Negative multinomial distribution

Notation \textrm{NM}(k_0,\,p)
Parameters

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”.

Support k_i \in \{0,1,2,\ldots\}, 1\leq i\leq m
PDF \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.
Mean \tfrac{k_0}{p_0}\,p
Variance \tfrac{k_0}{p_0^2}\,pp' + \tfrac{k_0}{p_0}\,\operatorname{diag}(p)
CF \bigg(\frac{p_0}{1 - p'e^{it}}\bigg)^{\!k_0}

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]

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 the 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 sites cancer rates and help measure some of the cancer type dependencies within each location.

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

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):

\hat{\mu}_{i,j} = \frac{x_{i,.}\times x_{.,j}}{x_{.,.}}
x_{i,.} = \sum_{j=0}^{3}{x_{i,j}}
x_{.,j} = \sum_{i=0}^{2}{x_{i,j}}
x_{.,.} = \sum_{i=0}^{2}\sum_{j=0}^{3}{{x_{i,j}}}
Example: \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 X=\left \{X_1=5, X_2=1, X_3=5\right \} and X \sim NM(k_0=10, \{p_1=0.2, p_2=0.1, p_3=0.2 \}). Then:

p_0 = 1 - \sum_{i=1}^3{p_i}=0.5
NM(X|k_0,\{p_1, p_2, p_3\})= 0.00465585119998784
cov[X_1,X_3] = \frac{10 \times 0.2 \times 0.2}{0.5^2}=1.6
\mu_2=\frac{k_0 p_2}{p_0} = \frac{10\times 0.1}{0.5}=2.0
\mu_3=\frac{k_0 p_3}{p_0} = \frac{10\times 0.2}{0.5}=4.0
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, 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 k_0 increases, the paired correlations tend to zero! Thus, for large k_0, the Negative Multinomial counts X_i behave as independent Poisson random variables with respect to their means \left ( \mu_i= k_0\frac{p_i}{p_0}\right ).

The marginal distribution of each of the X_i variables is negative binomial, as the X_i count (considered as success) is measured against all the other outcomes (failure). But jointly, the distribution of X=\{X_1,\cdots,X_m\} is negative multinomial, i.e., X \sim NM(k_0,\{p_1,\cdots,p_m\}) .

Parameter estimation

Hutchinson's melanomic freckle type of cancer (X_0) is \hat{\mu}_0 = 34/3=11.33.
Superficial type of cancer (X_1) is \hat{\mu}_1 = 185/3=61.67.
Nodular type of cancer (X_2) is \hat{\mu}_2 = 125/3=41.67.
Indeterminant type of cancer (X_3) is \hat{\mu}_3 = 56/3=18.67.
\Chi^2 = \sum_i{\frac{(x_i-\mu_i)^2}{\mu_i}}, we can replace the expected-means (\mu_i) by their estimates, \hat{\mu_i}, and replace denominators by the corresponding negative multinomial variances. Then we get the following test statistic for negative multinomial distributed data:
\Chi^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 k_0 parameter by varying the values of k_0 in the expression \Chi^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 (\mu_j) for the 4 different cancer types are:
\hat{\mu}_1 = 185/3=61.67; \hat{\mu}_2 = 125/3=41.67; and \hat{\mu}_3 = 56/3=18.67.
Thus, we can solve the equation above \Chi^2(k_0) = 5.261948 for the single variable of interest -- the unknown parameter k_0. In the cancer example, suppose x=\{x_1=5,x_2=1,x_3=5\}. Then, the solution is an asymptotic chi-squared distribution driven estimate of the parameter k_0.
\Chi^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 )}}.
\Chi^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 k_0 provides the desired estimate for the last parameter.
Mathematica provides 3 distinct (k_0) solutions to this equation: {50.5466, -21.5204, 2.40461}. Since k_0>0 there are 2 candidate solutions.
\frac{61.67}{k_0}p_0=31p_0=p_1
20p_0=p_2
9p_0=p_3
Hence, 1-p_0=p_1+p_2+p_3=60p_0, and p_0=\frac{1}{61}, p_1=\frac{31}{61}, p_2=\frac{20}{61} and p_3=\frac{9}{61}.
Therefore, the best model distribution for the observed sample x=\{x_1=5,x_2=1,x_3=5\} is X \sim NM\left (2, \left \{\frac{31}{61}, \frac{20}{61},\frac{9}{61}\right\} \right ).

Related distributions

References

  1. ↑ 1.0 1.1 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.

Further reading

Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1997). "Chapter 36: Negative Multinomial and Other Multinomial-Related Distributions". Discrete Multivariate Distributions. Wiley. ISBN 0-471-12844-9.