While each covariate may provide some unique information about changes in the outcome, there are many settings where the covariates provide overlapping information. This is common in psychiatry where reports on a subject’s psychopathology are gathered from multiple informants. Multiple informant data can be particularly useful in studies of child psychopathology since gathering information from parents, teachers, and mental health workers may give a more accurate assessment of the underlying state of the child (Achenbach et al., 1987). In this setting, it may be of interest to estimate and compare regression parameters that are specific to each informant (or source). The use of multiple sources is also common in the field of diagnostic testing, where more than one type of test may be available to detect the presence of a risk factor, symptom, or disease. (Leisenring et al., 2000). In the latter setting, it will be of interest to determine the predictive value of each test unconditional on the results of the other tests under consideration. This will be the analytic method of choice when the goal is to choose a single diagnostic or screening instrument from a group of candidate instruments. A similar type of example arises in studies of obesity. In this setting the goal may be to determine the best marginal predictor of obesity in adulthood (see, for example, Whitaker et al., 1997). Thus interest is not in prediction of obesity in adulthood as a function of many childhood risk factors, but rather in determining its best single predictor early in life. We explore the latter two applications in more detail in the following paragraphs and return to a multiple informant application in Section 4.

To better illustrate this non-standard application of regression in the diagnostic testing setting, consider the following study reported in Leisenring et al. (2000). This study was designed to compare two binary screening measures for coronary artery disease (CAD). The true presence or absence of CAD for each subject was known from the results of an angiogram—the gold standard. While the angiogram is the gold standard, the procedure is invasive and expensive, making screening procedures valuable tools for assessing the need to administer more rigorous testing. The two screening measures under consideration were presence or absence of chest pain (CP), and the passing or failing of an exercise stress test (EST). Note that if these data were jointly analyzed in a standard regression modelling framework, then CAD status would be regarded as the outcome, with CP and EST being the predictors in, for example, a multiple logistic regression model. The results would provide probabilities or risk of CAD conditional on values for both CP and EST. However, when the goal is to choose a single screening measure, the regression parameter relating CP (or EST) to CAD has a somewhat unappealing interpretation due to conditioning on EST (or CP), and vice versa.

Here, the question of interest is which of the two screening measures is more strongly associated with the presence of CAD unconditional of the value of the screening measure not under consideration. While this comparison could be made in terms of the misclassification rates, Pepe et al. (1999) recommend obtaining and comparing estimates of the association (expressed in terms of an odds ratio) of each of the screening measures with the outcome (unconditional on the results of the other screening measure). Interestingly, when the marginal probabilities of the two screening measures are very similar, the apparent misclassification rate can be shown to be a monotonic function of the odds ratio between the screening measure and CAD status. Note that interest is not in prediction of CAD as a function of CP and EST, but rather in comparing the unconditional associations of each of the predictors with CAD. While the screening measures used in this particular example may be relatively inexpensive to obtain, in general it would not be desirable to gather information from more than one diagnostic test or screen. Thus, the goal is to select the screening measure with the strongest (unconditional) association with the outcome. The comparison of the regression coefficients from the two separate analyses required to assess this is a non-standard problem because the estimates are correlated, and there is no widely accepted method for jointly obtaining and comparing estimates of such marginal associations.

Another example where information is obtained from multiple sources with the goal of relating each to the outcome marginally occurs in obesity research. A study conducted by Whitaker et al. (1997) sought to determine the best childhood predictors of adult obesity. Data were collected retrospectively about each subject from three sources: a measure of obesity obtained on the subject during the her or his childhood, a measure of obesity on the subject’s mother, and a measure of obesity on the subject’s father. These three predictors were measured at five occasions during the subject’s childhood. It is important to reiterate that interest was not in prediction of obesity in adulthood as a function of these three childhood predictors, but was rather to determine which of the three was the most strongly associated with obesity in adulthood. As noted by Pepe et al. (1999), such marginal associations are generally of primary interest in clinical applications due to their interpretability. In this example, obtaining such associations allows one to assess whether a child’s risk of obesity is most strongly influenced by her or his own obesity, or by his or her parents’ obesity status. Standard regression analyses that include all sources do not allow a direct assessment of this, however, since the associations obtained are conditional on the other sources.

