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- Abstract
- LINEAR MIXED MODEL
- MODELS FOR BINARY RESPONSES
- EXAMPLES: CRIMINAL OFFENDING DATA
- METHOD
- RESULTS
- DISCUSSION
- WHICH MODEL SHOULD BE USED?
- COVARIATES OF CHANGE
- SUMMARY
- REFERENCES

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Multivariate Behav Res. Author manuscript; available in PMC 2010 June 29.

Published in final edited form as:

Multivariate Behav Res. 2009 January 1; 44(1): 28–58.

doi: 10.1080/00273170802620071PMCID: PMC2893373

NIHMSID: NIHMS203981

Jeffrey D. Long

University of Minnesota

Rolf Loeber

University of Pittsburgh

Cambridge University

Correspondence concerning this article should be addressed to Jeffrey D. Long, Quantitative Methods in Education, University of Minnesota, 56 East River Road, Minneapolis, MN 55455. Email: ude.nmu@jgnol

Two models for the analysis of longitudinal binary data are discussed: the marginal model and the random intercepts model. In contrast to the linear mixed model (LMM), the two models for binary data are not subsumed under a single hierarchical model. The marginal model provides group-level information whereas the random intercepts model provides individual-level information including information about heterogeneity of growth. It is shown how a type of numerical averaging can be used with the random intercepts model to obtain group-level information, thus approximating individual and marginal aspects of the LMM. The types of inferences associated with each model are illustrated with longitudinal criminal offending data based on *N* = 506 males followed over a 22-year period. Violent offending indexed by official records and self-report were analyzed, with the marginal model estimated using generalized estimating equations and the random intercepts model estimated using maximum likelihood. The results show that the numerical averaging based on the random intercepts can produce prediction curves almost identical to those obtained directly from the marginal model parameter estimates. The results provide a basis for contrasting the models and the estimation procedures and key features are discussed to aid in selecting a method for empirical analysis.

Recent advances in statistical methods have made a number of tools available for the analysis of nonnormal longitudinal data. Among these tools are longitudinal extensions of the logistic regression model for binary data (see, e.g., Molenberghs & Verbeke, 2006, chap. 5). A practical problem for applied research is that multiple classes of such models exist and it is not always clear which model should be used. In contrast to methods based on the normality assumption, each binary model class has its unique parameters with specific interpretations. It is important for the applied researcher to understand the key features in order to use the model appropriate for the desired type of inference.

This article addresses fundamental issues of applied analysis using the two most common classes of longitudinal models for binary data: the marginal model and the random intercepts model. To provide motivation and illustration, we consider two real data examples from the Pittsburgh Youth Study (PYS), which is a longitudinal study of inner-city boys that began in 1987 (Loeber et al., 2003). The data consist of the yearly binary classification of violent offending from official records and self-report collected on the same cohort of males over a 22-year period. Consistent with the theme of the real data examples, we relate aspects of the models to growth curve analysis in criminology research. Discussion focuses on model interpretation and the types of research questions that can be addressed. Incommensurability of the parameters is considered and a numerical averaging method based on the fitted random intercepts model is suggested to address this issue. To provide familiar context, we contrast the models for binary data with the normal-based linear mixed model for longitudinal data.

Longitudinal methods based on the normality assumption have the advantage that individual-level and group-level information is subsumed under the same model. As an example, consider one of the most popular models for longitudinal data, the linear mixed model (LMM; Laird & Ware, 1982). Suppose *y _{ij}* is the response score for the

$${\mathit{y}}_{i}={\mathit{X}}_{i}\beta +{\mathit{Z}}_{i}{\mathit{b}}_{i}+{\mathit{e}}_{i},$$

(1)

where * X_{i}* is a

In this article we are primarily concerned with models that have no covariates of change other than time transformations. Additionally, the criminal offending data to be considered later have nonlinear trajectories, so we will also be preoccupied with quadratic polynomial growth curves. For such models, the first column of *X _{i}* is a ones vector and the remaining columns are the

$${y}_{ij}=({\beta}_{0}+{b}_{i0})+({\beta}_{1}+{b}_{i1}){t}_{ij}+({\beta}_{2}+{b}_{i2}){t}_{ij}^{2}+{e}_{ij}.$$

(2)

Here * X_{i}* =

Equation (2) is an individual-level model as it is a growth curve for the *i*th person (unit), with the *k*th person parameter being ${\stackrel{~}{\beta}}_{ik}={\beta}_{k}+{b}_{ik}$. The collection of ${\stackrel{~}{\beta}}_{ik}$ defines a person's growth curve and *e _{ij}* indicates the scatter of points about a person's curve. Heterogeneity in growth among people is indexed by the variance of ${\stackrel{~}{\beta}}_{ik}$, for example, Var(

In addition to individual-level information, the LMM also provides information about the group-level growth curve. Taking the expectation of the individual-level parameters in Equation (2),^{1} we obtain the group-level parameters, *E*(β_{k} + *b _{ik}*) = β

$$E\left({y}_{ij}\right)={\beta}_{0}+{\beta}_{1}{t}_{ij}+{\beta}_{2}{t}_{ij}^{2},$$

(3)

which can be interpreted as the group-level growth curve that results from averaging the individual growth curves. Equation (3) is referred to as a marginal model because *E*(*y _{ij}*) is analogous to a marginal mean.

The marginal model of Equation (3) provides insight into the individual-level model as it shows that the collection of random effects for a person represents the individual's deviation from the group-level growth curve. When a person's collection of *b _{ik}* are all zero, the individual's growth curve is identical to the group-level growth curve. When at least one random effects is not zero, the individual's growth curve in some respect deviates from the group growth curve.

The last section illustrates that the LMM subsumes both an individual-level model and a marginal model. The ability to aggregate over the individual-level model to obtain the marginal model is based on the normality assumption and the fact that *y _{ij}* is a linear function of the predictors. In the case of binary data, the unity of the individual-level and marginal model is lost because the relation between the response and the predictors is nonlinear. Consequently, different approaches must be taken to develop the two types of models. We begin with the marginal model for binary data as it is a direct extension of the logistic regression model for cross-sectional data.

