Sensitivity analysis of longitudinal binary data with non-monotone missing values (2004)(en)(14s)

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Biostatistics (2004), 5, 4, pp. 531–544 doi: 10.1093/biostatistics/kxh006

Sensitivity analysis of longitudinal binary data with non-monotone missing values PASCAL MININI Laboratoire GlaxoSmithKline, Unit´e M´ethodologie et Biostatistique, 100 route de Versailles, 78163 Marly le Roi, France, and INSERM U472, 16 avenue Paul Vaillant-Couturier, 94807 Villejuif, France MICHEL CHAVANCE∗ INSERM U472, 16 avenue Paul Vaillant-Couturier, 94807 Villejuif, France [email protected] S UMMARY This paper highlights the consequences of incomplete observations in the analysis of longitudinal binary data, in particular non-monotone missing data patterns. Sensitivity analysis is advocated and a method is proposed based on a log–linear model. A sensitivity parameter that represents the relationship between the response mechanism and the missing data mechanism is introduced. It is shown that although this parameter is identifiable, its estimation is highly questionable. A far better approach is to consider a range of plausible values and to estimate the parameters of interest conditionally upon each value of the sensitivity parameter. This allows us to assess the sensitivity of study’s conclusion to assumptions regarding the missing data mechanism. The method is applied to a randomized clinical trial comparing the efficacy of two treatment regimens in patients with persistent asthma. Keywords: Binary data; EM; Ignorance; Longitudinal study; Missing; Multiple imputation; Non-monotone; Sensitivity analysis; Uncertainty.

1. I NTRODUCTION We consider longitudinal studies designed to repeatedly observe a binary response at n prespecified occasions. In practice, successful completion of all planned measurements from all subjects is extremely rare. Two main sources of missing data can be distinguished. On the one hand, some subjects will drop-out from the study; for example as a result of an adverse event, the lack of efficacy of the study treatment, or simply the refusal of the subject to continue the study. This will result in a monotone pattern of missing data (Little and Rubin, 1987). On the other hand, some data will be missing intermittently, for example because of an illness, an invalid measurement or forgetfulness. This will result in a non-monotone pattern. Longitudinal studies generally suffer from both types of missingness, and the collected data are often incomplete with a nonmonotone structure. The classification proposed by Little and Rubin (1987) is based on the relationship between the mechanism leading to complete or incomplete data (the missing data process) and the mechanism ∗ To whom correspondence should be addressed.

c Oxford University Press 2004; all rights reserved. Biostatistics Vol. 5 No. 4 

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controlling the actual value of the response of interest (the response process). Data are missing at random (MAR) when the missing data process depends only on observed responses, and missing not at random when it depends on unobserved responses. In the framework of likelihood-based inference, if the missing data are MAR and if the parameters of the missing data process and those of the response process are distinct, then the missing data process is termed to be ignorable. Otherwise it is nonignorable. Over the past few years, considerable attention has been given to the modelling of longitudinal binary data with nonignorable missing values, via generalized linear mixed models (e.g. Follmann and Wu, 1995; Ibrahim et al., 2001) or generalized estimating equations (e.g. Paik, 1997; Lipsitz et al., 2000; Fitzmaurice and Laird, 2000). However, a paradigm has emerged: handling incomplete observations necessarily requires assumptions that cannot be assessed from the observed data (Little, 1994a; Rubin, 1994; Verbeke and Molenberghs, 2000). In these circumstances, the need for sensitivity analyses has been clearly recognized. Molenberghs et al. (2001); Kenward et al. (2001); Vansteelandt et al. (2000) and Vansteelandt and Goetghebeur (2001) have developed the concepts of ignorance and uncertainty. On the one hand, the usual imprecision is due to the finite random sampling, which is acknowledged via confidence intervals, the width of which approaches zero as the sample size grows. On the other hand, ignorance is due to the incompleteness of data and can be reflected by the interval of ignorance. Ignorance due to a given proportion of missing data would not disappear even with an infinite sample size. Imprecision and ignorance are combined into the concept of uncertainty, which acknowledges both sources. In controlled clinical trials, it has been recommended by the Committee for Proprietary Medicinal Products (2001) to conduct a sensitivity analysis in order to assess the impact of different missing data assumptions regarding the conclusion of a study. With binary responses, a best-case/worst-case analysis can be performed assigning a positive response to all missing data in the control group and a negative response in the experimental group. Although the assumptions of this approach are unrealistic, this is the most convincing analysis if the conclusion of the study is not qualitatively modified. However, in most cases, the benefit of the new treatment would be annihilated by such an extreme analysis (Unnebrink and Windeler, 1999). In the case of a single binary measure, Hollis (2002) proposed a simple and attractive method that consists in examining all possible allocations of missing data. In another framework, Copas and Li (1997) used a first-order Taylor expansion to perform a sensitivity analysis around the MAR assumption. However, Skinner (1997) suggested that a better approach would be to estimate the parameter of interest conditionally on the sensitivity parameter. The strategy previously proposed by Little (1994b) will be used here. This consists of drawing inferences about the parameters of interest under a range of plausible values for a sensitivity parameter, i.e. under different assumptions regarding the missing data mechanism. This has been widely developed for sensitivity analyses (see for example Rotnitzky et al., 1998, 2001; Scharfstein et al., 1999; Birmingham et al., 2003). These methods deal mainly with quantitative data subject to dropout, the comparison being restricted to the value measured at the end of the study. Here, we will consider longitudinal binary responses with non-monotone missing data, all measurements being considered as equally valuable. A joint modelling of the response process and the missing data process, based on a log–linear model is proposed in Section 2. A sensitivity parameter is introduced that represents the relationship between the response process and the missing data process. An important feature of this modelling is that it does not require a monotone missing data structure. In Section 3, it is shown that although the sensitivity parameter is identifiable, its estimation is highly questionable. A far better approach is to consider a range of plausible values, and to estimate the parameters of interest conditionally upon these plausible values. When the objective of the study is to describe the association between explanatory variables and the response of interest, the log–linear model introduced in Section 2 may not be satisfactory. In this case, it is proposed in Section 4 to perform multiple imputations of missing data, and to analyse the completed data using the multiple imputation estimator (Rubin, 1987).

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Longitudinal binary data with missing values

In Section 5, the method is applied to a randomized clinical trial comparing the efficacy of two treatment regimens in patients with persistent asthma. 2. N OTATION AND DISTRIBUTIONAL ASSUMPTIONS 2.1

Modelling the response and missing data processes

We assume that N subjects are to be observed at n different times. Let Y = (Y1 , . . . , Yn ) denote the 1 × n vector of complete binary data for a given subject, i.e. data that would have been observed if no measurement was missing. Let M j denote the missing data indicator with M j = 1 if the jth response is missing and M j = 0 otherwise, and form the 1 × n vector M = (M1 , . . . , Mn ). The joint distribution of Y and M can be expressed using a log–linear model (Bishop et al., 1975) as log P [Y = y, M = m] = µ +

n 

λj yj +



λ jk y j yk +

j