Poisson regression model (created 1999-09-21)
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Dear Professor Mean, I have just received feedback on a manuscript under review in which one reviewer recommended use of Poisson regression. I am not familiar with this technique--when it is appropriate and/or recommended, what assumptions the data must meet, whether the procedure in SAS? SPSS? I would appreciate a reference and/or citation to article(s) in which it has been used. Thanks! -- Denied Denise
I always distrust reviewers who insist on a specific statistical method. It's probably that they used this technique for their dissertation and they think that everyone else should follow their pioneering lead. This is not unlike the saying that when your only tool is a hammer, everything looks like a nail to you.
Is your data a nail or not? Well, Poisson regression assumes that your data follows a Poisson distribution, a distribution that we frequently encounter when we are counting a number of events. The distribution was first used to characterize deaths by horse kicks in the Prussian army. Let's hope that your application is not as unpleasant.
Poisson distributions have three special problems that make traditional (i.e., least squares) regression problematic.
In contrast, the Poisson regression model is not troubled by any of the above conditions. In particular, Poisson regression implicitly uses a log transformation which adjusts for the skewness and prevents the model from producing negative predicted values. Poisson regression also models the variance as a function of the mean.
Alternatives to Poisson regression
There are at least two good alternatives to the Poisson regression model. The negative binomial distribution is also a good model for counts and you can derive this distribution quite naturally as an extension to the Poisson distribution. I think you can get this distribution by placing a prior distribution on the mean parameter of the Poisson. The negative binomial distribution has a variance which is larger than the mean. In contrast, the Poisson distribution has a variance which is equal to the mean.
There are also models that incorporate Poisson probabilities but then allow the probability of a zero to be a bit larger or a lot larger than what Poisson might determine. These are sometimes called ZIP (Zero Inflated Poisson) models. Think of this as a mixture distribution where you choose zero with a certain probability and a Poisson random variable otherwise. This is also a quite natural extension of the Poisson distribution.
If you're just starting out, try the Poisson model first, as it is much simpler than the others.
Poisson regression is a special case of the Generalized Linear Model. This model deserves the name "Generalized" because it also includes traditional regression and logistic regression under its umbrella. If you want to understand Poisson regression, you need to understand the Generalized Linear Model.
The classic reference book is McCullagh P. and Nelder, J.A. (1983) Generalized Linear Models. London England: Chapman and Hall. This is still the best reference, in my humble opinion, in spite of its age.
Can you use SAS or SPSS?
SAS has a procedure GENMOD that will compute a generalized linear model. SPSS does not yet have a module for generalized linear models, but can fit a Poisson regression using the GENLOG procedure. There are a few tricks that you need to worry about in SPSS if your independent variable is continuous or if you have zero counts for some of your data. Details can be found at the SPSS web site:
What if my data is a rate and not a count?
Poisson regression can also be used to analyze rate data. Rates are simply counts divided by a measure like area or time. For example, infection rates are often measured as a number per patient day of exposure. To fit a model using rates, you need to have the original counts (the numerator of the rate) and the measure of time/area (the denominator of the rate). The Poisson model is fit to the counts and uses the log of the denominator as an offset variable. The details are a bit messy, so refer to the appropriate software manual.
Single group count
Suppose we have a hospital floor and we count a total of 25 nosocomial infections in a month. Our best estimate, then, of the infection rate is 25 per month or 240 per year. We can place confidence limits around this value two ways.
For the Poisson distribution, the variance equals the mean. The standard deviation would then be equal to the square root of the mean. So you can compute an approximate 95% confidence interval by going up and down by two standard deviations. The square root of 25, of course, is 5, so an approximate 95% confidence interval would be 15 to 35.
Second, you can compute an exact confidence interval using Poisson probabilities. The details appear at two of the web sites listed below. For 25 events, the exact 95% confidence interval is 16.2 to 36.9.
The are at least two other approaches based on research by Daly [PubMed citation] and Byar (I could not find a good citation).
Single group rate
Let's make the example a bit more complicated. Suppose there were a total of 500 patient days of exposure during that month. Then the rate of nosocomial infections would be 0.05 per patient day or 50 per thousand patient days. We can get confidence intervals for this rate by simply adjusting the confidence intervals above in proportion. So the exact confidence interval for the rate would be
16.2 / 500 = 0.0324, or 32 per thousand patient days
36.9 / 500 = 0.0738 or 74 per thousand patient days.
Two group counts
Suppose we have two groups and we measure a count on both groups. How would you test whether these groups are similar? To make this question more precise, assume that
and that these two variables are independent. We want to test the hypothesis
There are several approaches that work well. The normal approximation statistic
tests the hypothesis that
You can also get a normal approximation for the term
which has an approximate standard deviation of
so the test statistic
tests the hypothesis that
You can also use a conditional argument to show that
is a binomial proportion and the hypothesis
is equivalent to the hypothesis that the two counts come from the same Poisson distribution.
Two group rates
Suppose you have two counts, but they have to be adjusted for different amounts of time or different numbers of patients at risk. Let C1 and C2 are the counts for the two groups and T1 and T2 represent the amount of time at risk Then you would compute rates R1 and R2 as
You can rely on the normal approximation to the Poisson distribution again.
or you can use a log transformation. Interestingly, the denominator for the log of the ratio of rates is identtical to the denominator for the log of the ratio of counts because
The last term is a constant and does not affect measures of variability. Thus, the test statistic would be
Finally, you can apply the same binomial argument to the rates, but there is a slight variation. It is easy to show that
Conditioning on the total, the left hand side is a binomial random variable. But rather than testing whether this binomial proportion is equal to 0.5, you would test for a value that would be larger than 0.5 if T1 > T2 and a value less than 0.5 if T1 > T2. For example if T1 is twice as large as T2 then the two rates are equal if C1 accounts for 2/3 of the total counts and C2 .accounts for 1/3 of the total counts.
