Modeling longitudinal and multilevel data

Clearly, classical inferential statistics e.g., ANOVA have little or no ambiguity in themodel when they are applied to fully experimental data i.e., the adequacy of the implicit model un

Modeling Longitudinal and Multilevel Data Practical Issues, Applied Approaches and Specific

Examples

Edited byTodd D LittleYale University

Kai U SchnabelJürgen BaumertMax Planck Institute for Human Development

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

8

Latent Transition Analysis As a Way of Testing Models ofStage-Sequential Change in Longitudinal Data

Linda M Collins, Stephanie L Hyatt, and John W.

Graham

147

as the presenters, we felt that a comprehensive volume on these issueswas needed We are particularly grateful to Dagmar Stenzel for editingand type-setting the contributions We also appreciate the support andadvice of Larry Erlbaum and his crack team at LEA throughout thisprocess Finally, the diligence and timeliness of all contributors cannot be thanked enough The efforts, patience, and expertise of all

involved has brought a volume that we hope will become a standardreference for social sciences researchers

TODD D LITTLENEW HAVEN, CT, USAKAI U SCHNABELBERLIN, GERMANYJÜRGEN BAUMERTBERLIN, GERMANY

We would like to dedicate this volume to the memory of our friend and colleague Magret Baltes.

longitudinal or multilevel data are collected For example, the

puzzling number of different ways to analyze longitudinal data is

likely to frustrate many researchers who neither consider themselvesexperts in statistics nor intend to become one And multilevel designsalso offer a plethora of complexities when it comes to decomposingthe various sources of variability in the participants' responses

Although researchers in the behavioral and social sciences are quitesophisticated when it comes to methodological and statistical issues,keeping up with the rapid advances and understanding the inherentcomplexities in the various analytic techniques for addressing

longitudinal and multilevel data can be daunting Such frustrations aremade even more salient because more and more research questions

applications to address such data

Many practical and theoretical issues are involved when addressinglongitudinal and multilevel data Some of these issues include (a)what information should be

evaluated, (b) what decision heuristics should be used, and (c) whatprocedures are most appropriate In addressing these issues, a primarygoal of each contribution is to highlight a specific set of issues and todemonstrate clear procedures for addressing those issues Each

contribution shares the theme that making strong tests of underlyingtheoretical models (i.e., bringing implicit assumptions into the explicitrealm of model specification) is critical for drawing veridical

conclusions from one's data They also share the common theme thatstatistical procedures are not mechanistic ends in themselves (i.e.,fixed and rule-bound), but rather are flexible tools that should be

adjusted and adapted into an appropriate means for testing a givensubstantive theory

Why Address Longitudinal and Multilevel Analysis Problems

Simultaneously?

At first sight, our decision to integrate longitudinal and multilevel dataanalysis in this volume may seem surprising However, when one

goes beyond a "cookbook understanding" of statistical procedures andexamines the basic rationale behind longitudinal and multilevel

procedures, many linkages between both perspectives become clear.Hox (chap 2, this volume), for instance, demonstrates a number ofuses of the multiple-group option in structural equation programs tomodel hierarchically structured data Similarly, MacCallum and Kim(chap 4, this volume) show how both analytic perspectives can beintegrated to investigate correlates of change Little, Lindenberger,and Maier (chap 10, this volume) also describe ways to integrate

multilevel and longitudinal analyses to examine selectivity effects inlongitudinal data