Much attention has recently been given to the statistical analysis of data arising from multiple sources (Horton and Fitzmaurice, 2004). One method for jointly obtaining estimates of the separate marginal associations relating each predictor to the response was simultaneously developed by Horton et al. (1999) and Pepe et al. (1999). Both used a non-standard application of generalized estimating equations (GEEs), under a so-called “working independence” assumption. In this approach, the univariate outcome is duplicated as many times as there are numbers of predictors, producing a cluster of identical responses for each subject. A separate marginal model is specified for each predictor but all regression parameters are jointly estimated. Note that the empirical (or “sandwich”) variance estimator (Huber, 1967) must then be used to obtain valid standard errors of these coefficients. While this GEE method produces valid estimates of the regression parameters, it may potentially lose efficiency in certain circumstances. Although in general, the efficiency losses associated with GEE methods are often not substantial (Liang and Zeger, 1986), the potential loss of efficiency has not been investigated in this setting. In this paper, we present a method for obtaining estimates of the separate pairwise associations with a binary outcome via maximum likelihood (ML) estimation. We compare the asymptotic relative efficiency of the working independence GEE-based method of Horton et al. (1999) and Pepe et al. (1999) to ML—particularly when the marginal associations can be assumed to be of similar magnitude.

The joint models for the marginal associations considered in this paper are a subset of those proposed by Glonek (1996) and later extended and generalized by Bergsma and Rudas (2002) as well as Rudas and Bergsma (2004). In particular, we consider a multinomial model for the binary response and binary predictors whose parameters are a mixture of marginal parameters that describe the marginal logits and pairwise log odds ratios among the response-predictor pairs separately, and conditional parameters for all other pairwise and higher-order associations. Thus, the model described here is a hybrid of log-linear models (e.g., Bishop et al., 1975) and the multivariate logistic models described by Glonek and McCullagh (1995) and Bergsma and Rudas (2002). A similar type of “mixed parameter” model was developed by Fitzmaurice and Laird (1993) for longitudinal binary outcomes. Such models are desirable due to the marginal interpretation of the parameters of primary interest. In addition, ML estimation of these parameters is robust to misspecification of the conditional pairwise and higherorder associations when the data are complete (Fitzmaurice and Laird, 1993; Glonek, 1996). That is, even if the model for the complementary set of conditional parameters for all other pairwise and higher-order associations is misspecified, the marginal parameter estimates remain consistent provided the model for these marginal associations has been correctly specified.

The resulting data can be summarized in a 2^K+1 contingency table for the response and the K binary predictors. For each subject a set of joint probabilities, π, describes the probability of belonging to any given cell of the contingency table. We denote the vector of all first-, second-, and higher-order products among by .

Standard log-linear models for the multinomial probabilities, π, are based on a transformation from π to a set of conditional parameters - through the expression,where C₁ is a matrix that takes appropriate contrasts of the log transformed probabilities. Log-linear models enjoy a number of advantages over other model formulations. Foremost among these is that estimation methods are computationally simple and are available in standard generalized linear model software packages (e.g., PROC GENMOD in SAS; SAS Institute, Cary, NC). However, the interpretation of parameters in such models are conditional in nature. That is, the parameters describe conditional logits, conditional log odds ratios, etc. Thus, log-linear models are most useful when there is interest in conditional independence restrictions among the variables (Glonek, 1996). In the setting considered here though, we are primarily interested in the marginal associations relating the outcome to each predictor separately, and in making comparisons among these associations. Thus, the conditional associations, especially those for higher-order associations, are considered nuisance characteristics of the data.

The marginal parameters of interest could be estimated by fitting a multivariate logistic model of the type considered by McCullagh and Nelder (1989). Such models are based on a transformation of π to a set of marginal parameters, ψ, described by,where L and C₂ are matrices that appropriately marginalize, and take contrasts of the first-, second-. and higher-order marginal probabilities, respectively. These parameters describe the marginal logits, marginal log odds ratios, etc. Thus, the key advantage to using such models in the multiple source predictor setting of interest in this paper is that the interpretation of the pairwise outcome-predictor parameters is not conditional on the remaining predictors. However, these models are computationally more difficult to employ since the inverse transformation from η₂ to π is not as straightforward as it is in the case of the log-linear models described by (1). Methods that avoid the inverse transformation, e.g., Lang and Agresti (1994) and Bergsma and Rapcsák (2005), can reduce some of the computational burden. Also, consistent estimates of the outcome-predictor pair-wise associations are obtained only if the entire joint distribution is correctly specified, i.e., the model for all higher-order associations must be correctly specified.

