong etiological theory for taxa).

9 Key Issue: The Interpretation of Groups Nagin (5) advocates use of groups as an approximating device even when groups are artificial. of RCD Key Issue: The Interpretation of Groups Conclusions Nagin (5) advocates use of groups as an approximating device even when groups are artificial. Drawbacks to Groups: Artificial groups prone to reification. Use of RCD and RCNM models may reduce power. Alternatives to Groups: RCN models relatively robust to violation of normality assumption for random effects. Other semiparametric RCGMs (Chen, Zhang & Davidian, ; Zhang & Davidian, 1) Perhaps better to focus less on groups and more on overall trends. RCGMs offer many conceptual and statistical advantages for modeling individual change Many different RCGMs (HLM, LCM, GMM, LCGA, SPGBA, etc) can be organized into three categories RCN assumes continuously distributed individual differences. RCD assumes discretely distributed individual differences. RCNM assumes continuously distributed individual differences within a small number of discrete groups. Key issue in choosing between models is utility of groups Groups correspond nicely to taxonomic theory Groups often improve model flexibility and fit. Interpretation of latent groups can be risky.