A Repeated Measures Approach for Simultaneous Modeling of Multiple Neurobehavioral Outcomes in Newborns Exposed to Cocaine in Utero

Abhik Das1 , W. Kenneth Poole1 and Henrietta S. Bada2

1 Statistics Research Division, Research Triangle Institute, Rockville, MD.
2 University of Kentucky Chandler Medical Center, Lexington, KY.

Received for publication May 22, 2003; accepted for publication October 16, 2003.


    ABSTRACT
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 INTRODUCTION
 BACKGROUND: THE MATERNAL...
 METHODS
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Multiple binary outcomes are encountered frequently in epidemiologic research. This work was motivated by the Maternal Lifestyle Study, 1993–1995, where newborns exposed prenatally to cocaine and a comparison cohort were examined for the presence of central and autonomic nervous system (CNS/ANS) signs. Thus, each infant contributed to multiple, possibly interrelated, binary outcomes that may collectively constitute one syndrome (even though specific outcomes that are affected by cocaine are of scientific interest). Because it is neither scientifically appropriate nor statistically efficient to fit separate models for each outcome, here we adopt a multivariate repeated measures approach to simultaneously model all the CNS/ANS outcomes as a function of cocaine exposure and other covariates. This formulation has a number of advantages. First, it implicitly recognizes that all the CNS/ANS outcomes may together constitute one syndrome. Second, simultaneous modeling boosts statistical efficiency by allowing for correlations among the outcomes, and it avoids multiple comparisons. Third, it allows for outcome-specific exposure effects, so that the specific signs that are affected by cocaine exposure can be identified.

binary data; generalized estimating equation; generalized linear mixed model; multiple comparisons; repeated measures; simultaneous inference; syndrome

Abbreviations: Abbreviations: CNS/ANS, central and autonomic nervous system; GEE, generalized estimating equation; GLMM, generalized linear mixed model.


    INTRODUCTION
 TOP
 ABSTRACT
 INTRODUCTION
 BACKGROUND: THE MATERNAL...
 METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
Binary outcome measures that indicate the presence or absence of certain medical conditions are a widely used tool in epidemiologic investigations. Moreover, public health studies frequently measure an array of such indicators for different medical conditions to make an overall assessment about an individual’s health status. For instance, overall health status in newborns is widely assessed through the Apgar score, which is a scale that is derived by eliciting answers to a series of categorical questions regarding five outcomes (color, reflex irritability, muscle tone, respiratory effort, and heart rate). Each of these scales generates a score of 0–2, which may be dichotomized according to a hypothesized threshold for poor outcomes.

The traditional approach for statistical analyses of such binary outcomes is logistic regression, where the probability of observing an "event" (i.e., the prevalence of some medical condition) is modeled as a function of the principal risk factor/treatment of interest and other covariates (1). However, the situation where multiple binary outcomes are simultaneously assessed on the same individual presents some basic methodological problems in that proper statistical modeling here should account for the following features of the data.

A. Each individual contributes to multiple outcomes. Thus, the different outcomes for each individual are likely to be correlated.

B. These multiple outcomes broadly purport to measure the same underlying condition or construct.

C. While the multiple outcomes may be measuring the same underlying phenomenon, outcome-specific effects (i.e., which specific outcomes are associated with the effect of interest) may still be of scientific interest in many situations.

Many researchers have attempted to address issues A and B by constructing scores that add up the various outcome indicators (which can sometimes also be interpreted as the total number of adverse medical conditions) and then regressing the total score on covariates using linear or Poisson regression (2). Others have used latent variable models to infer about the underlying, yet unobserved, process that these multiple outcomes purport to measure (3). However, the principal limitation in both these approaches is that, by focusing on the aggregate, they lose the ability to discern outcome-specific effects (i.e., which specific outcomes are associated with the effect of interest) that may be of substantive interest, thus failing to address issue C. This is critical, since aggregation of multiple outcomes risks combining indicators of distinct processes, which could mask subtle relations between specific outcomes and risk factors. Moreover, latent variable models generally entail strong (often untestable) modeling assumptions that may significantly impact analytical findings in ways that may not be obviously discernible.

