Centre for Biostatistics and Genetic Epidemiology, Department of Health Sciences, University of Leicester, Adrian Building, University Road, Leicester LE1 7RH, UK
School of Health and Population Sciences, University of Birmingham, Birmingham B15 2TT, UK
The Studies Coordinating Centre, Division of Hypertension and Cardiovascular Rehabilitation, Department of Cardiovascular Research, University of Leuven, Campus Sint Rafaël, Kapucijnenvoer 35, Block D, Box 7001, Leuven BE3000, Belgium
Department of Epidemiology, Maastricht University, Peter Debyeplein 1, Box 616, Maastricht, MD NL6200, The Netherlands
Centre for Epidemiological Studies and Clinical Trials, Ruijin Hospital, Shanghai Jiaotong University School of Medicine, Ruijin 2nd Road 197, Shanghai 200025, China
INSERM, CIC201, Lyon F69000, France
Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Box 281, Stockholm S171 77, Sweden
Abstract
Background
An Individual Patient Data (IPD) metaanalysis is often considered the goldstandard for synthesising survival data from clinical trials. An IPD metaanalysis can be achieved by either a twostage or a onestage approach, depending on whether the trials are analysed separately or simultaneously. A range of onestage hierarchical Cox models have been previously proposed, but these are known to be computationally intensive and are not currently available in all standard statistical software. We describe an alternative approach using Poisson based Generalised Linear Models (GLMs).
Methods
We illustrate, through application and simulation, the Poisson approach both classically and in a Bayesian framework, in twostage and onestage approaches. We outline the benefits of our onestage approach through extension to modelling treatmentcovariate interactions and nonproportional hazards. Ten trials of hypertension treatment, with allcause death the outcome of interest, are used to apply and assess the approach.
Results
We show that the Poisson approach obtains almost identical estimates to the Cox model, is additionally computationally efficient and directly estimates the baseline hazard. Some downward bias is observed in classical estimates of the heterogeneity in the treatment effect, with improved performance from the Bayesian approach.
Conclusion
Our approach provides a highly flexible and computationally efficient framework, available in all standard statistical software, to the investigation of not only heterogeneity, but the presence of nonproportional hazards and treatment effect modifiers.
Background
Metaanalysis methods are used to integrate quantitative findings from a set of related research studies with the aim of providing more reliable and accurate estimates of a treatment effect
An approach often considered the
IPD metaanalyses of timetoevent data can use either a twostage or onestage approach. The most commonly used, the twostage, is achieved by first fitting individual survival models to each trial. The chosen estimates of effect are then combined in a standard metaanalysis framework, now equivalent to an AD metaanalysis. In a onestage IPD metaanalysis, patient data from all studies are analysed simultaneously within a hierarchical framework. This draws parallels with the analysis of IPD from multicentre clinical trials, accommodating clustering within treatment centres; however, in a multicentre trial the treatment effect is not often random, whereas in a metaanalysis it often is. This is because in a multicentre trial we can achieve greater consistency in inclusion/exclusion criteria and other aspects of trial protocol, indicating that a fixed treatment effect is likely to be more appropriate. Senn discusses these issues in more detail
The aim of this paper is to explore the use of Poisson regression, and the generalised mixed model extensions, to incorporate random effects to perform one and twostage IPD metaanalyses of timetoevent outcomes, as an alternative to hierarchical Cox models, and to extend the models to incorporate nonproportional hazards and treatmenteffect modifiers.
Methods
The Poisson approach to survival analysis
Poisson regression is used in the modelling of count data and contingency tables; however, the extension to modelling survival data via a piecewise exponential model
A standard approach when choosing interval lengths is to use yearly splits