Fitzmaurice and Laird (1993) noted this conundrum and developed a class of models that includes a mixture of log-linear and multivariate logistic parameters in such a way as to realize advantageous features of both. In their formulation, where interest is in the marginal means, the first-order logits were marginal while all two- and higher-order parameters were conditional. We, however, consider an alternative mapping,where the ψ parameters are functions of marginal probabilities and the ω parameters are functions of conditional probabilities. Glonek (1996) described such models in a general way, illustrating a closely related model in which the first-order logits and second-order log odds ratios were marginal, while all third- and higher-order parameters were conditional. Note that in our mapping, only the subset of pairwise associations between the outcome and the predictors are marginal (i.e.,); the remaining pairwise associations (among the predictors), and the third- and higher-order associations are conditional.

First, we consider the case where there are two predictors. The values for the first-order marginal parameters (e.g., ) were varied between -3 to 3 in steps of 1, corresponding to marginal probabilities that range from 0.047 to 0.953. The values for the second- and higher-order conditional parameters (e.g., ) ranged from -10 to 10 in steps of 1. Note that these values are on the log scale, and thus cover a wide grid of marginal and conditional odds ratios. The minimal ARE was calculated for values of the marginal pairwise outcome-predictor log odds ratios ranging from 0 to 4 (corresponding to marginal odds ratios ranging from 1.0 to 54.6). We consider both the case where each outcome-predictor marginal association is unique , and the case where there is assumed to be a shared parameter for these associations . We have restricted the marginal predictor-predictor odds ratios to be greater than 1 since it is assumed that the multiple predictors provide overlapping information about the univariate outcome.

In the first scenario, where the two coefficients for the outcome-predictor marginal log odds ratios are unique, the ARE=1, and asymptotically, there is no efficiency loss associated with independence GEE estimation when compared to ML estimation. This occurs because the GEE and ML estimators for the marginal outcome-predictor associations are equivalent when the multinomial model is saturated (see Appendix A). Thus, the ARE=1 when there are no shared parameters, regardless of the strength of the marginal pairwise associations or the values of the conditional parameters considered ( and ). However, when a common outcome-predictor parameter (i.e., ) is assumed, the multinomial model is no longer saturated and the minimal ARE is approximately 0.90. Of note, the ARE=1 when the two conditional parameters are both set to 0, regardless of the values of the marginal parameters; however, such conditional independencies are unlikely to arise in practice in this setting.

Table 1 shows the minimum ARE across a wide range of values for ψ and ωfor various strengths of the outcome-predictor log odds ratio . The minimum ARE always occurs when the conditional associations take on their maximum values (i.e., ). This is illustrated in Figure 1 which shows the ARE as a function of ω_X1X2 and ω_YX₁X₂ for a fixed set of first-order marginal logits and ψ_YX = 4. Here the values, ψ_Y = -3, ψ_X1 = 3, and ψ_X2 = 2, were chosen since that is where the minimum ARE occurs for the value of ψ_YX considered. However, the pattern is similar regardless of the values of the marginal parameters. Table 1 also shows the minimum ARE obtained when values of the marginal predictor-predictor odds ratios ranged from 1 to 2, as well as those obtained for marginal predictor-predictor odds ratios that ranged from 1 to 5. This indicates that, even though extreme values of ψ and ω were needed to obtain the minimum ARE, values quite close to this minimum were seen for reasonable values of these marginal predictor-predictor odds ratios.