Another reason for using multilevel comparisons in the context oflongitudinal analysis is implicitly addressed in several chapters of thisvolume and is related to the common vagaries of most theories in the

derive an exact a priori hypothesis about the shape of the

developmental function describing the nature of the changes overtime In this case, rules of parsimony then become an auxiliary

guideline to help choose a developmental function that is

mathematically feasible and is a reasonable approximation of the data(e.g., using simple models such as linear-growth functions, or

polynomial functions of limited degrees such as quadratic and cubictrends; Schnabel, 1996) Although investigating interindividual

differences in development is a fundamental issue in longitudinal

research (Baltes & Nesselroade, 1979), many research questions thatcome up in this context often have an exploratory character to them.McArdle and Bell (chap 5, this volume) offer clear recommendationswhen an analysis is more exploratory in nature, that is, to utilize ahighly parameterized (i.e., saturated) time-to-time-change model (e.g.,spline models; Meredith & Tisak, 1990), which does not fit a

particular curve function across the sequence of measurements, inorder to describe the degree of deviation between this solution and aless parameterized, more parsimonious, model The crux of the

problem with weakly justified decisions about the underlying changemodel is that they can have a strong influence on the estimation of thecentral parameters

Checking the stability of the results across relevant subgroups is

another heuristic to test the robustness of the model

The merits of pragmatism is one lesson to learn across the chapters ofthis volume, including the realm of inferential statistics For example,from the perspective of the general linear model, Widaman (chap 9,this volume) emphasizes the idea of testing a hierarchically orderedsequence of alternative models that differ according to theoreticallymeaningful constraints (e.g., across time and/or groups) rather thanrelying on traditional hypothesis testing procedures when one is

testing complex models Such a shift in the empirical rationale is awell-known feature of structural equation modeling, but it is less

common to traditional inferential approaches Clearly, classical

inferential statistics (e.g., ANOVA) have little or no ambiguity in themodel when they are applied to fully experimental data (i.e., the

adequacy of the implicit model underlying the data analysis is

determined by the degree it correctly reflects the experimental design,with the only uncertainty being the assumptions about the distribution

of the variable or variables in the population) However, as soon asone collects data using a nonexperimental design, the underlying

statistical model needs to be used adaptively in order to test the

assumptions about the processes generating the data In other words,the shift from strict hypothesis testing to the logic of model testing isnot made because one uses structural equation modeling procedures,but rather because one uses quasi-experimental data (Cook &

Campbell, 1979)

Ambiguity in Longitudinal Research:

Testing and Modeling

Analyzing longitudinal data sometimes renders the researcher with theuncomfortable feeling of ambiguity and relativity of the findings

However, such ambiguity can also be seen as a chance to investigate

procedures are only analytic tools that one uses to try to model theunderlying dynamic of a set of variablesin this sense, there is no

single best way for analyzing a given data set

The long-standing debate about the epistemological value of structuralequation models and the related debate about whether it is permissible

to make causal inferences from cross-sectional data is not

automatically solved by using longitudinal data The problem mayeven be acerbated in a multiwave study when, for instance, the

variance of a variable (or a latent construct) decreases over time,

indicating a negative correlation between intercept and slope that isnot easy to detect with fallible measures (Raykov, 1994b; Rogosa,Brandt, & Zimowski, 1982) Coleman (1968), for example, providedtheoretical arguments why this is likely for many psychological

variables that are embedded in a complex homeostatic system wherethe organism actively works to reduce extreme system states in order

to regain equilibrium It can easily be shown that any existing causalinfluence on the dependent variable from an independent variable

measured on a prior occasion (cross-lagged coefficient) can only add

variance to the dependent variableirrespective of its sign Thus, a

shrinkage in vari-

ance does not indicate that other relevant explanatory variables weremissing This casts a slightly different light on the dichotomy of

stability and change and the widely held belief that change needs to beexplained because it is an active process whereas stability is

considered the trivial nonactive behavior of the system At least in therealm of psychological and educational research, understanding theprocesses related to stability should be given the same theoretical andempirical importance as understanding the processes related to change(Nesselroade, 1991)

