 # Negative Binomial I

### Input.

Input for the mean and average, top 'blue' boxes, can be any positive (decimal) value. Input for the number of trials, 'green' box, must be an integer value, a whole positive number without decimals.

If the mean and variance are exactly the same, the program will not work. Use the Poisson instead.

A windows version of the negative binomial procedure is available here

### Explanation.

This version of the negative binomial distribution is a generalization of the Poisson as used to study the distribution of accidents and events at the individual level. For the Poisson it is assumed that the chance of having an accident or a disease is randomly distributed: all individuals have an equal chance, of having one, two or more accidents. However, this assumption may be incorrect; you may observe that relatively more people have a higher number of accidents than the Poisson predicts. Mathematically the correctness of the assumption can be checked by seeing if in the Poisson the variance and the mean are equal. If the variance is larger than the mean, the assumption is incorrect.

The negative binomial distribution does not assume randomness: there is a possibility of 'proneness', i.e. certain groups of individuals in the population have a higher chance of having accidents or diseases than others. A third parameter is now included, the variance of the distribution. The variance can be interpreted as a factor that expresses the level of proneness. The larger the variance is relative to the mean, the higher the level of proneness in the population. Note that the variance is an expectation value, it is related to the expected value. The variance is the square of the standard deviation. Lastly, the mean to which the variance is related is in this case always a mean, even if the value of the mean is between zero and one. There is therefore no mathematical relationship between the variance and the mean, as is the case with proportions.

One of the problems with the negative binomial distribution is that there does not seem to be a clear meaning which can be easily given to the variance (Arbous AG, Kerrich JE, 1951). Thus, after fitting the empirical distribution to the Negative Binomial distribution, and estimating the variance, as a measure of accident or disease proneness in the population, you are confronted with the question of what this measure means for individuals or groups in the population, or the development of policy or decisions.

Input is the same as for the Poisson, only the variance of the distribution is added in the middle, blue cell. One would normally expect the variance to be larger than the mean. If this is not the case use the Poisson (and if the variance and mean are the same the two distributions produce the same results). The variance and mean are imported parameters, you have to calculate or estimate them yourself from the distribution you study, using either a dedicated computer packages or a hand held calculator (which is absolutely fine for the purpose). For the variance one squares the sample standard deviation, on the calculator it is the one with the "s" symbol, not the one with the funny little "sigma" symbol. The DOS-version of this SISA program has an option which does all the calculations for you.

### Further Reading.

Arbous AG, Kerrich JE. Accident Statistics and the Concept of Accident-Proneness. Part 1: A Critical Evaluation. Part 11. The Mathematical Background. Biometrics 1951;341-433. 