Conway–Maxwell–Poisson distribution
Topic of Standard errors, and models for counts, part of Generalised linear models · Also in Underdispersion, positive responses, and quasi-likelihood
Conway–Maxwell–Poisson distribution
| Type | Topic |
| Taught in | Standard errors, and models for counts · Underdispersion, positive responses, and quasi-likelihood |
| Part | Generalised linear models |
A count distribution with probability function proportional to \lambda^y/(y!)^\nu. The extra parameter allows overdispersion (\nu<1), the Poisson (\nu=1) and underdispersion (\nu>1); reparameterised by its mean it can be fitted as a GLM-like model.
In Standard errors, and models for counts
That mixture cannot produce underdispersion. Counts that are tighter than a Poisson need a different family. The Conway–Maxwell–Poisson distribution has been available since Conway and Maxwell (1962), from queueing theory with dependent waiting times, and it was brought back into statistical use by Shmueli and coauthors in 2005. In the original scaling, f(y;\lambda,\nu)\propto \frac{\lambda^y}{(y!)^\nu}, \qquad y=0,1,2,\ldots, with \nu<1 overdispersed, \nu=1 exactly Poisson, and \nu>1 underdispersed. Calling \nu a dispersion runs the name backwards: larger \nu means a tighter distribution. The serious defect is that \lambda is not the mean. There is no closed form for \mathrm{E}(Y) or \mathrm{Var}(Y), so a regression on \lambda cannot be compared with a Poisson or negative-binomial regression on \mu.
In Underdispersion, positive responses, and quasi-likelihood
Cameron and Johansson (1997) recorded the number of takeover bids received by 126 US firms between 1978 and 1985. The covariates split into three groups. Defensive actions by the target: a lawsuit (leglrest), a proposed change in asset structure (rearest), a proposed change in ownership structure (finrest), and an invitation to a friendly third party, a white knight (whtknght). Characteristics of the firm: the bid premium (bidprem), the percentage of stock held by institutions (insthold), book value in billions (size), and its square. And one indicator for Department of Justice intervention (regulatn).
After those covariates, the counts are tightly bunched. A Poisson mean of 7 would happily produce anything from about 3 to 13. These firms, when the fit said 7, mostly produced 6, 7, or 8. That is underdispersion, which neither the Poisson nor the negative binomial can represent. The mean-parameterised Conway–Maxwell–Poisson can.
The R package that fitted it was withdrawn from CRAN in 2022, ahead of a replacement that did not arrive. The fit given here is a MATLAB fit of a log-linear mean. There is no formula interface. You build {\color{blue}\boldsymbol{X}} yourself: a column of ones and the nine covariates, 126 rows. The call returns the coefficient vector, the dispersion \nu, the implied rates \lambda(\mu,\nu), the maximised log-likelihood, and standard errors for \boldsymbol{\beta}. The estimates, standard errors, and ratios are
| term | estimate | std. error | z |
|---|---|---|---|
| intercept | 0.9897 | 0.4352 | 2.27 |
| legal defence | 0.2679 | 0.1228 | 2.18 |
| asset restructure | -0.1732 | 0.1547 | -1.12 |
| ownership change | 0.0677 | 0.1743 | 0.39 |
| white knight | 0.4813 | 0.1317 | 3.66 |
| bid premium | -0.6849 | 0.3075 | -2.23 |
| institutional holding | -0.3679 | 0.3466 | -1.06 |
| size | 0.1793 | 0.0476 | 3.77 |
| size squared | -0.0076 | 0.0025 | -3.05 |
| regulation | -0.0376 | 0.1302 | -0.29 |
The estimated dispersion is \hat\nu=1.754. Because \nu>1, the counts are tighter than a Poisson with the same mean, by the convention of this parameterisation.
The legal-defence coefficient is the easiest sentence. Two firms with the same size, the same premium, and the same other actions, one of which sues and one of which does not, differ by 0.2679 on the log-mean scale. The firm that sues is estimated to receive \exp(0.2679)=1.31 times as many bids. The ratio of estimate to standard error is 2.18, which clears 2 and so sits past the usual two-sided 5% cutoff on the z scale. The same reading applies to every other row. Size enters as a curve: the linear term is positive and the square is negative, so the mean rises with book value and then bends back. A one-line slope for size would hide that bend.
Why the underdispersion is plausible is the same selection story as the mines, in a different market. A private person cannot bid for a large firm. The pool of possible bidders shrinks as the target grows, and for the largest targets the count is often zero or one. Conditionally on the covariates, the count is more deterministic than a Poisson process. The Poisson model is the boundary case \nu=1. The fitted \hat\nu=1.754 lies above that boundary, which is the comparison a test of \nu=1 makes. The point to keep is that the mean is on the same scale as a Poisson log-linear model, so the coefficient 0.2679 means the same kind of thing.
Used in
See also
Overdispersion · Underdispersion · Quasi-Poisson · Negative binomial distribution · Standard errors in a GLM · Binomial counts in a GLM · Count responses · Reading a log-linear coefficient