Individual analyses of each outcome, with separate logistic regression models being fitted for each indicator, would satisfy issue C but at the expense of issues A and B. Specifically, outcomes assessed on the same individual are likely to exhibit intracluster correlation, since they are subject to shared influences that are particular to that individual (or cluster). If data on these underlying shared influences (detailed individual-level information such as genetic makeup, environmental exposures, and so on) are available, identifying and including these factors as covariates in the regression model may account for this clustering to some extent. However, in practice, it is highly unlikely that all such factors can be identified and measured to the extent that intracluster correlation gets reduced to a negligible level.

Ignoring correlations among the multiple outcomes essentially wastes valuable information by not exploiting this distinctive covariance structure in the data, whereby the overall variability in the outcomes can be broken down into that between individuals and that within individuals. This approach will generally produce inefficient estimates, and the loss in precision resulting from not utilizing information across the multiple outcomes may be substantial (4).

A related problem with individual analyses for each outcome is that of multiple comparisons. Specifically, fitting separate logistic regression models for each individual outcome would leave the analysis vulnerable to a multiple comparisons problem, whereby performing several tests on the same kinds of variables in the same data inflates the type I error to magnitudes that are unacceptably higher than the nominal significance level.

In this paper, we adopt a multivariate modeling approach to analyze multiple binary outcomes in a statistical framework that addresses all three issues (A, B, and C) presented earlier. Note that a solution already exists for paired binary data through the use of bivariate logistic regression models (5). However, that methodology is limited to two binary outcomes assessed for each individual. Our approach builds upon this work and extends to the multiple (i.e., >2) outcomes setting, where we borrow from statistical methods for longitudinal and repeated measures data to simultaneously model all the outcomes assessed on an individual. Thus, we assume that each individual contributes a vector of correlated binary outcomes, all of which may be jointly influenced by his/her treatment/risk factor status and other personal traits.

There are several advantages to such a repeated measures approach. First, it explicitly recognizes that all the outcomes for an individual may be correlated. Second, the fact that a single model captures all the outcomes underlines that they may collectively constitute one overarching condition or construct. Third, by simultaneously modeling all the outcomes, this approach avoids the multiple comparisons problem. Fourth, in spite of the simultaneous modeling formulation, by appropriately nesting the effect of interest (treatment or principal risk factor) within the multiple outcomes, we can test for the outcome-specific effects in which we are interested. At the same time, by constructing proper linear contrasts, we can test whether such separate effects for each outcome are, in fact, supported by the data. If not, we may choose to examine a simpler global effect. Finally, this technique is statistically more efficient (in terms of conserving degrees of freedom and enhancing the power to observe significant effects) than fitting separate models for each outcome.

This paper is organized as follows. The next section provides a brief background on the public health study that motivated this research. Details on the statistical methods used to put our multivariate modeling approach into operation are presented next, followed by the results of applying these methods to actual data from our motivational study. Finally, we conclude with some general observations on the substantive and methodological implications of this exercise.


    BACKGROUND: THE MATERNAL LIFESTYLE STUDY
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 INTRODUCTION
 BACKGROUND: THE MATERNAL...
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This paper is motivated by data from the Maternal Lifestyle Study, which is a multisite prospective cohort study that aims to evaluate the relation between maternal cocaine and/or opiate use during pregnancy and the presence of acute neonatal complications and long-term adverse neurodevelopmental outcomes in infants (6). The Maternal Lifestyle Study is the largest prospective study of its kind to date, with 11,811 mother-infant dyads enrolled at baseline. Maternal cocaine and/or opiate use during the index pregnancy was determined for all the enrolled subjects using a combination of self-report and subsequent laboratory verification (infant meconium toxicology). The 8,351 subjects, for whom cocaine/opiate exposure status could be conclusively confirmed, are the focus of this paper.

