A generalized linear mixed model for longitudinal binary data with a marginal logit link function

Summary

Longitudinal studies of a binary outcome are common in the health, social, and behavioral sciences. In general, a feature of random effects logistic regression models for longitudinal binary data is that the marginal functional form, when integrated over the distribution of the random effects, is no longer of logistic form. Recently, Wang and Louis (2003) proposed a random intercept model in the clustered binary data setting where the marginal model has a logistic form. An acknowledged limitation of their model is that it allows only a single random effect that varies from cluster to cluster. In this paper, we propose a modification of their model to handle longitudinal data, allowing separate, but correlated, random intercepts at each measurement occasion. The proposed model allows for a flexible correlation structure among the random intercepts, where the correlations can be interpreted in terms of Kendall’s τ. For example, the marginal correlations among the repeated binary outcomes can decline with increasing time separation, while the model retains the property of having matching conditional and marginal logit link functions. Finally, the proposed method is used to analyze data from a longitudinal study designed to monitor cardiac abnormalities in children born to HIV-infected women.

1 Introduction

Longitudinal studies of a binary outcome are common in the health, social, and behavioral sciences. For example, in the Pediatric Pulmonary and Cardiac Complications (P²C²) of Vertically Transmitted HIV Infection Study (Lipshultz et al., 1998), a longitudinal study designed to monitor heart disease and the progression of cardiac abnormalities in children born to HIV-infected women, a key outcome was the binary variable ‘pumping ability of the heart’ (normal/abnormal). Previous results (Lipshultz et al., 1998; Lipshultz et al., 2000; Lipshultz et al., 2002) from the P²C² study have shown that sub-clinical cardiac abnormalities develop early in children born to HIV-infected women, and that they are frequent, persistent, and often progressive. In the P²C² study a birth cohort of 401 infants born to women infected with HIV-1 were followed over time for up to six years. Of these 401 infants, 74 (18.8%) were HIV positive, and 319 (81.2%) were HIV negative. It is of interest to model the effect of HIV status of the child on the marginal probability of abnormal pumping ability of the heart over time. Additional covariates include mother’s smoking status during pregnancy; gestational age and birth-weight standardized for age (1=abnormal, 0=normal). Table 1 shows data from 10 of the 401 subjects on file.

Table 1.

Data from P²C² longitudinal study for 10 randomly selected children

CASE	HIVb	Mom Smokedc	Gest. Age(wks)	Birth Weightd	Heart Pumping ability at agea:
					Birth	1	2	3	4	5	6
1	1	0	41	0	0	0	0	0	0	0	.
2	1	1	34	0	1	.	0	0	1	.	.
3	0	1	40	0	1	0	0	.	.	.	.
4	1	0	40	0	0	.	0	0	0	1	.
5	0	1	39	0	.	1	0	.	.	.	.
6	0	1	35	0	1	.	.	.	.	.	.
7	0	0	36	0	.	0	0	.	.	.	.
7	1	0	33	1	.	1	1	1	.	.	.
8	0	0	36	1	0	0	.	.	.	.	.
9	0	0	41	1	.	.	.	.	0	.	.
10	0	1	34	1	.	0	0	.	0	1	0

(. = missing)

^a1= abnormal, 0= normal;

^b1= HIV positive, 0= not HIV positive;

^c1 = mother smoked during pregnancy, 0 mother did not smoke;

^d1 = low birth-weight for age, 0 = normal birth-weight;

We consider likelihood-based estimation of the logistic regression model for the marginal probabilities of the repeated binary responses. This, of course, requires a fully parametric likelihood approach based on the joint multinomial distribution of the repeated binary outcomes from each subject. In practice, full likelihood-based methods for fitting of marginal models for discrete longitudinal data have proven to be very challenging for the following reasons: (i) it can be conceptually difficult to model higher-order associations in a flexible and interpretable manner that is consistent with the model for the marginal expectations (e.g., Bahadur, 1961), (ii) given a marginal model for the vector of repeated outcomes, the multinomial probabilities cannot, in general, be expressed in closed-form as a function of the model parameters, and (iii) the number of multinomial probabilities grows exponentially with the number of repeated measures.

Although various likelihood approaches have been proposed, e.g., models based on two- and higher-order correlations (Bahadur, 1961; Zhao and Prentice, 1990), models based on two- and higher-order odds ratios (McCullagh and Nelder, 1989; Lipsitz et al., 1990; Molenberghs and Lesaffre, 1994), none of these likelihood-based models have proven to be of real practical use unless the number of repeated measures is relatively small (say, less than 5). As the number of repeated measures increases, the number of parameters that need to be specified and estimated proliferates rapidly for any of these joint distributions, and a solution to the likelihood equations quickly becomes intractable.

Other full likelihood approaches have been formulated as generalized linear mixed models (GLMM). For example, Heagerty (1999) and Heagerty and Zeger (2000) have developed a likelihood-based approach that combines the versatility of GLMMs for modelling the within-subject association with a marginal logistic regression model for the marginal probability of response. They refer to their general class of models as marginalized random effects models. Recall that in the standard GLMM for binary outcomes, the marginal probabilities, obtained by integrating over the random effects, in general, no longer follow a generalized linear model, due to the non-linearity of the link function typically adopted in regression models for discrete responses. In contrast, the marginalized random effects model can be specifically formulated such that the marginal probabilities follow a logistic regression model. Unlike the standard generalized linear mixed model, the marginalized random effects models of Heagerty (1999) has no closed form expression for the conditional probability of response (conditional on the random effects). When the main interest is in the marginal model parameters, the latter feature has no impact on the interpretability of the model; however, it can be a drawback when trying to implement an algorithm to obtain the maximum likelihood (ML) estimates using commonly available software, e.g., PROC NLMIXED in SAS (V9.2).

