Informative Drop-Out in Longitudinal Data Analysis

Abstract

SUMMARY A model is proposed for continuous longitudinal data with non‐ignorable or informative drop‐out (ID). The model combines a multivariate linear model for the underlying response with a logistic regression model for the drop‐out process. The latter incorporates dependence of the probability of drop‐out on unobserved, or missing, observations. Parameters in the model are estimated by using maximum likelihood (ML) and inferences drawn through conventional likelihood procedures. In particular, likelihood ratio tests can be used to assess the informativeness of the drop‐out process through comparison of the full model with reduced models corresponding to random drop‐out (RD) and completely random processes. A simulation study is used to assess the procedure in two settings: the comparison of time trends under a linear regression model with autocorrelated errors and the estimation of period means and treatment differences from a four‐period four‐treatment crossover trial. It is seen in both settings that, when data are generated under an ID process, the ML estimators from the ID model do not suffer from the bias that is present in the ordinary least squares and RD ML estimators. The approach is then applied to three examples. These derive from a milk protein trial involving three groups of cows, milk yield data from a study of mastitis in dairy cattle and data from a multicentre clinical trial on the study of depression. All three examples provide evidence of an underlying ID process, two with some strength. It is seen that the assumption of an ID rather than an RD process has practical implications for the interpretation of the data.

Use of OSWALD for analyzing longitudinal data with informative dropout

“…logit P k ( y 1 , y 2 , …y k−1 , y k ) = θ 0 + θ 1 y k + ∑ j=2 q θ j y k+1− j where P k = conditional probability of a subject dropping out at time t k given the previously recorded history of y 1 , y 2 , ...,y k-1 and underlying unobserved value of y k; θ = (θ 0 ,θ 1 ,...,θ q ) is a vector of (q+1) parameters relating the dropout process to the unobserved (y k ) and previously observed responses [3].…”

“…With the dropout process modeled, OSWALD draws inference based on the likelihood function [3]. The ID model does not provide standard error estimates.…”

“…For longitudinal data with dropouts, a statistical model is usually considered for the dropout process in addition to the model for the repeatedly measured outcomes, or the complete data model, where maximum likelihood method can be used to perform inference [1,2,3,4,5]. When the dropout process is RD and the parameters for the dropout process are distinct from the parameters for the complete data model, the dropout process is called ignorable for the maximum likelihood method and statistical analysis can be based solely on the observed values as if there were no missing values [1,2].…”

“…The widely used statistical software SAS Proc Mixed [6] assumes this process; however, estimates of standard errors for fixed effect parameters are usually underestimated and this leads to smaller p-values when the dropout process is not CRD [7]. When the dropout process is ID, a model for the dropout process needs to be specified, usually as a function of the covariates, observed responses, unobserved responses, or even some random effects from the complete data model [2,3,4,5]. Logistic or probit regression is often applied on an ID process.…”

An illness–death stochastic model in the analysis of longitudinal dementia data

“…The approaches taken can be broadly classified into the selection model approach (Diggle and Kenward [3]), and the pattern mixture model approach, as reviewed by Little [4]. The selection model approach has been applied to dementia studies with complex sampling by Gao and Hui [5].…”