Newborns in the Maternal Lifestyle Study were examined within the first few days after birth for the presence of a constellation of abnormal neurobehavioral manifestations, known collectively as central and autonomic nervous system (CNS/ANS) signs, to determine whether the prevalence of such signs was disproportionately higher in children exposed prenatally to cocaine (7). The recording of these signs was considered clinically meaningful because they comprise the manifestations reported with neonatal "abstinence syndrome" or neonatal narcotic withdrawal syndrome (8, 9), and previous studies have reported that cocaine-exposed infants exhibited more stress behavior and central nervous system signs, as well as increased tone, motor activity, jerky movements, and tremors, than did nonexposed controls (10, 11).

The significant feature of CNS/ANS signs that is important for this study is that, essentially, they are a collection of several presence/absence type (i.e., binary) outcomes that are assessed on the same individual simultaneously. Moreover, even though there is speculation in the medical literature that these signs collectively constitute one syndrome (12), researchers are at the same time interested in knowing specifically which CNS/ANS signs are associated with prenatal cocaine exposure, and whether cocaine and opiates differentially affect different sets of signs. This necessitates the examination of outcome-specific effects in our study while bearing in mind that the various signs from each infant are highly interrelated.

Table 1 presents an association matrix for the different CNS/ANS signs, where the pairwise associations between these binary outcomes are measured in terms of odds ratios that characterize their interrelatedness. The high degree of association between these signs demonstrated here (for instance, an infant with jitteriness/tremors was 16.6 times more likely to manifest with hypertonia, 11.3 times more likely to be difficult to console, 9.3 times more likely to be irritable, and so on) seems to support the contention that these signs may collectively constitute one syndrome. The table also shows that data analyses for this study may be further complicated by the low prevalence rates for most of these signs. (Note that, of the 12 CNS/ANS signs originally measured in the study, three (difficult to arouse, hyperalert, and hyperactive) had to be dropped because of extremely low prevalence, which made meaningful statistical analyses impossible.)


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TABLE 1. Prevalence and interrelatedness of central and autonomic nervous system signs (as expressed by odds ratios) in newborns, the multisite{dagger} Maternal Lifestyle Study, 1993–1995
 
In summary, table 1 demonstrates that the odds ratios, which capture the statistical dependence between signs from the same child, are in general both substantial in magnitude and highly significant. Ignoring this clearly apparent dispersion structure in the data would lead to inefficient estimates and inaccurate inferences. These factors underscore the necessity of adopting the multivariate modeling/repeated measures approach outlined in the previous section for the analysis of these data.


    METHODS
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 INTRODUCTION
 BACKGROUND: THE MATERNAL...
 METHODS
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There are several different approaches available for implementing a multivariate repeated measures approach that models clustered binary data, that is, methods that simultaneously model all the binary outcomes elicited from an individual (13). Most of these approaches can be grouped into two classes: population-averaged marginal modeling using generalized estimating equations (GEEs) (5, 14, 15) and cluster-specific hierarchical modeling using generalized linear mixed models (GLMMs) (16, 17). In the following subsections, we discuss the application of these two methods in the context of our study and then seek to justify the approach that we finally adopted.

Population-averaged approach
Here we would model average CNS/ANS prevalence over cocaine-exposed and -unexposed children who share common explanatory features, while accounting for correlation among multiple CNS/ANS outcomes from the same child. Note that, if there is heterogeneity in these prevalence rates across the different signs, the overall cocaine effect would be the average of individual cocaine effects for each sign.