In this paper, the goal of our approach is to develop a generalized linear mixed model which has a straightforward interpretation of the effect of the covariates, both conditionally and marginally. For a generalized linear mixed model, conditional on the random effects, the regression parameters have a simple interpretation, such as differences in means (linear regression), log-odds ratios (logistic regression), or log rate ratios (Poisson regression). Often, though, one is also interested in the effects of the covariates on the population-averaged or marginal mean, obtained by integrating the conditional mean over the distribution of the random effects. However, there is typically no closed form expression for the marginal mean as a function of the covariates. As such, there is no simple expression for the marginal model. For example, for a binary outcome, we would want to formulate a table of the odds ratios for a one unit increase in each covariate, given the other covariate values. The typical generalized linear mixed (logistic regression) model with normal random effects does not provide a simple expression for the marginal odds ratios.

As an alternative to the marginalized random effects models of Heagerty (1999), but restricted to the setting of clustered binary data, Wang and Louis (2003) proposed a random intercept generalized linear mixed model in which both the conditional model (conditional on the random effect) and the marginal model (integrated over the distribution of the random intercept) follow a logistic regression model, with model parameters proportional to each other. The random intercept in the model of Wang and Louis (2003) follows a ‘bridge’ distribution. The results of Wang and Louis (2003) hold for a model with only a single random intercept for all responses within a cluster. The restriction to models with only a random intercept is somewhat unappealing for longitudinal studies as the degree of association among a pair of repeated measures from two different time-points typically depends on their time separation. To take the declining correlation into account one could extend the model to have a random intercept plus a random slope with time, where the random intercept and slope follow a bridge distribution. Unfortunately, a linear combination of random variables from the bridge distribution no longer follows a bridge distribution, so that the desired property that the marginal model is of logistic form no longer holds.

In this paper, we propose a modification of the bridge random intercept model to handle longitudinal data. In particular, we propose separate, but correlated, random intercepts at each occasion. A multivariate density using a copula model for the random intercepts from different time points assures that the marginal density of each random effect follows a bridge distribution. The proposed model allows for a flexible marginal correlation among the repeated binary outcomes, including a declining association with increasing time separation while retaining the property that the marginal probabilities follow a logistic regression model. Further, the within-subject association has an appealing interpretation in terms of Kendall’s τ between pairs of random intercepts as well as Kendall’s τ between any pair of repeated responses. The proposed model can also be thought of as a modification of the correlated random normal intercepts generalized linear mixed model for longitudinal binary proposed by Albert et al. (2002); however, the marginal model of Albert et al. (2002) is not logistic. The proposed model is more analogous to probit-normal marginal models for longitudinal binary data (Caffo et al., 2007; Caffo and Griswold, 2006).

Except for the linear mixed model, there is typically no closed form expression for the marginal likelihood (integrated over all possible values of the random effects) for any generalized linear mixed model, Thus, numerical integration techniques must be used to approximate the likelihood, including the likelihood based on our proposed approach here. These numerical integration techniques include Laplace approximations, and Gauss-Hermite quadrature, and Monte Carlo integration algorithms. Poor numerical approximations to the likelihood will lead to biased estimates for the fixed effects and variance components. Pinheiro and Bates (1995) showed that their Monte Carlo importance sampling algorithm had good properties, and it has been implemented in standard generalized linear mixed models software, including PROC NLMIXED in SAS or the NLME function in R.

The method-of-moments based generalized estimating equations (GEE) is an alternative approach that can be used to estimate the marginal regression parameters. Often, however, both the subject-specific conditional (on the random effects) and the marginal regression parameters are of interest; with GEE, only the latter are estimated. In addition, because GEE techniques (Liang and Zeger, 1986; Fitzmaurice et al. 1993; Diggle et al. 2002) for estimation of marginal regression parameters are not likelihood based, these methods cannot be used for prediction of the joint probability of the responses over time. For making inferences about the regression parameters, likelihood ratio tests are not available for hypotheses testing, and likelihood based model diagnostics cannot be used with the GEE approach. Although beyond the scope of this paper, with missing data, a full likelihood method typically gives less bias than GEE methods; the latter require the restrictive assumption that outcomes are missing completely at random. Lee and Nelder (2004) document the drawbacks of GEE methods even in cases when the main interest lies only in the marginal regression parameters.

2 Random effects model with a bridge random effects distribution

Although longitudinal data are clustered, there is in addition an implicit ordering of the repeated measures on each subject. For ease of presentation, we assume that n independent subjects are observed at a common set of t = 1, …, m times. Note, the model and associated methodology can be used when the observation times t₁ < ··· < t_mi are unequally spaced, and when the grid of observation times as well as number of observations m_i vary from subject to subject. The outcome at time t is binary, i.e., we let Y_it = 1 if subject i has response 1 (say, success) at time t, and Y_it = 0 otherwise. Each individual has a J × 1 covariate vector x_it, measured at time t, which includes both time-stationary and time-varying covariates. Our approach can be used with time-varying covariates, but it is assumed that the covariates are non-random; in particular, all time-varying covariates are assumed to be external covariates in the sense described by Kalbfleisch and Prentice (1980). Random time-varying covariates can potentially lead to bias for any GLMM as described by Fitzmaurice (1995). We are primarily interested in making inference about the marginal distribution of Y_it, which is Bernoulli with probability p_it = p_it(β) = E(Y_it|x_it, β) = pr(Y_it = 1|x_it, β) indexed by unknown parameter vector β.