Structural Equation Modeling and Hierarchical Linear ModelsTheUnequal Twins

The present volume focuses considerably on structural equation

modeling (SEM) and hierarchical linear modeling (HLM) procedures

to analyze longitudinal data This emphasis is due primarily to the factthat both approaches are flexible tools for examining complicated datastructures in a feasible way In several chapters, the contributors showthat both approaches yield exactly or approximately the same results.For example, Hox (chap 2, this volume) shows that SEM can be usedfor nested data Chou, Bentler, and Pentz (chap 3, this volume)

demonstrate their similarity in the context of latent growth modeling,showing that a less complexalthough less efficienttwo-step approachproduces approximately the same point estimates As MacCallum andKim (chap 4, this volume) demonstrate, it is possible to analyze

simultaneously more than one dependent variable in the HLM

framework, thus enabling the researcher, when the variables are

combined in a latent growth model, to test hypotheses about

correlations in change components McArdle and Bell (chap 5, thisvolume) similarly demonstrate the equivalence of both approaches,demonstrating that analysis based on raw data may be a robust

alternative to classical SEM that analyzes covariance matrices when

modeling error structures (Steyer, Partchev, & Shanahan, chap 6, thisvolume) and the possibility of modeling mutual influences over time,

it is not very easy to analyze nested data structures with it (or it needs

a fairly cumbersome setup) This is the domain of HLM, which, inturn, does not allow structuring error components according to a

complex measurement structure (or it needs a fairly cumbersome

setup) HLM is also more flexible when the repeated measurementoccasions vary between individuals

From a practical standpoint, another important difference between theSEM and HLM approaches is related to the handling of missing dataafeature where HLM was thought to be the more appropriate tool

However, as Wothke (chap 12, this volume) demonstrates, SEM

procedures have narrowed the gap by using full information

maximum likelihood estimation of the covariance matrix (as

implemented in the latest versions of Amos and Mx) Neale (chap 14,this volume) extends these consider-

(chap 11, this volume) about new techniques for data imputation, onemight speculate about future integration of imputation techniques instatistical softwarenot restricted to SEM or HLM programs, of course

In this regard, Arbuckle (chap 13, this volume) describes some futuredirections for creating specific procedures that are tailored to handlethe specific type of analysis problem of a given design, including

issues of missingness On the other hand, not all data structures can behandled in SEM or HLM procedures In this respect, the latent

transition analyses presented by Collins, Hyatt, and Graham (chap 8,this volume) provide important alternative statistical procedures that

do not assume interval scalesin particular for the latent variables

As mentioned, a primary goal of this volume is to assist researchers inmaking decisions, such as what technique to use for what kind of

question, when is one technique more appropriate than another, andhow does one handle the numerous technical details involved in suchprocedures Based on papers that were presented and discussed at aconference in conjunction with a summer school workshop held inBerlin, Germany, in June, 1997, all chapters in this volume addressnumerous practically relevant questions for empirical social scientistswho desire to have an appropriate way to analyze their longitudinaland/or multilevel data We are indebted to all the contributors to thisvolume for the tremendous work and gracious generosity that theyhave extended to this volume

population For example, in educational research there may be a

sample of schools and within each school, a sample of pupils Thisstructure results in a data set consisting of pupil data (e.g.,

socioeconomic status [SES], intelligence, school career) and schooldata (e.g., school size, denomination, but also aggregated pupil

variables such as mean SES) In this chapter, the generic term

multilevel is used to refer to analysis models for hierarchically

structured data, with variables defined at all levels of the hierarchy.Typically, such research problems include hypotheses of relationshipsbetween variables defined at different levels of the hierarchy

A well-known multilevel model is the hierarchical linear regressionmodel, which is essentially an extension of the familiar multiple

"multilevel regression analysis" and "multilevel SEM," for example

The Multilevel Regression Model for Grouped Data

The multilevel regression model is a hierarchical linear regression

model, with a dependent variable defined at the lowest (usually theindividual) level and explanatory variables at all existing levels Usingdummy coding for categorical variables, the multilevel regressionmodel can be used for analysis of variance (ANOVA), and it has beenextended to include dependent variables that are binary, categorical, orotherwise non-normal data and generalized to include multivariateresponse models and cross-classified data (cf Bryk & Raudenbush,1992; Longford, 1993; and especially Goldstein, 1995)

