Introduction

In the foregoing chapters, different techniques for longitudinal data analysis were described and illustrated. In these chapters a distinction was made between the analysis of continuous, dichotomous, and categorical outcome variables. In this chapter, some “new” features for longitudinal data analysis will be introduced. In Section 13.2 longitudinal data analysis with outcome variables with upper or lower censoring will be discussed, while in Section 13.3 a short introduction will be given of the analysis aiming to classify individuals with the same developmental trajectories. Because it basically goes beyond the scope of this book, both explanations will be relatively short and basic, but many references will be given in which more detailed and/or mathematical information can be found.

Outcome variables with upper or lower censoring

Introduction

In Section 9.2.1 several possibilities were discussed to define the change in a particular continuous outcome variable over time. It was suggested that for a longitudinal study in which the outcome variable reaches either a ceiling or a floor a different definition of change should be used. When the development over time is analyzed in a particular continuous outcome variable, it is quite common that the variable reaches either a ceiling or a floor. For instance, in rehabilitation research, most of the patients will recover after a certain amount of time. On the instrument to measure the rehabilitation process, these patients cannot score any higher than the maximum. It is also possible that so-called floor effects occur. For instance, when pain medication is investigated and the outcome variable pain is measured on a visual analog scale, some patients will report “no pain” after a certain amount of time. They cannot score lower than the “no pain” level.

Introduction

Experimental (longitudinal) studies differ from observational longitudinal studies in that experimental studies (in epidemiology often described as trials) include one or more interventions. In general, before the intervention (i.e. at baseline) the population is (randomly) divided into two or more groups. In the case of two groups, one of the groups receives the intervention of interest and the other group receives a placebo intervention, no intervention at all, or the “usual” treatment. The latter is known as the control group. Both groups are monitored over a certain period of time, in order to find out whether the groups differ with regard to a particular outcome variable. The outcome variable can be continuous, dichotomous, or categorical.

In epidemiology, the simplest form of an experimental longitudinal study is one in which a baseline measurement and only one follow-up measurement are performed (Figure 9.1). If the subjects are randomly assigned to the different groups (interventions), a comparison of the follow-up values between the groups will give an answer to the question of which intervention is more effective with regard to the particular outcome variable. The assumption is that random allocation at baseline will ensure that there is no difference between the groups at baseline (in fact, in this situation a baseline measure is not even necessary).

Introduction

One of the main methodological problems in longitudinal studies is missing data, i.e. the (unpleasant) situation when not all N subjects have data on all T measurements. When subjects have missing data at the end of a longitudinal study they are often referred to as drop-outs. It is, however, also possible that subjects miss one particular measurement, and then return to the study at the next follow-up. This type of missing data is often referred to as intermittent missing data (Figure 10.1). It should be noted that, in practice, drop-outs and intermittent missing data usually occur together.

Besides the distinction regarding the missing data pattern (i.e. intermittent missing data versus drop-outs), in the statistical literature a distinction is made regarding the missing data mechanism. Three types of missing data are distinguished: (1) missing completely at random (MCAR: missing, independent of both unobserved and observed data); (2) missing at random (MAR: missing, dependent on observed data, but not on unobserved data, or, in other words, given the observed data, the unobserved data are random); and (3) missing not at random (MNAR: missing, dependent on unobserved data) (Little and Rubin, 2003). Missing at random usually occurs when data are missing by design. An illustrative example is the Longitudinal Aging Study Amsterdam (Deeg and Westendorp-de Serière, 1994). In this observational longitudinal study, a large cohort of elderly subjects was screened for the clinical existence of depression. Because the number of non-depressed subjects was much greater than the number of depressed subjects, a random sample of non-depressed subjects was selected for the follow-up, in combination with the total group of depressed subjects. So, given the fact that the subjects were not depressed, the data were missing at random (Figure 10.2).

Introduction

Epidemiological studies can be roughly divided into observational and experimental studies (Figure 2.1). Observational studies can be further divided into case-control studies and cohort studies. Case-control studies are never longitudinal, in the way that longitudinal studies were defined in Chapter 1. The outcome variable Y (a dichotomous outcome variable distinguishing “case” from “control”) is measured only once. Furthermore, case-control studies are always retrospective in design. The outcome variable Y is observed at a certain time-point, and the covariates are measured retrospectively.