Wang and Louis (2003) proposed the following random intercept logistic regression model for the conditional subject-specific probability

$$p_{it}=p_{it}(b_i)=\text{pr}(Y_{it}=1\mid b_i,x_{it},\mathit{\beta })=\frac{exp(b_i+{\phi }^{-1}{x}_{it}^{\prime }\beta )}{1+exp(b_i+{\phi }^{-1}{x}_{it}^{\prime }\beta )},$$ (1)

where, given the subject-specific random effect b_i the Y_it’s from the same subject are assumed independent Bernoulli random variables, i.e., Y_it|b_i ~ Bern(p_it). When b_i follows a ‘bridge’ distribution,

$$f_b(b_i\mid \phi )=\frac{1}{2\pi }\frac{\mathit{sin}(\phi \pi )}{\mathit{cosh}(\phi b_i)+\mathit{cos}(\phi \pi )},(-\infty <b_i<\infty ),$$ (2)

indexed by unknown parameter φ (0 < φ < 1), the marginal probability of success (Wang and Louis, 2003) equals

$$\text{pr}[Y_{it}=1\mid x_{it},\beta ]=E_b[p_{it}(b_i)]=\frac{exp[x_{it}^{\prime }\beta ]}{1+exp[x_{it}^{\prime }\beta ]},$$ (3)

where E_b denotes the expectation evaluated with respect to the density of the b_i. Thus, the marginal probabilities follow a logistic regression model similar to the conditional model given in (1), except with parameter β instead of parameter φ^{− 1}β. The bridge random variable in (2) has mean 0 and φ is the rescaling parameter. In particular,

$$\mathit{Var}(b_i)=\frac{{\pi }^2}{3}\left(\frac{1}{{\phi }^2}-1\right)$$

so that the larger the value of φ, the smaller the variance. The bridge distribution is symmetric about 0 and has heavier tails than the Gaussian distribution but lighter tails than the Logistic distribution. It can also be shown to be a scale mixture of Gaussian random variables. The rescaling parameter φ ∈ (0, 1) can be interpreted as the attenuation parameter that controls attenuation of the marginal regression effect due to integration of the random effects (Neuhaus et al., 1991). For a random effects logistic model, the only disadvantage to the choice of the bridge over the normal density for the random effects is that the bridge is not the default for any packaged computer programs. The bridge density has a closed form that is easily programmed, although it still requires numerical integration to obtain the MLE. Thus, the computation necessary to obtain the MLE is on a par with other random effects distributions (e.g., the normal), but the interpretability of the marginal model parameters makes the bridge distribution an attractive choice. For a more in depth description of properties of the bridge distribution, see Wang and Louis (2003) and Wang and Louis (2004).

Here, we propose a model with distinct, but correlated, random bridge intercepts at each time point, i.e., b_i in (1) is replaced by a separate random intercept at time t, say b_it, where each b_it follows a bridge distribution and the b_it’s from the same subject have a flexible association structure. Specifically, we now let b_i = (b_i1, ···, b_im) denote the vector of random intercepts at the m time points for subject i. Given the vector of random effects b_i, the Y_it’s for subject i are assumed to be independent Bernoulli random variables, i.e., Y_it|b_i ~ Bern(p_it), where

$$p_{it}=\frac{exp(b_{it}+{\phi }^{-1}{x}_{it}^{\prime }\beta )}{1+exp(b_{it}+{\phi }^{-1}{x}_{it}^{\prime }\beta )},$$ (4)

and the m × 1 dimensional b_i has a multivariate density such that the marginal density of each b_it is a bridge distribution as in (2). For simplicity, we assume the parameter φ of the bridge distribution is the same for all times. Since b_it has a bridge distribution, the marginal success probability will be of the logistic form in (3). For the purpose of building a flexible association among b_i, as well as assuring the desired marginal density of each b_it, we use a Gaussian copula (Nelson, 1999) for b_i. Mathematically, a copula is a simple way of formulating an m-dimensional multivariate distribution, and is specified as a function of the marginal CDF’s. If F₁(w₁), F₂(w₂), ···, F_m(w_m) are the cumulative distribution functions of the random variables W₁, W₂, ···, W_m, respectively, then there exists a function C such that the joint CDF is F(w₁, ···, w_m) = C(F₁(w₁), ···, F_m(w_m)), with one dimensional marginal distributions given by F₁(w₁), ···, F_m(w_m). The concept and application of copulas are illustrated in Nelsen (1998) and Joe (1997).