The Basic Multilevel Regression Model

In most applications, the first (lowest) level consists of individuals,the second level of groups of individuals, and higher levels of sets ofgroups Conceptually, the model can be viewed as a hierarchical

system of regression equations For example, assume that data has

been collected in j schools, with a different number of pupils n j in

each school On the pupil level, there are the dependent variable y ij

(e.g., school career) and the explanatory variable x ij (e.g., pupil SES).One can set up a regression equation to predict the dependent variable

y from the explanatory variable x:

In Equation 1, x ij and y ij are the scores of pupil i in school j, b 0j is the

regression intercept, b 1j the regression slope, and e ij the residual errorterm The multilevel regression model depicted in Equation 1

the school level Assume one school-level explanatory variable z j

(e.g., school size) Then, the model for the b 's becomes:

In Equation 2, g 00 and g 01 are the intercept and slope of the

regression equation used to predict b 0j from z j ; u 0j is the residual

error term in the equation for b 0j Thus, if g 01 is positive and

significant, one concludes that the school-career outcome is higher in

large schools than in small schools Similarly, in Equation 3, g 10 and

g 11 are the intercept and slope to predict b 1j from z j , and u 1j is the

residual error term in the equation for b 1j Thus, if g 11 is positive and

possibility becomes clearer if the model is written as a single

equation, by substituting Equations 2 and 3 into Equation 1:

In the multilevel regression model depicted in Equation 4, which is aspecial case of the general mixed-linear model (Harville, 1977), twoparts can be distinguished The fixed part contains the regression

their overall mean The interaction term z j x ij is sometimes referred to

as a cross-level interaction, because it involves explanatory variables from different levels The individual-level errors, e ij , are assumed to

Estimation and Significance Testing in the Multilevel Regression

Model

The parameters (regression coefficients and variance components) ofthe multilevel regression model are commonly estimated using

maximum likelihood (ML) methods Asymptotic standard errors areavailable for hypothesis testing The usual significance test in

Student distribution with J q 1 degrees of freedom (J = number of groups; q = number of fixed parameters) and to use a chi-square test

for the random effects For a discussion of the issues involved inchoosing between such tests, see Hox (1998)

The likelihood function can be used to test the significance of thedifference between two nested models Most multilevel programs

output a value that is called the deviance (computed as: deviance = 2

the log likelihood) If a smaller model is a nested subset of a largermodel, which means that it is obtained by either dropping parameters

or imposing constraints on the larger model, the difference betweenthe two deviances can be tested against a chi-square distribution Thedegrees of freedom for this test is the difference in the number ofparameters estimated This test can be used instead of the Wald testfor multivariate tests of groups of parameters and for tests of

Two different maximum likelihood functions are used in the available

software: Full ML (FML) and Restricted ML (RML) FML includes the fixedparameters in the likelihood function; RML does not Most software offers achoice between the two methods Because RML does not include the fixedparameters in the likelihood function, a deviance test based on RML can only

be used to test for differences in the random part

Example of Multilevel Regression Analysis of Grouped Data

The multilevel regression model is most appropriate for data structures thathave many groups, because it is more flexible and more parsimonious thananalysis-of-variance-type models For instance, assume a study of school

careers in 50 schools In each school, take one class and measure the pupils'achievement scores, SES, gender, class size, and how experienced their

teacher is The study has a total of 979 pupils from 50 classes, with an averageclass size of just under 20 Note that by taking one class per school the schooland the class level are collapsed: It is impossible to distinguish between

school and class effects Table 2.1 presents the results of a sequence of

multilevel regression models with the achievement score as the dependentvariable

TABLE 2.1 Results of the Analysis of the School-Achievement Example

In Table 2.1, several different models are presented The first model,the intercept-only model, serves as a baseline; it shows that the totalvariance is divided into two parts, 88.8 at the pupil level and 21.7 atthe school level This information gives the intraclass correlation,

which is the proportion of variance accounted for at the group level

In the school data, in acchievement the intraclass correlation is 0.20(i.e., 21.7/[88.8 + 21.7]) In other words, 20% of the variance is at theschool level Turning to the next model in Table 2.1, one sees thatsome of this variation is explained by pupil and school characteristics.Pupil SES turns out to have a regression coefficient with significantvariance across schools, which is partly explained by the interactionwith the teachers' experience For a more detailed introduction to

multilevel regression, see Bryk and Raudenbush (1992) and Hox

(1995)

