Negative multinomial distribution - AbsoluteAstronomy.com

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the negative multinomial distribution is a generalization of the negative binomial distribution

Negative binomial distribution

In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified number of failures occur...

(NB(r, p)) to more than two outcomes.

Suppose we have an experiment that generates m+1≥2 possible outcomes, {X₀,…,X_m}, each occurring with non-negative probabilities {p₀,…,p_m} respectively. If sampling proceeded until n observations were made, then {X₀,…,X_m} would have been multinomially distributed. However, if the experiment is stopped once X₀ reaches the predetermined value k₀, then the distribution of the m-tuple {X₁,…,X_m} is negative multinomial.

Negative multinomial distribution example

The table below shows the an example of 400 Melanoma

Melanoma

Melanoma is a malignant tumor of melanocytes. Melanocytes are cells that produce the dark pigment, melanin, which is responsible for the color of skin. They predominantly occur in skin, but are also found in other parts of the body, including the bowel and the eye...

(skin cancer) Patients where the Type and Site of the cancer are recorded for each subject.

Type	Site			Totals
Type	Head and Neck	Trunk	Extremities	Totals
Hutchinson's melanomic freckle	22	2	10	34
Superficial	16	54	115	185
Nodular	19	33	73	125
Indeterminant	11	17	28	56
Column Totals	68	106	226	400

The sites (locations) of the cancer may be independent, but there may be positive dependencies of the type of cancer for a given location (site). For example, localized exposure to radiation implies that elevated level of one type of cancer (at a given location) may indicate higher level of another cancer type at the same location. The Negative Multinomial distribution may be used to model the sites cancer rates and help measure some of the cancer type dependencies within each location.

If

denote the cancer rates for each site (

) and each type of cancer (

), for a fixed site (

) the cancer rates are independent Negative Multinomial distributed random variables. That is, for each column index (site) the column-vector X has the following distribution:

Different columns in the table (sites) are considered to be different instances of the random multinomially distributed vector, X. Then we have the following estimates of expected counts (frequencies of cancer):

Example:

For the first site (Head and Neck, j=0), suppose that

and

. Then:

and therefore,

Notice that the pair-wise NM correlations are always positive, where as the correlations between multinomial counts are always negative. As the parameter

increases, the paired correlations tend to zero! Thus, for large

, the Negative Multinomial counts

behave as independent Poisson random variables

Poisson distribution

In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...

with respect to their means

.

The marginal distribution

Marginal distribution

In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. The term marginal variable is used to refer to those variables in the subset of variables being retained...

of each of the

variables is negative binomial, as the

count (considered as success) is measured against all the other outcomes (failure). But jointly, the distribution of

is negative multinomial, i.e.,

Parameter estimation

Estimation of the mean (expected) frequency counts () of each outcome () using maximum likelihood is possible. If we have a single observation vector , then If we have several observation vectors, like in this case we have the cancer type frequencies for 3 different sites, then the MLE estimates of the mean counts are , where is the cancer-type index and the summation is over the number of observed (sampled) vectors (I). For the cancer data above, we have the following MLE estimates for the expectations for the frequency counts:

Hutchinson's melanomic freckle type of cancer (

) is

Superficial type of cancer (

) is

Nodular type of cancer (

) is

Indeterminant type of cancer (

) is

There is no MLE estimate for the NM parameter. However, there are approximate protocols for estimating the parameter using the chi-squared goodness of fit statistic. In the usual chi-squared statistic:

, we can replace the expected-means (

) by their estimates,

, and replace denominators by the corresponding negative multinomial variances. Then we get the following test statistic for negative multinomial distributed data:

Next, we can estimate the

parameter by varying the values of

in the expression

and matching the values of this statistic with the corresponding asymptotic chi-squared distribution. The following protocol summarizes these steps using the cancer data above.

DF: The degree of freedom for the Chi-squared distribution in this case is:

df = (# rows – 1)(# columns – 1) = (3-1)*(4-1) = 6

Median: The median of a chi-squared random variable with 6 df is 5.261948.

Mean Counts Estimates: The mean counts estimates (