To formulate the Gaussian copula for b_i, we form a m × 1 vector, Z_i = [Z_i1, …, Z_im]′, which is multivariate normal with mean vector 0 and covariance matrix Σ, where the diagonal elements of Σ equal 1 so that Σ is also the correlation matrix. Note, for identifiability, we restrict V ar(Z_it) to equal 1; if V ar(Z_it) is left as a parameter to estimate, then V ar(b_it) would be a function of both φ and V ar(Z_it), but only one of the two would be estimable. We let ρ_ist = Corr(Z_is, Z_it) denote the correlation between Z_is and Z_it; various choices for the structures of ρ_ist are discussed below. Using the probability integral transform (Hoel, et. al., 1971), $b_{it}=F_b^{-1}(\mathrm{\Phi }(Z_{it}))$ has CDF F_b(b_it), where Φ is the CDF of a standard normal density,

$$F_b^{-1}(u)=\frac{1}{\phi }\mathit{log}\left[\frac{\mathit{sin}(\phi \pi u)}{\mathit{sin}\{\phi \pi (1-u)\}}\right]$$

is the inverse cumulative distribution function of b_it for 0 < u_it < 1, and

$$F_b(b_{it})=1-\frac{1}{\pi \phi }\left[\frac{\pi }{2}-\mathit{arctan}\left\{\frac{exp(\phi b_{it})+\mathit{cos}(\phi \pi )}{\mathit{sin}(\phi \pi )}\right\}\right]$$ (5)

denotes the cumulative distribution function of the bridge distribution. Thus, $b_{it}=F_b^{-1}(\mathrm{\Phi }(Z_{it}))$ has the marginal bridge distribution of interest, and the ’s within a subject are correlated due to the correlation among the Z_it’s.

To fully specify the distribution of Z_i = [Z_i1, …, Z_im]′, we must specify the correlation matrix Σ. A popular longitudinal correlation structure is the autoregressive(1) (AR(1)) structure,

$${\rho }_{\mathit{ist}}=\mathit{Corr}(Z_{is},Z_{it})={\rho }^{\mid t-s\mid },$$ (6)

where −1 < ρ < 1. In principle, any suitable longitudinal correlation structure for the Z_it’s could be assumed, such as Toeplitz, ante-dependence or anisotropic exponential. Alternatively, as discussed by Hougaard (2000), Kendall’s τ is often recommended as a measure of association between a pair of continuous random variables since it is invariant to monotone transformations of the random variables. For a pair of normal random variables, Hougaard (2000) shows that Kendall’s τ equals

$${\tau }_{\mathit{ist}}=\frac{2\mathit{arcsin}({\rho }_{\mathit{ist}})}{\pi },$$ (7)

where arcsin(·) is the inverse sin function and −1 ≤ τ_ist ≤ 1. Because the bridge random variables b_is and b_it are monotone transformations of Z_is and Z_it, and Kendall’s τ is invariant to monotone transformations, then (7) is also Kendall’s τ between the bridge random variables b_is and b_it. This is important because (7) is easy to calculate and it shows that the copula model can capture the full range of possible association between b_is and b_it. One possibility we suggest is specifying the association model in terms of τ_ist, such as AR(1),

$${\tau }_{\mathit{ist}}={\tau }^{\mid t-s\mid },$$ (8)

and then transforming back to ρ_ist = sin(πτ_ist/2) to get the multivariate normal correlation matrix Σ. The relationship between the Kendall’s τ for b_is and b_it and the Kendall’s τ for Y_is and Y_it can only be computed numerically.

To explore the extent of the associations that the bridge random effects can induce, we considered a plot of the relationship between Kendall’s τ for (b_is, b_it) and Kendall’s τ for (Y_is, Y_it), calculated via Monte Carlo simulation (see Figure 1). For this illustration, we considered two time points with bridge model

Figure 1.

Plot of Kendall’s τ for (Y_is, Y_it) (denoted τ_Y) versus Kendall’s τ for (b_is, b_it) (denoted τ_B)

$$\text{pr}(Y_{it}=1\mid b_i,x_{it},\mathit{\beta })=\frac{exp[b_i+(3-2t){\phi }^{-1}]}{1+exp[b_i+(3-2t){\phi }^{-1}]},$$

for t = 1, 2, and let φ = 0.1, 0.3, 0.5, 0.7, 0.9. From Figure 1 we see that the curves follow closely along the 45 degree line, meaning that Kendall’s τ for (b_is, b_it) is a close approximation to Kendall’s τ for (Y_is, Y_it). Further, in terms of Kendall’s τ, the range of association is −1 to 1, and there are no constraints on the association. We have found that this is not true for the usual correlation coefficient, i.e., Corr(b_is, b_it) can be much different than Corr(Y_is, Y_it).

Here, we briefly discuss identifiability issues, which are similar to identifiability issues for a linear mixed model. With both φ and ρ_ist in the model, identifiability issues can arise, depending on the number of pairs of time points, and the model for the association over time. When there are only m = 2 times, the model is not identified if both ρ_ist and φ are left unspecified, i.e., with only two times points, the association between b_is and b_it is completely determined by either the variance of the random effects (a function of φ,) or the correlation between random effects (a function of ρ_ist), but not both. As is the case for a linear mixed model, for three or more repeated measures, the identifiability of the model will depend on the specified correlation structure. For example, for three time points, there are three pairs of times, so that we could have φ in the model, as well as a model for ρ_ist that has two parameters. The above identifiability issues do not arise when one models ρ_ist and/or φ as a function of cluster-level (time-stationary) covariates, although identifiability issues could arise, as in any regression model, if one models ρ_ist and/or φ as a function of too many cluster-level covariates.

