# Negative multinomial distribution

In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(r,p)) to more than two outcomes.[1]

Notation ${\displaystyle {\textrm {NM}}(k_{0},\,p)}$ k0 ∈ N0 — the number of failures before the experiment is stopped,p ∈ Rm — m-vector of "success" probabilities,p0 = 1 − (p1+…+pm) — the probability of a "failure". ${\displaystyle k_{i}\in \{0,1,2,\ldots \},1\leq i\leq m}$ ${\displaystyle \Gamma \!\left(\sum _{i=0}^{m}{k_{i}}\right){\frac {p_{0}^{k_{0}}}{\Gamma (k_{0})}}\prod _{i=1}^{m}{\frac {p_{i}^{k_{i}}}{k_{i}!}},}$ where Γ(x) is the Gamma function. ${\displaystyle {\tfrac {k_{0}}{p_{0}}}\,p}$ ${\displaystyle {\tfrac {k_{0}}{p_{0}^{2}}}\,pp'+{\tfrac {k_{0}}{p_{0}}}\,\operatorname {diag} (p)}$ ${\displaystyle {\bigg (}{\frac {p_{0}}{1-p'e^{it}}}{\bigg )}^{\!k_{0}}}$

Suppose we have an experiment that generates m+1≥2 possible outcomes, {X0,...,Xm}, each occurring with non-negative probabilities {p0,...,pm} respectively. If sampling proceeded until n observations were made, then {X0,...,Xm} 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,...,Xm} is negative multinomial. These variables are not multinomially distributed because their sum X1+...+Xm is not fixed, being a draw from a negative binomial distribution.

## Negative multinomial distribution example

The table below shows an example of 400 melanoma (skin cancer) patients where the Type and Site of the cancer are recorded for each subject.

 Type Site Totals Head and Neck Trunk Extremities 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 cancer rates at a given site and help measure some of the cancer type dependencies within each location.

If ${\displaystyle x_{i,j}}$ denote the cancer rates for each site (${\displaystyle 0\leq i\leq 2}$) and each type of cancer (${\displaystyle 0\leq j\leq 3}$), for a fixed site (${\displaystyle i_{0}}$) 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:

${\displaystyle X=\{X_{1},X_{2},X_{3}\}\sim NM(k_{0},\{p_{1},p_{2},p_{3}\})}$.

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):

${\displaystyle {\hat {\mu }}_{i,j}={\frac {x_{i,.}\times x_{.,j}}{x_{.,.}}}}$
${\displaystyle x_{i,.}=\sum _{j=0}^{3}{x_{i,j}}}$
${\displaystyle x_{.,j}=\sum _{i=0}^{2}{x_{i,j}}}$
${\displaystyle x_{.,.}=\sum _{i=0}^{2}\sum _{j=0}^{3}{x_{i,j}}}$
Example: ${\displaystyle {\hat {\mu }}_{1,1}={\frac {x_{1,.}\times x_{.,1}}{x_{.,.}}}={\frac {34\times 68}{400}}=5.78}$

For the first site (Head and Neck, j=0), suppose that ${\displaystyle X=\left\{X_{1}=5,X_{2}=1,X_{3}=5\right\}}$ and ${\displaystyle X\sim NM(k_{0}=10,\{p_{1}=0.2,p_{2}=0.1,p_{3}=0.2\})}$. Then:

${\displaystyle p_{0}=1-\sum _{i=1}^{3}{p_{i}}=0.5}$
${\displaystyle NM(X|k_{0},\{p_{1},p_{2},p_{3}\})=0.00465585119998784}$
${\displaystyle cov[X_{1},X_{3}]={\frac {10\times 0.2\times 0.2}{0.5^{2}}}=1.6}$
${\displaystyle \mu _{2}={\frac {k_{0}p_{2}}{p_{0}}}={\frac {10\times 0.1}{0.5}}=2.0}$
${\displaystyle \mu _{3}={\frac {k_{0}p_{3}}{p_{0}}}={\frac {10\times 0.2}{0.5}}=4.0}$
${\displaystyle corr[X_{2},X_{3}]=\left({\frac {\mu _{2}\times \mu _{3}}{(k_{0}+\mu _{2})(k_{0}+\mu _{3})}}\right)^{\frac {1}{2}}}$ and therefore, ${\displaystyle corr[X_{2},X_{3}]=\left({\frac {2\times 4}{(10+2)(10+4)}}\right)^{\frac {1}{2}}=0.21821789023599242.}$

Notice that the pair-wise NM correlations are always positive, whereas the correlations between multinomial counts are always negative. As the parameter ${\displaystyle k_{0}}$ increases, the paired correlations tend to zero! Thus, for large ${\displaystyle k_{0}}$, the Negative Multinomial counts ${\displaystyle X_{i}}$ behave as independent Poisson random variables with respect to their means ${\displaystyle \left(\mu _{i}=k_{0}{\frac {p_{i}}{p_{0}}}\right)}$.

The marginal distribution of each of the ${\displaystyle X_{i}}$ variables is negative binomial, as the ${\displaystyle X_{i}}$ count (considered as success) is measured against all the other outcomes (failure). But jointly, the distribution of ${\displaystyle X=\{X_{1},\cdots ,X_{m}\}}$ is negative multinomial, i.e., ${\displaystyle X\sim NM(k_{0},\{p_{1},\cdots ,p_{m}\})}$ .

