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Joint mean-correlation model and its application to NorStOP binary data

In this paper we propose a generalized estimating equations method to joint model the mean and correlation structures for longitudinal binary data based on Gaussian copula, and apply it to a large cohort study for the UK’s North Staffordshire Osteoarthritis Project (NorStOP), where the responses can be pre-processed to binary variables. The resulting estimators for the mean and correlation parameters are proven to be consistent and asymptotically normally distributed. Since the theory and simulation results were studied in our previous manuscript [1], we give a brief introduction of our approach but mainly focus on its application to the NorStOP data.


Introduction
The UK North Staffordshire Osteoarthritis Project (NorStOP) [2-3] for the local population whose age is over 50 years was a clinical research aiming to investigate physical and psychological health for older people. The NorStOP study was a health questionnaire survey which lasted for 6 years and included the subject information like age, gender, alcohol consumption, cost of living, body mass index (BMI), pain interference degree (PI) as well as anxiety and depression status. We switch the depression status to the binary responses and explore its dependence on other explanatory variables. Meanwhile, characterizing the correlation relationship between pairs of binary responses is another important objective. Many methods have been studied for modeling mean and correlation for binary data. Standard methods including logistic regression, GEE methods and random intercept models were used by Kuchibhatla and Fillenbaum (2003) [4] to model correlated binary outcomes, but these methods cannot deal with the issue on how the correlations between pairs of binary outcomes depend on certain covariates. A class of marginal models which separately model the mean and correlations of binary responses with explanatory covariates were proposed by Dale (1986) [5], Prentice (1988) [6] and Liang et al. (1992) [7]. Lipsitz and Laird (1991) [8] proposed generalized estimating equation to estimate the odds ratio which is used to model the association between binary outcomes, and Carey et al. (1993) [9] also introduced alternating logistic regressions based on pairwise odds ratios for modeling correlated binary data. Cessie and Houwelingen (1994) [10] discussed and compared different association measures for the dependence among correlated binary responses. More recently, Chaganty and Joe [11] studied the efficiency of GEE for binary responses when choosing various working correlation matrices.
Nevertheless, since binary data contains much less information than any other kinds of data and many parameters in covariance matrices need to be estimated, modeling correctly the covariance structures is still a big challenge. In order to resolve this issue, on the basis of latent Gaussian copula  [12], we introduce a joint mean-correlation model for analyzing longitudinal binary data. Assume that the binary observations are generated through dichotomizing a latent Gaussian variable at an unknown cutoff, the association between each pair of outcomes can be characterized by a function of the correlation between their latent variables. It is worth mentioning that, the above assumption is a natural mechanism for producing correlated binary variables, which is also easy to implement in terms of computation. In this paper, we consider a semiparametric regression model for estimating the mean parameters while the two commonly-used correlation structures Compound Symmetry (CS) and AR(1) for latent correlation matrices are considered. Furthermore, the technique of generalized estimating equations (GEE, Liang and Zeger 1986, Prentice 1988, Prentice and Zhao 1991) [6,13,14] is used to estimate all the parameters in the models, and misspecification of the working correlation matrix does not lead to inconsistency of the parameter estimators obtained by GEE (Wang and Carey 2003) [15].
The rest of the article is organized as follows. In section 2, we simply introduce the models and the algorithm used for estimating the unknown parameters. In section 3, we employ the proposed method to analyze the NorStOP data. Section 4 gives some concluding remarks.  [12], we use to denote the latent correlation matrix of , and the bridge function which links the latent correlation coefficient to the covariance for binary observations is given by

