## 1 Introduction

In a review of individual participant data (IPD) meta-analyses published during the years 1999 to 2001, Simmonds *et al.* [1] found that in practice meta-analysis of IPD was most frequently conducted using simple two-stage methods. For example, in the meta-analysis of survival IPD, hazard ratios can be estimated for each study individually in the first stage using a proportional hazards model, or approximated by a log-rank statistic (for example, [2]). Individual study results can then be combined using standard random-effects or fixed-effects meta-analysis in the second stage.

In the analysis of a single survival study, the hazard ratio is a commonly used measure of treatment effect and is therefore a natural quantity to consider when undertaking a meta-analysis. However, an assumption of proportional hazards is even more restrictive in meta-analysis because it is imposed on multiple studies. If the proportional hazards assumption does not hold for some studies, then the estimated hazard ratio depends on the length of follow-up in those studies, and meta-analysis may not be appropriate. Methods that relax the proportional hazards assumption, the majority of which focus on a measure or measures of treatment effect that are based on the hazard ratio, have been proposed. For example, Moodie *et al.* [3] meta-analysed a function that can be interpreted as a log-transformed weighted average of hazard ratios over time for each trial. Arends *et al.* [4] extended the model of Dear *et al.* [5] by modelling the survival probabilities in treatment and control groups using a multivariate, mixed-effects model. The treatment effect in the model of Arends *et al.* is the hazard ratio, which can be time varying when treatment-by-time interaction terms are included in the model. Fiocco *et al.* [6] described a piecewise-constant hazard model, where hazards for the treatment and control arms are modelled using a bivariate frailty model. And more recently, Ouwens *et al.* [7] have proposed a meta-analysis model in which hazard ratios are time varying and expressed in terms of the shape and the scale parameters of parametric survival curves. The methods we propose here provide an alternative approach to the use of time-varying hazard ratios in this context.

In a recent paper, Siannis *et al.* [8] considered the percentile ratio as an alternative measure of treatment effect in survival IPD meta-analysis when proportional hazards assumptions are not appropriate. The percentile ratio *q*_{k} at percentile level *k* comparing the survival distributions of two groups is defined as

The median ratio at *k* = 0.5 represents the expected ratio of times at which half of the population will fail in the treatment group compared with the control group. Similarly, for other percentile levels *k*, the percentile ratio is the expected ratio for the time at which 100*k%* of the population will fail in the treatment group compared with the control group. Because the interpretation of percentile ratios can be so easily explained, their use could lead to clearer, more practical understanding of survival differences between treatment groups. In general, percentile ratios may vary across percentile levels. If the survival distributions in question are accelerated failure time models, however, the percentile ratio *q* is constant across all values of *k* [8].

Siannis *et al.* estimated percentile ratios using a one-stage parametric model with data at the individual study level being modelled using either accelerated failure time or proportional hazards distributions. In the simplest version of the model, accelerated failure time distributions were used to model the data at the study level. In this case, the combined percentile ratio *q* is constant across percentile levels and can be modelled using either fixed or random effects. The proposed framework is very general, however, and could be used to model any choice of distribution at the study level. This was illustrated using a combination of accelerated failure time and proportional hazards models.

Motivated by the popularity of simple two-stage analyses, we propose an alternative, two-stage method for meta-analysis of percentile ratios, which in addition avoids all distributional assumptions in the first stage. In stage 1, we use Kaplan–Meier estimates of the survivor functions for the treatment and control groups to estimate percentile ratios and their variance–covariance matrix. In stage 2, percentile ratios are combined using either univariate or multivariate, random-effects meta-analysis (see [9] for an overview of multivariate meta-analysis). The pros and cons of using multivariate meta-analysis in this context are explored in the analysis of an example data set.

The layout of the paper is as follows. In Section 2, we describe the new two-stage method and explore its properties using a simulation study in Section 3. We apply the method to an example data set in Section 4 and conclude with a discussion in Section 5.