When the development over time is analysed in a particular continuous outcome variable, it is quite common that the variable reaches either a ceiling or a floor. When floor or ceiling effects occurs, standard regression-based methods are not suitable for the longitudinal data analysis. To analyse outcome variables with floor or ceiling effects, two-part models can be used. In Chapter 9 it is explained that a distinction can be made between the standard two-part models and the joint two-part models. For the standard two-part models, the process which is studied is seen as two separate processes. One process reaching the floor or ceiling or not and one process for the observations not reaching the floor or ceiling. For the joint two-part models the process which is studied is seen as one process. When the outcome variable has a latent normal distribution, tobit mixed model analysis can be used. A latent normal distribution means that the outcome variable has a normal distribution, but part of that normal distribution cannot be measured and will have the same (floor or ceiling) value for all observations. The advantage of a tobit mixed model analysis is that the regression coefficient of the analysis has the same interpretation as the regression coefficient obtained from a standard mixed model analysis .Although tobit mixed model analysis is not used extensively in medical studies, it has very nice features. Also in this chapter, all methods are accompanied by extensive real-life data examples.

One of the most complicated parts of a longitudinal data analysis is the interpretation of the regression coefficient. The regression coefficient is a weighted average of the between-subjects relationship and the within-subjects relationship. In Chapter 5, first hybrid models are introduced. Hybrid models are developed to disentangle the between- and within-subjects relationship. Hybrid models can be performed by calculating: (1) the individual average value of the covariate, which is used to obtain the between-subjects part of the relationship and (2) the deviation score, which is the difference between the observed values and the individual mean value and is used to obtain the within-subjects part of the relationship. In this chapter the modelling of changes and the autoregressive model are also discussed. Both models intend to estimate only the within-subjects part of the longitudinal relationship. All methods are accompanied by extensive real-life data examples.

In Chapter 8, longitudinal data analysis with a categorical outcome variable is discussed. The discussion includes simple methods which are mostly based on the change in proportions as well as regression-based methods, such as multinomial logistic mixed model analysis. Inn addition, longitudinal data analysis with a count outcome variable is also discussed. Regarding this, both Poisson GEE analysis and Poisson mixed model analysis can be used. When the count outcome variable suffers from overdispersion, negative binomial GEE analysis and negative binomial mixed model analysis can be used. Both longitudinal Poisson regression and longitudinal mixed model regression give rate ratios as effect estimates and the results of both methods are comparable. All methods in this chapter are accompanied by extensive real-life data examples.

It is often assumed that one of the main features of a longitudinal study is the fact that causality can be detected. This is, however, only partly true for observational longitudinal studies. In Chapter 6, several methods are discussed, which claim to detect causal relationships. In a time-lag model, the covariate measured at a certain time-point is related to the outcome measured one time-point further. Because of the temporality between the outcome and the covariate, the observed relationship is assumed to be causal. A relatively new method to detect causal relationships in longitudinal studies is longitudinal mediation analysis. In this chapter, several longitudinal mediation models are discussed. In the last part of this chapter, some (more) sophisticated methods that claim to estimate causal relationships (i.e. G-methods and joint models) are examined. All methods are accompanied by extensive real-life data examples.

In Chapter 7, longitudinal data analysis with a dichotomous outcome variable is discussed. The discussion includes simple methods such as the change in proportions, the McNemar test and Cochrane’s Q as well as regression-based methods such as logistic mixed model analysis and logistic GEE analysis. An important part of this chapter is related to the different results obtained from a logistic GEE analysis and a logistic mixed model analysis. The difference is caused by the fact that GEE analysis is a population average approach, while mixed model analysis is a subject-specific approach. This difference has no influence on the results of a linear mixed model or GEE analysis, but has influence on the results of a logistic mixed model or GEE analysis. It is shown that the results obtained from a logistic GEE analysis are more valid than the results obtained from a logistic mixed model analysis. Also in this chapter, all methods are accompanied by extensive real-life data examples.

In Chapter 12 sample size calculations for longitudinal studies are discussed. It is shown that a sample size calculation for longitudinal studies is based on the standard sample size calculation formula with an additional multiplication factor which includes the number of follow-up measurements and the estimated correlation between the repeated measurements. Sample size calculations for both a continuous and a dichotomous outcome variable are illustrated with an example. Besides the practical part of the sample size calculation, the chapter also includes a discussion regarding the sense or nonsense of sample size calculations.

In Chapter 3, regression-based methods to analyse longitudinal data are introduced. Linear mixed models analysis and linear GEE mixed model analysis are explained in detail, while the adjustment for covariance method is explained in less detail. It is shown that the different regression-based methods adjust for the correlated observations within the subject in a different way; linear mixed model analysis by allowing different regression coefficients for different subjects (i.e. random intercept and random slope(s)), GEE analysis by estimating directly the correlation between the repeated observations within the subject by assuming a priori a certain correlation structure. It is explained that a linear mixed model analysis with only a random intercept is basically the same as a linear GEE analysis with an exchangeable correlation structure. In this chapter, special attention is given to the interpretation of the regression coefficient, which is a weighted average of the between-subjects relationship and the within-subjects relationship. All methods are accompanied by extensive real-life data examples.

In Chapter 10 the analysis of longitudinal intervention studies is discussed. Most attention is given to the analysis of data from a randomised controlled trial (RCT). In light of the discussion, a distinction is made between an RCT with one follow-up measurement and an RCT with more than one follow-up measurement. For both situations, it is argued that a (longitudinal) analysis of covariance must be used for the analysis of RCT data. With an analysis of covariance, an adjustment is made for the baseline value of the outcome variable and therefore, an adjustment is made for regression to the mean. Besides the analysis of RCT data, the chapter also includes a discussion about stepped wedge trials and about the analysis of intervention effects in observational longitudinal studies. Finally, a discussion of the difference in difference method is provided. Also in this chapter, all methods are accompanied by extensive real- life data examples.

In Chapter 11 the problem of missing data is discussed. Missing data always occurs in longitudinal studies and can be divided based on the missing data mechanism: missing completely at random (MCAR), missing at random (MAR) and missing not at random (MNAR). The problem of the distinction in missing data mechanisms is that it is highly theoretical. More important is the distinction between informative and non-informative missing data. An important part of this chapter deals with imputation methods, such as last value carries forward and multiple imputation. An important conclusion of example studies shown in this chapter is that multiple imputation is, in general, not necessary for missing data in longitudinal studies. It is even better not to impute the missing data and us mixed model analysis for the longitudinal data analysis. In this chapter it is also shown that mixed model analysis deals slightly better with missing data than GEE analysis, although the differences between the two methods are not as great as often suggested.