Let us for the moment consider a single cross section of time. Suppose for the cross section we have the binary response variable, *y _{i}*, which can take the value of 0 or 1. In the context of criminology research, we might have

Now consider the longitudinal case in which the binary response is collected at repeated cross sections. Similar to the LMM, the binary response is denoted as *y _{ij}* and we have

$$E\left({y}_{ij}\right)=Pr({y}_{ij}=1)=\frac{\mathit{exp}\left({\mathit{X}}_{i}\gamma \right)}{1+\mathit{exp}\left({\mathit{X}}_{i}\gamma \right)},$$

(4)

where *X*_{i} is a *n _{i}* ×

When the time metric is age in years and *Pr*(*y _{ij}* = 1) represents prevalence, the growth curve is known as the

One of the most replicated findings regarding the age-crime prevalence curve is that it is nonlinear increasing from late childhood, peaking in adolescence, and dropping afterward (Fergusson & Horwood, 2002; Lacourse, Nagin, Tremblay, Vitaro, & Claes, 2003; Nagin, Farrington, & Moffitt, 1995). Given this backdrop, we primarily discuss marginal models with nonlinear growth curves. The quadratic polynomial is one example:

$$Pr({y}_{ij}=1)=\frac{\mathit{exp}({\gamma}_{0}+{\gamma}_{1}{t}_{ij}+{\gamma}_{2}{t}_{ij}^{2})}{1+\mathit{exp}({\gamma}_{0}+{\gamma}_{1}{t}_{ij}+{\gamma}_{2}{t}_{ij}^{2})}.$$

(5)

With the age-crime curve it is common for γ_{2} to be less than zero so that the growth curve is concave with respect to time (i.e., ∩). The upper panel of Figure 1 shows an example of Equation (5) denoted as a dashed line (the solid lines are discussed later).

Simulated individual growth curves (solid lines) and mean growth curve (dashed line). Upper panel is probability metric and lower panel is logit metric.

The curve in Figure 1 shows asymptote-like behavior as it approaches zero. This is advantageous for modeling age-crime relations as prevalence is expected to be zero at ages prior to the ascent up to the peak and zero at ages after the descent down from the peak (Nagin et al., 1995). The asymptotic behavior does not occur in the logit metric because the model is linear in the predictors,

$$\mathit{log}it\left\{Pr({y}_{ij}=1)\right\}={\mathit{log}}_{e}\left\{\frac{Pr({y}_{ij}=1)}{1-Pr({y}_{ij}=1)\phantom{\rule{thickmathspace}{0ex}}}\right\}={\gamma}_{0}+{\gamma}_{1}{t}_{ij}+{\gamma}_{2}{t}_{ij}^{2},$$

(6)

which produces a parabola as shown in the lower panel of Figure 1, denoted as a dashed line (the solid curves are discussed later).

When the age-crime curve is considered, it is often better to work with the probability metric than the logit metric as the former can be interpreted as prevalence and allows zero values in actual data that one can graph against a prediction curve (odds and logits based on zero prevalence are not defined). The probability and logit metric have a nonlinear but monotonic relation so that among other things, the time value at which the peak occurs is the same (see the upper and lower panels of Figure 1).

In contrast to the LMM, the marginal model for binary data cannot be derived from an individual-level model. An implication is that there is no information available about individual heterogeneity in growth analogous to the variance of the random effects in the LMM. For this reason, the marginal model is most appropriate when the applied researcher wants to make group-level inferences. For example, Equation (5) is useful for modeling change in prevalence over time because this is a group-level probability. The marginal model is not appropriate for studying heterogeneity in growth, nor is it appropriate for tracking changes of individual-level probabilities. For these types of inferences, we must use a random effects model for binary data, which we now discuss.

The random effects model is an individual-level model and as such describes change for separate people (units). Similar to the LMM, individual growth curves are specified by the inclusion of random effects. However, in the random effects model for binary data only individual-specific information is available. Individual growth curve parameters cannot be averaged over to yield group-level parameters, though as we see, a type of averaging can be used with the individual probabilities.

Multiple random effects can be included in models for binary data, but it is common to include only random intercepts. A practical reason for this is that models with more than random intercepts can be very difficult to estimate with current numerical methods (Diggle, Heagerty, Liang, & Zeger, 2002, chap. 11). Perhaps more compelling is that binary data often provide relatively little information about individual heterogeneity beyond intercept variability. An illustration is provided by the criminal offending data presented later. Individuals in the sample were randomly drawn from the community, and as a consequence the majority of participants had no criminal offenses for the duration of the data collection. The preponderance of time series with zero values provided limited information about individual differences in growth trajectories and models with more than random intercepts could not be estimated (i.e., the computer algorithms did not converge). What information did exist about heterogeneity was probably captured by random intercepts. Because this situation seems to be the rule rather than the exception (Fitzmaurice, Laird, & Ware, 2004, chap. 12), only random intercepts models are considered here.

The random intercepts model in effect “individualizes” the marginal model by incorporating the random intercepts term in its probability definition. Suppose *d _{i}* is the random intercepts value for the

The general form of the random intercepts model is

$$E({y}_{ij}{d}_{i})=Pr({y}_{ij}=1{d}_{i})=\frac{\mathit{exp}({d}_{i}+{\mathit{X}}_{i}\delta )}{1+\mathit{exp}({d}_{i}+{\mathit{X}}_{i}\delta )},$$

(7)

where * X_{i}* is as previously defined, β is a vector of unknown individual-level regression coefficients, and we assume ${d}_{i}~\mathcal{N}(0,{\sigma}_{d}^{2})$. Similar to the LMM, ${\sigma}_{d}^{2}$ is an important parameter as it indexes heterogeneity in growth. ${\sigma}_{d}^{2}$ is especially useful when examining time-static covariates of change (e.g., sex). The proportional decrease in ${\sigma}_{d}^{2}$ when a time-static covariate is added to the model can be used as a measure of effect size.

As a particular example of Equation (7), consider the quadratic polynomial model

$$Pr({y}_{ij}=1{d}_{i})=\frac{\mathit{exp}({d}_{i}+{\delta}_{0}+{\delta}_{1}{t}_{ij}+{\delta}_{2}{t}_{ij}^{2})}{1+\mathit{exp}({d}_{i}+{\delta}_{0}+{\delta}_{1}{t}_{ij}+{\delta}_{2}{t}_{ij}^{2})}.$$

(8)

The logit form expresses $\mathit{log}it\{Pr({y}_{ij}{d}_{i}=1)\}={\mathit{log}}_{e}\{\frac{Pr({y}_{ij}=1{d}_{i})1-Pr({y}_{ij}=1{d}_{i})\}}{}$ as a linear function of the predictors (the details are left to the reader).

