The analysis of randomised controlled trial data with more than one follow-up measurement. A comparison between different approaches

When a continuous outcome variable from a randomised control trial (RCT) is analysed, it is recommended to use analysis of covariance in order to obtain a valid estimation of the intervention effect [1‐4]. The idea behind using analysis of covariance is that a correction is made for regression to the mean. Regression to the mean at follow-up is expected to occur when the mean baseline values of the intervention and control group differ from one another. Essentially, differences in mean baseline values between the intervention and control group are not expected, as these groups come from the same population. If differences at baseline do indeed occur, these differences are due to chance (i.e. random fluctuations and/or measurement error). Correction for regression to the mean using analysis of covariance can be achieved by addition of the baseline value as a covariate in an analysis in which the follow-up measurement is the outcome variable and a group allocation is the independent variable. Not correcting for baseline differences can lead to either over- or underestimation of the estimated intervention effect [4].

In most RCT’s, however, more than one follow-up measurement is conducted. With the availability of sophisticated statistical analysis, such as generalised estimating equations (GEE-analysis) and mixed effects modelling, it is common to analyse the whole development of the outcome variable over time in one analysis, instead of analysing separate measurements. These longitudinal designs are frequently analysed using a longitudinal analysis of covariance [5‐8]. However, it is questionable whether this is the right approach, because the result of this analysis is highly influenced by the first period of the study [9]. Therefore, it is sometimes suggested to use a so-called autoregressive approach [9‐11]. In this approach, the outcome variable at a certain time-point is corrected for the value of the outcome variable one time-point earlier. So, the measurement at the first follow-up measurement is corrected for the baseline value of the outcome variable, while the outcome variable at the second follow-up measurement is corrected for the value of the outcome variable at the first follow-up measurement, etc. Although this autoregression approach seems to be an adequate solution, a new problem arises. The correction for differences in the outcome variable at baseline is performed because it is assumed that the groups to be compared are equal at baseline. At the first follow-up measurement the situation is different, though, because differences between the groups can now be caused by the fact that one group received the intervention and the other group did not [12, 13]. So except for the first follow-up measurement, correction for previous measurements does not make sense and is therefore not recommended [3]. Instead, we are searching for an analytic approach that combines a correction for the baseline value for the outcome variable at the first follow-up measurement, with a procedure that does not correct for any values of the outcome variable at the next follow-up measurements.

The purpose of this paper is, therefore, to propose such an approach and to compare the results of this new combination approach with the approaches that are normally used in longitudinal analyses of data from RCT’s.

In this paper, five different approaches are compared. In the first approach, possible regression to the mean is ignored and the changes between subsequent measurements are used as outcome variables. This approach is referred to as the longitudinal analysis of changes.

Second, analysis of covariance is used. By using longitudinal analysis of covariance, the outcome variable at all follow-up measurements is corrected for the baseline value (formula 1).Where: y _it  = outcome variable for subject i at time t; b ₀ = intercept; int = intervention variable for subject i; b ₁ = regression coefficient for the intervention variable; y _it0 = baseline value of the outcome variable subject i; b ₂ = regression coefficient for the baseline value and ε _it  = ‘error’ for subject i at time t.

The third approach is autoregression. In this approach the outcome variable at the different follow-up measurements is corrected for the value of the outcome variable one time-point earlier (formula 2).Where: y _it  = outcome variable for subject i at time t; b ₀ = intercept; int = intervention variable for subject i; b ₁ = regression coefficient for the intervention variable; y _it−1 = value of the outcome variable for subject i at t − 1; b ₂ = regression coefficient for the outcome variable at t − 1 and ε _it  = ‘error’ for subject i at time t.

The fourth and the fifth approach are combined approaches in line with our aim, namely to correct for regression to the mean by only adjusting the first follow-up measurement of the outcome variable for its baseline value. The fourth approach is a combination of the ‘residual change’ method, first described by Blomquist [14], and ‘normal’ change scores. The first step in the residual change method is to perform a linear regression analysis between the follow-up measurement and the baseline value. The second step is to calculate the difference between the observed value at the follow-up measurement and the predicted value (i.e. the predicted value by the earlier mentioned regression analysis). This difference is called the ‘residual change’. The third step is to use these residual change scores as the outcome variable in a linear regression analysis in which the effect of the intervention is analysed. The advantage of this method is that no further correction has to be performed, because the correction for the baseline value is already reflected in the outcome variable. In the combination approach to analyse longitudinal data, the ‘residual change’ score is used as outcome for the first follow-up measurement, while the ‘normal’ change scores are used as outcome for the other follow-up measurements. This approach will be further referred to as the ‘residual change’ combination.

The fifth approach is a combination of analysis of covariance and ‘normal’ change scores. In this approach, the change between baseline and the first follow-up is corrected for the baseline value. For the remaining follow-up measurements, the change scores between subsequent measurements can be used without any correction. However, due to the fact that the first change has to be corrected for the baseline value, a correction factor for the remaining change scores must be added to the longitudinal regression model. Because this correction is only necessary for computational reasons without having any influence on the estimated regression coefficient, a correction must be made for the mean value of the outcome variable at the beginning of the specific period over which the change score is calculated. This approach will be further referred to as the ‘analysis of covariance’ combination.

For all approaches, the intervention effect is estimated with GEE-analysis and all GEE-analyses were performed with STATA (version 10).

In this paper, two combination approaches were proposed in which the first follow-up measurement was adjusted for differences at baseline and in which the next follow-up measurement was not corrected for either baseline or the earlier follow-up measurement. Although the ‘residual change’ score is commonly assumed to be equal to analysis of covariance, it is known that the use of ‘residual change’ scores always leads to lower intervention effects, compared to analysis of covariance [15, 16]. This can also be seen in Tables 1 and 2, where the ‘residual change’ combination leads to lower intervention effects compared to the ‘analysis of covariance’ combination approach. Although this underestimation of the use of the ‘residual change’ score is generally small, Forbes and Carlin [16] showed that it can be non-trivial when the number of participants reduces, and/or baseline differences increase, and/or the ‘true’ intervention effect gets larger.

Because of this drawback of the ‘residual change’ scores, the ‘analysis of covariance’ combination approach is introduced in the present study as a very nice alternative. In the presented analyses, the change between baseline and the first follow-up measurement was used as first outcome in the longitudinal analyses. It is also possible to use the value of the outcome variable at the first follow-up measurement as first outcome variable in the longitudinal analyses. When both are corrected for baseline, the estimated effect of the intervention will be exactly the same [9].

In the example datasets, continuous outcome variables were considered. In RCT’s where the outcome variable is dichotomous, there is generally no problem of regression to the mean. This has to do with the fact that normally no baseline differences regarding the outcome variable exist between the intervention and control group. Regarding the two examples used in the present study, for instance, hypertension could have been the outcome variable instead of systolic blood pressure. At the beginning of the study all patients must have hypertension, because otherwise they could not be included in the study. Likewise, participant’s level of physical activity in comparison to the recommended level could be the outcome, rather than the exact level of physical activity. One of the inclusion criteria would have been to have an activity level below the recommended level and again, no group differences would be present, hence, no baseline correction would be necessary.