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Jordan Bennett
Jordan Bennett

Survival Methods Part-1.rar

Many randomised trials have count outcomes, such as the number of falls or the number of asthma exacerbations. These outcomes have been treated as counts, continuous outcomes or dichotomised and analysed using a variety of analytical methods. This study examines whether different methods of analysis yield estimates of intervention effect that are similar enough to be reasonably pooled in a meta-analysis.

survival methods part-1.rar

Data were simulated for 10,000 randomised trials under three different amounts of overdispersion, four different event rates and two effect sizes. Each simulated trial was analysed using nine different methods of analysis: rate ratio, Poisson regression, negative binomial regression, risk ratio from dichotomised data, survival to the first event, two methods of adjusting for multiple survival times, ratio of means and ratio of medians. Individual patient data was gathered from eight fall prevention trials, and similar analyses were undertaken.

All methods produced similar effect sizes when there was no difference between treatments. Results were similar when there was a moderate difference with two exceptions when the event became more common: (1) risk ratios computed from dichotomised count outcomes and hazard ratios from survival analysis of the time to the first event yielded intervention effects that differed from rate ratios estimated from the negative binomial model (reference model) and (2) the precision of the estimates differed depending on the method used, which may affect both the pooled intervention effect and the observed heterogeneity.

Information about the differences in treatments is lost when event rates increase and the outcome is dichotomised or time to the first event is analysed otherwise similar results are obtained. Further research is needed to examine the effect of differing variances from the different methods on the confidence intervals of pooled estimates.

The variety of analytic methods used in RCTs with count outcomes causes difficulties when carrying out a meta-analysis. In addition to the usual problems of heterogeneity arising from populations and treatments, there is heterogeneity in outcomes and analysis methods used across RCTs to evaluate the effect of the intervention. This raises a key question of whether the results from these alternative methods of analysis are comparable enough (exchangeable) to be combined in a meta-analysis.

Data sets for a two-group parallel RCT with varying parameters were created. The size of each group was randomly chosen from a normal distribution with a mean of 100 and a standard deviation of 2. This kept the sample sizes of the two arms approximately equal and was large enough to provide stable estimates of the difference between the groups. The number of events experienced for each individual was randomly selected from a Poisson distribution with various means covering the range of potential rates that might be expected in practice. There were two groups of simulations, one with a moderate effect of the intervention, with a rate reduction of approximately 30 % in each of the chosen base values, and one with only a small random difference between groups. There were four groups of simulations ranging from a mean of 0.15 to a mean of 7 for the Poisson distributions when there was no overdispersion. As it is common for count data to have some overdispersion, this was built into the data sets by including 0, 20 and 40 % of the numbers of events drawn from Poisson distributions with a higher mean (representing no, moderate and high overdispersion, respectively). Overdispersion was built into both arms of the trial so that the relative differences between the arms would remain approximately the same. As all methods were used on each data set, these minor differences are not problematic. The combinations of parameters for the simulation scenarios with the approximately 30 % reduction in event rate are shown in Table 1. The simulations with no reduction used the rates, with minor random perturbations, in the control group for both arms.

Where possible, information about the 20 % of observations lost to follow-up was included in the analysis [12]. Simple rate ratios were calculated using person days of follow-up, and the Poisson and negative binomial regression models allowed for varying lengths of follow-up through inclusion of an offset in the model. The survival models allow for varying lengths of follow-up through censoring. However, it is not possible to allow for follow-up time for intervention effect estimates 4, 8 and 9.

Each data set was analysed to estimate the effect of exercise versus no exercise using a (1) simple RaR, (2) RR calculated from the dichotomised outcome (fallers and non-fallers), (3) RaR estimated from Poisson regression, (4) RaR estimated from the negative binomial regression and (5) the ratio of means. The median number of falls in all groups was zero, so the ratio of medians could not be computed. Nor was it possible to undertake survival analyses because most studies either did not collect the times of the falls or did not provide this data. One trial was cluster randomised and so the Poisson regression and negative binomial regression were allowed for the potential within-cluster correlation [15].