The Multilevel Regression Model for Longitudinal Data

In the previous section, individuals are considered to be the lowestlevel of the hierarchy In longitudinal research, one has a series ofrepeated measures for each individual One way to model such data is

to view the series of repeated measures as a separate level below theindividual level The individual level becomes the second level, and it

is possible to add a third and higher levels for possible group

structures Multilevel models for longitudinal data are discussed by,among others, Bryk and Raudenbush (1987, 1992) and Goldstein

response model (Goldstein, 1995) with dummy variables indicatingthe different occasions The standard multilevel model in Equation 9

(Goldstein, 1995; Maas & Snijders, 1997) Multilevel analysis is

useful for fixed-occasion data when there are missing observationsbecause of the absence of individuals at specific occasions or panelattrition Because multilevel models do not assume equal numbers ofidentical occasions for all individuals, such missing data pose no

on individual attributes Quadratic and higher functions can be used tomodel nonlinear dependencies on time, and both time-varying andperson-level covariates can be added to the model For a more detaileddiscussion of such models, see MacCallum and Kim (chap 4, thisvolume)

As mentioned, multilevel regression models for growth curves

commonly assume residual errors that are uncorrelated over time

Especially for growth curves with closely spaced observations, thisassumption may be implausible Models that are more complex are

possible for the residual errors, e ti For instance, one can specify an

autocorrelation structure for the residuals or model the variance of theresiduals as a function of time or age Some such models are

discussed by Gibbons et al (1993) and Goldstein (1995), and the

program MixReg (Hedeker & Gibbons, 1996) has built-in options forcorrelated errors

Example of Multilevel Analyses of Longitudinal Data

The example data have been generated by Rogosa and Saner (1995),with 200 individuals measured at five equidistant time-points and, atthe person level, one time-invariant covariate The model for thesedata is given by:

The time points, c, are coded as t = 0, 1, 2, 3, 4, and the covariate, z, is

centered around its overall mean As a result, the intercept can be

interpreted as the expected value at the first occasion for individuals

with an average value of z Although using time points t = 1, 2, 3, 4, 5 and raw scores for z would be completely equivalent, the estimates are

slightly more difficult to interpret

Table 2.2 presents the results of a multilevel analysis of these

longitudinal data Model 1 (intercept only) contains only an interceptterm; this model serves as a null or baseline model The intercept-onlymodel estimates the repeated measures variance as 84.4 and the

person level variance as 42.7 Thus, the intraclass correlation or theproportion of variance accounted for at the person level is estimated

thirds of the variance of the measures is variance over time and one-third is variance between individuals In Model 2 (intercept + time),the time variable is added as a predictor with varying coefficients fordifferent persons The model predicts a value of 44.0 at the first

as 0.34 (i.e., 42.7/[84.4 + 42.7]) In other words, approximately two-occasion, which increases by 5.0 on each succeeding occasion Thevariance components for the intercept and the regression slope for thetime variable are both significant The significant intercept variancemeans that individuals have different initial states, and the significantslope variance means that individuals also have different growth rates.There is a negative correlation between the initial status and the

growth rate (r = 24); individuals who start high tend to grow at a

coefficients can be modeled by the covariate Note that the type ofcorrelation between the intercept and slope is different; in Model 2 it

is an ordinary (zero-order) correlation, but in Model 3, it is a partial

correlation, conditional on the covariate z.

cut values for the amount of variance explained by the various effects.Bryk and Raudenbush (1992) suggested using the residual error

