Negative Binomial distribution

Format: NegBin(s, p)

 

The Negative Binomial distribution estimates the number of failures there will be before s successes are achieved where there is a probability p of success with each trial. Examples of the Negative Binomial distribution are shown below:

 

Uses

Binomial examples

The Negative Binomial distribution has two applications for a binomial process:

• The number of failures in order to achieve s successes = NegBin(s,p);

• The number of failures there might have been when we have observed s successes = NegBin(s+1,p)

The first use is when we know that we will stop at the sth success. The second is when we only know that there had been a certain number of successes.

For example, a hospital has received a total of 17 people with a rare disease in the last month. The disease has a long incubation period. There have been no new admissions for this disease for a fair number of days. The hospital knows that people infected with this problem have a 65% chance of showing symptoms. It is also known that all people with symptoms will turn up at the hospital. They are worried about how many people there are infected in the outbreak who have not turned up in hospital and may therefore infect others. The answer is NegBin(17+1,65%). IF we knew (we don't) that the last person to be infected was symptomatic, the answer would be NegBin(17,65%). The total number infected would be 17+NegBin(17+1,65%).

Poisson example

The Negative Binomial distribution is frequently used in accident statistics and other Poisson processes because the Negative Binomial distribution can be derived as a Poisson random variable whose rate parameter lambda is itself random and Gamma distributed, i.e.:

Poisson(Gamma(a,b)) = NegBin(a, 1/(b+1))

The Negative Binomial distribution therefore also has applications in the insurance industry, where for example the rate at which people have accidents is affected by a random variable like the weather, or in marketing. This has a number of implications: it means that the Negative Binomial distribution must have a greater spread than a Poisson distribution with the same mean; and it means that if one attempts to fit frequencies of random events to a Poisson distribution but find the Poisson distribution too narrow, then a Negative Binomial can be tried and if that fits well, this suggests that the Poisson rate is not constant but random, and can be approximated by the corresponding Gamma distribution (see here ).

In most software implementations, the a parameter for the Negative Binomial is restricted to being integer to better correspond with its relationship to the binomial process. A more general statement without this restriction is that:

Poisson(Gamma(a,b)) = Pólya(a, b)

Comments

The Negative Binomial distribution is affected by the same restrictions as those described for the Geometric i.e. p remains constant and cannot be altered by knowledge or skill gained in the tries and the distribution assumes that as many tries will be made as are found necessary to achieve s successes: it makes no allowance for those who would cut their losses and give up. The Pascal distribution, or Binomial Waiting-Time distribution, is a Negative Binomial distribution shifted s units along the x-axis, i.e. a distribution that runs from s to infinity.

The NegBin is often a good approximation to the Inverse Hypergeometric (s<<D), and is itself sometimes approximated by a Normal (s large) or a Gamma distribution (p very small).

The Negative Binomial distribution gets its name because the equation that is produced from an expansion of the expression [Q-P]-s equates to its terms.

An alternative formulation of the Negative Binomial distribution has the distribution modeling the total number of trials to observe s successes (instead of total number of failures). In this formulation, the probability mass function is given by:

 

The Beta-Negative Binomial is an extension of the Negative Binomial distribution where there the probability p is itself a random variable.

Zero-modified version

When modeling or analyzing counting data, it is often desirable to modify probability of zero of the discrete distribution we use, to more accurately model the probability of "no event occurring". We can make two types of modifications to our distribution for this:

• Zero-inflated model - we increase the probability of zero.

• Zero-truncated model - we entirely remove the probability of zero events occurring.

ModelRisk functions added to Microsoft Excel for the Negative Binomial distribution

VoseNegBinom generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseNegBinomObject constructs a distribution object for this distribution.

VoseNegBinomProb returns the probability mass or cumulative distribution function for this distribution.

VoseNegBinomProb10 returns the log10 of the probability mass or cumulative distribution function.

VoseNegBinomFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseNegBinomFitObject constructs a distribution object of this distribution fitted to data.

VoseNegBinomFitP returns the parameters of this distribution fitted to data.

 

ModelRisk functions added to Microsoft Excel for the Zero-Inflated Negative Binomial distribution

VoseZINegBinom generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseZINegBinomObject constructs a distribution object for this distribution.

VoseZINegBinomProb returns the probability mass or cumulative distribution function for this distribution.

VoseZINegBinomProb10 returns the log10 of the probability mass or cumulative distribution function.

VoseZINegBinomFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseZINegBinomFitObject constructs a distribution object of this distribution fitted to data.

VoseZINegBinomFitP returns the parameters of this distribution fitted to data.

 

ModelRisk functions added to Microsoft Excel for the Zero-Truncated Negative Binomial distribution

VoseZTNegBinom generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseZTNegBinomObject constructs a distribution object for this distribution.

VoseZTNegBinomProb returns the probability mass or cumulative distribution function for this distribution.

VoseZTNegBinomProb10 returns the log10 of the probability mass or cumulative distribution function.

VoseZTNegBinomFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseZTNegBinomFitObject constructs a distribution object of this distribution fitted to data.

VoseZTNegBinomFitP returns the parameters of this distribution fitted to data.

 

Negative Binomial distribution equations

 

Zero-Inflated Negative Binomial distribution equations

 

Zero-Truncated Negative Binomial distribution equations