SIMULTANEOUS MODELING OF MULTIPLE BINARY OUTCOMES: A REPEATED MEASURES APPROACH

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1 SIMULTANEOUS MODELING OF MULTIPLE BINARY OUTCOMES: A REPEATED MEASURES APPROACH Abhik Das, W. Kenneth Poole, Research Triangle Institute, 6110 Executive Blvd., Suite 420, Rockville, MD and Henrietta S. Bada, University of Kentucky Chandler Medical Center, 800 Rose Street, Room MS473, Lexington, KY KEY WORDS: Syndrome; Multiple comparisons; Multivariate modeling; Correlated outcomes; Outcome-specific effects; GEE; GLMM Abstract Multiple binary outcomes are frequently encountered in epidemiologic research. This work was motivated by the Maternal Lifestyle Study, where newborns prenatally exposed to cocaine and a comparison cohort, were examined for the presence of autonomic and central nervous system (ANS/CNS) 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). Since 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 ANS/CNS outcomes as a function of cocaine exposure and other covariates. This formulation has a number of advantages. First, it implicitly recognizes that all the ANS/CNS outcomes may together constitute one syndrome. Second, simultaneous modeling boosts statistical efficiency by allowing for correlations among the outcomes, and avoids multiple comparisons. Third, it allows for outcomespecific exposure effects, so that we can identify the specific signs that are affected by cocaine exposure. Introduction 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 test, which is often composed of a series of five binary questions about a child s physical state at birth. 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 (McCullagh and Nelder, 1989). However, the situation where multiple binary outcomes are simultaneously assessed on the same individual presents some basic statistical problems, since proper statistical modeling in this case needs to reflect that: 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. Outcome-specific effects (i.e., which specific outcomes are associated with the effect of interest) may be of interest. Many researchers have attempted to address A and B by adding outcome indicators into scores (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 (Stewart and Ware, 1992). Others have used latent variable models to infer about the underlying, yet unobserved, process that these multiple outcomes purport to measure (Bandeen- Roche et al, 1997). However, the principal limitation in both these approaches is that, by focusing on the aggregate, they loose the ability to discern outcomespecific effects (i.e., which specific outcomes are associated with the effect of interest) which may be of interest, thus failing to address C. This is critical, since aggregation of multiple outcomes risks combining indicators of distinct processes, which could mask subtle relationships between specific outcomes and risk factors. In addition, latent variable models generally entail strong modeling assumptions which may critically impact analytic findings. 677

2 Individual analyses of each outcome, with separate logistic regression models being fitted for each indicator, would satisfy C, but at the expense of A and B. Specifically, outcomes assessed on the same individual are likely to exhibit intra-cluster correlation, since they are subject to shared influences that are peculiar to that individual (cluster). If detailed individual-level information (such as genetic make-up, environmental exposures, etc.) is available, identifying and including these factors as covariates in the regression model may account for intra-cluster correlation to some extent. However, in practice it is highly unlikely that all such factors can be identified and quantified to the extent that intra-cluster correlation becomes analytically negligible. Ignoring correlations among the multiple outcomes essentially wastes information by not exploiting this dispersion structure in the data, whereby the overall variability in the outcomes can be decomposed into that between individuals and that within individuals. This produces inefficient estimates and the loss in precision from not utilizing information across the multiple outcomes may be substantial (Liang and Zeger, 1993). 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 describe multiple binary outcomes in a statistical framework that addresses all three issues A-C raised earlier. Our approach borrows from statistical methods for longitudinal and repeated measures data analysis to simultaneously model all the outcomes assessed on an individual as a function of covariates within a regression framework. 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 implicitly 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 collectively constitute one underlying condition or construct. Third, by simultaneously modeling all the outcomes, this approach avoids the multiple comparisons problem. 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 briefly describes the public health study that motivated this research. Details on the statistical methods employed to operationalize our multivariate modeling approach are presented next, followed by the results of applying these methods to our motivational study. Finally, we conclude with some general observations on the substantive and methodological implications of this exercise. Background: The Maternal Lifestyle Study This paper is motivated by data from the Maternal Lifestyle Study (MLS), which is a multi-site prospective cohort study whose objective is to evaluate the relationship between maternal cocaine and/or opiate use during pregnancy and the presence of acute neonatal complications and long-term adverse neuro-developmental outcomes in infants (Bauer et al, 2002). Maternal cocaine and/or opiate use during the index pregnancy was determined by self-report and/or a laboratory examination (infant meconium toxicology). MLS is the largest prospective study of its kind till date, with 11,811 mother-infant dyads enrolled at baseline. The 8,351 subjects among these, for whom cocaine/opiate exposure status could be confirmed, are the focus of this paper. Newborns in MLS 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 prenatally exposed to cocaine (Bada et al, In Press). Recording of these signs was considered clinically meaningful because they comprise the manifestations reported with neonatal "abstinence syndrome" or neonatal narcotic withdrawal syndrome (Rosen and Pippenger, 1976; Zelson et al, 1973), and previous studies have reported that cocaine-exposed infants exhibited more stress behavior, increased tone, motor activity, jerky movements, tremors, and more CNS signs than nonexposed controls (Napiorkowski et al, 1996; Oro and Dixon, 1987). The significant feature of CNS/ANS signs that is important for this study is that, essentially, it is a collection of several presence/absence type (i.e., binary) outcomes that are assessed on the same individual simultaneously. Moreover, though there is speculation in the medical literature that these signs collectively constitute one syndrome (Roberts, 1984), researchers are also interested in knowing which CNS/ANS signs are associated with prenatal cocaine 678