One deficit of multilevel approaches is that they do not provide clear-variance of the intercept-only model as a benchmark and examininghow much this goes down when explanatory variables are added tothe model In Table 2.2, this strategy leads to inconsistencies, because

in Model 2 the residual error variance for the intercept actually goes

up when the time variable is added to the model The reason is that inmultilevel models with random coefficients the notion of amount ofvariance explained at a specific level is not a simple concept As

Snijders and Bosker (1994) explained in detail, the problem arisesbecause the statistical model behind multilevel models is a

hierarchical sampling model:

Page 22 TABLE 2.2

measurements, which in many cases are (almost) the same for all

individuals in the sample Thus, the variability between persons in thetime series variable is in fact much lower than the hierarchical

sampling model assumes Snijders and Bosker (1994) described

procedures to correct the problem A simple approximation is to usethe occasion-level error variance from Model 1, and the person-levelerror variance of Model 2, which includes the time variable Then,observe that the error variance at the repeated-measures level goesdown from 84.4 to 11.9, which means that the time variable explainsabout 86% of the variance between the occasions To see how much

variance the person-level variable, z, explains, regard the intercept

the covariate z explains about 17% of the initial variation among the persons Likewise, one can calculate that z explains about 39% of the

initial variance of the time slopes These values are rough indications;however, more precise procedures are given by Snijders and Bosker(1994)

As the example makes clear, applying multilevel regression models tolongitudinal data is straightforward, especially if investigators restrictthemselves to a single dependent variable and to a linear or

polynomial function of the time variable More complicated modelsare possible, such as nonlinear models or multivariate models for

several dependent variables; for illustrations, see MacCallum and Kim(chap 4, this volume) and Goldstein (1995)

Multilevel Structural Equation Models

Structural equation models (SEMs) for multilevel data are described

by, among others, Goldstein and McDonald (Goldstein & McDonald,1988; McDonald & Goldstein, 1989), Muthén and Satorra (Muthén,1989; Muthén & Satorra, 1989), Longford and Muthén (Longford,1993; Longford & Muthén, 1992), and Lee and Poon (1992)

Nontechnical introductions are given by Muthén (1994), McDonald(1994), and Hox (1995) This section describes an approximation

If the population data is decomposed into between-groups variablesand within-groups variables, three population covariance matrices are

distinguished: the total covariance matrix, SB; the between-groups

covariance matrix, SB; and the within-groups covariance matrix, SW.Just as the group means and the individual deviation scores

themselves, the covariance matrices, SB and SW, are orthogonal andsum to the total covariance matrix, S:

, is given by the pooled within-Equation 13 corresponds to the conventional equation for the

covariance matrix of the individual deviation scores, with N G in thedenominator instead of the usual N 1 Thus, one can model the

population within-group structure by constructing and testing a

structural model for SPW

The between-groups covariance matrix for the disaggregated groupmeans, SB, calculated in the sample, is given by:

Equation 14 corresponds to the conventional equation for the

covariance matrix of the disaggregated group means, with G in thedenominator instead of the usual N 1 Unfortunately, SB is not a

simple estimator of the population between-groups covariance matrix,

SB Instead, SB is an estimator of the sum of two matrices:

where d is a scaling factor equal to the common group size (Muthén,

1989, 1994) Thus, to model the between-groups structure, specifytwo models for SB: one for the within-groups structure and one for thebetween-groups structure Muthén (1989, 1994) proposed using themultigroup option of standard SEM software to analyze these models.There are two groups, with covariance matrices, SPW and SB, based

on N G and G observations The model for SW must be specified forboth SPW and SB, with equality restrictions between both groups Themodel for SB is added to the model for SB, with the scale factor, d,

estimates are close enough to the full maximum likelihood estimates

to be useful in their own right Comparisons of pseudobalanced

estimates with full maximum likelihood estimates or with knownpopulation values have been made by Muthén (1991b, 1994), Hox(1993), and McDonald (1994) They all conclude that the

pseudobalanced estimates are generally accurate

The multilevel part of the covariance structure model outlined issimpler than that of the multilevel regression model It is comparable

to the multilevel regression model