This study indicates that a large proportion of papers fail to recognise the issue of missing
data, and many gave insufficient information to ensure that an accurate method of missing
data handling was used. The majority of papers failed to explain their reasons for the method of missing data handling employed within their trial. As well as this, less than 40% of papers gave detailed reasons for the missingness. Collecting and presenting the reasons for missing data can prove a valuable and important asset when establishing the consistency of trials, as well as drawing accurate conclusions. There was very little con-
sistency in the levels that di↵erent trials considered the problems caused by missing data.
In general, a greater awareness is needed in order to ensure that clinical investigators can obtain clinically accurate results from the trial in question by making informed choices and using appropriate methods of missing data handling.
Joint modelling of longitudinal and time-to-event data was not used in any of the papers included in this systematic review, despite the benefits highlighted in Section 1.4. Many papers fail to address the issues caused by dropout appropriately so it could be recom- mended that joint modelling should be employed more often. In the correct circumstances, the amount of “guesswork” required when establishing successful handling of missing data techniques can be reduced by using these models, and details of dropout can also be mod- elled alongside the longitudinal outcome.
It is possible that the reason no identified papers used joint modelling is due to a lack of awareness or understanding of the model, or that some important trial design properties for joint models are yet to be addressed in detail in published literature. In the next chapter, we focus on the methodological and trial design aspects of joint modelling by investigating sample size and power calculations for the random slope and intercept joint model.
Chapter 4
Sample Size and Power
4.1
Introduction
In Chapters 1 and 2, a methodology for joint models has been presented and the benefits of applying these methods to simultaneously monitor a longitudinal outcome and dropout has been justified for certain trial designs. In the field of joint modelling, the published literature has focused primarily on the development of model specifications [42], [44], [10], while the topic of trial design for joint longitudinal and event time outcomes is rarely dis- cussed. When planning a trial to be analysed using joint modelling methods, the same design considerations should be addressed prior to a study as with any other type of statis- tical analysis [1]. In particular it is important to generate a sample size which ensures that
enough individuals are included in the study to detect a clinically significant di↵erence,
but that also minimises the risk of patients being unnecessarily exposed to an experimental treatment [99].
Currently, there are several sample size formulae derived in literature for separate lon- gitudinal and time-to-event data. A summary of the work done in the area of longitudinal data is highlighted in Diggle (2002) [22], while Schoenfeld [100] originally derived a sample size formula for the Cox-Proportional Hazards model in 1983, with the other developments and specifications since presented in Therneau and Grambsch (2000) [101]. However, for models which account for both types of data, greater considerations must be given when deriving and estimating the power and sample size within a study.
While Chen et al (2011) [48] derived a sample size formula for the estimation of overall
treatment e↵ect in the general polynomial joint model [48], little work has been done on
the development of sample size formulae in joint modelling for other specifications. In particular no sample size formulae or power formulae have been generated for the Hen- derson et al. [10] random slope and intercept joint model, which is the primary focus of this thesis. When fitting this model, there are generally three main parameters of interest;
the longitudinal treatment e↵ect, 1, the time-to-event treatment e↵ect, 2 and the link
between the longitudinal and time-to-event outcomes, .
The systematic review carried out in Chapter 3 showed that joint modelling of longitu- dinal and time-to-event data is rarely used in practice. Currently, sample size calculations for this type of modelling are done using simulations [140]. However, this may be one of the reasons that joint modelling is not used more often. Clinicians may be unaware of how to
conduct these pre-trial simulations or be put o↵by what they perceive to be complicated
statistical programming.
For the Henderson et al. specification of the joint model [10] , no research has focused
on how di↵erent parameters and trial properties a↵ect the power for 1, 2 and . In
this chapter, sample size formulae for and 2 are derived for the aforementioned random
slope and intercept joint model using the distribution of the Rao score statistic [102]. Fur-
thermore, a discussion is invoked about the potential factors a↵ecting the sample size for 1.
Using a simulation study based on the parameters of MAGNETIC, the success of the newly generated sample size and power formulae are tested, and the properties of the power
for 1 investigated. As a final task, the power for each parameter in the MAGNETIC trial
will be calculated.