The maximum likelihood estimates for the marginal likelihood, integrated over the random effects, say,

$$L(\beta ,\phi ,\rho )=\prod _{i=1}^n{\int }_{b_i}\left[\prod _{t=1}^m{p}_{it}^{y_{it}}{(1-p_{it})}^{(1-y_{it})}\right]f({\mathbf{b}}_i\mid \phi ,\rho )d{\mathbf{b}}_i,$$

can be obtained using a simulation maximization method such as the Monte Carlo importance sampling algorithm described by Pinheiro and Bates (1995), and implemented in PROC NLMIXED in SAS (V9.2) or the NLME function in R; the estimated covariance matrix is obtained using the Pinheiro and Bates (1995) numerical approximation to the inverse of the negative second derivative (information) matrix. A SAS macro for fitting the model is available upon request from the first author. If there are missing outcome data that are missing at random (Rubin, 1976; Laird, 1988), each individual contributes m_i ≤ m conditionally independent (given the random effects) Bernoulli random variables with success probabilities given by (4) to the overall likelihood, and the marginal likelihood is again formed by integrating over the random effects. Appealing to large sample theory for generalized linear mixed models (Fahrmeir and Tutz, 2001), if the likelihood is correctly specified, the maximum likelihood estimates are consistent, asymptotically normal, and the large sample variance of the maximum likelihood estimates can be consistently estimate by the inverse of the negative second derivative (information) matrix.

In order for the Monte Carlo importance sampling algorithm of Pinheiro and Bates (1995) to provide a computationally stable and efficient way of approximating the marginal likelihood, one must carefully choose the importance sampling distribution from which to sample. We have found that the Pinheiro and Bates (1995) suggestion of a multivariate normal approximation for $\left[{\mathrm{\Pi }}_{t=1}^m{p}_{it}^{y_{it}}{(1-p_{it})}^{(1-y_{it})}\right]f({\mathbf{b}}_i\mid \phi ,\rho )$ produces stable results. Further, once the likelihood is approximated, we suggest using the Newton-Raphson algorithm to obtain the maximum likelihood estimate, which requires good starting values for the parameter estimates. We have found that using the ordinary logistic regression estimates of β as the starting values leads to computational stability. In the present study (discussed in the next section), with seven time points, the algorithm is stable and converged quite fast (within 2 minutes). In general, an increase in the dimension of the integration has both positive and negative trade-offs. First, with an increase in the number of time points (or dimension of the integration), there is more information from which to estimate the association parameters φ and τ (or ρ), so that the chances of a flat or multimodal likelihood is far less than it might be with fewer time points. However, with an increase in the dimension of the random effects, the computation required to maximize the likelihood increases. Similar to the approach recommended by Albert et al. (2002), we suggest performing at most 50 iterations of the Newton-Raphson algorithm, with 50 Monte Carlo samples drawn for iterations 1–19, 100 Monte Carlo samples drawn for iterations 20–39, and 1000 iterations for iterations 40–50.

3 Example: Longitudinal study of cardiac function in children born to women infected with HIV-1

In this section, we illustrate the application of the proposed methodology to the analysis of the data from children born to women infected with HIV-1 described in the Introduction. In the P²C² study, a birth cohort of 401 infants born to women infected with HIV-1 were to have cardiovascular function measured approximately every year from birth to age 6; giving up to 7 measurements on each child. Of these 401 infants, 74 (18.8%) were HIV positive, and 319 (81.2%) were HIV negative. The main scientific interest is in determining if HIV-1 infected children are more likely to have abnormal ‘pumping ability of the heart’ at time t, (1=yes, 0=no). The main covariate of interest is the effect of HIV infection in the child; other covariates that could be potential confounders are mother’s smoking status during pregnancy (1=yes, 0=no); gestational age (in weeks) and birth-weight standardized for age (1=abnormal, 0=normal). A child of a mother who smokes is expected to have worse heart function. Children with younger gestational age and lower birth-weight (standardized for gestational age) may also be at risk for cardiac problems.

Thus, to examine the effect of HIV infection in the infants, we considered the following marginal logistic regression model,

$$log\left[\frac{p_{it}}{1-p_{it}}\right]=b_{it}+{\beta }_0+{\beta }_{1}t+{\beta }_2{\text{HIV}}_i+{\beta }_{12}t\ast {\text{HIV}}_i+{\beta }_3{\text{smoke}}_i+{\beta }_4{\text{age}}_i+{\beta }_5{\text{wt}}_i,$$ (9)

for t = 0, 1, …, 6, where HIV_i equals 1 if the i^th child is born with HIV-1 and equals 0 if otherwise; smoke_i equals 1 if the mother smoked during pregnancy, and 0 otherwise; age_i is the gestational age (in weeks); and wt_i equals 1 if the child’s birth-weight for gestational age was abnormal, and 0 otherwise.

Here, we compare our proposed estimation technique with four alternative approaches: 1) the bridge random effects model of Wang and Louis (2003) with a single bridge random effect; 2) Heagerty’s (1999) marginalized random effects model with a linear term for time in the random effects variance, as implemented using the R-macro:

http://faculty.washington.edu/heagerty/Software/LDA/;

3) the maximum likelihood estimates assuming a parametric Bahadur representation of the multinomial distribution (Bahadur, 1961) with an AR(1) correlation structure between Y_is and Y_it, i.e.,

$$\mathit{Corr}(Y_{is},Y_{it})={\mathrm{\Gamma }}^{\mid t-s\mid },$$ (10)

and 4) generalized estimating equations (GEE) with an AR(1) correlation structure for Corr(Y_is, Y_it). For the proposed approach, we use two association models for the bridge random intercepts, one is AR(1) on the Corr(b_is, b_it), and the other is AR(1) on the Kendall’s τ between b_is and b_it. All approaches assume the same marginal model, but different association structures. With the exception of the random effects model with a single bridge random effect, the association between pairs of outcomes decreases as the time separation increases.

