{"id":499,"date":"2018-09-25T19:05:10","date_gmt":"2018-09-26T02:05:10","guid":{"rendered":"https:\/\/blogs.ubc.ca\/datawithstata\/?page_id=499"},"modified":"2019-05-08T18:40:20","modified_gmt":"2019-05-09T01:40:20","slug":"poisson-regression-2","status":"publish","type":"page","link":"https:\/\/blogs.ubc.ca\/datawithstata\/home-page\/regression\/poisson-regression-2\/","title":{"rendered":"Regression with Count Variable"},"content":{"rendered":"

When the dependent variable is a non-negative count variable, the standard OLS<\/abbr> regression is no longer valid. Typically, the Poisson regression or some variation of it is used to analyze such count data. Here we will discuss the Poisson<\/strong>, Negative Binomial<\/strong>, and Zero-Inflated Negative Binomial<\/strong> regressions along with their STATA<\/abbr> implementations.<\/p>\n

Poisson Regression<\/h3>\n

Poisson regression fits models of the number of occurrences (counts) of an event where it is assumed that the number of occurrences follow a Poisson distribution. The Poisson distribution has been applied to diverse events with the following basic assumptions:<\/p>\n

    \n
  1. There is a quantity called the incidence rate<\/em> that is the rate at which events occur, e.g., 5 per second, 20 per square metre, etc.<\/li>\n
  2. The incidence rate can be multiplied by exposure<\/em> to obtain the expected number of observed events, e.g., a rate of 5 per second multiplied by 30 seconds means that 150 events are expected.<\/li>\n
  3. Over very small exposures, the probability of finding more than one event is small compared with the exposure length.<\/li>\n
  4. Non-overlapping exposures are mutually independent.<\/li>\n<\/ol>\n

    With these assumptions, to find the probability of k<\/em> events in an exposure of size E<\/em>, one divides E<\/em> into n <\/em>sub-intervals, and approximate the answer as the binomial probability of observing k<\/em> successes in n<\/em> trials. By letting n <\/em>to be arbitrarily large, one obtains the Poisson distribution.<\/p>\n

    In the Poisson regression model, the incidence rate of the j<\/em>th<\/sup> observation is assumed to be given by rj<\/sub> = exp(b0<\/sub>+b1<\/sub>x1j<\/sub>+b2<\/sub>x2j<\/sub>)\u00a0<\/em>if\u00a0x1<\/sub> and\u00a0x2<\/sub> were the two explanatory variables. If Ej<\/sub><\/em> is the exposure, the expected number of events will be Cj<\/sub>\u00a0= exp[ln(Ej<\/sub>)+ b0<\/sub>+b1<\/sub>x1j<\/sub>+b2<\/sub>x2j<\/sub>]<\/em>.<\/p>\n

    The parameters are computed using a log-likelihood function which is:<\/p>\n

    lnL<\/em> = \u2211 [ yj<\/sub>*\u03b2*x – rj<\/sub> \u2013 lnyj<\/sub>!]<\/p>\n

    Example:<\/span><\/h4>\n

    We have data on the number of deaths, average age in months and the total number of person-years of exposure to risk\u00a0 of the children between birth and age 10 for three cohorts of children born in 1941\u201349, 1960\u201367, and 1968\u201376.\u00a0 We want to understand how the deaths of the children changes with age of the children. This falls under running a regression with Count variable and\u00a0\u00a0a Poisson regression can be implemented (to install the data in Stata<\/abbr>, type: webuse\u00a0<\/strong>rod93, clear). Before we run a Poisson regression, generate logexposure<\/em> as natural log of exposure.<\/p>\n

    <\/a>poisson<\/strong> deaths age_mos, exposure(<\/strong>logexposure<\/em>)
    \n<\/strong><\/p>\n

    \"\"<\/p>\n