Simulations with a very small mean and no overdispersion yielded estimates for all analytical methods that were similar to the negative binomial rate ratio (Fig. 1, Table 2). The percentile-based confidence intervals (CIs) around the estimates are very similar for all the methods (Fig. 1). The full distributions of the results are given in Additional file 1: Figures 1a to 1d. Dichotomising the data into event/no event yielded the largest difference, with an average RR of 0.8088 compared with 0.7941 from the negative binomial regression (an increase of 3 % (Additional file 1: Table S1)), but this is unlikely to change the interpretation of the result. The 95 % CIs become narrower as overdispersion increased but this was less so for the three survival analysis methods (Fig. 1). While the estimates from the other methods seem to be little affected by overdispersion, dichotomising the outcome yields a RR closer to 1 by 0.0481 for high overdispersion (an increase of 8 %), which may impact on clinical interpretation. The HR for time to the first event is the second most different result (0.7998 vs negative binomial 0.7941, 1 % increase), and this increases slightly with overdispersion (to a 4 % increase). Unsurprisingly, the estimates from the Poisson regression and the simple RaR are always exactly the same. Dichotomising the data, and the three survival analysis methods, have the largest standard deviations, suggesting that for any particular data set, the estimates computed by these analytical methods may differ substantially from the negative binomial RaR. The ratio of medians is impossible to calculate for low means as the median is zero if fewer than 50 % of participants in either group do not have the event.

When the mean in the control group is 2, dichotomising the data produces an estimate of effect much closer to 1 than the negative binomial RaR (Fig. 3, Table 4). The percentile-based confidence interval around the dichotomised RR is narrower than the other CIs (Fig. 3). The HR estimates from time to the first event analyses also move closer to 1, and the differences between the HR estimates and the RaR are more variable. Adjusting for multiple survival times, by using the marginal or Andersen-Gill method, seems to lessen the difference, but the differences have relatively large standard deviations, so individual intervention effects may be quite discrepant. The Andersen-Gill method yields estimates that are slightly further from 1 than the negative binomial regression estimates, but usually not enough to influence interpretation. The ratio of medians produces an average estimate closer to 1 than the negative binomial RR. The distribution of the ratio of medians is highly concentrated at discrete values (Additional file 1: Figure S3a, b, and c). The differences have the largest standard deviation across all levels of overdispersion, so that results from individual studies using this method may differ importantly from the negative binomial estimate.

With a mean in the control of 7, dichotomising these data sets yielded, on average, RRs close to 1 (Fig. 4, Table 5). The percentile-based CI is very narrow (Fig. 4). The time to the first event analysis was also suggestive of no intervention effect. The confidence intervals around the negative binomial RaR are wider than those around the simple RaR and the Poisson RaR. The percentile-based confidence intervals for the three survival analyses were wider than the other confidence intervals (Fig. 4). Adjusting for multiple survival times gave estimates closer to the negative binomial RaR, with the marginal model closer than Andersen-Gill. The results for the Andersen-Gill method are even further from one than the negative binomial RaR than with a smaller mean, yielding estimates that are importantly different between the two analytical methods. The standard deviations for the differences between the HR and the estimate from the negative binomial model in both of the marginal and Andersen-Gill methods were large. The ratio of medians yields similar estimates, on average, compared with the negative binomial RaR in scenarios with large means with confidence limits similar to those from the negative binomial regression (Fig. 4). This is despite the distribution of the ratios of medians being very non-normal.

Meta-analysing risk ratios from dichotomised outcomes yielded estimates of intervention effect that differed from those from the other analytical methods (Fig. 5). The other methods all provided very similar results. There was little variation in the precision of the fixed effect estimate across the analytical methods. However, the 95 % CI of the random effects estimate based on the dichotomised outcomes was notably narrower than the 95 % CI for other random effects analyses. 041b061a72


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