In contrast to the marginal model, Equation (8) is a growth curve for an individual. Though the δ* _{k}* are referred to as fixed effects, they are not analogous to the fixed effects in the LMM. The δ

As an illustration of the random intercepts model and its relation to the marginal model, consider some simulated criminal offending data based on Equation (8). Suppose the random intercepts scores for 3 individuals are *d*_{1} = −2, *d*_{2} = 0, and *d*_{3} = 2, and the parameter values are δ_{0} = −25, δ_{1} = 3, and δ_{2} = −0.1. Substituting these values into Equation (8) and solving for values of *t _{ij}* in the age range of [0, 30], the individual probability of offending,

The value of a person's random effect determines the height of his or her growth curve with *d*_{3} = 2 having the curve with the highest peak and *d*_{1} = −2 having the curve with the lowest peak. Each person has the same rate of change because δ_{1} and δ_{2} are constant among individuals. Because *E*(*E*(*y _{ij}* |

The lower panel of Figure 1 shows the same growth curves but in the logit metric. The growth curves are vertically shifted, illustrating the random intercepts differences. Variability among the intercepts is indexed by ${\sigma}_{d}^{2}$, which in this case is equal to ${\scriptstyle \frac{1}{3}}(-{2}^{2}+{0}^{2}+{2}^{2})={\scriptstyle \frac{8}{3}}$. ${\sigma}_{d}^{2}$ is in the logit metric and expression in the probability metric is involved as the variance changes with the probabilities. It has been noted that even seemingly small values of ${\sigma}_{d}^{2}$ can reflect important heterogeneity (Duchateau & Janssen, 2005).

The δ* _{k}* of the random effects model are always person-specific, even when considered in isolation. To see this, consider the case of

$$Pr({y}_{ij}=1{d}_{i}=0)=\frac{\mathit{exp}({\delta}_{0}+{\delta}_{1}{t}_{ij}+{\delta}_{2}{t}_{ij}^{2})}{1+\mathit{exp}({\delta}_{0}+{\delta}_{1}{t}_{ij}+{\delta}_{2}{t}_{ij}^{2})}.$$

(9)

Though the right-hand side of this formula has the exact form as the marginal model it is merely the growth curve for an individual with a random intercepts value of zero, not any type of averaged curve. This is verified in Figure 1 where it can be seen that the *d _{i}* = 0 growth curve does not correspond to the marginal growth curve.

The only time Equation (9) will have a group-level interpretation and be at least approximately equivalent to a marginal model is when ${\sigma}_{d}^{2}=0$. In this case all individuals share the same random intercept value and the same individual growth curve. The concept is formalized in the following approximate relation.

Suppose the estimation procedures described in the next section are used for each model. Then it can be shown (Zeger, Liang, & Albert, 1988) that the parameters of the marginal and random intercepts models have the approximate relation

$$\delta \approx \gamma \sqrt{1+0.346{\sigma}_{d}^{2}}.$$

(10)

Equation (10) indicates that when the variance of the random intercepts is zero, the random effects parameters (δ* _{k}*) are approximately equivalent to the marginal model parameters and thus have a group-level interpretation. However, when the variance is greater than zero, the parameters are not approximately equivalent and the δ

Note that Equation (10) refers to the entire vector of parameters. This verifies the point made earlier that none of the δ* _{k}* have a group-level interpretation, whether or not they are associated with the random intercepts. Equation (10) is also helpful for explaining why the parameter estimates of the different models can be so different in empirical analysis. The approximate relation indicates the δ

The incommensurability of the random effects model and the marginal model is inconvenient for applied analysis. The marginal model provides inferences about the group-level growth curve but has the limitation of not providing information about individual variability. The random intercept model provides this information in the form of ${\sigma}_{d}^{2}$, but only individual-specific conclusions can be made. It would be desirable to have both group-level and individual-level information available in a single model.

In principle, marginal information can be obtained from the random intercepts model by taking the expectation over the random intercepts. In terms of Equation (7) this is

$$E\left({y}_{ij}\right)=E[E({y}_{ij}{d}_{i})]=E\left\{\frac{\mathit{exp}({d}_{i}+{\mathit{X}}_{i}\delta )}{1+\mathit{exp}({d}_{i}+{\mathit{X}}_{i}\delta )}\right\}.$$

(11)

It turns out the last equation on the right is very problematic to work with because it does not have an exponential form or an analytic solution (Fitzmaurice et al., 2004, chap. 13). An alternative is to work with the sample version of *E*(*y*_{ij} | *d*_{i}) and obtain marginal information from numerical averaging in a similar fashion to what was done in constructing the mean curve in Figure 1 (Lindsey, 1999, chap. 7; Molenberghs & Verbeke, 2006, chap. 16).

Numerical averaging is carried out in the following steps. First, based on empirical data the random intercepts parameters are estimated, including an estimate of σ_{d}, denoted as ${\widehat{\sigma}}_{d}$. Then a large number of simulated random effects values, ${\widehat{d}}_{i}$, are drawn from a normal distribution with mean zero and standard deviation ${\widehat{\sigma}}_{d}$. We substitute ${\widehat{d}}_{i}$, $\widehat{\delta}$, and *t*_{ij} (where *t*_{ij} *t*_{j}) into Equation (7) and compute the estimated individual probabilities for each participant. For the *j* th cross section (e.g., age), the average of the estimated individual probabilities is computed yielding the estimated marginal probability, $\widehat{E}\left({y}_{ij}\right)$.

More formally, suppose *S* is the number of simulated ${\widehat{d}}_{i}$ and the selected model is the random effects quadratic polynomial of Equation (8). Then,

$$\widehat{E}\left({y}_{ij}\right)=\sum _{i=1}^{S}\frac{\mathit{exp}({\widehat{d}}_{i}+{\widehat{\delta}}_{0}+{\widehat{\delta}}_{1}{t}_{j}+{\widehat{\delta}}_{2}{t}_{j}^{2})}{1+\mathit{exp}({\widehat{d}}_{i}+{\widehat{\delta}}_{0}+{\widehat{\delta}}_{1}{t}_{j}+{\widehat{\delta}}_{2}{t}_{j}^{2})}.$$

(12)

As we see in the real data example, Equation (12) can yield marginal probabilities that are nearly identical to those obtained directly from the marginal model.

Therefore, at least some limited marginal information can be obtained from the random intercepts model. The information is limited in that the averaging method does not yield parameters that have group-level interpretations. To obtain approximate group-level parameters, one could use Equation (10) based on the arameter estimates, that is, ${\widehat{\delta}}_{k}={\widehat{\gamma}}_{k}\sqrt{1+0.346{\widehat{\sigma}}_{d}^{2}}$. However, it turns out this is not a very accurate approximation in many instances when ${\widehat{\sigma}}_{d}^{2}>0$ (Neuhaus, Kalbfleisch, & Hauck, 1991).

An additional advantage of the numerical averaging is that the variability of the simulated individual probabilities can be examined. For example, boxplots of the estimated probabilities at each timepoint can be constructed. Unlike the conditional variance, *Var*(*y*_{ij}) = *E*(*y*_{ij})(1 - *E*(*y*_{ij})), the variability based on the estimated probabilities is grounded in the individual-level model.

