Do alternative methods for analysing count data produce similar estimates? Implications for metaanalyses
 Peter Herbison^{1}Email author,
 M. Clare Robertson^{2} and
 Joanne E. McKenzie^{3}
DOI: 10.1186/s136430150144x
© Herbison et al. 2015
Received: 6 July 2015
Accepted: 26 October 2015
Published: 17 November 2015
Abstract
Background
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 metaanalysis.
Methods
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.
Results
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.
The results of the case study of individual data from eight trials evaluating exercise programmes to prevent falls in older people supported the simulation study findings.
Conclusions
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.
Keywords
Count outcomes Metaanalysis Methods of analysis RatesBackground
Often the outcomes measured in medical research are count outcomes. Typically, these measure the number of times a particular event happens to an individual in a defined period. Examples of count outcomes include the number of falls by the individual, the number of asthma exacerbations or the number of incontinence episodes. These outcomes are commonly measured in randomised controlled trials (RCTs) to determine the effect of an intervention.

A simple rate ratio—the ratio of the number of events per person time at risk in each of the treatment groups.

A rate ratio calculated from the Poisson regression family—such as Poisson and negative binomial.

A risk ratio after the data are dichotomised into those with and without the event.

A hazard ratio using the time to the event—either the time to the first event or using a method that copes with multiple times to events.

A difference in means that treats the data as continuous and is compared using a t test or linear regression. More recently, the ratio of means has been used [4]. These analyses cause few problems for count outcomes with a high mean, such as pulse rate, as the Poisson distribution with a high mean approximates a normal distribution. In practice, however, this approach is often used on data with lower means.

A difference in medians tested by a nonparametric test such as the Wilcoxon rank sum test or the ratio of medians.
The variety of analytic methods used in RCTs with count outcomes causes difficulties when carrying out a metaanalysis. 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 metaanalysis.
This paper describes a simulation study designed to see whether mixing the results of different methods of analysis could give reasonable answers in a metaanalysis.
Falling is a major health problem for older people, with approximately 30 % of people over the age of 65 falling each year, with many falls resulting in injury and hospitalisation. The 2009 Cochrane systematic review “Interventions for preventing falls in older people living in the community” included 43 trials that assessed the effect of exercise programmes [5]. The two primary outcomes in this review were the rate of falls and the proportion of fallers. Twentysix of the 43 studies contributed to the rate of falls metaanalysis, and 31 to the number of fallers. Some studies could not be used because of the way the data were analysed and presented. We asked for individual patient data from randomised trials included in this systematic review, analysed them in different ways and compared the resulting metaanalyses.
Methods
The simulation study
Means of the poisson distributions used in the simulations
Base values  Overdispersed values (0 %—no overdispersion, 20 %—moderate overdispersion and 40 %—high overdispersion from these Median, 2.5 and 97.5 centilesdistributions)  