In general, observational cohort studies can be divided into prospective, retrospective, and cross-sectional cohort studies. A prospective cohort study is the only cohort study that can be characterized as a longitudinal study. Prospective cohort studies are usually designed to analyze the longitudinal development of a certain characteristic over time. It is argued that this longitudinal development concerns growth processes. However, in studies investigating the elderly, the process of deterioration is the focus of the study, whereas in other developmental processes growth and deterioration can alternately follow each other. Moreover, in many epidemiological studies one is interested not only in the actual growth or deterioration over time, but also in the longitudinal relationship between several characteristics over time. Another important aspect of epidemiological observational prospective studies is that sometimes one is not really interested in growth or deterioration, but rather in the “stability” of a certain characteristic over time. In epidemiology this phenomenon is known as tracking (Twisk et al., 1994, 1997, 1998a, 1998b, 2000).

Introduction

With a paired t-test and multivariate analysis of variance (MANOVA) for repeated measurements it is possible to investigate changes in one continuous variable over time and to compare the development of a continuous variable over time between different groups. These methods, however, are not suitable for analysis of the longitudinal relationship between a continuous outcome variable and several covariates (which can be either continuous, dichotomous, or categorical). Before the development of “sophisticated” statistical techniques such as generalized estimating equations (GEE) analysis and mixed model analysis, “traditional” methods were used to analyze longitudinal data. The general idea of these “traditional” methods was to reduce the statistical longitudinal problem into a cross-sectional problem. Even nowadays these (limited) approaches are sometimes used in the analysis of longitudinal data.

“Traditional” methods

The greatest advantage of the “traditional” methods is that cross-sectional statistical techniques can be used to analyze the longitudinal data. The most commonly used technique for reducing the longitudinal problem to a cross-sectional problem is analysis of the relationships between changes in different parameters between two points in time (Figure 4.1). Because of its importance and its widespread use, a detailed discussion of the analysis of changes will be given in Chapter 9.

Introduction

Longitudinal studies are defined as studies in which the outcome variable is repeatedly measured; i.e. the outcome variable is measured in the same subject on several occasions. In longitudinal studies the observations of one subject over time are not independent of each other, and therefore it is necessary to apply special statistical techniques, which take into account the fact that the repeated observations of each subject are correlated. The definition of longitudinal studies (used in this book) implicates that statistical techniques like survival analyses are beyond the scope of this book. Those techniques basically are not longitudinal data analysing techniques because (in general) the outcome variable is an irreversible endpoint and therefore strictly speaking is only measured at one occasion. After the occurrence of an event no more observations are carried out on that particular subject.

Why are longitudinal studies so popular these days? One of the reasons for this popularity is that there is a general belief that with longitudinal studies the problem of causality can be solved. This is, however, a typical misunderstanding and is only partly true. Table 1.1 shows the most important criteria for causality, which can be found in every epidemiological textbook (e.g. Rothman and Greenland, 1998). Only one of them is specific for a longitudinal study: the rule of temporality. There has to be a time-lag between outcome variable Y (effect) and covariate X (cause); in time the cause has to precede the effect. The question of whether or not causality exists can only be (partly) answered in specific longitudinal studies (i.e. experimental studies) and certainly not in all longitudinal studies. What then is the advantage of performing a longitudinal study? A longitudinal study is expensive, time consuming, and difficult to analyze. If there are no advantages over cross-sectional studies why bother? The main advantage of a longitudinal study compared to a cross-sectional study is that the individual development of a certain outcome variable over time can be studied.

The most important feature of this book is the word “applied” in the title. This implies that the emphasis of this book lies more on the application of statistical techniques for longitudinal data analysis and not so much on the mathematical background. In most other books on the topic of longitudinal data analysis, the mathematical background is the major issue, which may not be surprising since (nearly) all the books on this topic have been written by statisticians. Although statisticians fully understand the difficult mathematical material underlying longitudinal data analysis, they often have difficulty in explaining this complex material in a way that is understandable for the researchers who have to use the technique or interpret the results. Therefore, this book is not written by a statistician, but by an epidemiologist. In fact, an epidemiologist is not primarily interested in the basic (difficult) mathematical background of the statistical methods, but in finding the answer to a specific research question; the epidemiologist wants to know how to apply a statistical technique and how to interpret the results. Owing to their different basic interests and different level of thinking, communication problems between statisticians and epidemiologists are quite common. This, in addition to the growing interest in longitudinal studies, initiated the writing of this book: a book on longitudinal data analysis, which is especially suitable for the “non-statistical” researcher (e.g. the epidemiologist). The aim of this book is to provide a practical guide on how to handle epidemiological data from longitudinal studies. The purpose of this book is to build a bridge over the communication gap that exists between statisticians and epidemiologists when addressing the complicated topic of longitudinal data analysis.

