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Groups Change Too: Analyzing Repeated Measures on Individuals Embedded Within Dynamic Groups Daniel J. Bauer

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Goal: To offer a more realistic model for repeated measures data when individuals are clustered within groups that undergo structural or functional change over time Roadmap: Causes and consequences of clustered data Multilevel modeling Analyzing change over time Stable versus dynamic groups Two applications of dynamic group models Outline of Talk

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Clustering is a Natural Feature of Data Humans exist within a social ecology including both natural and constructed groups (e.g., family and school)

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Observations on individuals from the same group tend to be correlated Peer groups subject to selection effects (homophily) and socialization effects (group norms) Schools include students drawn from similar sociodemographic backgrounds, and students are exposed to common teachers and curricula Family members have common genes, environmental exposures, and social influences Yet most statistical models assume independence of observations (more specifically, independent residuals) Clustering Usually Implies Correlation

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Consequences of Ignoring Dependence What happens if we erroneously analyze the data as if they were independent? Standard errors, test statistics, degrees of freedom, p-values, and confidence intervals are all incorrect Tests tend to be too liberal, inflating Type I errors Most importantly, we neglect important processes in the data How similar are individuals within groups? How strong are group effects on individuals? What predictors account for within- versus between-group differences?

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Appropriately Analyzing Clustered Data There are several possible ways to analyze cluster-correlated data Fixed-effects approaches Generalized estimating equations Multilevel models with random effects Multilevel models offer unique insights into both individual and group-level processes

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A Classic Example Science achievement scores for student from different schools:

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A Simple Multilevel Model A basic two-level model for clustered data: Overall average (fixed effect) Group-level influences (random effect) Individual-level influences that are independent of group (random effect) Variance component associated with each random effect Correlation between individuals scores

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The Variance Components Here we see how each component of variability maps onto our plot of the data 0 vjvj r ij

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Extending the Model Normally, our next step would be to incorporate predictors at the individual and group level to explain each source of variability in the data Suppose, however, we didnt just measure our outcome once, but multiply over time, with the goal of capturing individual trajectories of change

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Modeling Individual Trajectories Person i=1 in Group j=1 Time y

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Modeling Individual Trajectories Person 1, Group 1 Person 2, Group 1 Person 4, Group 2 Time y 012 Person 3, Group 2 Mean 3 Trajectory Individual Differences Time-Specific Residual Group Effect

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Taking a Closer Look This is a typical three-level model for capturing individual change when individuals are clustered within groups Note that the group effect, v j, is constant over time Is this consistent with theory? Mean Trajectory Individual Differences Time-Specific Residual Group Effect

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Chronosystem Dynamic Groups Just an individuals change, so too does the social ecology

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Dynamic Groups We refer to dynamic groups as those that undergo structural and/or functional change over time yet maintain their core integrity as units Examples: Rockbridge and Hickman High Schools both experience turnover in students, teachers, administrators, and curricula, yet continue to be characterized by distinctive school cultures The Jones Family undergoes structural changes as a consequence of divorce, remarriage, and child birth, and undergoes functional changes as a consequence of parental addiction and unemployment, yet the Jones Family remains distinct from the Smith Family

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Rewriting the Model With dynamic groups, we would expect group effects to be correlated over time, but not necessarily constant A more realistic model might thus be The group effect is now time-varying Key is then to discern its temporal structure Mean Trajectory Individual Differences Time-Specific Residual Group Effect

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Temporal Structure of Group Effects Say we have 4 time points How should we structure the covariance matrix of the group effects over time? ? 17

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Temporal Structure of Group Effects Traditional stable groups model Correlated 1.0 over time Say we have 4 time points How should we structure the covariance matrix of the group effects over time?

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Temporal Structure of Group Effects Say we have 4 time points How should we structure the covariance matrix of the group effects over time? Toeplitz model Banded Covariance Matrix

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Temporal Structure of Group Effects Say we have 4 time points How should we structure the covariance matrix of the group effects over time? Stabilization model Stabilizing Banded Covariance Matrix

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Temporal Structure of Group Effects Say we have 4 time points How should we structure the covariance matrix of the group effects over time? Compound symmetric model Equal covariances

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Temporal Structure of Group Effects Say we have 4 time points How should we structure the covariance matrix of the group effects over time? AR( 1 ) model Exponential Decay

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ARMA( 1,1 ) model Temporal Structure of Group Effects Say we have 4 time points How should we structure the covariance matrix of the group effects over time? Rapid Decay

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Temporal Structure of Group Effects Unstructured model Say we have 4 time points How should we structure the covariance matrix of the group effects over time? No Structure Imposed

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Why It Matters Specifying a poor temporal structure for the group effects risks Incorrect tests of regression coefficients for predictors Biased estimates of variance components at each level of the model Occluding important findings regarding the nature and stability of group effects over time Goal is thus to identify a theoretically plausible structure that fits the data well Some structures are nested and can be compared using LRT Others are non-nested and can be compared by BIC, AIC, etc.