The population-averaged marginal GEE model may be expressed as follows. If Yij indicates the presence of the jth CNS/ANS sign in the ith child, (i = 1, ..., n; j = 1, ..., m) and pij = Pr(Yij = 1), and OR represents "odds ratio," then

logit(pij) = {alpha}j + ß1j coci + ß2j opii + xiß,

and

log[OR(Yij, Yik)] = {gamma}, j != k. (1)

Here, the {alpha}s are sign-specific log odds, the ßs are log odds ratios, and "coc" and "opi" are indicator variables for prenatal cocaine and opiate exposures, respectively, while x denotes other subject-level covariates and {gamma} is the log odds ratio characterizing within-subject dependence between signs from the same child. Note that, in this formulation, we can easily construct a test for the null hypothesis H0: ß1j = ß1 (H0: ß2j = ß2), j = 1, ..., m, that examines whether it is indeed reasonable to assume separate cocaine (opiate) effects for each sign. Failure to reject this test would indicate that the cocaine (opiate) effect is the same for all the CNS/ANS signs, and a simpler model reflecting the same may better describe the data at hand.

Interpretation of regression coefficients from this model is straightforward. Thus, ß1j (ß2j) is the average log odds ratio for the jth CNS/ANS sign in children exposed to cocaine (opiates) relative to children who were not exposed to cocaine (opiates). Further, since preliminary exploratory analyses indicated a wide range of prevalence rates for the different CNS/ANS signs, we incorporated sign-specific intercepts in the regression model displayed in equation 1 to account for this feature in the data.

There are several advantages to adopting the above GEE modeling approach for our data. First, the population-averaged marginal modeling formulation in equation 1 provides population average estimates of overall trends that are important from a public health perspective. Second, compared with cluster-specific approaches that generally involve multidimensional integration (numerical or otherwise) of random effects, the GEE estimation algorithm is computationally efficient for relatively large data sets such as ours (where n = 8,351 and m = 9). It is also widely implemented in standard-use statistical packages for correlated data analysis, such as SAS (SAS Institute, Inc., Cary, North Carolina), SPSS (SPSS, Inc., Chicago, Illinois), STATA (Stata Corporation, College Station, Texas), and SUDAAN (Research Triangle Institute, Research Triangle Park, North Carolina) software (18, 19). Third, note that the model depicted in equation 1 uses odds ratios rather than correlation coefficients to model dependence between signs from the same child. This is preferable, since odds ratios provide a more natural framework for modeling the within-subject dependencies that are induced by the binary responses we have in this study (20).

Cluster-specific approach
Here, we adjust for multiple outcomes by allowing for child-specific regression coefficients, that is, coefficients that vary from one child to another. This technique acknowledges the distinctive characteristics of each child that could make his/her responses differ. The cluster-specific approach thus uses a flexible modeling formulation that allows each child to have his/her own unique CNS/ANS profile, which may depend on the child’s cocaine exposure status (and other covariates).

A cluster-specific GLMM consists of a two-step hierarchical regression framework, where the first step models the prevalence of CNS/ANS signs for a child as a function of his/her cocaine and opiate exposure, as well as an overall child-specific effect {theta}i. In the next step, this unobserved child-specific effect is related to observed child-level covariates through a linear regression equation. Given the notation developed earlier, this model can thus be written as follows:

Step I: (Yij½{theta}i) ~ Bernoulli(pij),

where

logit(pij) = {theta}i + {alpha}j + ß1j coci + ß2j opii.

Step II: {theta}i = xiß + ei,

where

ei ~ N(0,{sigma}2). (2)

Here, the {theta}i denote cluster (i.e., child)-specific effects that are unique for the ith child. The variance for the residual error terms ei (i.e., {sigma}2) reflects natural heterogeneity among children in terms of unmeasured characteristics such as individual temperament, genetic makeup, environmental exposure, and so on. Note that, similar to the GEE model, this formulation also allows us to easily test whether separate exposure effects are merited for each CNS/ANS sign or whether a global effect may suffice.

Compared with the population-averaged model, there are some subtle differences in interpretation for the regression coefficients in equation 2. Here, ß1j (ß2j) may be interpreted as the log odds ratio for the jth CNS/ANS sign when a child is exposed to cocaine (opiates), relative to when that same child is not exposed to cocaine (opiates).

The principal motivation in adopting this cluster-specific approach for our study is that it is more appropriate than population-averaged GEE methods for estimating the effects of within-cluster covariates, that is, covariates that change within the subject (12). This is especially relevant for our study, since the sign-specific exposure effects captured by ß1 and ß2 (in equation 2) that are our primary focus are covariates of this type. Moreover, software that enables the fitting of such models is also gradually becoming available for general use (21, 22).