Control  Treatment  Control  Treatment  
Very low  0.2  0.15  0.6  0.4 
Low  0.5  0.35  0.8  0.6 
Medium  2  1.5  3  2.5 
High  7  5  10  7 
The treatment period was set at 365 days. About 20 % of observations were randomly chosen to be lost to followup after a randomly chosen number of days. The number of days to the events experienced was chosen from a uniform distribution. The simulation started by choosing the sample size and then the number of events in each arm and the followup period. Then the time to each of the events was generated. This was followed by the different analyses, and the results were stored on a file. There were 10,000 replications of each simulation scenario. Stata code for one of the simulations is available in the Additional file 1.
 1.
Simple rate ratio (RaR) calculated from count data (the ratio of the number of events divided by person time of followup in the intervention arm to the control arm)
 2.
 3.
Rate ratio estimated from the negative binomial regression [7]
 4.
Risk ratio (RR) calculated from dichotomised data
 5.
Hazard ratio (HR) estimated from survival analysis for time to the first event [8]
 6.
Hazard ratio for multiple events estimated using the marginal model [9]
 7.
 8.
The ratio of the mean number of events [4]
 9.
The ratio of the median number of events.
The negative binomial regressions used the meandispersion model. The marginal model for repeated time to events assumes that individuals are at risk for every event from the time of study entry, whereas the AndersenGill method assumes that people are at risk for the second event only from the time they had the first event. People with n events have n + 1 records in the file, n ending in the event and the n + 1th censored at the end of the followup. Adjustment is made for multiple records per person in both of these models. See Robertson et al. [11] for more details.
Where possible, information about the 20 % of observations lost to followup was included in the analysis [12]. Simple rate ratios were calculated using person days of followup, and the Poisson and negative binomial regression models allowed for varying lengths of followup through inclusion of an offset in the model. The survival models allow for varying lengths of followup through censoring. However, it is not possible to allow for followup time for intervention effect estimates 4, 8 and 9.
Ratios of means and medians (estimates 8 and 9) were used in preference to differences in means and medians, since these ratio measures were comparable with the other estimates of intervention effect although they would not usually be used in practice.
The results of the simulations were examined in several ways. Histograms of the results were produced, with the median, 2.5th and 97.5th centiles of these distributions plotted using forest plots. In addition, to examine the exchangeability of the methods, we compared the estimated intervention effect from each method of analysis with the estimated rate ratio from the negative binomial regression using two metrics. The first metric compared the mean of the differences in the two estimates (for ease of interpretation) across the 10,000 replicates, while the second metric compared the mean of the ratios of the two estimates (since the underlying scale is relative). The rate ratio from the negative binomial regression model was chosen as the reference estimate. The negative binomial model is a more general model compared with the Poisson regression model that relaxes the strong assumption that the underlying rate of the outcome is the same for each included participant. The negative binomial model requires the additional estimation of a dispersion parameter (which will make it less efficient than the Poisson model in the absence of overdispersion); however, the model is theoretically more plausible [11, 13, 14]. The mean and standard deviation of the differences in the estimates are presented in the “Results” section and of the ratios in the Additional file 1. The interpretation of a positive mean difference is that the estimate for the comparison method was closer to 1 compared with the estimate from the negative binomial regression. Mean differences enable a judgement about whether there is, on average, an important difference between the estimates calculated from each of the analytical methods and the comparison method. The standard deviation of the differences gives an indication of how close the average result is to the negative binomial rate ratio, with large values indicating that the estimates are not always close and that for any particular trial, the use of an alternative method may result in an importantly different estimate of intervention effect.
Stata V13 (StataCorp, College Station, TX, USA) was used for all simulations.
The case study
All of the corresponding authors of the trials that contributed to the comparison of multiple component group or homebased exercise programmes versus no exercise programme in the 2009 Cochrane systematic review were invited by email to contribute to this part of the study.
The email provided authors with information and preliminary results of the simulation study and informed them of the purpose of this empirical study. Authors were then asked whether they would consider taking part in the empirical study. Those opting to participate in the empirical study were sent a second email and given the option of undertaking a series of analyses themselves (and contributing the results) or providing deidentified individual participant data sets, which we would analyse. All authors who agreed to take part chose the latter option.
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 nonfallers), (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 withincluster correlation [15].
For each analytical method, estimates of the intervention effect were pooled using both fixed and random effects metaanalytic models using the metan routine in Stata [16]. Fixed effect metaanalyses used the method of Mantel and Haenszel [17], and the random effects models used the method of DerSimonian and Laird [18].
Results
The following are results for the simulations with the approximately 30 % reduction in treatment effect.
Simulations with a very low mean
Simulation results for the very low mean (control 0.2, treatment 0.15)
No overdispersion  Moderate overdispersion  High overdispersion  

RaR^{a} = 0.79  RaR^{a} = 0.73  RaR^{a} = 0.72  
Mean difference^{b}  SD  Mean difference  SD  Mean difference  SD  
Dichotomise RR  0.01  0.09  0.05  0.09  0.05  0.09 
Poisson RaR  <0.01  0.01  <0.01  0.01  <0.01  0.01 
Simple RaR  <0.01  0.01  <0.01  0.01  <0.01  0.01 
Time to the first event HR  0.01  0.09  0.02  0.10  0.02  0.09 
Marginal model HR  0.01  0.09  0.02  0.09  0.02  0.09 
AndersenGill HR  <0.01  0.09  0.02  0.09  0.02  0.09 
Ratio of means  <0.01  0.03  0.02  0.03  <0.01  0.03 
Ratio of medians  Not possible as both medians are zero 
Simulation scenarios with a low mean
Simulation results for the low mean (control 0.5, treatment 0.35)
No overdispersion  Moderate overdispersion  High overdispersion  