To illustrate the methods we present two real data examples. In terms of growth curve form, we consider the quadratic models presented earlier, that is, the marginal model of Equation (5) and the random intercepts model of Equation (8). We consider binary classifications of violent offending from official records and self-report. The data set of the former has a very low proportion of missing data whereas the data set of the latter has a relatively high proportion of missing data. Because missing data are almost inevitable in age-crime analysis spanning many years (Brame & Piquero, 2003), we thought it important to examine the methods under two different missing data scenarios.

Participants were the oldest cohort of males (*N* = 506) in the PYS who were measured beginning in 1987. Details regarding participant characteristics and methods of data collection can be found in Loeber, Farrington, Stouthamer-Loeber, Moffitt, and Caspi (1998) and Loeber et al. (2003). The official records measure was a yearly binary classification of conviction of a violent offense (e.g., strong-arming, attacking to seriously hurt or kill), with 0 = no conviction, 1 = conviction. The official records measure was culled from various court sources and spanned 9 to 30 years of age. All participants had complete data up to 18 years of age, and 488 participants (96%) had complete data for all ages. Nearly all missing data was due to death and thus was of the dropout variety (i.e., not intermittent).

The self-report measure indexed whether the person committed a violent act at a given age (0 = did not commit the act, 1 = committed the act), based on interviews with the participant, his parent(s), and his teacher(s). The cohort had self-report data for the ages 12–27 years. In contrast to official records, self-report missing was mostly intermittent with the vast majority due to inability to interview the respondents, though 4% of the participants dropped out due to death as previously noted. None of the participants had complete data and 88% had data for half or fewer of the timepoints (i.e., less than or equal to eight timepoints). In terms of cross sections, 50% of the data was missing at 12 years, an average of 12% was missing for 13–24 years, and an average of 76% was missing for 25–27 years.

The marginal model was estimated using generalized estimating equations (GEE; Liang & Zeger, 1986). GEE is a quasi-likelihood estimation method that accounts for pairwise dependency among the repeated measures through a “working” correlation matrix. For consistency with the random effects model, a working correlation matrix with an exchangeable structure was used, which had a single correlation parameter, ρ. Liang and Zeger show that under certain conditions GEE yields consistent and asymptotically normal estimates of γ and standard errors can be based on a robust “sandwich” estimator, which was used in the analysis. Parameter estimates were obtained with the R function geeglm of geepackage (Halekoh, Höjsgaard, & Yan, 2006), but we note that similar analysis can be performed using SAS PROC GENMOD and other statistical software.

The random intercepts model was estimated using maximum likelihood (ML) methods. Analogous to the compound symmetry structure of LMM, the dependencies due to repeated measures are accounted for through the variance of the random intercepts under the assumption, ${d}_{i}~\mathcal{N}(0,{\sigma}_{d}^{2})$. Likelihood estimation is generally more difficult for the binary random effects model than the normal-based LMM due to the lack of analytic integral formulas (Diggle et al., 2002, chap. 11). Integrals must be approximated with computer-intensive methods, one of the most popular being Gaussian quadrature, which was used here. In Gaussian quadrature, the accuracy of the integral approximation is controlled by the number of quadrature points, which was 50 in the analysis.^{3} Based on the normality assumption, asymptotic and nonasymptotic standard errors can be estimated with the former being presented later. The R function and package glmmML were used for the analysis (Broström, 2008), but we note that similar estimation can be performed with SAS PROC NLMIXED and other statistical software.

Additional issues regarding estimation are discussed in the following section. R syntax for the analysis is presented in the Appendix along with code for the numerical averaging previously described. Similar SAS syntax is available from Jeffrey D. Long.

Observed prevalence was computed for each variable at each age and is shown in Figures 2 and and3.3. Official records observed prevalence is denoted by the filled circles in Figure 2 and similarly for self-report in Figure 3. The vertical axes of the two figures have different scales to assess differences (or lack thereof) in the prediction curves to be discussed later. The peak value of the self-report prevalence was higher than official records, which is consistent with research showing official sources of offending tend to underestimate prevalence whereas self-report does not (Espiritu, Huizinga, Crawford, & Loeber, 2001; Loeber et al., 1998).

Official records observed violence prevalence (filled circles), and predicted growth curves from the GEE solution (dashed line) and maximum likelihood solution (solid line).

Self-report observed violence prevalence (filled circles), and predicted growth curves from the GEE solution (dashed line) and maximum likelihood solution (solid line).

The GEE results for the marginal model are shown in Table 1 and the ML results for the random intercepts model are shown in Table 2. The left-hand side of the tables shows the official records results and the right-hand side shows the self-report results. Each Wald-type Z value is computed as the parameter estimate divided by its estimated standard error (*SE*) and is used to test the null hypothesis that the parameter in question is zero. For the case of σ_{d}, the Z test tends to yield too large a *p* value (it tends to be “conservative”) because the null parameter value is at the lower limit and the sampling distribution is not normal (Stram & Lee, 1994). To address this problem, a larger Type I error rate such as α = 0.10 (rather than α = 0.05) may be used when evaluating the variance component (Berkhof & Snijders, 2001).

The last row of Table 1 contains the estimates of the constant value in the working correlation matrix (ρ). The estimate for official records was not statistically significant (|Z| < 1:64 < 1:96), whereas that for self-report was significant. The last row of Table 2 contains estimates of σ_{d}, which were statistically significant for both official records and self-report.

As for the growth curve parameter estimates, Table 1 and Table 2 indicate that ${\widehat{\delta}}_{k}{\widehat{\gamma}}_{k}$ for every *k*. The standard errors associated with the ${\widehat{\delta}}_{k}$ also were larger except for the case of the official records quadratic coefficient. The Z values of the growth curve parameter estimates in both tables were relatively large (|Z_{k}| > 6, *p _{k}* < 0.0001), indicating rejection of the null hypothesis in question.

Predicted prevalence curves were computed based on the parameter estimates. For the marginal model, the GEE estimates were substituted into Equation (5) along with integer values of *t _{j}* (

The real data examples illustrate analysis based on the marginal model and the random intercepts model. GEE was used to estimate the former whereas ML based on Gaussian quadrature was used to estimate the latter. Two longitudinal binary response variables were analyzed: violent offending from official records and self-report measured on the same *N* = 506 participants over the ages of 9–30 years (self-report spanned 12–27 years). The official records offending data was complete for almost all the participants whereas the self-report offending had incomplete data for all the participants.

