![]() The variables are resp, the number of victims the respondent knows, and race, the race of the respondent (black or white). The data are from a survey of 1308 people in which they were asked how many homicide victims they know. The data are presented in Table 13.6 in section 13.4.3. Just as we did for a geometric random variable, on this page, we present and verify four properties of a negative binomial random variable. A geometric distribution is a special case of a negative binomial distribution with \ (r1\). To illustrate the negative binomial distribution, let’s work with some data from the book, Categorical Data Analysis, by Alan Agresti (2002). Any specific negative binomial distribution depends on the value of the parameter \ (p\). Say our count is random variable Y from a negative binomial distribution, then the variance of Y isĪs the dispersion parameter gets larger and larger, the variance converges to the same value as the mean, and the negative binomial converges to a Poisson distribution. The variance of a negative binomial distribution is a function of its mean and has an additional parameter, k, called the dispersion parameter. ![]() This suggests it might serve as a useful approximation for modeling counts with variability different from its mean. Unlike the Poisson distribution, the variance and the mean are not equivalent. The negative binomial distribution, like the Poisson distribution, describes the probabilities of the occurrence of whole numbers greater than or equal to 0. One approach that addresses this issue is Negative Binomial Regression. Performing Poisson regression on count data that exhibits this behavior results in a model that doesn’t fit well. When we see this happen with data that we assume (or hope) is Poisson distributed, we say we have under- or overdispersion, depending on if the variance is smaller or larger than the mean. A distribution of counts will usually have a variance that’s not equal to its mean. While convenient to remember, it’s not often realistic. ![]() However, one potential drawback of Poisson regression is that it may not accurately describe the variability of the counts.Ī Poisson distribution is parameterized by \(\lambda\), which is both its mean and variance. Poisson regression is a limiting case as approaches zero. For example, we might model the number of documented concussions to NFL quarterbacks as a function of snaps played and the total years experience of his offensive line. Negative binomial regression 1, 2 allows to model overdispersion introducing a shape parameter in the variance specification, so that, for count mean response i, i 1,, n, the inflated variance has the form i + i 2, where > 0. This is a generalized linear model where a response is assumed to have a Poisson distribution conditional on a weighted sum of predictors. When it comes to modeling counts (i.e., whole numbers greater than or equal to 0), we often start with Poisson regression.
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