3 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. In fact, Table 1 below shows that the correlation problem is particularly acute in our data. Odds ratios, which capture the statistical dependence between signs from the same child, are in general both substantial in magnitude and highly significant. Ignoring such clear structure in the data would lead to inefficient estimates and inaccurate inferences. These factors underscore the need to adopt the multivariate modeling / repeated measures approach outlined in the previous section to analyze this data. Methods There are several different approaches available for implementing a multivariate repeated measures approach that models clustered binary data, i.e., methods that simultaneously model all the binary outcomes elicited from an individual (Neuhaus, 1992). Most of these approaches can be grouped into two classes: population-averaged marginal modeling using GEE (Fitzmaurice et al, 1995; Legler et al, 1995; Liang and Zeger, 1986), and cluster-specific hierarchical modeling using GLMM (Breslow and Clayton, 1993; Stiratelli et al, 1984). In the following subsections we discuss the application of these two methods in the context of our study and seek to justify the approach that we finally adopted. Population-averaged approach Here we model average CNS/ANS prevalence over cocaine exposed and unexposed children who share common explanatory features, while accounting for correlation among multiple ANS/CNS outcomes from the same child. Note that if there is heterogeneity in these prevalence rates across signs, the overall cocaine effect is the average of individual cocaine effects for each sign. The marginal GEE model may be expressed as follows. If Y ij indicates presence of the j th CNS/ANS sign in the i th child, (i = 1,,n; j = 1,,m) and p ij = Pr(Y ij = 1), then logit(p ij ) = α j + β1 j coc i + β2 j opi i + x i β, log[or(y ij,y ik )] = γ, j k, (*) where the α s are sign-specific log odds, β s are log odds ratios, coc and opi are indicators for prenatal cocaine and opiate exposure respectively, γ is the log odds ratio that captures the within-subject dependence between signs from the same child, and x denotes other subject-level covariates. Interpretation of regression coefficients from this model is straightforward. Thus, β1 j (β2 j ) is the log odds ratio for the j th CNS/ANS sign in cocaine (opiates) exposed children relative to children who were not exposed to cocaine (opiates). Further, note that the regression equation for model (*) allows for sign-specific intercepts in order to accommodate the wide range of prevalence rates for the different CNS/ANS signs that was indicated by preliminary exploratory analyses. There are several advantages to adopting the above GEE modeling approach for our data. First, the population-averaged marginal modeling formulation in (*) provides population average estimates of overall trends that are important from a public health perspective. Second, the GEE estimation algorithm is computationally efficient for relatively large data sets Table 1. Interrelatedness of CNS/ANS signs as expressed by Odds Ratios (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) Jitteriness/tremors (1) 9.3* 11.1* 16.6* 12.8* 12.6* 11.3* * 4.8* 5.0* 4.0* Irritability (2) 27.5* 15.9* 29.6* 27.6* 142.9* * 2.1* 7.4* 2.6* High-pitched cry (3) 14.8* 4.7* 30.4* 31.9* * 3.6* 3.1* 4.9* Hypertonia (4) 44.9* 38.4* 27.2* 6.7* 13.2* 7.0* * Hyperalert (5) 76.6* 36.0* * 11.6* 4.3* 2.5 Hyperactive (6) 58.0* * 18.8* * Difficult to console (7) * 5.1* 6.5* 2.6* Difficult to arouse (8) * Excessive suck (9) * 4.0* Poor suck (10) 2.5* 2.9* Nasal stuffiness (11) 4.8* (12) Sneezing *p value < OR cannot be calculated with 0 in one cell. 679