Example: In a study of ankle sprains, athletes wearing cushioned soles sustained 41 sprains while athletes wearing uncushioned soles sustained only 27 sprains. There were not an equal number of athletes in each group and these athletes did not play in exactly the same number of games. The researchers defined an exposure as a single athlete wearing shoes of a particular type in a game or in a practice. There were 30,724 exposures among athletes wearing cushioned soles and 13,767 exposures among athletes wearing uncushioned soles.
You can compute the rates per thousand exposures as
41 / 30.724 = 1.33and
27 / 13.767 = 1.96.
The difference in rates is 0.63 and the standard error is 0.43.
Determining sample sizes
[To be added]
Exact confidence interval for Poisson count. Tomas Aragon and Travis Porco. Accessed on October 29, 2002. http://www.medepi.org/epitools/rfunctions/cipois.html
Confidence Intervals for the Mean of a Poisson Distribution. P.D. M. Macdonald. Accessed on October 29, 2002. http://www.math.mcmaster.ca/peter/s743/poissonalpha.html
Denied Denise had a manuscript rejected. The reviewers suggested that she use Poisson regression. Professor Mean explains that you should consider using Poisson regression when you are trying to predict a count or a rate.
Determining the size of a total purchasing site to manage the financial risks of rare costly referrals: computer simulation model. M. O. Bachmann, G. Bevan. Bmj 1996: 313(7064); 1054-7. [Medline] [Abstract] [Full text]
Influence of changing travel patterns on child death rates from injury: trend analysis. C. DiGuiseppi, I. Roberts, L. Li. Bmj 1997: 314(7082); 710-3. [Medline] [Abstract] [Full text]
Generalized Linear Models. McCullagh, P. and Nelder, J.A. (1983). London, England, Chapman and Hall, Inc. ISBN: 0-412-23850-0.
Risk ratio and rate ratio estimation in case-cohort designs: hypertension and cardiovascular mortality. E. G. Schouten, J. M. Dekker, F. J. Kok, S. Le Cessie, H. C. Van Houwelingen, J. Pool, J. P. Vanderbroucke. Stat Med 1993: 12(18); 1733-45. [Medline]
Criticism of a hierarchical model using Bayes factors. J. H. Albert. Statistics in Medicine 1999: 18(3); 287-305. [Medline]
Permutation Tests for Joinpoint Regression with Applications to Cancer Rates. Hyune-Ju Kim, Michael P. Fay, Eric J. Feuer, Douglas N. Midthune. Statistics in Medicine 2000: 19(3); 335-351. [Medline]
Exact confidence interval for Poisson count. Tomas Aragon, Travis Porco. Accessed on 2002-11-27. www.medepi.org/epitools/rfunctions/cipois.html
Coping with extra poisson variability in the analysis of factors influencing vaginal ring expulsions [letter; comment]. CG Demetrio, MS Ridout. Statistics in Medicine 1994: 13(8); 873-76. [Medline]
The comparison of two poisson-distributed observations. K Detre. Biometrics 1970: ?(?); 851-54.
Regression analyses of counts and rates: poisson, overdispersed poisson, and negative binomial models. William Gardner. Psychological Bulletin 1995: 118(3); 392-404. [Medline]
Poisson regression analysis in clinical research. F Kianifard, P. P. Gallo. Journal of Biopharmaceutical Statistics 1995: 5(1); 115-29. [Medline]
Maximum (Max) and Mid-P Confidence Intervals and p Values for the Standardized Mortality and Incidence Ratios. Pandurang M. Kulkarni, Ram C. Tripathi, Joel E. Michalek. American Journal of Epidemiology 1998: 147(1); 83-86. [Medline]
Estimating the ratio of two Poisson rates. Robert Price. Computational Statistics & Data Analysis 2000: 34345-56.
The application of poisson random-effects regression models to the analyses of adolescents; current level of smoking. Ohidul Siddiqui. Preventive Medicine 1999: 2992-101. [Medline]
Negative binomial and mixed Poisson regression. JF Lawless. The Canadian Journal of Statistics 1987: 15(3); 209-25.
Linear and nonlinear techniques for the deconvolution of hormone time-series. G. De Nicolao, D. Liberati. IEEE Trans Biomed Eng 1993: 40(5); 440-55.
Additional power computations for designing comparative Poisson trials. C. C. Brown, S. B. Green. American Journal of Epidemiology 1982: 115(5); 752-8. [Medline]
Power computations for designing comparative poisson trials. M Gail. Biometrics 1974: 30(?); 231-37.
A more powerful test for comparing two Poisson means. K. Krishnamoorthy, Jessica Thomson, University of Louisiana at Lafayette. Accessed on 2003-02-10. www.mathpreprints.com/math/Preprint/krishna/20021020/1/?=&coll=Selection
Power in comparing Poisson means: I. One-sample test. LS Nelson. Journal of Quality Technology 1991: 23(1); 68-70.
Power in comparing Poisson means: II. Two-sample test. LS Nelson. Journal of Quality Technology 1991: 23(2); 163-66.
Sample size for Poisson regression. DF Signorini. Biometrika 1991: 78(2); 446-50.
Application of sample survey methods for modelling ratios to incidence densities. L. M. Lavange, L. L. Keyes, G. G. Koch, P. A. Margolis. Stat Med 1994: 13(4); 343-55.