# odTest: likelihood ratio test for over-dispersion in count data In pscl: Political Science Computational Laboratory

## Description

Compares the log-likelihoods of a negative binomial regression model and a Poisson regression model.

## Usage

 `1` ```odTest(glmobj, alpha=.05, digits = max(3, getOption("digits") - 3)) ```

## Arguments

 `glmobj` an object of class `negbin` produced by `glm.nb` `alpha` significance level of over-dispersion test `digits` number of digits in printed output

## Details

The negative binomial model relaxes the assumption in the Poisson model that the (conditional) variance equals the (conditional) mean, by estimating one extra parameter. A likelihood ratio (LR) test can be used to test the null hypothesis that the restriction implicit in the Poisson model is true. The LR test-statistic has a non-standard distribution, even asymptotically, since the negative binomial over-dispersion parameter (called `theta` in `glm.nb`) is restricted to be positive. The asymptotic distribution of the LR (likelihood ratio) test-statistic has probability mass of one half at zero, and a half chi-square (1) distribution above zero. This means that if testing at the alpha = .05 level, one should not reject the null unless the LR test statistic exceeds the critical value associated with the 2 alpha = .10 level; this LR test involves just one parameter restriction, so the critical value of the test statistic at the p = .05 level is 2.7, instead of the usual 3.8 (i.e., the .90 quantile of the chi-square (1) distribution, versus the .95 quantile).

A Poisson model is run using `glm` with family set to `link{poisson}`, using the `formula` in the negbin model object passed as input. The `logLik` functions are used to extract the log-likelihood for each model.

## Value

None; prints results and returns silently

## Author(s)

Simon Jackman simon.jackman@sydney.edu.au. John Fox noted an error in an earlier version.

## References

A. Colin Cameron and Pravin K. Trivedi (1998) Regression analysis of count data. New York: Cambridge University Press.

Lawless, J. F. (1987) Negative Binomial and Mixed Poisson Regressions. The Canadian Journal of Statistics. 15:209-225.

`glm.nb`, `logLik`

## Examples

 ```1 2 3 4 5``` ```data(bioChemists) modelnb <- MASS::glm.nb(art ~ ., data=bioChemists, trace=TRUE) odTest(modelnb) ```

### Example output

```Loading required package: MASS
Classes and Methods for R developed in the

Political Science Computational Laboratory

Department of Political Science

Stanford University

Simon Jackman

hurdle and zeroinfl functions by Achim Zeileis

Theta(1) = 2.268830, 2(Ls - Lm) = 1004.930000
Theta(2) = 2.264410, 2(Ls - Lm) = 1004.280000
Theta(3) = 2.264400, 2(Ls - Lm) = 1004.280000
Theta(4) = 2.264390, 2(Ls - Lm) = 1004.280000
Theta(5) = 2.264390, 2(Ls - Lm) = 1004.280000
Theta(6) = 2.264390, 2(Ls - Lm) = 1004.280000
Theta(7) = 2.264390, 2(Ls - Lm) = 1004.280000
Theta(8) = 2.264390, 2(Ls - Lm) = 1004.280000
Likelihood ratio test of H0: Poisson, as restricted NB model:
n.b., the distribution of the test-statistic under H0 is non-standard
e.g., see help(odTest) for details/references

Critical value of test statistic at the alpha= 0.05 level: 2.7055
Chi-Square Test Statistic =  180.196 p-value = < 2.2e-16
```

pscl documentation built on March 26, 2020, 7:36 p.m.