Undertaking a onestage IPD metaanalysis within a Poisson framework is beneficial due to the highly developed area of GLMs. Random effects GLMs are available within all commonly used statistical software packages (e.g. Stata, SAS and R), allowing models to be applied without the need for specialist software.
Model fitting in a single trial
Consider the analysis of timetoevent data from a single trial, investigating the effect of a treatment. For the
where
where
Twostage IPD metaanalyses models for survival data
The twostage approach can be thought of as more traditional, with individual survival models applied to each trial, and appropriate summary statistics extracted to allow AD metaanalysis techniques to be applied.
We extract from the
Onestage IPD metaanalyses models for survival data
We now describe onestage IPD metaanalyses models using the framework of proportional hazards models. The following models, if fitted using the Cox proportional hazards model, correspond to those developed by TudurSmith et. al.
Model A: Fixed treatment effect with proportional trial effects
For the
where
The treatment group coding of 0.5/0.5 is used in all onestage models presented in this paper. Using this coding of the treatment group indicator, we assume equal variability in the log hazard rate across trials for both treatment groups. If we chose the 0/1 coding, this imposes the restrictive assumption that the variability in the log hazard rate of the treatment group coded 0, is zero
Model B: Fixed treatment effect with baseline hazard stratified by trial
In reality, the assumption that the hazard functions in all trials are proportional is likely to be inappropriate. By allowing separate baseline hazard functions for each trial we can relax this assumption, whilst still assuming proportional hazards between treatment groups within each trial. Allowing separate baseline hazards, we have:
where
Model C: Random treatment effect with proportional trial effects
Models which allow for betweentrial heterogeneity in the treatment effect are now considered. The following formulations assume an underlying mean treatment effect, coming from a population of treatment effects. The hazard function for the
where
Model D: Random treatment effect with baseline hazard stratified by trial
Finally, separate baseline hazards are allowed, with a random treatment effect:
where
Models A to D, within a hierarchical Cox framework, were applied by TudurSmith et al.
The Poisson approach to onestage IPD metaanalysis models of survival data
We now introduce Poisson based GLM formulations of the models shown above. Techniques to increase the computational efficiency of the models are described in Section titled "Model fitting" below.
Onestage IPD Poisson generalised linear survival models
Models A and C: Fixed/random treatment effect with proportional trial effects. For time intervals,
where λ_{
k
}represents the constant hazard rate in the
Models B and D: Fixed/random treatment effect with baseline hazard stratified by trial. Models B and D are similarly altered. For trials,
where λ_{
jk
}represents the constant hazard rate in the
Model fitting
We present Model A in the form of a Poisson GLM:
where
Fixed effect Models A and B can be implemented using any GLM software package, such as
It is widely known that within a mixed effects framework, maximum likelihood performs poorly when estimating variance parameters when there are a small number of studies
If we have
When handling sparse event data, the situation may arise when no events occur within a split time interval. In this case, when applying the models described in this section, we obtain nuisance estimates of the baseline hazard rate for any time interval in which no events occur. This can be remedied by the merging of time intervals.
Simulation study
To fully assess the performance of these methods a simulation study was devised. Data is simulated consisting of a random treatment effect and proportional trial effects. We investigate the impact of the number of studies and time interval length by simulating either 5, 10 or 30 trials, and applying Poisson onestage models with time intervals of length 0.25, 0.5 or 1 year. Each trial is simulated under the following steps:
1. Generate 2000 patients; 50% assigned to treatment, 50% to control.
2. Simulate a random treatment effect (on the log scale) with mean, α = 0.4, and inherent betweentrial heterogeneity,
3. Generate a fixed trial effect,
4. Generate survival times from a Weibull distribution using a formulation proposed by Bender et al.
