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## Poisson regression

Stata’s poisson fits maximum-likelihood models of the number of occurrences (counts) of an event. In a Poisson regression model, the incidence rate for the jth observation is assumed to be given by

   r_j = exp(b_0 + b_1*x_(1,j) + ... + b_k*x_(k,j)


If E_j is the exposure, the expected number of events C_j will be

   C_j = E_j * r_j
= exp[ ln(E_j) + b_0 + b_1*x_(1,j) + ... + b_k*x_(k,j) ]


This is the model fitted by poisson. E_j may be specified or, if not specified, is assumed to be 1.

. poisson deaths smokes i.agecat, exposure(pyears) irr

Iteration 0:   log likelihood = -33.823284
Iteration 1:   log likelihood = -33.600471
Iteration 2:   log likelihood = -33.600153
Iteration 3:   log likelihood = -33.600153

Poisson regression                                Number of obs   =         10
LR chi2(5)      =     922.93
Prob > chi2     =     0.0000
Log likelihood = -33.600153                       Pseudo R2       =     0.9321

deaths          IRR   Std. Err.      z    P>|z|     [95% Conf. Interval]

smokes     1.425519   .1530638     3.30   0.001     1.154984    1.759421

agecat
45-54      4.410584   .8605197     7.61   0.000     3.009011    6.464997
55-64       13.8392   2.542638    14.30   0.000     9.654328    19.83809
65-74      28.51678   5.269878    18.13   0.000     19.85177    40.96395
75-84      40.45121   7.775511    19.25   0.000     27.75326    58.95885

_cons     .0003636   .0000697   -41.30   0.000     .0002497    .0005296
ln(pyears)            1  (exposure)



The syntax of all estimation commands is the same: the name of the dependent variable is followed by the names of the independent variables, which are followed by a comma and any options. In this case, we controlled for the exposure (person-years recorded in the variable pyears) and asked that results be displayed as incidence-rate ratios rather than as coefficients.

svy: poisson can be used to analyze complex survey data, and the mi estimate: poisson command performs estimation using multiple imputations. Also, Stata provides Cox regression, exponential, Weibull, and other parametric survival models, as well as logistic regression, and all can be used to analyze complex survey data or to perform estimation using multiple imputations.