Guidelines for the design of clinical trials with longitudinal outcomes

Abstract

A common objective of longitudinal clinical trials is to compare rates of change in a continuous response variable between two groups. The power realized for such a study is a function of both the number of people recruited and the planned number of measurements for each participant. By varying these two quantities in opposite directions, power can be kept at the desired level. We consider the problem of how best to choose the sample size and frequency of measurement, with a view to minimizing either the total number of measurements or the cost of a study. Some general guidelines are first developed for the situation in which all participants have complete observations. In practice, however, longitudinal studies often suffer from dropout, where a participant leaves the study permanently so that no further observations are possible. We therefore consider the impact of unanticipated dropout on power and also ways of allowing for dropout at the design stage. Based on our results, we propose some general design guidelines for longitudinal trials comparing rates of change when dropout is present.

Introduction

In this paper we consider design issues for clinical trials in which the objective is to compare rates of change of a continuous longitudinal response between two treatment arms. Such comparisons are common in studies assessing the change in a disease marker, such as lung function in respiratory studies, blood pressure in cardiovascular studies, and viral load in HIV (human immunodeficiency virus) studies. The power realized for such a study depends on the number of participants, the planned number and timing of measurements for each participant, and the extent to which planned measurements are missing. Of these variables, the sample size and measurement schedule can often be controlled, subject to cost and other constraints. By varying the number of participants (n) and the planned number of measurements per participant (m) in opposite directions, power can be kept at the desired level. In this paper, we consider the problem of how best to choose m and n, with a view to minimizing either the total number of measurements or the cost of a study.

We begin by discussing some general guidelines for the situation in which all participants have complete observations. Sample size requirements for longitudinal studies with no missing data have been examined previously in various contexts. For example, Diggle et al. [1] give a general introduction to design considerations for longitudinal studies, including sample size formulae for several types of hypotheses. Kirby et al. [2] consider sample sizes for longitudinal trials comparing rates of change between two groups, where the within-individual correlations are assumed to have a damped exponential structure. Frison and Pocock 3, 4 and Schouten [5] consider methods for determining sample sizes using summary statistics, whereas Liu and Liang [6] propose a method based on generalized estimating equations. In practice, however, longitudinal studies often suffer from participants prematurely leaving the study permanently, so that no further observations are possible. For the remainder of the paper we will refer to this as dropout, although this term should not be confused with nonadherence to experimental therapy, which some authors have also referred to as dropout. It will often be difficult at the design stage to predict the level of dropout and how it will occur over the course of the study. Hence we consider the consequences, in terms of loss of power, of not allowing for dropout at the design stage. An alternative strategy would be to make some explicit allowance for dropout when designing the study, and we investigate possible approaches to this problem. Based on our results, we propose some general design guidelines for longitudinal studies affected by dropout.

The above questions are considered within the framework of a linear random effects model for the response profiles over time. Random effects models are a common method for modeling the dependence in repeatedly measured outcomes, and the assumption that the responses can be adequately modeled by a linear function of time is often a valid approximation for design purposes. Wu [7] has also used this model to investigate sample size requirements for three different estimators when comparing rates of change in the presence of dropout. In that paper, sample size formulae were derived under the implicit assumption that the dropout is deterministic. Using the methodology proposed by Verbeke and Lesaffre [8], we regard the dropout process as stochastic and study the resulting power under various dropout processes. Verbeke and Lesaffre applied their methodology to animal experiments involving anesthesia, in which the measurement itself induced dropout through death. Although such dropout patterns are also possible in clinical trials on humans, here we focus on dropout models that are more common in clinical trials, namely, where taking the measurement does not induce dropout, and the pattern of dropout may be constant over the trial or skewed toward the beginning or end.

An important assumption is that dropout occurs through noninformative mechanisms, or in other words, the tendency for a measurement to be missing due to dropout is not influenced by the value of the measurement. While this will be true in some circumstances, there will also be many circumstances where this is not true. It is important, however, to bear in mind the distinction between modeling assumptions used in study design and those used to analyze a completed study. Complex modeling assumptions such as informative dropout will often be difficult to specify at the design stage and may be of little benefit if their complexity is out of line with the extent of prior knowledge about likely parameter values. It is true that power calculations and other design characteristics based on noninformative dropout cannot be guaranteed to be accurate if such an assumption does not hold. However, such designs will be more robust to power loss than those that are based on the assumption of no dropout, and furthermore they will not compromise the ability to carry out informative dropout analyses at the end of the study, if such models are warranted. For these reasons we believe it is useful to consider the effect of noninformative dropout on longitudinal study design.