RaR^{a} = 0.71  RaR^{a} = 0.73  RaR^{a} = 0.74  
Mean difference^{b}  SD  Mean difference  SD  Mean difference  SD  
Dichotomise RR  0.05  0.08  0.05  0.08  0.06  0.08 
Poisson RaR  <0.01  0.06  <0.01  0.01  <0.01  0.01 
Simple RaR  <0.01  0.06  <0.01  0.01  <0.01  0.01 
Time to the first event HR  0.01  0.08  0.01  0.09  0.02  0.09 
Marginal model HR  0.01  0.08  0.01  0.08  0.01  0.08 
AndersenGill HR  0.01  0.08  0.01  0.08  0.01  0.08 
Ratio of means  <0.01  0.03  <0.01  0.03  <0.01  0.03 
Ratio of medians  Only possible for 292, 717 and 2087 of the 10,000 simulations, respectively 
Simulation scenarios with a moderate mean
Simulation results for the moderate mean (control 2, treatment 1.5)
No over dispersion  Moderate overdispersion  High overdispersion  

RaR^{a} = 0.75  RaR^{a} = 0.77  RaR^{a} = 0.79  
Mean difference^{b}  SD  Mean difference  SD  Mean difference  SD  
Dichotomise RR  0.15  0.07  0.14  0.08  0.14  0.08 
Poisson RaR  <0.01  0.01  <0.01  0.01  <0.01  0.01 
Simple RaR  <0.01  0.01  <0.01  0.01  <0.01  0.01 
Time to the first event HR  0.07  0.11  0.06  0.11  0.07  0.12 
Marginal model HR  0.02  0.09  0.01  0.09  0.01  0.10 
AndersenGill HR  −0.01  0.08  −0.02  0.09  −0.02  0.09 
Ratio of means  <0.01  0.03  <0.01  0.04  <0.01  0.04 
Ratio of medians  0.05  0.19  0.09  0.17  0.07  0.14 
Simulation scenarios with a large mean
Simulation results for the high mean (control 7, treatment 5)
No overdispersion  Moderate overdispersion  High overdispersion  

RaR^{a} = 0.71  RaR^{a} = 0.70  RaR^{a} = 0.70  
Mean difference^{b}  SD  Mean difference  SD  Mean difference  SD  
Dichotomise RR  0.29  0.06  0.30  0.07  0.30  0.07 
Poisson RaR  0.01  0.02  0.01  0.03  0.01  0.03 
Simple RaR  0.01  0.02  0.01  0.03  0.01  0.03 
Time to the first event HR  0.28  0.14  0.29  0.14  0.30  0.15 
Marginal model HR  0.03  0.10  0.04  0.10  0.04  0.10 
AndersenGill HR  −0.07  0.09  −0.05  0.10  −0.05  0.10 
Ratio of means  0.01  0.04  0.01  0.05  0.01  0.05 
Ratio of medians  0.01  0.06  0.02  0.07  0.01  0.06 
Convergence
Rates of nonconvergence for negative binomial regression from the 10,000 simulations
Very low mean (%)  Low mean (%)  Moderate mean (%)  High mean (%)  

No overdispersion  8.15  6.99  1.37  0.04 
Moderate overdispersion  3.77  5.28  0.33  0 
High overdispersion  2.62  4.29  0.20  0 
The ratio of medians was impossible to calculate for the very low mean, and for the low mean, the ratio of medians was only possible for 292, 717 and 2087 of the 10,000 simulations for no, moderate and high overdispersion, respectively.
The empirical case study
Characteristics of studies in the empirical study
Study name  Followup  Interventions  Number in arm  Number of falls 