## Parameter estimation

• Estimation of the mean (expected) frequency counts (${\displaystyle \mu _{j}}$) of each outcome (${\displaystyle X_{j}}$) using maximum likelihood is possible. If we have a single observation vector ${\displaystyle \{x_{1},\cdots ,x_{m}\}}$, then ${\displaystyle {\hat {\mu }}_{i}=x_{i}.}$ 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 ${\displaystyle {\hat {\mu }}_{j}={\frac {x_{j,.}}{I}}}$, where ${\displaystyle 0\leq j\leq J}$ 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 (${\displaystyle X_{0}}$) is ${\displaystyle {\hat {\mu }}_{0}=34/3=11.33}$.
Superficial type of cancer (${\displaystyle X_{1}}$) is ${\displaystyle {\hat {\mu }}_{1}=185/3=61.67}$.
Nodular type of cancer (${\displaystyle X_{2}}$) is ${\displaystyle {\hat {\mu }}_{2}=125/3=41.67}$.
Indeterminant type of cancer (${\displaystyle X_{3}}$) is ${\displaystyle {\hat {\mu }}_{3}=56/3=18.67}$.
• There is no MLE estimate for the NM ${\displaystyle k_{0}}$ parameter.[1][2] However, there are approximate protocols for estimating the ${\displaystyle k_{0}}$ parameter using the chi-squared goodness of fit statistic. In the usual chi-squared statistic:
${\displaystyle \mathrm {X} ^{2}=\sum _{i}{\frac {(x_{i}-\mu _{i})^{2}}{\mu _{i}}}}$, we can replace the expected-means (${\displaystyle \mu _{i}}$) by their estimates, ${\displaystyle {\hat {\mu _{i}}}}$, and replace denominators by the corresponding negative multinomial variances. Then we get the following test statistic for negative multinomial distributed data:
${\displaystyle \mathrm {X} ^{2}(k_{0})=\sum _{i}{\frac {(x_{i}-{\hat {\mu _{i}}})^{2}}{{\hat {\mu _{i}}}\left(1+{\frac {\hat {\mu _{i}}}{k_{0}}}\right)}}}$.
Next, we can estimate the ${\displaystyle k_{0}}$ parameter by varying the values of ${\displaystyle k_{0}}$ in the expression ${\displaystyle \mathrm {X} ^{2}(k_{0})}$ 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 (${\displaystyle \mu _{j}}$) for the 4 different cancer types are:
${\displaystyle {\hat {\mu }}_{1}=185/3=61.67}$; ${\displaystyle {\hat {\mu }}_{2}=125/3=41.67}$; and ${\displaystyle {\hat {\mu }}_{3}=56/3=18.67}$.
Thus, we can solve the equation above ${\displaystyle \mathrm {X} ^{2}(k_{0})=5.261948}$ for the single variable of interest -- the unknown parameter ${\displaystyle k_{0}}$. In the cancer example, suppose ${\displaystyle x=\{x_{1}=5,x_{2}=1,x_{3}=5\}}$. Then, the solution is an asymptotic chi-squared distribution driven estimate of the parameter ${\displaystyle k_{0}}$.
${\displaystyle \mathrm {X} ^{2}(k_{0})=\sum _{i=1}^{3}{\frac {(x_{i}-{\hat {\mu _{i}}})^{2}}{{\hat {\mu _{i}}}\left(1+{\frac {\hat {\mu _{i}}}{k_{0}}}\right)}}}$.
${\displaystyle \mathrm {X} ^{2}(k_{0})={\frac {(5-61.67)^{2}}{61.67(1+61.67/k_{0})}}+{\frac {(1-41.67)^{2}}{41.67(1+41.67/k_{0})}}+{\frac {(5-18.67)^{2}}{18.67(1+18.67/k_{0})}}=5.261948.}$ Solving this equation for ${\displaystyle k_{0}}$ provides the desired estimate for the last parameter.
Mathematica provides 3 distinct (${\displaystyle k_{0}}$) solutions to this equation: {50.5466, -21.5204, 2.40461}. Since ${\displaystyle k_{0}>0}$ there are 2 candidate solutions.
• Estimates of probabilities: Assume ${\displaystyle k_{0}=2}$ and ${\displaystyle {\frac {\mu _{i}}{k_{0}}}p_{0}=p_{i}}$, then:
${\displaystyle {\frac {61.67}{k_{0}}}p_{0}=31p_{0}=p_{1}}$
${\displaystyle 20p_{0}=p_{2}}$
${\displaystyle 9p_{0}=p_{3}}$
Hence, ${\displaystyle 1-p_{0}=p_{1}+p_{2}+p_{3}=60p_{0}}$, and ${\displaystyle p_{0}={\frac {1}{61}}}$, ${\displaystyle p_{1}={\frac {31}{61}}}$, ${\displaystyle p_{2}={\frac {20}{61}}}$ and ${\displaystyle p_{3}={\frac {9}{61}}}$.
Therefore, the best model distribution for the observed sample ${\displaystyle x=\{x_{1}=5,x_{2}=1,x_{3}=5\}}$ is ${\displaystyle X\sim NM\left(2,\left\{{\frac {31}{61}},{\frac {20}{61}},{\frac {9}{61}}\right\}\right).}$

## References

1. Le Gall, F. The modes of a negative multinomial distribution, Statistics & Probability Letters, Volume 76, Issue 6, 15 March 2006, Pages 619-624, ISSN 0167-7152, 10.1016/j.spl.2005.09.009.
2. Zelterman, Daniel (2002). Advanced log-linear models using SAS. SAS Publishing. p. 196. ISBN 978-1-59047-080-0.

Waller LA and Zelterman D. (1997). Log-linear modeling with the negative multi- nomial distribution. Biometrics 53: 971-82.

Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1997). "Chapter 36: Negative Multinomial and Other Multinomial-Related Distributions". Discrete Multivariate Distributions. Wiley. ISBN 978-0-471-12844-1.