Notes on Poisson Regression and Some Extensions

packages. They are very useful for particular applications. When this model is fit to these data,the overdispersion parameter is no longer statistically significant.Exercise 6: Fit a common model specification (i.e., same set of variables) for all these models andcompare on the basis of likelihood ratio tests (when possible) and BIC criteria (for non-nestedmodels). Which model emerges as best for this data.Models for Rates in Time. Let t 1 , t 2 , . . . , t n denote the waiting times until an event occursfor a sample of n individuals, with distribution function F (t) = Pr(T < t) and probability densityfunction f(t), where we assume that T is a continuous random variable (T > 0). The hazard rate,intensity function, or failure rate is denoted by µ(t). The hazard rate is the instantaneousprobability of an event in the interval [t, t + ∆t], given that the event has not already occurredbefore the beginning of the interval. More formally, the hazard rate is the limit of a conditionalprobability (or transition probability)1µ(t) = lim Pr[t ≤ T < t + ∆t | T ≥ t]. (4)∆t→0 ∆t∆t>0The probability of surviving the interval [t, t + ∆t] is given by the survival function.S(t) = Pr[T > t] = 1 − F (t) =∫ ∞tf(u)du. (5)If we assume that the random variable for waiting-time (T ) follows an exponential distributionwith density function, f(t) = µ exp(−µt), the expressions for the survival function in Eq. 5 isS(t) = exp(−µt). The expression for the hazard rate of Eq. 4 is defined is the ratio, µ(t) =f(t)/S(t) = µ. For the exponential distribution, this implies a constant hazard over time (i.e., notdepending on t).Consider a Poisson variate d that denotes the number of events occurring in an observationwindow (exposure period) of length t. If events can occur repeatedly over time, and if the timesbetween events are independent exponential variables with mean time to event occurrence is givenby E(T ) = 1/µ (i.e., a time-homogeneous Poisson process), then the probability of d events in atime-interval of length t follows a Poisson distribution,Pr(d | µ, t) = (tµ)d exp(−tµ). (6)d!The mean number of events in time-interval t is, λ = tµ. For a sample of size n, we can model theconditional mean count as a function of independent variables, so that for the ith individual, theexpected number of events in time-interval t i isµ i = t i λ i = t i exp(x ′ iβ).The likelihood is a product of the individual Poisson probabilities in Eq. 6, and is proportional toL =n∏(t i λ i ) d iexp(−t i λ i ), (7)i=1which is the kernel of a Poisson likelihood.It will often be the case that we do not observe the event times for some individuals in thesample. In this case, the event times are said to be right censored, and the Poisson variate is16