4 such as ours (where n = 8,351 and m = 9), specially compared to cluster-specific approaches. It is also widely implemented in standard-use statistical packages (such as SAS, SUDAAN, etc.). Third, model (*) uses odds ratios rather than correlation coefficients to model the dependence between signs from the same child. For the binary responses that we have here, the odds ratio is the more natural framework for modeling such within-subject dependencies (Lipsitz et al, 1991). Cluster-specific approach Here we adjust for repeated measures by allowing for regression coefficients that vary from one child to another, acknowledging the distinctive characteristics of each child that could make their responses vary. This flexible modeling formulation allows each child to have its own unique CNS/ANS profile, which may depend on its cocaine exposure status. A cluster-specific generalized linear mixed model (GLMM) consists of a two-step hierarchical regression framework, where the first step models the prevalence of CNS/ANS signs as a function of cocaine and opiate exposure, as well as an overall child-specific effect θ 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 Step I: (Y ij θ i ) ~ Bernoulli(p ij ), where logit(p ij ) = θ i + α j + β1 j coc i + β2 j opi i. Step II: θ i = x i β + e i, e i ~ N(0,σ 2 ). (**) Note that here, the θ i denote cluster (i.e., child) specific effects that are unique for the i th child. The variance for the residual error terms e i (i.e., σ 2 ) reflects natural heterogeneity among children in terms of unmeasured characteristics such as genetic make-up, environmental exposure, etc. Compared to the population-averaged model, there are some subtle differences in the interpretation of regression coefficients from model (**). Here β1 j (β2 j ) may be interpreted as the log odds ratio for the j th 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 clusterspecific approach for our study is that it is more appropriate than population-averaged GEE methods for estimating the effects of within-cluster covariates (i.e., covariates that change within subject), such as the sign-specific exposure effects captured by β1 and β2 in model (**) that are the primary focus of this study (Neuhaus, 1992). 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 (σ 2 ) ensures that we allow for regression heterogeneity by accommodating varying regression models across children. Third, the multi-level model in (**) guards against aggregation bias by decomposing any structure in the data into within-child and betweenchildren 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 GEE (Diggle et al, 1994). Our Approach The literature 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. Though the cluster-specific approach is considered more appropriate for assessing the effects of within-cluster covariates, it also makes untestable assumptions about the effects of cluster-level covariates (Neuhaus, 1992). On the other hand, though a marginal modeling approach may not properly estimate the within-cluster effects that are of interest in this study, it does provide estimates that are robust to model misspecification (especially in terms of the correlation structure in the data). These competing factors present a real 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 appealing from a public health perspective. Given these concerns, we decided to apply both approaches to our data to enable the answering of different kinds of research questions, and to investigate whether the two approaches produce different conclusions in this respect. Results Table 2 below presents the results from the fitting of models (*) and (**) to our data. Of the twelve CNS/ANS signs listed in Table 1, three had to be dropped because of very low prevalence which made model fitting impossible. Potential confounders such as study site, race, gestational age, socio-economic status (with use of Medicare insurance as a proxy for the same), and prenatal exposure to tobacco, alcohol 680

5 and marijuana were included as covariates in both these models. The GEE model (*) was fitted in SAS version 8.2 (SAS Institute, 2001) using the alternating logistic regression (ALR) algorithm (Carey, Zeger and Diggle, 1993). This model estimated the odds ratio for within-subject dependence (i.e., e γ ) to be 4.2, with a 95% confidence interval (CI) given by ( ), indicating substantial within-child (or intracluster) dependence in the data. The large sample size for this study (8351 subjects with 9 repeated measures each, for a total of 75,159), made fitting of the GLMM (**) impossible with existing software such as the GLIMMIX macro in SAS (Wolfinger and O Connell, 1993), or HLM version 5 (SSI, 2001). To circumvent this problem, we adopted a delete-d jackknife estimation method (with d = 66,159) with 200 replications to reduce the Table 2. Results from simultaneous modeling of multiple CNS/ANS signs using GEE and GLMM Effect Level GEE Model GLMM Odds Ratio 95% CI Odds Ratio 95% CI Clinic 1 # Site A 1.53 * * Site B 0.1 * * Site C 0.38 * * Gestational Age Race 2 Black Other Medicaid Smoking 3 # < ½ packs/day 1.27 * * ½ packs/day 1.36 * * Drinking 4 < 1/month /month > 1/week Marijuana Opiate effects on Hypertonia 5.28 * * ANS/CNS Signs # Jitteriness 3.07 * * High pitched cry 4.52 * * Diff to console 3.59 * * Irritability 3.45 * * Sneezing Excessive suck 5.54 * * Nasal stuffiness * Cocaine effects on ANS/CNS Signs # Poor suck Hypertonia 1.87 * Jitteriness 2.38 * * High pitched cry Diff to console Irritability 1.96 * * Sneezing Excessive suck 2.92 * * Nasal stuffiness Poor suck Compared to Site D, which is the reference group 2 Compared to Whites, who are the reference group 3 Compared to non-smokers, who are the reference group 4 Compared to non-drinkers, who are the reference group * Odds Ratio significantly different from 1 (p<0.05 from Wald test) # Significant overall effect for both GEE and GLMM (p<0.05 for an overall Type 3 test) Note: ANS signs are shaded, CNS signs are not. 681