Barnett et al. 2003 [19]  12 months  Group exercise with home exercise plan  76  40 
No exercise  74  70  
Campbell et al. 1997 [20]  24 months  Individualised supervised home exercise  116  88 
No treatment  117  152  
Campbell et. al 1999 [21]  10 months  Individualised supervised home exercise  45  22 
No treatment  48  35  
Green et al. 2002 [22]  9 months  Individualised physiotherapy exercise in the community  85  74 
No treatment  85  51  
Lord et al. 1995 [23]  12 months  Exercise classes  75  44 
No treatment  94  75  
Lord et al. 2003 [19]  12 months  Group exercise  259  174 
No treatment or relaxation exercises  249  211  
Robertson et al. 2001 [24]  12 months  Individualised supervised home exercise  121  80 
No exercise  119  109  
Skelton et al. 2005 [25]  36 weeks  Individualised supervised group and home exercise  50  66 
Pamphlet on home exercises  31  119 
No difference between groups
When there was no difference in the effect of treatment in the groups, all methods gave very similar results for all scenarios.
Discussion
The results of this study suggest that it may well be possible in many situations to combine in a metaanalysis the estimates of intervention effects for count outcomes analysed in various ways, as the results from the different analysis methods were very similar. Apart from a few instances, most analyses gave estimates that were on average close to the RaR from a negative binomial regression. Further, examination of the range of data from the simulations showed that the confidence intervals of most of the methods were similar. Therefore, pooling intervention estimates calculated by different methods is likely to be generally reasonable. This has been shown using both simulations and actual data from a metaanalysis of RCTs. When events were rare, or there was no treatment effect, all methods of analysis provide a very similar estimate of intervention effect with similar variation. An exception to this is the ratio of medians, which is impossible to calculate unless both groups have more than 50 % of participants with events. As events become more common, dichotomising the results into those with the event and those without increasingly loses the ability to discriminate between treatments, and the confidence interval becomes narrower. Intuitively, as events become more common, it is likely that all, or almost all, of the participants will experience one or more events. Similarly, time to the first event loses the ability to discriminate with increasing event rates, but this happens more slowly than with dichotomising the data.
Poisson regression and negative binomial regression models gave very similar results for the RaR, even when there was a significant amount of overdispersion. This was expected given these distributions have the same expected value [13, 26]. The standard error of the RaR estimated from Poisson regression will be too small in the presence of overdispersion, which will have implications for the weights in metaanalytic models. In this simulation, the underestimation of the standard error was only slight but was most noticeable with both a high mean and a lot of overdispersion. Trials that are analysed using Poisson regression in the presence of overdispersion will receive too much weight in the metaanalysis. The impact of not allowing for overdispersion, and subsequent underestimation of the variance of the intervention effect, was evident when comparing the fixed effect metaanalysis confidence intervals calculated from using Poisson regression compared with the negative binomial regression in the empirical study.
Adjusting the survival analyses for multiple events also gave estimates close to those from the negative binomial regression, although the confidence intervals were wider, especially as the mean increases. An exception to this was the AndersenGill method that gave an estimate of the HR that was, on average, slightly further from 1 than the negative binomial RaR. The difference between the estimates increases as the mean increases, which may lead to a different interpretation of the intervention effect and make it unreasonable to combine AndersenGill HR estimates with those estimated from the negative binomial regression. All survival models in these simulations make the assumption of proportional hazards. In our simulations, the proportional hazards assumption is likely to be true because of the way the data was generated but may not be so for any particular RCT.
The ratio of medians is clearly inappropriate where the event rate is low as the medians in one or both groups are likely to be zero. As the event rate increases, the average difference between estimates calculated from the ratio of medians and negative binomial regression is small. However, in any particular trial, the difference could be large, as indicated by the large standard deviation of the differences. Especially when the mean is low, the distribution of the ratio of medians is highly concentrated at discrete values but becomes smoother as the mean increases. This could lead to different variances compared with the other models. In practice, it is difficult to use the ratio of medians as the standard error cannot be computed from commonly reported statistics. There is a formula for the 95 % confidence interval of the ratio of medians, but calculation requires the original data [27]. An alternative to using this formula, but still requiring the original data, is to use a method such as bootstrapping to compute the standard errors. More commonly, trial authors will report one of the other effect measures, such as the simple RaR (or at least the raw data that allows this ratio to be calculated). Calculation of the ratio of means is likely to be possible from many studies where the means are reported. There is a standard formula that calculates an approximate standard error from the mean, standard deviation and number of individuals in each of the arms of the study [4].
It is perhaps unsurprising that the estimates and their distributions are similar. The simple RaR and Poisson regression estimate the same parameter; any differences are likely to be due to rounding errors, as the Poisson regression requires more calculations to be performed. The expected values of the estimates from Poisson regression and negative binomial regression are the same. Survival analysis and Poisson regression estimate the same parameter when the baseline hazard is constant [28], which in these simulations will hold, and should for many RCTs. The ratio of means is the coefficient from a linear regression of group assignment on the log of the count outcomes. This is similar to the coefficient in a Poisson regression, except that linear regression does not cope well with zero scores in the outcome, the error structure is different and it is unable to adjust for different followup periods.
We chose the negative binomial model as the reference model as it seems appropriate for this sort of data, especially in the presence of overdispersion. This does not allow for the estimation of bias in any of the methods, as we do not have the “true” value. As the question we wanted to answer was whether the results of the different methods could be combined in a metaanalysis looking at the difference from one of the methods was more appropriate.
There are other possibilities for the analysis of count outcomes, such as zero inflated Poisson, zero inflated negative binomial and Poisson regression with robust errors which allows for overdispersion by relaxing the requirement that the mean and variance are equal. However, we did not evaluate these methods since they are not used very often in practice.
Previously, it has been established that, to prevent bias, it is important to account for the length of exposure, which may differ because of dropouts that are not missing at random [12, 29]. The simple rate ratio, Poisson regression and negative binomial regression are all able to adjust for varying followup times, as do the survival analysis methods. Thus, it is surprising that the ratio of means and the ratio of medians yield similar effect estimates to those estimated from the negative binomial regression. This may be a result of the data sets generated assuming similar attrition across groups, and the missing data mechanism being participants missing completely at random. Under various scenarios (e.g. varying attrition rates and different missing data mechanisms (e.g. not missing at random)), the ratio of means and ratio of medians may yield effect estimates that differ compared with those estimated from negative binomial regression.
The choice of a uniform distribution to pick the times that the events occurred may not be the most realistic option. Events may be more likely to occur closer together or further apart than a uniform distribution would give. They also may not be independent of each other, particularly as having an event may increase or decrease the time to the next event and this may depend on the nature of the event.
The fact that intervention effect estimates from RCTs using different analytical methods can, in some circumstances, be pooled in a metaanalysis should not make the method of analysis a random choice in any particular trial. The analysis should match the hypothesis and the study design. We have previously advocated for the use of negative binomial regression in evaluating falls prevention studies [11], as have others for this type of data [14]. Negative binomial regression allows for all events to be included (thus using all information) and the length of exposure to vary and more appropriately accounts for overdispersed data. But it does treat individuals who have multiple events in quick succession, and then none for the rest of the followup period the same as those who have the same number of events evenly spread out throughout the period.
We have concentrated on the point estimates, with no detailed examination of the variances of these. Thus, more questions remain to be answered about metaanalysis of count data outcomes analysed using alternative methods. The impact of the trial analytical method on metaanalytic intervention effects, their standard errors and heterogeneity needs to be investigated. The impact is likely to vary by the chosen metaanalysis model (random effects versus fixed effect), so any investigation should examine both models. This simulation only examined data that were missing completely at random. This is overly simplistic, and research examining the impact of different missing data mechanisms and how these interact with the trial and metaanalysis methods would be valuable.
The focus of this paper is on RCTs, but these methods of analysis are used for other types of studies (nonrandomised trials, observational studies), which may also be included in metaanalyses. For study types other than RCTs, it would be critical to examine the impact of covariates and missing data, in addition to the examination we have undertaken in this paper.
Conclusions
We have shown in this simulation study, that analysing outcomes using different methods yielded estimates of intervention effect that were similar in both average estimates and variances. When the mean of the counts is more than 0.5, analyses using dichotomisation or time to the first event should not be pooled with intervention effects estimated from other methods. Dichotomising, when the event rate is at this level or higher, may not be an appropriate method for analysing individual studies as it is likely to underestimate treatment differences as well as giving confidence intervals that are too narrow.
Abbreviations
 CI:

confidence interval
 HR:

hazard ratio
 RaR:

rate ratio
 RCT:

randomised controlled trial
 RR:

risk ratio
Declarations
Acknowledgements
This study was funded, in part, by a Health Research Council of New Zealand Grant (12256). We are grateful to Emeritus Professor A. John Campbell for his valuable contribution to this paper and to the trial authors who provided falls data from their randomised controlled trials for these analyses.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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