This results in 9 scenarios, in which 1000 repetitions were simulated. For each simulated dataset, Model C was applied both classically using
Extensions to the onestage approach
Treatment effect modifiers
It is becoming increasingly accepted that variation in treatment effects, as a source of heterogeneity, can only be sufficiently detected and explained when IPD are available
The discrimination between withintrial and acrosstrial treatmentcovariate interactions is a current issue in IPD metaanalysis
Fixed treatment effect with separate trial effects
Let
Summary statistics for the IPD metaanalysis investigating effectiveness of antihypertension drugs
Trial
Total number of patients
AllCause Deaths
Percent Overweight (%)
Control
Treatment
Control
Treatment
Control
Treatment
ATMH
754
785
13
9
64.24
65.69
COOP
199
150
22
20
51.25
56.00
EWPH
82
90
25
24
62.20
63.33
HDFP
2371
2427
82
81
74.02
71.86
MRC1
3445
3546
63
67
67.52
69.57
MRC2
1337
1314
156
138
61.11
60.81
SHEP
2371
2365
229
210
67.95
68.84
STOP
131
137
7
4
58.78
63.50
SYCH
1121
1239
77
56
39.77
38.66
SYSE
2285
2380
126
115
68.39
68.31
where
Nonproportional hazards for the treatment effect
It has been shown that the benefits of a treatment can be deemed greater during the initial period of followup time in certain contexts
Fixed treatment effect with separate trial effects
Extending Model B, we first dichotomise followup time at time
This can be extended by further splitting of followup time; however, the time variable,
The hypertension data
The example dataset used to illustrate the models in this paper comes from an IPD metaanalysis investigating the effects of antihypertension drugs in lowering systolic and diastolic blood pressure as determinants of cardiovascular outcomes
Summary statistics for the timetoevent outcome allcause death and an overweight covariate are presented in Table
Results
Single trial application
Comparing approaches, we apply a proportional hazards model investigating the effect of the treatment. The SHEP trial is used as an example, with outcome allcause death. Estimated hazard ratios for the treatment effect are presented in Table
Estimates of treatment effect in the SHEP trial
Method
Hazard ratio
95% CI
Cox
0.913
0.757
1.101
Poisson (1)
0.913
0.757
1.101
Poisson (0.5)
0.913
0.757
1.101
Poisson (0.25)
0.913
0.757
1.101
Twostage IPD metaanalyses models for survival data
We now apply twostage random effects metaanalyses models to the hypertension data. In the first step we compare the Cox and Poisson models to obtain the estimates of the treatment effect in each trial,
Table
Results from twostage random effects metaanalyses.
Model
Pooled Hazard Ratio
95% CI
Cox
0.880
0.796
0.974
0
Poisson (0.25)
0.881
0.796
0.974
0
Poisson (0.5)
0.880
0.796
0.974
0
Poisson (1)
0.880
0.796
0.973
0
Outcome is allcause death
Twostage metaanalyses with outcome allcause death
Twostage metaanalyses with outcome allcause death. Cox models are used in the first step.
Twostage metaanalyses with outcome allcause death
Twostage metaanalyses with outcome allcause death. Poisson GLMs are used in the first step.
Onestage IPD metaanalyses models for survival data
We now apply each of the models described in the methods section "Onestage IPD metaanalyses models for survival data" to the hypertension data, using the Poisson method both classically and under a Bayesian approach. Further comparison of Models A(fixed treatment and fixed proportional trial effects) and B (fixed treatment and baseline stratified by trial) are made using Cox proportional hazards models, under a classical approach. Under Bayesian Models A, B, C and D all parameters are assigned a vague prior of N(0,1000^{2}), excluding the heterogeneity parameter in Models C and D, where τ ~ N(0,1) with τ > 0. A burnin of 1000 was used, with 100,000 samples and thinning at every 20th sample to remove autocorrelation.
Estimates of the treatment effect and 95% confidence/credible interval are seen in Table
Estimates of the treatment effect from applying Models A to D both classically and under a Bayesian approach
Framework
Model
Treatment effect
Trial effect
Cox
Poisson (1)
Poisson (0.5)
Poisson (0.25)
Hazard ratio
95% CI
Hazard ratio
95% CI
Hazard ratio
95% CI
Hazard ratio
95% CI
Classical
A
Fixed
Proportional
0.877
0.793
0.970
0.877
0.793
0.970
0.877
0.793
0.970
0.877
0.793
0.970
B
Fixed
Stratified
0.880
0.795
0.973
0.879
0.795
0.973
0.880
0.796
0.973
0.880
0.796
0.973
C
Random
Proportional