We next consider the case when there are three predictors. Given that the set of conditional parameters, ω, is 8-dimensional, we place some simplifying constraints on the values investigated. Specifically, all second-order predictor-predictor associations were constrained to be equal (ω_X1X2 = ω_X1X3 = ω_X2X3), as well as all third-order associations among the outcome and pairs of predictors (ω_YX1X2 = ω_YX1X3 = ω_YX2X3). The remaining three-way (ω_X1X2X3) and four-way (ω_YX1X2X3) associations were allowed to vary independently. While these conditional parameters are not required to be constrained, and such equality constraints are undesirable in practical applications, we do so here solely to reduce the dimensionality of the study of ARE in the three-predictor variable case. The ARE was calculated for a grid of values for the marginal logits ranging from -3 to 3, and values of the conditional parameters ranging from -5 to 5, again at steps of 1. The range of the conditional parameters was more restricted than in the 2-predictor case due to extremely small joint probabilities encountered. The values for the ψ_YX associations again ranged from 0 to 4. As described in the two-predictor case, the GEE and ML estimators for the marginal outcome-predictor associations are equivalent when the model is saturated. This is reflected by the ARE being equal to 1 when the coefficients for the marginal outcome-predictor associations are unique .

The ARE also equals 1 when ω = 0 (even for a shared ψ_YX association). However, when a common parameter for all three marginal outcome-predictor associations was considered with ω ≠ 0, the minimum ARE ranged from approximately 0.78 to 0.82. The largest losses in efficiency of the independence GEE estimator were observed for large positive values of ω_X1X2X3 and ω_YX1X2X3 and large absolute values of ψ_YX. Table 2 lists the minimum relative efficiencies seen for varying values of a shared outcome-predictor association coefficient. It also shows the minimum ARE obtained for values of all marginal predictor-predictor odds ratios (OR_X1X2, OR_X1X3, and OR_X2X3) ranging from 1 to 2, as well as those obtained for all marginal predictor-predictor odds ratios ranging from 1 to 5. Again, while the minima occurred for large marginal predictor-predictor odds ratios (on the order of 10² to 10⁴), most of the loss occurs when these marginal odds ratios take on less extreme values. When the minima were obtained, one or more cell probabilities approach zero. In order for all the cell probabilities to be of modest size ARE values of 0.90 or larger were seen.

The data were obtained prospectively from 1952 to 1968, with one outcome of interest being mortality during this 16-year period. Information on psychiatric distress was obtained from two predictors, and it is of interest to relate each of these predictors to the outcome, marginally. One predictor was a self-report measure called the DPAX (depression-anxiety scale) and was processed via computer algorithm (see Murphy et al., 1985, for more information about the DPAX). This measure is designed to detect the presence of anxiety and/or depression through a self-report questionnaire. The second predictor (denoted GP) contained information about the presence of psychiatric distress as determined by a general physician. Each physician diagnosis was validated by a psychiatrist. Note that while the DPAX detects the presence of depression and/or anxiety, the GP data indicates any general mental disturbance deemed relevant by the physician. These data are summarized in Table 3.

Note that since we do not have covariates in addition to the two binary predictors (X₁ for DPAX; X₂ for GP), we can calculate the joint probabilities directly. However, it is of interest to determine whether X₁ and X₂(DPAX and GP) have marginal pair-wise associations with the outcome (mortality) that are significantly different. Standard regression techniques would not readily allow such a comparison since the coefficients from a regression analysis including both X₁ and X₂(DPAX and GP) have conditional interpretations. In order to assess this, we first fit the saturated model,The matrix, I, is the identity matrix of indicated dimension. Note that β₁, . . . , β₅ correspond directly to the marginal parameters (p = 5) with primary interest in β₄ and β₅, while β₆ and β₇ correspond directly to the conditional parameters (q = 2). The construction of C and L are shown in Appendix B for this two-predictor example.

We next fit a reduced model in which we assume a shared association parameter, with β₄ = β₅(describing the marginal log odds ratio between the outcome with each of the two predictors separately). The ML estimates of β and their model-based standard errors are shown in Table 4 for the saturated and reduced models. In order to test the hypothesis that β₄ = β₅, we performed a likelihood ratio test (LRT) of H₀: β₄ = β₅ and found that the model with a shared parameter for the marginal pairwise associations between mortality and DPAX, and mortality and GP, is tenable (LRT = 0.03; p > 0.85). We caution the reader that such a comparisons is meaningful only when the predictors have the same underlying scales of measurement (as they do in this example).