Introduction

In Chapter 4, GEE analysis and mixed model analysis were introduced as two (sophisticated) methods that can be used to analyze the longitudinal relationship between an outcome variable Y and several covariates X. In this chapter, the models described in Chapter 4 (which are known as standard models) are slightly altered in order to answer specific research questions.

Alternative models

Time-lag model

It is assumed that the greatest advantage of a longitudinal study design in epidemiological research is that causal relationships can be detected. However, in fact this is only partly true for experimental designs (see Chapter 9). In observational longitudinal studies in general, no answer can be given to the question of whether a certain relationship is causal or not. With the standard models described in Chapter 4, it is only possible to detect longitudinal associations between an outcome variable Y and one (or more) covariate(s) X. When there is some rationale about possible causation in observational longitudinal studies, these associations are called “quasi-causal relationships.” In every epidemiological textbook a list of arguments can be found which can give an indication as to whether or not an observed relationship is causal (see Table 1.1). One of these concerns the temporal sequence of the relationship. When the covariate X precedes the outcome variable Y, the observed relationship may be causal (Figure 6.1).

Introduction

In the foregoing chapters many research questions have been addressed and many techniques for the analysis of longitudinal data have been discussed. In the examples, the relatively “simple” statistical techniques were performed with SPSS, while the generalized estimating equations (GEE) analyses and the mixed model analyses were performed with Stata. This chapter provides an overview of a few major software packages (i.e. SPSS, SAS, R, and MLwiN) and their ability to perform sophisticated longitudinal data analysis. In this chapter, only GEE analysis and mixed model analysis will be discussed. Multivariate analysis of variance (MANOVA) for repeated measurements can be performed with all major software packages, and can usually be found under the repeated measurements option of the generalized linear model (GLM) or as an extension of the (M)ANOVA procedure. The emphasis of this overview is on the output and syntax of the sophisticated longitudinal analysis in the different software packages, and especially is on the comparison of the results obtained with the various different packages. In this overview, only an-alysis with a continuous and a dichotomous outcome variable will be discussed in detail.

Introduction

Before performing a longitudinal (experimental) study, it is “necessary” to calculate the number of subjects needed to ensure that a certain predefined effect will be significant. Sample size calculations are also a prerequisite for research grants and are used by (medical) ethical committees in their evaluation of study design protocols. Besides this, sample size calculations are part of the CONSORT statement, meaning that without a sample size calculation, a paper reporting the results of an experimental study will not be published. The importance of sample size calculations is basically a very strange phenomenon. First of all, sample size calculations are based on many assumptions which can easily be changed, in which case the number of subjects needed will be totally different. Secondly, sample size calculations are related to the importance of “significance levels” (i.e. How many subjects are needed to make a certain “effect” significant?), and that is strange because in epidemiological research the importance of significance is becoming more and more questionable. Nevertheless there is a large amount of literature discussing sample size calculations in longitudinal studies (e.g. Lui and Cumberland, 1992; Snijders and Bosker, 1993; Diggle et al., 1994; Lee and Durbin, 1994; Lipsitz and Fitzmaurice, 1994; Liu, G. and Liang, 1997; Hedeker et al., 1999).

In general, the sample size calculations used for a longitudinal experimental study are the same as for “standard” experimental studies. It should be noted that the “standard” sample size calculations are developed for experimental studies with one follow-up measurement. In fact, with the standard sample size calculations the difference in a certain outcome variable between several groups at the first follow-up measurement is used as an effect size. This assumes that the baseline values for the groups to be compared are equal, which seems to be a reasonable assumption in a randomized trial, but which is not always true (see Chapter 9). Equation 11.1 shows how the sample size can be calculated in the “standard” situation for a continuous outcome variable.