Features of the two models can be seen by comparing Tables 1 and and2.2. Consistent with Equation (10), each random intercepts parameter estimate $\left({\widehat{\delta}}_{k}\right)$ was larger in absolute value than the corresponding marginal parameter estimate $\left({\widehat{\gamma}}_{k}\right)$. Note this difference was true for the linear and quadratic terms illustrating that the introduction of the random intercept causes an “offset” of the entire vector of the ${\widehat{\delta}}_{k}$ relative to the marginal model. Interestingly, the larger ${\widehat{\delta}}_{k}$ were accompanied by larger standard error estimates in all but one case (official records quadratic), which produced *Z* values that were similar across the models. Such consistency in *Z* values has been shown in other contexts (Jansen, Beuchens, Molenberghs, Verbeke, & Mallinckrodt, 2006; Neuhaus, 1993; Neuhaus et al., 1991; Zeger et al., 1988).

Though *Z* values are provided for all the growth curve parameter estimates in Tables 1 and and2,2, it should be noted that only the *Z* value of the quadratic coefficient are interpreted. The quadratic coefficient is the only coefficient invariant to arbitrary linear transformations of the time scale (Griepentrog, Ryan, & Smith, 1982; Morrell, Pearson, & Brant, 1997). For both models and both variables, the estimate of the quadratic parameter (${\widehat{\gamma}}_{2}$ or ${\widehat{\delta}}_{2}$) had a negative sign, indicating all the growth curves were concave with respect to age. The corresponding *Z* values were relatively large, indicating the need for concave growth curves as opposed to linear growth curves. Additional analysis not presented indicated that cubic terms were not required.

Perhaps the most interesting result of the analysis was the agreement of the prevalence prediction curves, shown in Figures 2 and and3.3. The curves were nearly identical for both official records and self-report. The agreement indicates that in at least some situations, group-level information like that obtained from the marginal model can also be obtained from the random intercepts model using numerical averaging.

The agreement of the prevalence prediction curves is especially intriguing because the two variables had different missing data conditions and the estimation procedures treat missing data in different ways. As explained by Liang and Zeger (1986), GEE uses a type of pairwise deletion and thus requires the assumption of missing completely at random (MCAR) for valid inference (for a discussion of missing data mechanisms, see Little & Rubin, 1989). In contrast, the ML estimation of the random intercepts model requires the less stringent assumption of missing at random (MAR; Jansen et al., 2006). Assumptions about missing data are especially pertinent in age-crime research as some proportion of dropout is likely if the repeated measures span more than a few years (Brame & Piquero, 2003; Manski & Nagin, 1998). Because criminal offending is correlated with incarceration and premature death (Loeber & Stouthamer-Loeber, 1998), dropout is often predictable from the responses at previous timepoints meaning that MAR might be a more plausible assumption than MCAR.

With the aforementioned in mind, we computed prevalence aggregating over ages 9–18 for those who dropped out and those who did not (all participants had complete data for 9–18 years of age). The dropout group had a prevalence rate of 0:03 compared with a rate of 0:01 for those with complete data. Thus, dropout was correlated with prevalence, indicating missingness could have been MAR for official records. We hasten to add that there is no formal test of the MAR assumption and the true state of affairs is unknown for this analysis. Despite the possibility of MAR, the rate of dropout was very low (4%) and the sample size relatively large (*N* = 506), which may explain why the different treatments of missing data yielded similar results. Analytic and simulation studies suggest that GEE and ML can yield similar results under such conditions (Fitzmaurice, Laird, & Rotnitsky, 1993; Touloumi, Babiker, Pocock, & Darbyshire, 2001).

Self-report had a substantially greater proportion of missing data. The missing data mechanism might have been a combination of MAR and MCAR, with the latter dominating. Because self-report was collected on the same participants as official records, the small proportion of dropout beginning at 18 years (4%) was possibly MAR. The much larger proportion of nondropout missingness (up to 76% at some ages) was due to the inability to contact informants. Ability to contact may be influenced by quasi-random factors such as informant illness, and thus, an MCAR mechanism might have been responsible for the majority of missingness. When the missing mechanism is MCAR and there is a large sample size, pairwise methods and ML can yield similar results even for relatively large proportions of missing data (Shin, Davison, & Long, 2009). This might explain the agreement of predicted prevalence among the methods for the self-report data. Further research is needed, perhaps in the form of simulation studies, to help determine situations in which GEE and ML with numerical averaging might yield substantially different results.

Additional differences in the estimation methods are illustrated by the results in the last row of Tables 1 and and2.2. The last row of Table 1 shows the estimates of ρ, and the last row of Table 2 shows the estimates of σ* _{d}*. The estimate of ρ for official records was not statistically significant whereas the estimate of σ

Both the marginal model as realized in GEE and the random intercepts model account for dependencies due to repeated measures through a single parameter. But the underlying model for each method is very different. A key feature of the GEE method is that ρ is treated as a nuisance parameter as the working correlation matrix makes no attempt at providing a full or even accurate specification of the multivariate distribution of the repeated measures (Crowder, 1995; Fitzmaurice, 1995). The working correlation matrix is a means of accounting for dependencies without having to specify the large number of parameters required when basing the marginal model on discrete multivariate distributions, such as the multinomial (Molenberghs & Verbeke, 2006, chap. 7).

In contrast to ρ of the marginal model, σ* _{d}* of the random intercepts model is integral, providing an index of individual heterogeneity and a basis for the specification of the full underlying multivariate distribution (Larsen, Petersen, Budtz-Jorgensen, & Endahl, 2000; Neuhaus, 2001). For official records, the values of $\widehat{\rho}$ and ${\widehat{\sigma}}_{d}$ and their associated

Based on the result for $\widehat{\rho}$, we analyzed the official records data as a traditional logistic regression using the R function glm. We obtained the γ-parameter estimates $\widehat{\gamma}=[-21.4736,2.0403,-0.0579]\prime $, which were nearly identical to the GEE estimates in Table 1. In addition, we obtained the standard error estimates, ${\mathit{SE}}_{\widehat{\gamma}}=[2.5152,0.2871,0.0081]\prime $, which were also very close to the GEE estimates. These results indicate that from the GEE viewpoint, there is very little difference in statistical information when the repeated measures of the official records are completely ignored. This is consistent with analytic and simulation studies that show the GEE model may have no advantage over the traditional logistic regression model for small values of ρ (Fitzmaurice, 1995; McDonald, 1993; Sutradhar & Das, 1999).

