C2.4 Poisson Regression

There are several Stata commands which allow the analysis of data with follow-up time and rates.  These are broadly grouped into ‘stand-alone’ commands (e.g. ‘poisson’) and those which are used after the ‘stset’ command which tells Stata that you have ‘survival data’.  You can run a Poisson regression using the ‘stset’ command, followed by the ‘streg’ command, but we are not covering that here. For more information, see the recommended readings at the end of this module.

Poisson regression can be used to model the log(count of events) or the log(rate), since a rate is equivalent to the ‘count of events’ divided by a period of follow-up time.

Here, we show a Poisson regression modelling the log(rate of disease) as the outcome.  In Stata, we can use the ‘poisson’ command as follows:

poisson death currsmoker, e(fu_years) irr

The command poisson works similarly to other regression commands such as logistic and regress, but note we use the ‘e’ option to specify follow-up time (see the help file for more details on this). If you do not use the ‘e’ option to specify the period of follow-up time, then you will be analysing a log(count) [rather than rate].

If we run the ‘poisson’ command above, we get the following output:

poisson death currsmoker,e(fu_years) irr

Iteration 0:   log likelihood = -4087.6737 

Iteration 1:   log likelihood = -4087.6737 

Poisson regression                                      Number of obs =  4,321

                                                        LR chi2(1)    =   5.12

                                                        Prob > chi2   = 0.0237

Log likelihood = -4087.6737                             Pseudo R2     = 0.0006

——————————————————————————

       death |        IRR   Std. err.      z    P>|z|     [95% conf. interval]

————-+—————————————————————-

  currsmoker |   1.184012   .0866404     2.31   0.021     1.025815    1.366605

       _cons |   .0505112   .0013993  -107.77   0.000     .0478417    .0533296

ln(fu_years) |          1  (exposure)

——————————————————————————

Note: _cons estimates baseline incidence rate.

The ‘IRR’ coefficient in the row for ‘currsmoker’ is the rate ratio of death for current smokers compared to non-smokers. This output shows that the rate of death is 18% higher in current smokers compared to non-smokers (RR 1.18, 95% CI: 1.03-1.37). This association is statistically significant (p=0.02).

The row ‘ln(fu_years)’ is always set at 1, and this is correct (even though it looks odd on the output). This refers to the follow-up time, which we set at 1 so that the regression does not multiply the variable denoting the follow-up time by any sort of constant.

Assumptions of Poisson regression

Poisson regression requires that within an exposure group such as smokers or non-smokers, the rate of the event of interest (such as death) stays constant over time, a very strong assumption. A cohort study often involves follow-up over many years and it is unrealistic, because of changes in age, to assume that the rate stays unchanged over follow-up.

One way to deal with this assumption is to split follow-up time using the ‘stsplit’ command in Stata, and model the rate of death in separate time bands of follow-up. The rate of death for smokers vs non-smokers can vary between time bands (such as the first 5 years of follow-up, compared to follow-up over years 6-10), as long as the rate is generally constant within time bands. This is outside the scope of introductory module on Poisson regression to cover, so please see the recommended readings at the end of the module.

•  Question C2.4: Use a Poisson regression to assess if current smoking associated with the rate of death, once adjusted for age group and frailty?