There are other advantages of the cluster-specific approach as well. First, it allows for CNS/ANS signs from the same child to be interrelated by allowing each child to have a different mean prevalence for such signs. Second, the incorporation of an additional subject-specific level of variability ({sigma}2) ensures that we allow for regression heterogeneity by accommodating varying regression models across children. Third, the multilevel model presented in equation 2 guards against aggregation bias by breaking down any structure in the data into within-child and between-children components. Finally, the GLMM formulation provides estimates that are clinically more meaningful, since they purport to make inferences about the individual rather than a population average as in the GEE (23).

Our approach
The literature on modeling of longitudinal and repeated measures does not present any clear consensus on the relative statistical merits of either of the approaches discussed above. Given the subtle differences in interpretation between estimates derived from the two, either approach may be valid depending on the question one is interested in answering. Although the cluster-specific approach is considered more appropriate for assessing the effects of within-cluster covariates, at the same time it makes untestable assumptions about the effects of cluster-level covariates (12). On the other hand, although a marginal modeling approach may not be suitable for estimating the within-cluster effects that are of interest in this study, it does provide estimates that are more robust to model misspecification (especially in terms of the correlation structure in the data). However, this comes at a price—unlike the GLMM, the GEE model is not strictly parametric and hence does not produce a likelihood function—which makes it impossible to use traditional likelihood-based procedures for model selection or to determine goodness-of-model fit.

The competing considerations discussed above present a substantial dilemma for us, since within-cluster sign-specific exposure effects that are best estimated by cluster-specific approaches are the focus of our study, while the population-averaged approach provides robust estimates that are more intuitively appealing from a public health perspective. In light of these concerns and the consequent ambiguities that are involved, we decided to apply both approaches to our data to enable the answering of different kinds of research questions and, simultaneously, to investigate whether these two approaches produce different conclusions in this respect.


    RESULTS
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 INTRODUCTION
 BACKGROUND: THE MATERNAL...
 METHODS
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 DISCUSSION
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Figures 14 present the results from fitting the models presented in equations 1 and 2 to our data. Aside from sign-specific intercepts and cocaine and opiate effects, potential confounders (such as study site, maternal race, socioeconomic status (Medicare insurance used as a proxy for the same), infant gestational age, and prenatal exposure to tobacco, alcohol, and marijuana) were also included as covariates in both these models.



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FIGURE 1. Results from simultaneous modeling of multiple central and autonomic nervous system signs in newborns, the multisite (Detroit, Michigan, Memphis, Tennessee, Miami, Florida, and Providence, Rhode Island) Maternal Lifestyle Study, 1993–1995. Parameter estimates for sign-specific intercepts (presented in an exponentiated form as the odds for each sign) were derived using both the population-averaged generalized estimating equation (GEE) and the cluster-specific generalized linear mixed model (GLMM) approaches. A horizontal bar placed at odds = 0 indexes negligible prevalence. High cry, high-pitched cry; diff console, difficult to console; nasal stuff, nasal stuffiness.

 


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FIGURE 4. Results from simultaneous modeling of multiple central and autonomic nervous system signs in newborns, the multisite (Detroit, Michigan, Memphis, Tennessee, Miami, Florida, and Providence, Rhode Island) Maternal Lifestyle Study, 1993–1995. Parameter estimates for sign-specific cocaine effects (presented as odds ratios) were derived using both the population-averaged generalized estimating equation (GEE) and the cluster-specific generalized linear mixed model (GLMM) approaches. A horizontal bar placed at odds ratio = 1 indexes no effect. High cry, high-pitched cry; diff console, difficult to console; nasal stuff, nasal stuffiness.