Turning to the ML results for official records (Table 2), the value of ${\widehat{\sigma}}_{d}$ and its associated *Z* value suggest statistically significant heterogeneity of intercepts and correlation between timepoints. This is at odds with the GEE results as it suggests that the traditional logistic regression model is not appropriate for the data. Though the model that generated the data is unknown, we feel the random intercepts model has greater appeal in this case. One problem with the GEE model is that the possible value of ρ is limited by the values of the marginal probabilities (Chaganty & Joe, 2004; Demidenko, 2004, chap. 7). Because the observed official records prevalence values were relatively low $\left(\mathit{max}\widehat{E}\left({y}_{ij}\right)<0.04\right)$, this may have caused the small value of $\widehat{\rho}$ rather than a lack of true correlation. Longitudinal data by nature induces correlation between timepoints. The random intercepts model accounts for this explicitly without constraints dictated by the growth curve probabilities.

Having introduced and discussed two distinct models for binary longitudinal data, a natural question arises as to which model should be used in practice. One answer to this question is to analyze the data with both models as was done in this article (for an example in a social science journal see Pellegrini & Long, 2007). One benefit of using both models is that group- and individual-level information is provided. The mean growth curve parameter estimates are available directly and so is the estimate of the variance of the intercepts. Another benefit is the analyst obtains an idea of the robustness of results under different estimation methods and different missing data assumptions. Consistency of results across the models suggests we might have greater confidence in their validity relative to the situation in which only one or the other method is used.

There are some pitfalls in presenting results based on both models. A justification for the inclusion of both models is usually required, which can be extensive depending on the level of sophistication of the audience. Perhaps more problematic is the discussion that might be required if the results disagree. As we have seen, issues and assumptions regarding the models and their estimation must be discussed to account for differences in results. Such discussion might require substantial manuscript space.

In the event the analyst wants to use a single method, the key features of the models and their estimation procedures should be considered. A summary of these features appears in Table 3. Some of these features have already been discussed, but it is worthwhile to highlight the potential advantages and disadvantages of the methods so that an informed decision can be made.

An advantage of the marginal model is that it provides a group-level growth curve, which is often of interest in social science research. For example, prevalence rather than individual probabilities is typically examined in criminology research as the age-crime curve is used to make inferences about aggregates of people (Farrington, 1986). Because GEE provides estimates of the group-level growth curve, the estimates can be directly substituted into Equation (5) to compute the estimated prevalence as shown in Figures 2 and and3.3. The GEE estimates can also be used to compute such substantively interesting quantities as the instantaneous rate of change in prevalence at a particular timepoint and the age at which the peak of the curve occurs. For the latter, we take the first derivative of Equation (5) with respect to *t _{ij}*, set it to zero, and solve for

On the negative side of things, the marginal model has been criticized because its growth curve is based on a series of dependent but unconnected cross-sections (Lindsey, 1999, chap. 11; Lindsey & Lambert, 1998; Neuhaus, 2001). The cross sections are unconnected as there are no individual growth curves and no means of tracking a person over time. Because it is not possible to locate people in relation to the prevalence at any timepoint, individuals are simply units over which to aggregate. Not only is this inconsistent with the hierarchical formulation of the LMM but it may also call into question the benefit of having a longitudinal design. If the goal is to estimate prevalence for several cross sections of age, then it is more efficient to draw random samples from different age cohorts rather than follow the same individuals as they age. In fact, a number of age-crime studies have relied on this type of cross-sectional design, for example, Sampson and Laub (2003).

As we saw with the official records variable, the GEE results can be nearly identical to traditional logistic regression that completely ignores the repeated measures (i.e., treats them as independent). Therefore, from the GEE perspective, a longitudinal design may have no advantage over a cross-sectional design, especially when the repeated measures have weak dependence. An applied researcher may be uncomfortable with a model and estimation procedure that do not necessarily take advantage of the longitudinal nature of their data. On the other hand, GEE does have advantages when the dependencies are stronger, such as greater efficiency (Chaganty & Joe, 2004). Furthermore, nonstatistical considerations, e.g., history effects, are usually of greater importance in the selection of a longitudinal design than statistical criteria. Therefore, the possible equivalence or near equivalence of the GEE results to traditional logistic regression may be irrelevant.

On a blatantly practical level, the marginal model has the advantage that it is relatively easy to estimate with GEE. The GEE method uses a type of iterative weighted least squares, which provides a quick and stable algorithm. GEE does not require any special demands on the user other than the specification of the growth curve and the structure of the working correlation matrix. The latter is usually selected from among a small number of alternatives (e.g., exchangeable, autocorrelation) built into statistical software. In addition, GEE requires no distributional assumptions (e.g., normality) and its estimators have desirable large-sample properties such as consistency and normality (assuming some weak regularity conditions and correct specification of the growth curve; Liang & Zeger, 1986).

As mentioned, GEE does require MCAR for valid inferences with missing data, making it less attractive than ML methods. To address this problem, versions of GEE have been suggested that use weights based on the probability of dropout (e.g., Robins, Rotnitsky, & Zhao, 1994). A difficulty with these weighting methods is they generally require correct specification of both the growth curve and the dropout process. If either is not correctly specified, the weighting can produce more biased estimates than regular GEE (Beunckens, Sotto, & Molenberghs, 2008). A more promising solution is to use GEE with multiple imputation (MI). This amounts to an analysis assuming MAR as MI is performed under this assumption and the GEE analysis is based on complete data (with no assumptions regarding missing data required). There is evidence based on simulation studies that this approach works well in terms of providing relatively low bias and relatively high precision (Beunckens et al., 2008; Jansen et al., 2006).

Another possible disadvantage of GEE is that it provides no formal statistical index of global fit.^{5} This means that model selection is less flexible than with ML methods. Specifically, there is no statistical comparison for nonnested models such as models with the same number of parameters. For example, suppose we want to compare the quadratic polynomial model with a log model formed by replacing *t _{ij}* with

Finally, the adequacy of the working correlation as a basis for analyzing longitudinal binary data has been questioned by a number of researchers (Chaganty & Joe, 2004; Crowder, 1995; Demidenko, 2004, chap. 7; McDonald, 1993; Sutradhar & Das, 1999). Much of the criticism has focused on the range restrictions of ρ imposed by the marginal probabilities, as previously discussed. One alternative is to model the dependencies using conditional odds ratios, which are not constrained by the marginal probabilities (e.g., Fitzmaurice, Laird, & Lipsitz, 1994).

Another increasingly popular alternative is a latent variable approach based on longitudinal extensions of the probit regression model (e.g., Muthén, 1996). In the latent variable approach, dependencies are modeled with correlations among normally distributed latent variables and thus are not bounded by the observed marginal probabilities. Similar to GEE, estimation of these models is typically not feasible with full ML as this involves integration over multivariate normal probabilities (Le Cessie & Van Houweligen, 1994). Consequently, so-called limited-information or quasi-likelihood methods are typically used (Molenberghs & Verbeke, 2006, chap. 7).