0.877
0.793
0.970
0.877
0.793
0.970
0.877
0.793
0.970
D
Random
Stratified



0.879
0.795
0.973
0.880
0.796
0.973
0.880
0.796
0.973
Bayesian
A
Fixed
Proportional



0.877
0.796
0.971
0.878
0.792
0.969
0.876
0.792
0.970
B
Fixed
Stratified



0.880
0.796
0.971
0.879
0.793
0.975
0.879
0.794
0.971
C
Random
Proportional



0.874
0.756
0.994
0.871
0.747
0.994
0.873
0.748
0.998
D
Random
Stratified



0.876
0.755
0.996
0.876
0.755
1.002
0.873
0.760
1.000
Each mixed effects model also produces an estimate of heterogeneity in the treatment effect, seen in Table
Estimates of heterogeneity from applying Models C and D both classically and under a Bayesian approach
Framework
Model
Treatment effect
Trial effect
Poisson (1)
Poisson (0.5)
Poisson (0.25)
τ
95% CI
τ
95% CI
τ
95% CI
Classical
C
Random
Proportional
5.83E10
0
.
2.01E09
0
.
5.60E09
0
.
D
Random
Stratified
5.92E09
0
.
1.10E11
0
.
4.90E08
0
.
Bayesian
C
Random
Proportional
0.082
0.004
0.310
0.085
0.004
0.319
0.081
0.004
0.321
D
Random
Stratified
0.081
0.004
0.310
0.080
0.004
0.299
0.077
0.003
0.306
To illustrate the computational efficiency of the method, using interval lengths of 1 year; application of Models C and D to collapsed data under a classical approach took 4.6 seconds and 60 seconds, respectively, to achieve convergence. Under a Bayesian approach the equivalent models took 64 seconds and 63 seconds, respectively, to complete the sampling.
Example code to fit Model C both classically within Stata, and under a Bayesian approach in WinBUGS
Simulation results
Results from the simulation study, detailing mean estimates and coverages of the treatment effect and heterogeneity can be found in Tables
Results of simulation study.
Split time
Model
5 Studies
10 Studies
30 Studies
0.25
Classical
0.402
0.394
0.396
Bayesian
0.403
0.396
0.397
0.5
Classical
0.401
0.392
0.396
Bayesian
0.403
0.393
0.397
1
Classical
0.401
0.392
0.396
Bayesian
0.402
0.393
0.396
Bayesian estimates are means of median values. Classical estimates are mean values. True value, α = 0.4. Coverage in italics
Results of simulation study.
Split time
Model
5 Studies
10 Studies
30 Studies
0.25
Classical
0.147
0.177
0.193


Bayesian
0.230
0.213
0.205
0.5
Classical
0.147
0.176
0.193


Bayesian
0.230
0.212
0.205
1
Classical
0.147
0.176
0.193


Bayesian
0.231
0.212
0.207
Bayesian estimates are means of median values. Classical estimates are mean values. True value, τ = 0.2. Coverage in italics
Scatter plot matrix comparing classical and Bayesian estimates of treatment effect
Scatter plot matrix comparing classical and Bayesian estimates of treatment effect. True value, α = 0.4.
Scatter plot matrix comparing classical and Bayesian estimates of betweenstudy standard deviation
Scatter plot matrix comparing classical and Bayesian estimates of betweenstudy standard deviation. True value, τ = 0.2.
We also conducted the simulations described above using a treatment group coding of 0/1. The estimates of heterogeneity from the classical model had much larger downward bias. For example, when using 0.5 year intervals, estimates of τ for 5, 10 and 30 studies were 0.112, 0.138 and 0.165, respectively when using the 0/1 treatment coding, compared with 0.147, 0.176 and 0.193 seen in Table
We extended the simulation study to include application of Model D (random treatment effect with baseline hazard stratified by trial) to data simulated as described above. Unfortunately, due to excessive computation time, it proved infeasible to conduct the simulation study on all 9 scenarios. For example, a single run of the scenario including 10 trials with 0.25 year splits takes approximately 32 minutes. However, the 5 trial scenarios were completed and showed entirely consistent results to those described above. The computational difficulties are exclusively due to the classical approach, as each Bayesian model takes only seconds to execute the required number of MCMC samples.
Onestage approach extensions
Treatment effect modifier
We apply Model (10), both classically and in a Bayesian framework, to the hypertension data to examine whether treatment effect is modified by being overweight (as defined by a BMI value ≥ 25). Note we dichotomise BMI to illustrate the methodology here, but in practice continuous variables are better analysed on their continuous scale. All parameters in the Bayesian approach use the vague prior N(0,1000^{2}). Results are shown in Table
Onestage IPD metaanalyses investigating the interaction between treatment and overweight status
Framework
Covariate
Cox
Poisson (1)
Poisson (0.5)
Poisson (0.25)
Hazard Ratio
95% CI
Hazard Ratio
95% CI
Hazard Ratio
95% CI
Hazard Ratio
95% CI
Classical
Treatment when
0.858
0.736
1.001
0.858
0.736
1.001
0.858
0.736
1.001
0.859
0.736
1.001
Overweight, exp
0.726
0.630
0.835
0.725
0.630
0.835
0.726
0.630
0.835
0.726
0.630
0.835
Treatment when
0.896
0.784
1.024
0.896
0.784
1.023
0.896
0.784
1.024
0.896
0.784
1.024
Bayesian
Treatment when