 
The GEE model (equation 1) was fitted in SAS version 8.2 (SAS Institute, Inc.) software using the alternating logistic regression algorithm (24) that uses odds ratios rather than correlation coefficients to model dependence between signs from the same child. This model estimated the odds ratio for within-subject dependence (i.e., e{gamma}) to be 4.2 (95 percent confidence interval: 3.8, 4.7), indicating substantial within-child (or intracluster) dependence in the data. Further, it strongly rejected the possibility of a common global exposure effect for either cocaine (p < 0.0001) or opiates (p = 0.009), thus supporting our formulation of separate sign-specific effects for each.

The GLMM presented in equation 2 was fitted using the "xtlogit" command in STATA version 7.0 (Stata Corporation) software, which estimates the parameters in a GLMM using Gaussian quadrature. This approach yielded an estimated subject-level variability ({sigma}2) of 2.25 (95 percent confidence interval: 2.0, 2.6), which again attests to the presence of substantial within-child dependence in the data. As in the GEE model, the possibility of a common global exposure effect was strongly rejected here for both cocaine (p = 0.0001) and opiates (p < 0.0001), thus supporting our formulation of separate sign-specific effects for each.

The results presented in figures 14 show that the GEE and GLMM approaches produced similar results in most cases. However, there are some subtle differences that are worth mentioning. The cocaine effects for two central nervous system signs (difficult to console and high-pitched cry) are statistically significant (i.e., the 95 percent confidence intervals for the odds ratios exclude the value of 1.0) for the GLMM but not for the GEE model (figure 4). Moreover, compared with the GEE model, the within-cluster (i.e., sign-specific) cocaine/opiate effects estimated by the GLMM appear to have consistently higher magnitudes and wider confidence limits (figures 3 and 4). Interestingly, this pattern seems to be reversed for the other factors included in the model (intercepts and cluster/individual-level sociodemographic covariates), where the GEE estimates have consistently higher magnitudes and wider confidence limits for statistically significant factors (figures 1 and 2).



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FIGURE 3. Results from simultaneous modeling of multiple central and autonomic nervous system signs in newborns, the multisite (Detroit, Michigan, Memphis, Tennessee, Miami, Florida, and Providence, Rhode Island) Maternal Lifestyle Study, 1993–1995. Parameter estimates for sign-specific opiate effects (presented as odds ratios) were derived using both the population-averaged generalized estimating equation (GEE) and the cluster-specific generalized linear mixed model (GLMM) approaches. A horizontal bar placed at odds ratio = 1 indexes no effect. High cry, high-pitched cry; diff console, difficult to console; nasal stuff, nasal stuffiness.

 


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FIGURE 2. Results from simultaneous modeling of multiple central and autonomic nervous system signs in newborns, the multisite (Detroit, Michigan, Memphis, Tennessee, Miami, Florida, and Providence, Rhode Island) Maternal Lifestyle Study, 1993–1995. Parameter estimates for sociodemographic covariates (presented as odds ratios) were derived using both the population-averaged generalized estimating equation (GEE) and the cluster-specific generalized linear mixed model (GLMM) approaches. A horizontal bar placed at odds ratio = 1 indexes no effect. Gest age, gestational age; tob, tobacco; alc, alcohol; med, medium.

 
These results are largely along expected lines. They bear out the assertions made in the literature that, while population-averaged GEE approaches are most appropriate for assessing the effects of cluster-level covariates (for these effects the GLMM yields attenuated estimates and confidence intervals that are too narrow), within-cluster covariates are best estimated using cluster-specific GLMM (25). In the latter situation, it is the GEE approach that yields attenuated estimates and confidence intervals that are too narrow. This is expected because, by definition, the population-averaged approach averages individual cocaine/opiate effects across subjects, and this generally attenuates the estimated effect size (25). Score tests for the GEE model and likelihood ratio tests for the GLMM indicated substantial cocaine and opiate effects overall (p < 0.0001 in all cases), although not all signs were significantly, or similarly, affected (figures 3 and 4). Other significant covariates included study site and prenatal exposure to tobacco (figure 2).