Because the Bahadur representation is used, we briefly describe it here. In the Bahadur distribution, the marginal model is p_it in (3). Next, we define the standardized variable S_it to be

$$S_{it}=\frac{Y_{it}-p_{it}}{{\{p_{it}(1-p_{it})\}}^{1/2}},$$

The pairwise correlation between Y_is and Y_it is Γ_st = E(S_isS_it), and the M^th-order correlation between the first M responses is defined as Γ_12…M = E(S_i1S_i2 ··· S_iM). The M^th-order correlation between any M of the m repeated binary responses is defined similarly. Then the Bahadur representation of the 2^m − 1 multinomial probabilities corresponding to the joint distribution of (Y_i1, Y_i2, … Y_im) is

$$\text{pr}\{(Y_{i1}=y_1),(Y_{i2}=y_2),\dots ,(Y_{im}=y_m)\mid X_i,\beta ,\mathrm{\Gamma }\}=\left\{\prod _{t=1}^m{p}_{it}^{y_{it}}{(1-p_{it})}^{1-y_{it}}\right\}\left\{1+\sum _{st}{\mathrm{\Gamma }}_{st}{s}_{is}{s}_{it}+\sum _{\mathit{stu}}{\mathrm{\Gamma }}_{\mathit{stu}}{s}_{is}{s}_{it}{s}_{iu}+\dots +{\mathrm{\Gamma }}_{1\dots m}{s}_{i1}\cdots s_{im}\right\}.$$ (11)

In obtaining the MLE from the Bahadur representation, we assumed all fifth and higher correlations are 0 (Γ_stuvw = … = Γ_1…m = 0); we assumed all fourth-order correlations are the same, regardless of the sets of times (Γ_stuv = Γ_{s′t′u′v′} for all stuv ≠ s′t′u′v′); and we assumed all third-order correlations are the same, regardless of the sets of times (Γ_stu = Γ_s′t′u′ for all stu ≠ s′t′u′). The model for the pairwise correlations Γ_st is AR(1) as in (10).

The importance sampling algorithm of Pinheiro and Bates (1995) was used to calculate the MLE for the bridge random effects model, with the same starting seed and the same number of Monte Carlo draws (400) for each model. Performing a sensitivity analysis, we found very little difference in the estimates and standard errors with 100, 200, 300, or 400 Monte Carlo draws. To obtain the estimates, we wrote a SAS macro using PROC NLMIXED; the macro can be obtained from the first author. For the model with a single bridge random effect, the SAS macro takes approximately 30 seconds to calculate the estimates on a Dual Core, 2.7GHz, 4GB Ram computer; for either the AR(1) model on the correlation or Kendall’s τ, the SAS macro takes approximately 2 minutes to calculate the estimates.

Table 2 gives the estimates of β obtained using the different approaches. We see that the results are generally similar. Although well within sampling random error, if one chooses a .05 level of significance as a cutoff, the parameter of greatest scientific interest, the interaction between Time and HIV status, is significant using Heagerty’s approach as well as our proposed approach with an AR(1) model for ρ or Kendall’s τ, but not using the single bridge random intercept model, GEE or the Bahadur representation. With a significant interaction, the odds ratio for children with HIV versus those without HIV increases over time. For example, using results from the bridge model with AR(1)-τ, children with HIV have exp(β̂₂ + β̂₁₂t) = exp(−0.076 + 0.323t) times the odds of having an abnormal pumping ability than children without HIV at time t. Thus, at 6 years of age, children with HIV have approximately 6 times the odds (or e^{− 0.076+0.323×6} = 6.4) of having an abnormal pumping ability.

Table 2.

Comparison of parameter estimates under alternative models for the within-subject association

Effect	Model	Estimate	SE	Z-statistic	P-value
Intercept	Bridge	1.827	1.459	1.25	0.211
	AR(1)-corr	1.374	1.590	0.86	0.388
	AR(1)-τ	1.389	1.514	0.92	0.359
	Heagerty	2.073	1.407	1.47	0.141
	Bahadur	1.763	1.352	1.30	0.193
	GEE	1.959	1.506	1.30	0.193
Time	Bridge	−0.641	0.080	−8.02	< 0.001
	AR(1)-corr	−0.812	0.102	−7.98	< 0.001
	AR(1)-τ	−0.815	0.094	−8.67	< 0.001
	Heagerty	−0.612	0.063	−9.64	< 0.001
	Bahadur	−0.637	0.088	−7.28	< 0.001
	GEE	−0.642	0.098	−6.57	< 0.001
HIV	Bridge	−0.075	0.266	−0.28	0.777
	AR(1)-corr	−0.082	0.264	−0.31	0.756
	AR(1)-τ	−0.076	0.259	−0.29	0.769
	Heagerty	−0.038	0.249	−0.15	0.879
	Bahadur	−0.037	0.269	−0.14	0.891
	GEE	−0.073	0.264	−0.28	0.782
TIME*HIV	Bridge	0.234	0.135	1.73	0.084
	AR(1)-corr	0.336	0.170	1.97	0.049
	AR(1)-τ	0.323	0.160	2.02	0.044
	Heagerty	0.226	0.101	2.23	0.025
	Bahadur	0.213	0.140	1.53	0.128
	GEE	0.251	0.156	1.61	0.108
MOM SMOKE	Bridge	−0.182	0.176	−1.03	0.303
	AR(1)-corr	−0.170	0.185	−0.92	0.359
	AR(1)-τ	−0.179	0.177	−1.01	0.314
	Heagerty	−0.197	0.187	−1.05	0.292
	Bahadur	−0.206	0.172	−1.20	0.231
	GEE	−0.200	0.173	−1.15	0.248
GEST AGE	Bridge	−0.045	0.037	−1.22	0.225
	AR(1)-corr	−0.038	0.040	−0.93	0.352
	AR(1)-τ	−0.037	0.038	−0.95	0.341
	Heagerty	−0.052	0.036	−1.45	0.149
	Bahadur	−0.043	0.034	−1.26	0.207
	GEE	−0.048	0.038	−1.26	0.207
Low Birth Wt	Bridge	0.086	0.190	0.45	0.652
	AR(1)-corr	0.122	0.198	0.62	0.536
	AR(1)-τ	0.136	0.191	0.71	0.477
	Heagerty	0.078	0.191	0.41	0.683
	Bahadur	0.096	0.173	0.55	0.581
	GEE	0.083	0.193	0.43	0.667
Parameter	Model	Estimate	95% Confidence Interval
φ	Bridge	0.847	[0.788,0.906]
	AR(1)-corr	0.686	[0.556,0.815]
	AR(1)-τ	0.731	[0.634,0.827]
ρ	AR(1)-corr	0.841	[0.725,0.957]
τ	AR(1)-τ	0.749	[0.651,0.847]
Γ	Bahadur AR(1)	0.206	[0.107,0.304]

For the main parameter of interest, it appears from Table 2 that Heagerty’s approach yields a discernibly smaller standard error estimate for the interaction term; however, we caution that this result cannot be expected in general. Overall, there is no clear pattern for the magnitudes of standard errors from one approach versus another. The AR(1) associations from the bridge random effects models can be interpreted as follows. Random intercepts that are 1 year apart have a correlation estimated to be 0.84 and Kendall’s τ estimated to be 0.75; both estimates indicate a high correlation among the repeated binary responses. To compare the fit of the bridge models, one can examine the Akaike information criterion (AIC) for the models, where smaller AIC is defined as better. The AIC for the AR(1) model based on ρ is 5522.6 and for the model based on τ is 5520.6. The AIC for the Bridge model of Wang and Louis (2003) is 5534.2. This suggests that the AR(1) model based on τ provides a slightly better fit than the AR(1) model based on ρ; both provide better fits than the bridge random effects model with a single random effect. For all practical purposes, the fits of the two models are almost indistinguishable. Thus, either model is appropriate for these data and the choice between them can be made in terms of ease of interpretation of the AR(1) association parameter.

4 Simulation Study

We conducted a simulation study to explore the finite sample properties of the proposed bridge generalized linear mixed models. Specifically, we compared the ML estimator for the bridge random effects model, the ML estimator assuming a Bahadur distribution, and a GEE estimator of β. To ensure feasibility of the simulation study, we restricted the number of occasions to m = 3 and considered a simple two-group (50 : 50 mixture) study design configuration (e.g., active treatment versus placebo), with 50 subjects in each group. We simulated from two ’true’ models: 1) a generalized linear mixed model with a bridge distribution, and 2) the Bahadur representation.

Let x_i = 0, 1 indicate group membership, and Y_it again denote the binary outcome at time t, t=1,2,3. When simulating from the bridge or Bahadur models, we let the true marginal logistic model be

$$\text{logit}(\text{pr}[Y_{it}=1\mid x_{it}])={\beta }_0+{\beta }_x{x}_i+{\beta }_{\tau }t,$$

with β₀ = −1.0, β_τ = −0.5 and β_x = 1.0. For the bridge random effects model, we specified an AR(1) model for the correlation structure for the Z_it’s in (6), i.e.,

$${\rho }_{\mathit{ist}}=\mathit{Corr}(Z_{is},Z_{it})={\rho }^{\mid t-s\mid },$$

for three possible true values of ρ = 0.1, 0.3, 0.6, and we also let V ar(Z_is) = 1. For the Bahadur representation given in (11), we specified an AR(1) model for the correlation structure for the Y_it’s

$${\mathrm{\Gamma }}_{\mathit{ist}}=\mathit{Corr}(Y_{is},Y_{it})={\mathrm{\Gamma }}^{\mid t-s\mid },$$

for three possible true values of Γ = 0.1, 0.25, 0.4; we set Γ₁₂₃ = 0. The constraints for the Bahadur representation did not allow Γ > 0.4. For each simulation configuration, 1000 simulation replications were performed. Our simulations were performed using PROC NLMIXED in SAS, with 200 Monte Carlo draws.

For each simulation replication we estimated the β’s by fitting the bridge random effects model with an AR(1) structure on the underlying Z_is’s, a Bahadur model with an AR(1) structure on the Y_is’s, and GEE with an AR(1) structure on the Y_is’s. Note that the GEE will be asymptotically unbiased when data are simulated from either a bridge random effects model or a Bahadur distribution. The MLE from the bridge random effects model will be asymptotically unbiased when data are simulated from a bridge random effects model, but could be biased when data are simulated from the Bahadur. Similarly, the MLE when assuming a Bahadur distribution will be asymptotically unbiased when data are simulated from the Bahadur representation, but could be biased when data are simulated from a bridge random effects model. The purpose of the simulation was to explore the robustness of the MLE from the bridge random effects model under mis-specification of the likelihood. We explored the properties of the three estimators with respect to bias, mean square error (MSE), and coverage probability.

