Intra-cluster correlation structure in longitudinal data analysis: Selection criteria and misspecification tests

Abstract

Selection criteria and misspecification tests for the intra-cluster correlation structure (ICS) in longitudinal data analysis are considered. In particular, the asymptotical distribution of the correlation information criterion (CIC) is derived and a new method for selecting a working ICS is proposed by standardizing the selection criterion as the $p$-value. The CIC test is found to be powerful in detecting misspecification of the working ICS structures, while with respect to the working ICS selection, the standardized CIC test is also shown to have satisfactory performance. Some simulation studies and applications to two real longitudinal datasets are made to illustrate how these criteria and tests might be useful.

Introduction

The generalized estimating equation (GEE) method first proposed by Liang and Zeger (1986) is a very popular approach for modeling correlated data from longitudinal or cluster studies. It is known that under mild regularity conditions, the parameter estimates from the GEE approach are consistent, even when the working intra-cluster correlation structure (ICS) is misspecified. However, as pointed out by many other researchers, such as Rotnitzky and Jewell (1990), Fitzmaurice (1995) and Wang and Carey, 2003, Wang and Carey, 2004, misspecification of the ICS can affect the asymptotic relative efficiency of parameter estimation without impairing consistency. Therefore, the selection of working ICS in GEE is pertinent to longitudinal and clustered data analysis and the appropriate specification of the correlation structure may improve estimation efficiency, hence leading to more reliable statistical inferences. The importance of the working ICS selection has also been addressed by Zhou et al. (2012), Wang (2003), Wang and Hin (2010), Zhou and Qu (2012) and Chen and Lazar (2012), among others.

Generally, we often have to deal with the following two problems about the correlation structures selection for longitudinal data analysis. One is to select the best correlated structure from a pool of candidate correlated structures, where various selection criteria have been proposed in the literature. However, it should be noted that the selected best ICS by different selection criteria may not be the true one. This is because the true correlation structure may not be included in the candidate correlated structures. The other problem is to test the working ICS misspecification, namely to test whether the selected or given ICS is true or not. That is if the null hypothesis that the chosen working ICS equals the true ICS is not rejected, then the corresponding hypothesized working ICS could be regarded as the true ICS in the following discussions.

For the various working ICS selection criteria based on different approximations in the literature, Pan (2001) proposed a quasi-likelihood under the independence model criterion (QIC), which is actually a modification of the Akaike information criterion (AIC) by Akaike (1973). Hin and Wang (2009) modified the QIC and proposed a correlation information criterion (CIC). As shown by Hin and Wang (2009), the CIC criterion greatly improves the QIC criterion in selecting working correlation structures. Rotnitzky and Jewell (1990) proposed a two-dimensional criterion, which was investigated and denoted as ${\bar{c}}_1$ and ${\bar{c}}_2$ criteria by Wang and Carey (2004) and Carey and Wang (2011), respectively. For the misspecification tests of the working ICS, O’Hara Hines and Hines, 2007, O’Hara Hines and Hines, 2010 considered the standardized form of the ${\bar{c}}_2$ criterion based on the graphical symmetrizing power-law transformation. More recently, Zhou et al. (2012) proposed an information ratio (IR) test for the misspecification for general covariance structures. In addition, a working ICS selection criterion was introduced based on the IR test and they concluded that in comparison to the CIC criterion, the IR test-based working ICS selection method shows stronger evidence.

Note that the IR test statistic could actually be regarded as the standardized form of the ${\bar{c}}_1$ criterion. We introduce the CIC test statistic in this paper to investigate the working ICS misspecification, where the CIC test $p$-value is a natural extension of the corresponding IR test $p$-value. Particularly, we shall first derive the asymptotic distribution of the CIC test statistic, and then a correlation structure selection procedure is similarly established based on the CIC test. The main goal of this paper is to compare the CIC test $p$-value method with the IR test $p$-value method, CIC, ${\bar{c}}_1$ and RJ criteria with respect to working ICS selection. The rest of the paper is organized as follows. In Section  2, the theoretical background is described and the asymptotic distribution of the CIC test statistic is derived. In Section  3, some simulation studies are conducted to evaluate the behaviors of the CIC test and the IR test, as well as the existing CIC, ${\bar{c}}_1$, and RJ criteria for working ICS selection. Applications of the different approaches are also made to two real datasets in Section  4 and some concluding remarks are provided in Section  5.