    DISCUSSION
 TOP
 ABSTRACT
 INTRODUCTION
 BACKGROUND: THE MATERNAL...
 METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
In the largest multisite study of its kind to date, we found that prenatal exposures to both cocaine and opiates were strongly associated with elevated risk of CNS/ANS manifestations. These effects were present even after controlling for interdependence among multiple signs observed on the same child, as well as potential confounders such as race, gestational age, socioeconomic status, and prenatal exposures to tobacco, alcohol, and marijuana. Moreover, even though different signs were found to be differentially associated with cocaine and opiate exposure in terms of both magnitude and statistical significance, opiate effects, in general, were of a larger magnitude than those for cocaine. Further substantive details on these findings and their clinical and public health implications have been discussed at length by Bada et al. (7).

In terms of statistical methodology, we have presented here an approach that adapts repeated measures techniques used primarily for analyzing longitudinal binary data to model multiple clinical outcomes (of the presence/absence variety) manifest by the same subject at the same time simultaneously. This formulation recognizes implicitly that all the outcomes may be interrelated and could together constitute one syndrome. Simultaneous modeling of all the outcomes also boosts statistical efficiency and avoids the multiple comparisons problem, while allowing for the testing of outcome-specific effects, if necessary. There are subtle differences in conceptualization and interpretation between the population-averaged GEE and the cluster-specific GLMM methods that may be used for implementing this approach, and either method (or both) may be appropriate, depending on the research question(s) being posed. Though small in most cases, these differences are borne out by the results presented for this study. In sum, outcome-specific effects may be reliably estimated from a GLMM, while subject-level effects may be best estimated using a GEE model.

The methods described in this paper are applicable in more general settings involving multiple outcomes measured simultaneously on the same subject. In fact, they are suitable for any set of outcomes that can be modeled using the exponential family of distributions. For example, suppose we had continuous, categorical, or count responses instead of the binary outcomes that we have dealt with here. Such a situation can be modeled in both the population-averaged and cluster-specific approaches by suitably altering the logit link function for the expected value of the response that is specified in equations 1 and 2. For continuous outcomes this would be the identity link, for count data it would be the log link, and so on. Even for binary outcomes, other links such as the probit link may be explored. We do, however, caution that the STATA "xtlogit" command we have used to estimate our GLMM parameters cannot accommodate other link functions (such as log or probit) or more complicated correlation structures (e.g., a three-level hierarchical model with multiple binary outcomes nested in individuals nested in clinical sites). Other software options would need to be explored in such situations. In sum, since investigators in public health research frequently measure multiple clinical outcomes on the same subject at the same time, we believe that the principles underlying the methods described in this paper are of significant importance to public health research in general.


    ACKNOWLEDGMENTS
 
The Maternal Lifestyle Study is supported by the National Institute of Child Health and Human Development through cooperative agreements U10 HD 27856, U10 HD 21397, U10 HD 21385, U10 HD 27904, U01 HD 36790, and U01 HD 19897, as well as by interagency agreements with the National Institute on Drug Abuse, Administration on Children, Youth, and Families, and Center for Substance Abuse Treatment.

The following institutions and investigators participated in the Maternal Lifestyle Study: University of Miami: Dr. Charles Bauer (principal investigator); Wayne State University: Dr. Seetha Shankaran (principal investigator); University of Tennessee: Dr. Henrietta Bada (principal investigator); Brown University: Dr. Barry Lester (principal investigator); Research Triangle Institute: Dr. W. K. Poole (principal investigator); National Institute of Child Health and Human Development: Dr. Linda Wright (program officer); National Institute on Drug Abuse: Dr. Vincent Smeriglio (program officer); Administration on Children, Youth, and Families: Dr. Penny Maza (program officer); and Center for Substance Abuse Treatment: Dr. Loretta Finnegan (program officer).


    NOTES
 
Correspondence to Dr. Abhik Das, Research Triangle Institute, 6110 Executive Boulevard, Suite 420, Rockville, MD 20852-3903 (e-mail: adas{at}rti.org). Back


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