We note that the independence GEE (using empirical standard errors) and ML methods both result in estimates and standard errors for the regression coefficients that are nearly identical. This is not surprising given the ARE results for the 2-predictor case reported in Section 3. We also used the joint probabilities estimated from the Stirling County data to determine the corresponding ARE. We obtained an ARE of approximately 1 which agrees very closely with the comparison of the standard errors from the GEE and ML analyses.

In this paper, we have developed a likelihood-based framework in which estimates of the marginal pairwise associations can be obtained and compared. The model presented here is similar to models considered by Fitzmaurice & Laird (1993) and Glonek (1996). While such models require specification of the joint distribution for ML estimation, only the model for the marginal parameters needs to be correctly specified for consistent estimates of the pairwise marginal associations with the outcome to be obtained. ML estimation is generally advantageous relative to GEE-based estimation in that it is asymptotically efficient. However, we have shown that there is no loss in efficiency associated with independence GEE methods for estimation of β when there are no shared parameters among the marginal outcome-predictor associations. That is, when the model is saturated, the GEE and ML estimators for the marginal outcome-predictor associations are equivalent (see Appendix A). While the ARE results indicate there is some loss of efficiency when there is a shared parameter among these associations, this loss may be modest relative to the increased computational difficulty in implementing the ML estimation routine. We note that we have not investigated the efficiency loss when there are more than three predictors. It may be the case that this loss is greater; how-ever, the computational difficulty in implementing the ML estimation procedure would also be greater. One would thus need to weigh the relative advantage of the efficiency gain when using ML estimation against its added computational burden.

The lack of substantial efficiency gain is consistent with previous work. For example, Fitzmaurice (1995) investigated the ARE of GEE versus ML estimation for the analysis of multivariate binary data. He showed that there is some efficiency loss associated with GEE versus ML methods using a working independence assumption for covariates that vary within a subject, but very little loss in efficiency when there are no covariates that vary within a subject. While he considered multiple outcomes for any set of predictors, recall that an appealing feature of logistic regression models is that the odds ratio can be estimated from a prospective or retrospective study design. One implication of this result when X is binary is that if X is treated as the response in the logistic regression, and Y is treated as the predictor, the estimate of the pairwise log odds ratio remains the same. Considering this fact, the ARE results reported in Fitzmaurice (1995) have implications in our setting. If we were to regard X₁, . . . , X_K as multiple responses, and Y as a predictor, then the ARE results from Fitzmaurice (1995) suggest that the independence GEE is almost efficient when compared to ML. That is, Y is a between-subject “predictor” in the logistic regression model for the multiple correlated “responses”, X₁, . . . , X_K.

Aside from efficiency, ML estimation is advantageous over GEE methods since it can incorporate missing data that are either missing completely at random (MCAR; Little and Rubin, 1987) or missing at random (MAR; Little and Rubin, 1987), but at the cost of requiring assumptions about the joint distribution of the outcomes. Given that missing data are common in survey data, this method will allow consistent estimates of the separate pairwise associations to be obtained provided missingness is ignorable. However, incorporating missing data is not straightforward since only the multivariate logistic component (ψ) of the model is reproducible (i.e., it will require the use of the EM algorithm). Also, by utilizing ML estimation, it is possible to conduct likelihood-based inference. This is advantageous in the discrete data setting since likelihood ratio tests (LRT) are purported to have better finite-sample properties than Wald tests when the outcome is binary (Hauck and Donner, 1977). ML estimation also has the advantage of allowing modelling of the joint distribution in a variety of ways. In contrast, by focusing only on the marginal probability of the outcome, GEEs do not allow modelling of the joint distribution at all.

While the method presented in this paper enjoys the benefits of any ML-based estimation procedure, for discrete multivariate data it becomes cumbersome as the number of predictors increases. That is, as the number of second- and higher-order nuisance parameters (which number 2^K+1- 2K - 1) proliferates it becomes computationally demanding. When the data are complete, our ARE results suggest that using ML estimation over the GEE method of Horton et al. (1999) and Pepe et al. (1999) may yield modest gains from an efficiency perspective alone.