0.857
0.734
0.993
0.860
0.736
1.000
0.859
0.733
0.999
Overweight, exp



0.725
0.634
0.836
0.726
0.632
0.838
0.725
0.631
0.840
Treatment when



0.896
0.781
1.022
0.896
0.787
1.023
0.897
0.785
1.025
Nonproportional hazards
We now apply Model (11) to the hypertension data, letting
Onestage IPD metaanalyses investigating a nonproportional treatment effect
Framework
Covariate
Poisson (1)
Poisson (0.5)
Poisson (0.25)
Hazard Ratio
95% CI
Hazard Ratio
95% CI
Hazard Ratio
95% CI
Classical
Treatment when
0.657
0.515
0.839
0.657
0.515
0.838
0.657
0.515
0.838
Treatment when
0.935
0.837
1.045
0.936
0.838
1.045
0.936
0.838
1.045
Bayesian
Treatment when
0.657
0.521
0.839
0.656
0.508
0.837
0.657
0.521
0.845
Treatment when
0.934
0.833
1.049
0.936
0.841
1.045
0.935
0.835
1.042
Estimated hazard rate in the COOP trial allowing for nonproportional hazards in the treatment effect
Estimated hazard rate in the COOP trial allowing for nonproportional hazards in the treatment effect.
Discussion
The importance of having IPD available has been established, allowing a full exploration of betweenstudy heterogeneity
In this paper, our aim was to illustrate an effective alternative to hierarchical Cox models, minimising computational issues and providing further interpretational benefits. Through minimal splitting of followup time, reliable estimates of effect can be obtained. Choice of interval lengths will depend on the underlying shape of the hazard function; however, the hazard ratio may be insensitive to the baseline, as illustrated by consistent estimates of the treatment effect across the 3 choices of interval length used in this paper. By combining the Poisson approach with the collapsing technique described above, we can dramatically reduce computation time. When analysing data with rare events, such models may be further advantageous through the need of less intervals. Differential followup times between trials can also be accounted for through this approach. Our approach provides direct estimates of the baseline hazard rate which is clinically important. These estimates allow the calculation of risk differences, or number needed to treat
Investigation of random treatment effect models showed a marked underestimation of heterogeneity under the classical approach. This may in fact be explained by the tendency of maximum likelihood to underestimate variance parameters
It must be noted that if purely interested in a pooled treatment effect, then there is no advantage in pursuing a onestage over a twostage approach; however, investigation of treatment effect modifiers and modelling assumptions should be conducted simultaneously, which can only be done effectively through a onestage approach. Although previous work has provided effective methods to investigate heterogeneity in the metaanalysis setting
In our analysis of the hypertension dataset, we observed a 27.4% (95% CI: 16.55, 37.0%) reduction in the mortality rate when a patient is overweight compared to a nonoverweight patient, with treatment group held constant. Although this is a surprising result, it is one that has been identified previously
The approach detailed in this paper has the further benefit of allowing adjustment for confounders to be implemented simply. This becomes important when analysing IPD from observational studies, where the need to adjust for confounders is often paramount
The flexibility of the Poisson approach described may be extended through the use of splines to model not only the baseline hazard, but also any timedependent effects
Finally, we recognise that the IPD approach does not necessarily solve all the problems for metaanalysis
Conclusion
For an IPD metaanalysis of survival data, our approach provides a highly flexible and computationally efficient framework. The methods are available in all standard statistical software, allowing the investigation of not only heterogeneity, but the presence of nonproportional hazards and treatment effect modifiers.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
PL and RR conceived the project. MC carried out the analyses and conducted the simulation study. MC drafted the paper which was later revised by PL and RR through substantial contributions to the contents of the paper. JS, JW and FG were involved in conception, design and acquisition of the hypertension data. All authors read and approved the final manuscript.
Appendix
A.1. Model C: Random treatment effect with proportional trial effects
Classical model within Stata:
Bayesian model within WinBUGS:
Acknowledgements
The authors would like to thank Lutgarde Thijs for her helpful comments and contribution to the management of the hypertension dataset. Michael Crowther is funded by a National Institute of Health Research (NIHR) Methods Fellowship (RPPG040710314). Richard Riley is supported by the MRC Midlands Hub for Trials Methodology Research, at the University of Birmingham (Medical Research Council Grant ID G0800808). We thank the referees for their comments, which have greatly improved the paper.
Prepublication history
The prepublication history for this paper can be accessed here: