To the Editors

For statistical analysis of chronic obstructive pulmonary disease (COPD) exacerbations, one of the key arguments against the Poisson model with overdispersion correction in the study by Keene et al. 1 is that only the negative binomial model takes into account variability across the patients. However, the Poisson model with overdispersion can also be viewed as equivalent to each individual having their own rate of exacerbations and the rate varying across the population following a gamma distribution 2, 3. Another key argument by Keene et al. 1 against the Poisson model with overdispersion is that it assumes a common mean for the entire population and weighs each unit of time equally. The negative binomial model also assumes a common mean for the entire population but weighs each unit of time differently. The variance for the Poisson model with overdispersion is a linear function of the mean while the variance for the negative binomial model is a quadratic function of the mean. Therefore, large and small counts are weighted differently in the two models but one set of weights is not necessarily better than the other 4. Because of the comparable complexity of these two models, one should compare model fitting to select a better model. It is possible that for the specific trials the authors described 1 the negative binomial model fitted better; however, in other trials the Poisson model with overdispersion would fit better. In a trial of short duration, very few patients are expected to have exacerbations and the zero-inflated Poisson model 5 is likely to fit the data better than either of the two models previously described.

One of the problems with the analysis of multiple exacerbations is that the exacerbations are not of similar duration or severity. Some are resolved within a few days while others may continue for several months. In a trial of fixed duration, the number of exacerbations may depend on the length of exacerbations. If a patient has an exacerbation of long duration in the early stages of the trial there is less time remaining for the patient to have additional exacerbations. This could be erroneously considered as an advantage over a patient who has two exacerbations of short duration towards the end of the trial. Another issue with analysing multiple exacerbations using the Poisson or negative binomial model is that both models implicitly assume a constant rate of exacerbation over time, which is highly questionable as the reoccurrence of exacerbations depends on how a patient recovers from previous events. Keene et al. 1 question the assumption of proportional hazard in the analysis of time-to-first exacerbation but in fact the underlying assumption of the Poisson or negative binomial model is a Poisson process, i.e. for a patient the occurrences of exacerbations are independent and the time interval between two adjacent exacerbations follows an exponential distribution. Such assumption further implies not only proportional hazard but also a constant baseline hazard. Overall, the assumptions behind the Poisson or negative binomial model are much stronger than the assumption of proportional hazard. Therefore, in terms of relying on less stringent assumptions, time-to-first event is superior to analysis of number or rate of COPD exacerbations to compare clinical interventions. A clinical intervention that reduces the risk of the first moderate-to-severe COPD exacerbation should be of great clinical value. Subsequent exacerbations may depend on how the first exacerbation is treated and, therefore, the effect of the study drug would be confounded with the medical treatment of the first exacerbation.

In summary, the key difference between the Poisson model with overdispersion and the negative binomial model is the form of mean-variance function. The two models have different weighting schemes, but one is not always superior to the other. The time-to-first event analysis methods assume constant hazard ratio between treatments over time but that is a much weaker assumption than the assumptions for the Poisson and negative binomial models. Time-to-first exacerbation is a much cleaner end-point than number of exacerbations, and should be considered as the most appropriate way to analyse chronic obstructive pulmonary disease exacerbations.

Statement of interest

Statements of interest for D. Liu and S. Menjoge can be found at www.erj.ersjournals.com/misc/statements.shtml