The random intercepts model has the advantage of providing a growth curve for individuals and information about heterogeneity of growth. Because of its individual-level nature, the random intercepts model has been widely used in settings such as biomedical research where focus is often on person-specific effects, such as individual response to medication (Diggle et al., 2002, chap. 7; Lindsey & Lambert, 1998; Neuhaus, 1992). In fact, the random intercepts model has been strongly recommended over the marginal model in such contexts because it provides growth curves that correspond to actual empirical trajectories (Lindsey & Lambert, 1998). An illustration is provided in Figure 1, where it can be seen that the the marginal growth curve is a trajectory that is not followed by any of the individuals and thus is a kind of statistical creation.

Even if the average growth curve is a statistical creation, it is appealing to a number of social scientists because its estimation is often the goal of growth curve analysis.^{6} The lack of an average growth curve may be one reason the random intercepts model has not been used extensively in the social sciences. As illustrated in our analysis, it is possible to obtain group-level information with the random intercepts model through numerical averaging. In this respect, the random intercepts model is more closely aligned with the LMM than the marginal model. The ability to provide both group-level and individual-level information seems an attractive quality of the random intercepts model.

There is some inconvenience regarding the group-level information obtained from the random intercepts model. Numerical averaging requires an extra step in the data analysis and routines for carrying it out are not part of standard statistical software (though the programming is relatively straightforward; see the Appendix). Numerical averaging is a post hoc procedure and as such it does not affect the interpretation of the model parameters. This means the random intercept parameters cannot be used directly to compute a predicted prevalence curve; nor can the parameters be used to compute instantaneous rate of change or age at peak of the prevalence curve (these can be directly computed for individual curves). Rather, these things must be obtained from the predicted prevalence, if possible.^{7}

An advantage of the random intercepts model is that its random effects structure is the basis for a full specification of the multivariate dependencies between timepoints. Cross sections are linked together by the individual growth curves and people can be traced over time. In this respect, the random intercepts model is a “full-fledged” longitudinal model that takes advantage of the repeated measurements inherent in longitudinal data. Accordingly, ML methods can be used meaning that global fit indices can be computed, such as Akaike's information criteria. These fit indices make it possible to statistically compare nonnested models such as models with equal numbers of parameters, a feat that is not possible with GEE. Nested models can be tested in the same way as with GEE, using specific or omnibus tests of point estimates. Similar to GEE, the ML estimators have favorable large-sample properties such as consistency and normality, but the assumption of normality for the random intercepts is typically required. In theory, small sample methods can be based on ML theory, but to date, these methods are in their infancy and not widely available in statistical software (SAS PROC NLMIXED may be an exception; see Littell et al., 2006, chap. 14). As for missing data, valid inference is possible under the MAR assumption making it consistent with LMM analysis.

A practical disadvantage of ML estimation is that it is much more demanding on the user than GEE. Gaussian quadrature requires that starting values and the number of quadrature points be supplied. Numerical algorithms can be very sensitive to starting values and parameter estimates may not be computed if values are very far afield of the final estimates. Starting values for fixed effects can be obtained from a cross-sectional logistic regression solution as shown in the R code of the Appendix. A starting value for the variance of the random intercepts is more difficult to come by, which often means a number of values must be tried. Care must also be taken in the selection of the number of quadrature points. Severe inaccuracies can occur when the number is too small, but substantial computing time and resources might be required when the number is large (Jansen et al., 2006; Lesaffre & Spiessens, 2001).

A final issue important to mention is covariates of change. Our discussion to this point has dealt with models that do not have covariates of change other than polynomial transformations of age. Here we consider covariates other than time transformations, which we refer to as correlates of change. There are two types of correlates of change: time-static, which have constant values over time, and time-varying. In age-crime studies, an example of a time-static correlate is family's socioeconomic status at first measurement (FSES), and an example of a time-varying correlate is yearly drug use (see, e.g., Fabio et al., 2006). Similar to LMM, both types of correlates may be included in the marginal model or the random effects model through the *X** _{i}* matrix. In the case of time-static correlates, columns of single correlates and their products with the polynomial transformations are added to the

When at least one correlate of change is time-varying, analysis with the marginal model using GEE can be quite involved as a nontrivial assumption about the conditional expectations must be investigated (Pepe & Anderson, 1994). If the assumption does not appear plausible, then an independent working correlation structure should be adopted to avoid bias in the growth curve estimates (Diggle et al., 2002, chap. 12). As previously noted, such an analysis ignores the longitudinal nature of the data and might be unattractive to applied researchers. Because a correctly specified random effects structure automatically meets the assumption about the conditional expectations, the random intercepts model may be preferable when modeling time-varying correlates (Lindsey & Lambert, 1998).

The issue with time-varying correlates may not weigh heavily as the majority of correlates studied in the social sciences appear to be of the time-static type. With static correlates, no additional assumptions are required for using the marginal model, and the features listed in Table 3 and discussed earlier may be considered when selecting a model. Similar to what was previously discussed, the marginal and random intercepts model parameters have different interpretations when static correlates are included. As a consequence, differences in growth curves based on correlates have different meanings in the two models. We illustrate the differences with a basic example.

Generally, when there are static correlate effects growth curves are conditional on the levels of the correlate(s). For the marginal model the conditional growth curves are based on the prevalence of people who share the same level of the static correlate(s). For example, suppose FSES is our single static correlate, coded as 0 = “low,” and 1 = “high,” and additional columns of * X_{i}* consist of FSES and its product with the linear and quadratic age transformations. Then the marginal model with FSES is

$$Pr({y}_{ij}=1)=\frac{\mathit{exp}({\gamma}_{0}+{\gamma}_{1}{t}_{ij}+{\gamma}_{2}{t}_{ij}^{2}+{\gamma}_{3}{\mathrm{FSES}}_{i}+{\gamma}_{4}{t}_{ij}{\mathrm{FSES}}_{i}+{\gamma}_{5}{t}_{ij}^{2}{\mathrm{FSES}}_{i})}{1+\mathit{exp}({\gamma}_{0}+{\gamma}_{1}{t}_{ij}+{\gamma}_{2}{t}_{ij}^{2}+{\gamma}_{3}{\mathrm{FSES}}_{i}+{\gamma}_{4}{t}_{ij}{\mathrm{FSES}}_{i}+{\gamma}_{5}{t}_{ij}^{2}{\mathrm{FSES}}_{i})}.$$

(13)

Solving Equation (13) for FSES = 0 provides the prevalence curve for low FSES with parameters γ_{0}, γ_{1}, γ_{2}, and solving for FSES = 1 provides the prevalence curve for high FSES with parameters γ_{0} + γ_{3}, γ_{1} + γ_{4}, γ_{2} + γ_{5}. The growth curves of the groups are identical when γ_{3} = γ_{4} = γ_{5}, which provides a natural null hypothesis of no difference in prevalence curves based on FSES. Because the prevalence curve of each group is based on aggregating among those who share the same level of FSES, the γ_{k} (k > 2) index differences between average growth curves. This is directly analogous to the types of effects that are indexed in LMMs with static correlates.