6 size of the problem to a computationally feasible level (Efron and Tibshirani, 1993). Thus, we drew 200 random samples from our study population, each with 100 subjects having 9 repeated measures each (i.e., a size of 9,000). Individual GLMMs were then fit to each of these samples using GLIMMIX, which uses penalized quasi-likelihoods (PQL) to estimate the parameters in a GLMM (Breslow and Clayton, 1993). The methods developed by Shao (1989) were then used to combine the estimates from these individual models and compute appropriate standard errors. This model estimated subject-level variability (σ 2 ) to be 1.2 (95% CI ), which again attests to the presence of substantial within-child dependence in the data. The results presented in Table 2 above show that the two models produced largely similar results. However, compared to the GEE model, the withincluster cocaine/opiate effects estimated by the GLMM in general have a higher magnitude and wider confidence limits, which is expected (Neuhaus et al, 1991). Type 3 tests (with only approximate values available for GLMM) indicated substantial cocaine and opiate effects overall, though, not all signs were significantly, or similarly, affected. Other significant covariates included study site and prenatal exposure to tobacco. Discussion In the largest multi-site study of its kind till date, we found that prenatal exposures to both cocaine and opiates were strongly associated with elevated risk of CNS/ANS manifestations even after controlling for potential confounders such as race, gestational age, socio-economic status, as well as prenatal exposures to tobacco, alcohol and marijuana. Moreover, even though different signs were found to be differentially associated with cocaine and opiate exposure, both in terms of magnitude and statistical significance, opiate effects, in general, were of a larger magnitude than those for cocaine. Additional substantive details on these findings and their clinical and public health implications are presented in Bada et al (In Press). In terms of statistical methodology, we have presented here an approach that adapts repeated measures techniques for analyzing longitudinal binary data to simultaneously model multiple clinical outcomes (of the presence/absence variety) manifest by the same subject at the same time. This formulation implicitly recognizes 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. Though there are subtle differences in conceptualization and interpretation between the population averaged GEE and the cluster-specific GLMM methods for implementing this approach, in this study they produced largely similar results. In addition, we presented a jackknife approximation method that we developed to estimate parameters in a GLMM when the data may be too large and sparse for existing GLMM software. Because there is a critical lack of reliable industrial strength GLMM fitting algorithms that can process large datasets and fit relatively complex models in the current crop of widely available statistical software, we believe that it is useful to have such approximation techniques available. In public health research, investigators frequently measure multiple clinical outcomes (in the form of continuous, binary, categorical, or count responses) on the same subject. The principles underlying the methods used here are thus generally applicable to any set of outcomes that can be modeled using the exponential family of distributions. Acknowledgements The MLS 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 interagency agreements with the National Institute on Drug Abuse (NIDA), Administration on Children, Youth and Families (ACYF), and Center for Substance Abuse Treatment (CSAT). References Bada, H, Bauer CR, Shankaran S, Lester B, Wright LL, Das A, et al. Central and Autonomic Nervous Systems (CNS/ANS) Signs Associated with In-Utero Cocaine/Opiate Exposure. Archives of Diseases in Childhood, In Press. Bandeen-Roche K, Miglioretti DL, Zeger SL, Rathouz PJ. Latent variable regression for multiple outcomes. Journal of the American Statistical Association, 1997, 92: Bauer CR, Shankaran S, Bada HS, Lester B, Wright LL, Krause-Steinrauf H et al. The maternal lifestyle study: Drug exposure during pregnancy and shortterm maternal outcomes. Am J Obstet Gynecol. 2002; 186: Breslow, N.E., Clayton, D.G., Approximate Inference in Generalized Linear Mixed Models. Journal of the American Statistical Association, 1993, 88(421),

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