The results of the simulations reported in Table 3 indicate that all of the methods are approximately unbiased and have correct coverage probabilities, even when the likelihood is mis-specified. In general, the MLE from the correctly specified likelihood tends to have the smallest MSE, although the ratio of MSE’s for pairs of approaches is at least 90% for most configurations. For example, the largest difference in ratios of MSE’s when simulating from the bridge random effects model is for ρ = 0.6 and β_x = 1; in this case, the ratio of the bridge MSE to the Bahadur MSE is 90.4%, which suggests the Bahadur MLE is 90% efficient in this case.

Table 3.

Results of simulation study. The true marginal logistic model has parameters (β_τ, β_x) = (−0.5, 1.0).

True Distribution		APPROACH	βτ = − 0.5	βx = 1.0	βτ= − 0.5	βx = 1.0	βτ = − 0.5	βx = 1.0
Bridge			ρ = 0.10		ρ = 0.30		ρ = 0.60
	Simulation Average	Bridge ML	−0.505	1.001	−0.509	1.009	−0.507	1.012
		Bahadur ML	−0.508	1.019	−0.502	1.016	−0.514	1.020
		GEE	−0.517	1.024	−0.509	1.033	−0.506	1.001
	Simulation MSE	Bridge ML	0.0291	0.0790	0.0297	0.0771	0.0282	0.0829
		Bahadur ML	0.0296	0.0793	0.0301	0.0782	0.0294	0.0917
		GEE	0.0299	0.0823	0.0305	0.0842	0.0284	0.0851
	Coverage Probabilitya	Bridge ML	94.0	95.5	95.1	94.9	93.2	94.7
		Bahadur ML	94.8	95.1	93.9	96.0	95.7	93.8
		GEE	94.3	93.6	93.9	95.1	94.6	95.1
Bahadur			Γ = 0.10		Γ = 0.30		Γ = 0.60
	Simulation Average	Bridge ML	−0.510	1.027	−0.508	1.001	−0.518	0.966
		Bahadur ML	−0.509	0.997	−0.514	1.031	−0.507	1.021
		GEE	−0.513	1.024	−0.506	1.015	−0.505	1.025
	Simulation MSE	Bridge ML	0.0299	0.0867	0.0278	0.1053	0.0241	0.1115
		Bahadur ML	0.0290	0.0809	0.0265	0.1036	0.0233	0.1113
		GEE	0.0288	0.0888	0.0272	0.1057	0.0256	0.1366
	Coverage Probabilitya	Bridge ML	93.4	95.2	93.0	94.7	93.2	94.7
		Bahadur ML	94.4	95.7	95.2	95.1	92.8	94.8
		GEE	95.5	95.4	95.9	95.0	93.8	93.6

^aCoverage Probability for a 95% Confidence Interval

The results of this simulation study suggests that the MLE from the bridge random effects model is approximately unbiased, and has correct coverage probabilities, even when the likelihood is mis-specified. We caution, however, that when there are missing data and a mis-specified likelihood, the MLE from the bridge random effects model (and the GEE estimator and MLE from the Bahadur model) could yield biased estimates.

5 Discussion

In this paper, we have proposed a correlated random intercepts model for longitudinal binary data that leads to a marginal logistic regression model. Although the main focus of this paper is on a marginal logistic model for the probability of response at each time point, the model also has the appealing property that the probability of response at each time point conditional on the random effect, is also of logistic form. Specifically, the logistic regression parameters for the marginal and conditional models are proportional to each other, with the proportionality factor determined by an ’attenuation parameter’. Thus, the proposed approach can also be used if there is interest in the conditional model. As discussed in the Introduction, a variety of generalized linear mixed models have previously been proposed that yield logistic marginal models; however, none of them have the property that both the marginal and conditional models are of logistic form. We note that the proposed approach can be generalized to other link functions with an appropriate bridge distribution, such as the complimentary log-log link for longitudinal binary data with a positive stable random effect. Furthermore, the proposed model can easily be fit using existing software, e.g., PROC NLMIXED in SAS. For example, using the Gaussian copula, we can express the marginal likelihood L(β, φ, ρ) in terms of standard non-linear mixed-effects models with random effects b_it. Then the model can be fit using SAS Proc NLMIXED, the R function NLME, or any non-linear mixed-effects software program that is flexible enough to allow transformations of the normal random effects.

Finally, the proposed method can be extended in a number of ways. First, consider a joint longitudinal model for a binary and continuous outcome measured over time. For a joint analysis of both outcomes, the longitudinal binary data can be modeled as in Section 3 and the continuous outcome can be modeled using a standard linear mixed effects model. Correlation between the longitudinal binary and continuous outcomes can be induced by specifying correlations between the random effects in the linear mixed effects model for the continuous outcomes and the bridge random effects in the model for the longitudinal binary outcomes. The second potential extension applied to the problem of ‘informative’ dropout, with the probability of dropout related to possibly unobserved outcomes. One approach for handling informative dropout is to model the (continuous) dropout time process with a parametric frailty model (Hougaard, 2000), in which the frailty is correlated with the bridge random effects in the model for the longitudinal binary outcomes.