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Example: Attitudes About Science Data drawn from the Longitudinal Study of American Youth (LSAY) Cohort 1 (N=2091): 1987, 1988, 1989 Cohort 2 (N=1407): 1990, 1991, schools, 3498 students, 7756 observations from grades Outcome is an IRT developmental scale score of science ability: 26

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Goals for analysis Evaluate the relationship between religious attitudes towards science and science achievement Science undermines morality We need less science, more faith The theory of evolution is true (R) The bible is Gods word Separate within-school and between-school effects, while controlling for SES Obtain accurate tests of these effects by appropriately accounting for school effects Determine the temporal structure of school effects

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Fitted Model Captures average trajectory for each cohort Captures within- and between- school effects of SES and attitudes Captures individual differences in change over time, time-varying school effects, and residuals from the individual trajectories

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StructureParametersAICBIC Intercept Intercept + Slope Selecting a Temporal Structure StructureParametersAICBIC Intercept Intercept + Slope Toeplitz Stabilizing Lag Stabilizing Lag Stabilizing Lag CS AR( 1 ) ARMA( 1,1 ) StructureParametersAICBIC Intercept Intercept + Slope Toeplitz Stabilizing Lag Stabilizing Lag Stabilizing Lag CS AR( 1 ) ARMA( 1,1 ) Traditional models Dynamic group models

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Fixed Effects Stable Group ModelDynamic Group Model Estimate95% CIEstimate95% CI Intercept60.51*(59.61,61.42)60.54*(59.55,61.54) Grade2.58*(2.43,2.73)2.49*(2.17,2.81) Cohort1.44*(.74,2.14)1.32*(.37,2.27) Grade*Cohort-.78*(-1.04,-.53)-.61*(-1.11,-.11) Student Attitudes -2.67*(-3.37,-1.98)-2.68*(-3.37,-1.98) School Attitudes -8.20*(-16.01,-.38)-8.83 * (-16.47,-.29) Student SES.13*(.10,.15).13*(.10,.15) School SES.12(-.09,.32).12(-.09,.34) * p<.05

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Dynamic group model shows diminishing correlation of school effect over time ** Superior model fit School Effects Over Time Traditional model with random intercept for school assumes constant school effect over time

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Summary Though the regression coefficient estimates are similar, the dynamic groups model fits the data better and likely provides more accurate tests of these coefficients Suggests both within and between-school effects of fundamentalist religious attitudes on science achievement The dynamic groups model also provides insights into the stability and change of school effects over time School effects highly stable from one year to the next But the correlation decays to.62 over a period of 4 years, indicating some drift in nature of school effects over time

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Example: Family Effects on Psychopathology Data drawn from the Michigan Longitudinal Study (PI: Zucker) 280 families, 588 children, 2468 repeated measures Repeated measures included from age and span 12 calendar years Outcomes are IRT scores of self-reported externalizing and depression Primary goal is to examine temporal stability of family effects on psychopathology Ancillary goals are to estimate trajectories of externalizing and depression for boys and girls, and to evaluate added risk due to parental impairment (alcoholism, depression, ASP)

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Fitted Model Captures differences in average trajectories of girls and boys Captures effects of parental impairment Captures individual differences in change over time, time- varying family effects, and residuals from the individual trajectories

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Selecting a Temporal Structure For both outcomes, the AR( 1 ) dynamic group model fits best ExternalizingDepression StructureParametersAICBICAICBIC Intercept Intercept + Slope CS AR( 1 ) ExternalizingDepression StructureParametersAICBICAICBIC Intercept Intercept + Slope CS AR( 1 )

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Fixed Effects ExternalizingInternalizing Estimate95% CIEstimate95% CI Intercept -.133(-.272,.006) * (-1.495,-1.120) Age.055 * (.027,.084).070 * (.031,.110) Age * (-.047,-.021).010(-.000,.020) Male.217 * (.096,.338)-.273 * (-.410,-.135) Age × Male * (-.104,-.036)-.078 * (-.124,-.031) Age 2 × Male.028 * (.012,.043) Parent Alc.415 * (.291,.540).207 * (.024,.390) Parent Dep.098(-.027,.223).179(-.004,.363) Parent ASP.207 * (.049,.364).257 * (.028,.486) * p<.05

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Family Effects Over Time

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Summary Gender differences consistent with other literature Parental history of alcoholism elevates risk of both depression and externalizing, and this is compounded by history of ASP History of parental depression does not have a significant effect Family effects are highly fluid A family that is troubled in one year is likely to continue to function poorly in the next year or two, but may right itself over the longer term Conversely, a family functioning well at one point in time is not immune from later difficulties Family effects less stable for externalizing than depression

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Conclusions Standard multilevel models fail to account for the fact that groups undergo change over time The effect of the group on its members is unlikely to be constant We propose the use of dynamic group models to obtain new insights on the temporal structure of group effects At a time lag of five years, school effects on science achievement were correlated.62 In contrast, family effects were correlated.53 for depression and only.25 for externalizing behavior This difference in stability is perhaps not surprising. Schools are large institutions with a great deal of inertia, whereas families are small groups that are potentially more vulnerable to stochastic events

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Acknowledgements and Disclaimer The project described was supported by supported by National Institutes of Health grants R01 DA (PI: Antonio Morgan-Lopez), R37 AA (PI: Robert Zucker), and R01 DA (PIs: Andrea Hussong and Patrick Curran). The content is solely the responsibility of the author and does not represent the official views of the National Institute on Drug Abuse, National Institute on Alcohol Abuse and Alcoholism, or the National Institutes of Health. Nisha Gottfredson Danielle Dean Robert Zucker Antonio Morgan-Lopez Andrea Hussong

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