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Missingness frequently complicates the analysis of longitudinal data. A popular solution for dealing with incomplete longitudinal data is the use of likelihood-based methods, when, for example, linear, generalized linear, or non-linear mixed models are considered, due to their validity under the assumption of missing at random (MAR). Semi-parametric methods such as generalized estimating equations (GEEs) offer another attractive approach but require the assumption of missing completely at random (MCAR). Weighted GEE (WGEE) has been proposed as an elegant way to ensure validity under MAR. Alternatively, multiple imputation (MI) can be used to pre-process incomplete data, after which GEE is applied (MI-GEE). Focusing on incomplete binary repeated measures, both methods are compared using the so-called asymptotic, as well as small-sample, simulations, in a variety of correctly specified as well as incorrectly specified models. In spite of the asymptotic unbiasedness of WGEE, results provide striking evidence that MI-GEE is both less biased and more accurate in the small to moderate sample sizes which typically arise in clinical trials

Longitudinal binary, or in general non-Gaussian, data are common in biomedical research and beyond. A typical study, for instance, would consist of repeatedly observing the presence or absence of some characteristic, taken in relation to covariates of interest. Data arising from such investigations, however, are often prone to incompleteness, or missingness. In the context of longitudinal studies, missingness predominantly occurs in the form of dropout, in which subjects fail to complete the study for one reason or another. The focus of this paper will be on this type of missingness. In what follows, we will discuss methodology that applies to all non-Gaussian settings, but illustrations and simulations will be confined to the prevalent binary case. The nature of the dropout mechanism affects both the analysis and interpretation of the remaining data. Since one can almost never be certain about the cause of dropout, certain assumptions have to be made. Therefore, when referring to the missingness process, we will use the terminology introduced by Rubin (1976) and Little and Rubin (1987). A non-response process is said to be missing completely at random (MCAR) if the missingness is independent of both unobserved and observed data, and missing at random (MAR) if, conditional on the observed data, the missingness is independent of the unobserved measurements. A process that is neither MCAR nor MAR is termed non-random (MNAR). Note that specific names for these mechanisms for the case of longitudinal data were cornered by Diggle and Kenward (1994). Moreover, Little (1995) further splits the MCAR case in situations where missingness is independent of both outcomes and covariates on the one hand, and cases where missingness is covariate-dependent only. For reasons of simplicity and generality, we prefer to retain the generic MCAR–MAR–MNAR terminology. Full details can be found in Molenberghs and Kenward (2007). In the context of likelihood inference, and when the parameters describing the measurement process are functionally independent of the parameters describing the missingness process, MCAR and MAR are ignorable, while an MNAR process is non-ignorable. This is not the case for frequentist inference, where the stronger condition of MCAR is required to ensure ignorability (Rubin, 1976). Indeed, frequentist methods, such as standard generalized estimating equations (GEEs), for which dropout does not need to be modelled, are only valid under the restrictive MCAR assumption. Weighted generalized estimating equations (WGEEs) and multiple imputation based generalized estimating equations (MI-GEEs) are two possible alternatives that make it possible to model the data under the MAR missingness mechanism. However, in both methods, dropout needs to be addressed, either by means of a dropout model for WGEE or by an imputation model for MI-GEE, meaning the missing-data mechanism is then not ignorable. A general taxonomy of models for longitudinal non-Gaussian data consists of three families: marginal, random-effects, and conditional models. Within these model families, a broad set of methods are available, although the marginal and random-effect models are most often used in longitudinal non-Gaussian settings. Such random-effect models, known as generalized linear mixed models, are typically estimated through maximum likelihood, or variations to this theme, implying that ignorability under MAR can be invoked. This is not the case for non-likelihood marginal models, such as the semi-parametric method of GEEs (Liang and Zeger, 1986), henceforth GEE, which is a second prevalent modelling approach in this area. Such models give valid inferences under the restrictive assumption of MCAR. To be able to analyze the longitudinal non-Gaussian profiles under the weaker MAR assumption, Robins et al. (1995) extended GEEs by using inverse probability weights, resulting in weighted estimating equations, or WGEE. An alternative approach is MI, developed by Rubin (1987). A detailed account is given in Schafer (2003). Missing values are imputed several times, and the resulting complete data sets are analyzed using a standard method, such as GEE. Afterwards, the obtained inferences are combined into a single one (MI-GEE). Regarding the missingness process, standard MI requires MAR to hold, even though extensions exist. Pros and cons of inverse probability weighting methods with respect to MI have been the subject of some debate (the discussion of Scharfstein et al., 1999, Clayton et al., 1998 and Carpenter et al., 2006). In this paper, the focus will be on the comparison between the two GEE versions for incomplete data mentioned above: WGEE and MI-GEE. Comparisons will be made by means of a simulation study, including both small-sample simulations, as well as so-called asymptotic simulations (Rotnitzky and Wypij, 1994). The behavior of both methods in terms of mean squared error (MSE), variance and bias of the estimators will be studied, under correctly specified and misspecified models. In this way, robustness of both methods under misspecification of either the dropout model, the imputation model, or the measurement model, can be explored. The outline of this paper is as follows. In Section 2, an overview of methods for analyzing incomplete longitudinal non-Gaussian data is given, with main attention on WGEE and MI together with GEE as analysis method. A description of the asymptotic and small-sample simulation design, as well as the results of the simulation study, is provided in Section 3. We conclude with a discussion in Section 4.

