is: . The second alternate formulation somewhat simplifies the expression by recognizing that the total number of trials is simply the number of successes and failures, that is: . These second formulations may be more intuitive to understand, however they are perhaps less practical as they have more terms. The definition where X is the number of k failures that occur for a given number of r successes. This definition is very similar to the primary definition used in this article, only that k successes and r failures are switched when considering what is being counted and what is given. Note however, that p still refers to the probability of "success". The definition where X is the number of n trials that occur for a given number of r successes. This definition is very similar to definition #2, only that r successes is given instead of k failures. Note however, that p still refers to the probability of "success". The definition of the negative binomial distribution can be extended to the case where the parameter r can take on a positive real value. Although it is impossible to visualize a non-integer number of "failures", we can still formally define the distribution through its probability mass function. The problem of extending the definition to real-valued (positive) r boils down to extending the binomial coefficient to its real-valued counterpart, based on the gamma function: After substituting this expression in the original definition, we say that X has a negative binomial (or Pólya) distribution if it has a probability mass function: Here r is a real, positive number. In negative binomial regression, the distribution is specified in terms of its mean, , which is then related to explanatory variables as in linear regression or other generalized linear models. From the expression for the mean m, one can derive and . Then, substituting these expressions in the one for the probability mass function when r is real-valued, yields this parametrization of the probability mass function in terms of m: The variance can then be written as . Some authors prefer to set , and express the variance as . In this context, and depending on the author, either the parameter r or its reciprocal α is referred to as the "dispersion parameter", "shape parameter" or "clustering coefficient", or the "heterogeneity" or "aggregation" parameter. The term "aggregation" is particularly used in ecology when describing counts