In contrast, the random intercepts model with FSES is

$$Pr({y}_{ij}=1{d}_{i})=\frac{\mathit{exp}({d}_{i}+{\delta}_{0}+{\delta}_{1}{t}_{ij}+{\delta}_{2}{t}_{ij}^{2}+{\delta}_{3}{\mathrm{FSES}}_{i}+{\delta}_{4}{t}_{ij}{\mathrm{FSES}}_{i}+{\delta}_{5}{t}_{ij}^{2}{\mathrm{FSES}}_{i})}{1+\mathit{exp}({d}_{i}+{\delta}_{0}+{\delta}_{1}{t}_{ij}+{\delta}_{2}{t}_{ij}^{2}+{\delta}_{3}{\mathrm{FSES}}_{i}+{\delta}_{4}{t}_{ij}{\mathrm{FSES}}_{i}+{\delta}_{5}{t}_{ij}^{2}{\mathrm{FSES}}_{i})}.$$

(14)

Similar to the marginal model, we can solve Equation (14) for each value of FSES to obtain two growth curves. However, because the random intercepts model is a model for an individual, differences between the two growth curves are not similar to those in the marginal model or the LMM. The growth curve for FSES = 0 is a growth curve for an individual with low FSES. Likewise, the growth curve for FSES = 1 is a growth curve for an individual with high FSES. It follows that δ_{3}, δ_{4}, and δ_{5} represent differences between low and high FSES *for an individual*. The question is whether the same individual is referenced for both levels of FSES.

Some researches answer the aforementioned question in the affirmative and consider the δ_{k} (*k* > 2) to index differences that would occur if the same individual had experienced low and high FSES (Diggle et al., 2002, chap. 7; Zeger et al., 1988). This type of effect has no direct counterpart in LMM. No individual actually experiences both levels of FSES (or the correlate would be time-varying), so the differences between levels are considered an abstraction or a type of extrapolation (Zeger, Liang, & Albert, 1991). Such a view seems to discourage the use of the random intercepts model when comparisons among levels of a static covariate is the goal. An applied researcher probably would not want to estimate parameters that index effects that might never occur. For example, if sex is the correlate there is not even a theoretical possibility that an individual can experience both its levels.

Other researchers point out that, similar to the parameters in the marginal model, the δ_{k} (*k* > 2) are estimated based on individuals who actually do have different levels of FSES, which must allow for some type of between-group interpretation (Galbraith, 1991; Lindsey & Lambert, 1998). In fact, the δ_{k} (*k* > 2) can be viewed as indexing differences similar to those of their corresponding γ_{k}, with averaging over individuals within levels of FSES being implicit rather than explicit (Galbraith, 1991). This view is buttressed by the findings previously noted that the inferential statistics (e.g., *Z* values) of marginal and random intercepts models tend to be similar in practice. This similarity would not exist if the parameters of the models were indexing radically different correlate effects (Neuhaus, 1993; Pendergast et al., 1996).

A distinct advantage of the random intercepts model is that, similar to the LMM, the variability in the random intercepts can be interpreted as individual differences due to omitted static correlates (Pendergast et al., 1996). In theory, modeling this variability can help prevent attenuation of the regression coefficients due to random error (Fuller, 1987, chap. 1). In addition, this provides a basis for an effect size measure for the static correlate(s). The variance of the intercepts can be estimated without the static correlate(s) in the model, and then the proportional decrease in the variance after adding the static correlate(s) can be assessed yielding a proportion-reduction-in-error measure.

When static correlates are included, explicit averaging with the random effects model can be performed along the lines already outlined. This is necessary if one wants to obtain average growth curves for the levels of the correlate(s) as the random intercepts parameter estimates cannot be used directly for this purpose (the marginal model parameters estimates can be used directly). After obtaining parameter estimates one can simulate a large number of random effects and then take averages of the individual probability estimates within levels of the correlate(s) at each timepoint. For example, one could average over the individual probabilities for those with FSES = 0 and then separately for those with FSES = 1 to yield average growth curves for low and high FSES. In this way, a type of marginal information can be obtained from the random intercepts model that considers correlate effects.

In summary, two models for longitudinal binary data were presented: the marginal model and the random intercepts model. In contrast to LMM, the models are incommensurate in that the marginal model parameters pertain to group-level growth curves whereas the random intercepts model parameters pertain to individual-level growth curves. It was shown how group-level prediction curves can be obtained from the random intercepts model using numerical averaging based on a fitted model. The real data examples showed that the numerical averaging can produce almost identical predicted curves to those based directly on marginal model parameter estimates. The ability of numerical averaging to approximate the marginal model growth curves appears to be unaffected by missing data, but more research on this topic is warranted.

A number of key features of the models and their associated estimation methods was discussed. Based on the discussion, we recommend where possible to report results from both the marginal model and the random intercepts model. When this is not practical, one or the other model must be selected. If only group-level information is of interest to the analyst we lean toward using the marginal model with MI if there is missing data. If individual-level information is of interest in part or whole, we recommend using the random intercepts model with numerical averaging.

The work on this article was supported by Grant 96-MU-FX-0012 from the Office of Juvenile Justice and Delinquency Prevention, Grant 50778 from the National Institute of Mental Health, and Grant 411018 from the National Institute of Drug Abuse. Points of view or opinions in this document are those of the authors and do not necessarily represent the official position or policies of the U.S. Department of Justice, the National Institute of Mental Health, and the National Institute of Drug Abuse.

^{1}The expectation is analogous to an average if there are no missing data and a weighted average if there are missing data.

^{2}The marginal model is also known as the population-averaged model.

^{3}A number of quadrature points were tried and the parameter estimates appeared to become stable when greater than 10 point, were used. The value 50 was arbitrarily chosen to provide a high degree of accuracy in the results.

^{4}If the working correlation matrix is an identity matrix (not just diagonal), then GEE is identical to the traditional logistic regression model estimated with ML (Fitzmaurice et al., 2004, chap. 10).

^{5}Some ad hoc procedures have been suggested; see, for example, Pan (2001).

^{6}Statistics based on aggregates are, of course, used extensively to summarize information at the group level regardless if they represent empirical impossibilities (e.g., 3.5 children per household).

^{7}The age at peak can be determined by an inspection of the predicted prevalence values, but it is not clear if the instantaneous rate of change can be computed.

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