When the analysis of incomplete binary longitudinal data is envisaged, several routes are available. Apart from likelihood-based methods, such as the generalized linear mixed-effects model (Molenberghs and Verbeke, 2005), non-likelihood methods are attractive, especially when a so-called marginal model is of interest. Since standard generalized estimating equations (Liang and Zeger, 1986)) are unbiased only under MCAR, a variety of modifications and alternatives to GEE have been proposed. Undoubtedly the most popular route is through weighted estimating equations, proposed by Robins et al. (1995), and a number of later extensions. Also of attraction is a combination of GEE and MI (Rubin, 1987) methods, i.e., MI-GEE. Once MI is considered an option, it has the merit of allowing for a variety of imputation techniques, whereafter several analysis methods can be considered. Two such routes considered in this paper are MI-GEE and MI-transition. In this paper we have provided quantitative evidence, based on asymptotic, as well as small-sample, simulations, that can be usefully applied in the decision-making process. We have considered WGEE, MI-GEE, and MI-transition under a variety of scenarios. While simulations are necessarily limited, we believe both methods have been put to the test in a fair fashion. Although asymptotically WGEE exhibits the desirable properties that it theoretically is known to possess, these are barely reproduced for small samples, even when every aspect of the analysis is correctly specified. Moreover, the observed sensitivity of WGEE to misspecification in either the dropout or measurement model renders these asymptotic properties meaningless. MI-GEE and MI-transition, on the other hand, demonstrate a certain degree of robustness to misspecification in either the imputation or measurement model, this, despite a further marginalization for the MI-transition case. Furthermore, WGEE's applicability to the case where also covariates are missing is less straightforward, while application of MI is relatively easy. Moreover, one can do MI under MAR with intermittent missing data. Although the results of this study provide insight about the methods under consideration, and thus are useful in the decision-making process, whenever inference is critical, it is always wise to try a couple of different methods, by way of sensitivity analysis. In view of previous work on the merits of inverse probability weighting methods versus MI (the discussion of Scharfstein et al., 1999, Clayton et al., 1998 and Carpenter et al., 2006), we now compare our findings with theirs. Clayton et al. (1998) investigated the use of inverse probability weighting (IPW) and MI, among others, in the context of longitudinal binary data in a multi-phase sampling setting. They found that, while IPW was inefficient for such a 2×2-phase design, MI showed remarkable efficiency. Moreover, this, along with possible extension to data arising from other designs, indicates the substantial strengths of MI. Carpenter et al. (2006), on the other hand, used simulations to study a so-called doubly-robust IPW estimator, introduced by Scharfstein et al. (1999), in comparison with standard IPW, maximum likelihood, and MI. The doubly robust IPW estimator is a modified version of the usual IPW, proposed to improve the efficiency of IPW estimators. IPW estimators were again found to be inefficient and sensitive to the choice of the weight model, but the doubly robust version proves to be as efficient as MI and robust to misspecification. Although applied to continuous Gaussian data, they expect the results to generalize to the discrete case. Whereas Clayton et al. (1998) used actual data and Carpenter et al. (2006) used simulations of a small-sample nature, we complement a small-sample simulation study with asymptotic simulations. Through our simulations, we reinforce the strength of MI over IPW, specifically in application to GEE. WGEE can be viewed as a type of IPW scheme that uses as weights the inverse of the probability of dropout (taken from some dropout model), while MI-GEE uses imputations for the missing data. WGEE was found to be inefficient for small-samples, in line with the findings of these two papers regarding the inefficiency of such IPW schemes. However, this (lack of) efficiency might well be addressed by adopting the doubly-robust IPW version in obtaining the WGEE solutions. Misspecifications are common in practice and it is seldom the case that one would have an entirely correctly specified analysis model. This, along with the fact that the nice properties of WGEE are not attained for modest sample sizes, which is common in typical clinical trials, discourages its recommendation. On the other hand, although theoretically MI-GEE does not provide consistent results when there is a misspecification, overall, it still yields more precise estimates than WGEE. Thus, we provided evidence for the important fact that MI-GEE is less biased and more precise in small and moderate samples, in spite of the asymptotic unbiasedness of WGEE. As a consequence, in practice, MI-GEE would be the preferred method for analysis over WGEE. Moreover, although the focus of this paper is on missingness in the response, in real-life settings, missingness in covariates is often encountered. In such cases, the choice for MI-GEE is even more convincing, since the use of WGEE would be ruled out. Finally, with MI, the imputation model is not restricted to the use of covariates that will be conditioned upon in the measurement model. Other covariates that are available, without necessarily being of interest in the measurement model, can be incorporated in the imputation model, thereby yielding presumably better imputations as well as wider applicability. Importantly, it ought to be clear that in the case of the conditionally specified model, a so-called direct likelihood approach, exploiting ignorability results, is a very viable alternative and may well be the user's preferred one. However, we wanted to focus on a comparison between inverse probability weighting methods and MI. Hence, not to overly clutter the simulation setting, we have left direct likelihood out of the picture. Additionally, direct likelihood would not apply to the marginal model settings, given the prohibitive nature of fitting such models as the Bahadur in other than the simplest settings. As a final remark, recall that asymptotic simulations were done to obtain the asymptotic bias and asymptotic variance. These have theoretical use only, and may provide guidance as to what happens in large to very large samples. Supplementing them with small-sample simulations is therefore an attractive route. Needless to say the method is of no use with conventional data analysis.