Generalized linear models and GEE
By means of the semiparametric regression [16,17,18], we construct two models for the parameters of mean and latent correlation as follows, where and are the associated unknown parameters; is ( , )-th element of correlation matrix ; , and are × 1 vector of covariates, value of covariate in nonparametric model (•) and time point of -th observation of -th individual, respectively. In this article, for a fixed value of , the function ℎ(•; ) is assumed to be monotone and differentiable. Furthermore, we consider the stationary case, which means that the correlation only depends on the time lag. For example, the AR(1) correlation ℎ , ; = | | , where is a parameter in (-1,1). The nonparametric function can be parametrized by some basis functions. For instance, we can use the following trigonometric basis functions ( ) = cos( ) , = 1, ⋯ , , where is the number of trigonometric basis functions. The BIC criterion can be used to select the number of basis functions. We treat the nonparametric function as a linear regression function with those basis function as covariates. That is, ( ) can be approximated by ( )′ . Hence, the regression model can be written as where ′ = ( , ) and = ( , ) . Based on the aforementioned models, we propose the generalized estimating equations below: where is the column vector obtained by vectorizing the lower triangular part of estimated covariance matrix (not including the diagonal elements) through column by column and is the corresponding part of the matrix ; ( ) = ( ( , ; ( )) , . The solutions of the above two estimating equations, and say, are termed the GEE estimators of and . Note that, ( ) is conditional on the estimator so that the consistency of estimator can be guaranteed.

Main algorithm
The quasi-Fisher scoring algorithm is applied to obtain the numerical solutions for all the parameters in our models. The parameter estimators and can be iteratively updated using weighted generalized least squares. Specifically, given , the mean parameter estimates { ( ) , = 0,1,2, ⋯ } can be updated through On the other side, given and , the latent correlation parameter { ( ) , = 0,1,2, ⋯ } can be calculated by

Application
We apply the proposed estimation method to the NorStOP study, which is a large cohort study of middle-aged and elderly people in the The objective of this research is to find the dependence of depression status on other explanatory covariates and the association between depression status at different time points. Aiming to model jointly the mean and covariance structures for the NorStOP data, we use the following generalized semiparametric mean model: where the income status and the PI are dummy variables; is an unknown nonparametric function and we use the aforementioned trigonometric basis functions to estimate . The BIC suggests that the optimal number of basis functions is 2 and we present the fitting results in Figure 1 below. For simplicity, we only present the case for independence structure when modelling in estimating equation . And also, the correlation structures adopted in modelling the latent correlation matrices are as follows, for , = 1, ⋯ , ,   Table 1 lists the parameter estimates and standard errors for and by using our method. All the parameters are proven to be significant via the test of significance. According to Table 1, it is clear that a participant can be relieved from depression when his/her age grows or economic status improves. Moreover, female or intemperate person are likely to be depressive, which is in accordance with our 'stereotype'. Table 1 also shows that the estimates of are very similar in the two cases, which implies that the resulting estimators of mean parameters are robust against misspecification of or , so do their standard errors. Besides, the estimated values of the parameter in the two cases indicate that there exists a significant latent correlation among depression status at different time points. Aiming to choose the optimal structure, we compute QIC (Pan 2001) [19] for the parameter estimator

Conclusions
In this paper we analyse the NorStOP data and find some significant scientific results. The statistical technique used is the copula-based GEE2 approach proposed by [1] that provides a valid way to model correlation coefficients for clustered binary data. For the NorStOP data, the binary response, i.e. depression status, is influenced linearly by some covariates, such as income status and PI level, and non-linearly by the effect of BMI. Our study shows that as the BMI increases, the probability that a human being suffers from depression would first decrease gradually, reach the lowest point, and then rise up. Other important finding includes that the PI level is positively associated with the depression status, while the income status is negatively correlated with it. On the other hand, our method suggests that there exist strong within-subject correlations between the repeated measurements of patients. Meanwhile, the QIC shows AR(1) is the likely structure of the latent correlation matrix. The correlation coefficient is positive, meaning that depression can be a kind of chronic coherent disease. It is worth mentioning that, we also applied the proposed method to analysis of other real datasets arising in social science. The simulation studies in [1] and the real data analysis presented here prove the generalizability and effectiveness of our method. Due to space limit, we cannot present all the results here but will present the details if accepted to give a talk to the conference.