# donner: Test of Proportion Homogeneity using Donner's Adjustment In aod: Analysis of Overdispersed Data

## Description

Tests the homogeneity of proportions between I groups (H0: p_1 = p_2 = ... = p_I ) from clustered binomial data (n, y) using the adjusted chi-squared statistic proposed by Donner (1989).

## Usage

 ```1 2``` ```donner(formula = NULL, response = NULL, weights = NULL, group = NULL, data, C = NULL) ```

## Arguments

 `formula` An optional formula where the left-hand side is either a matrix of the form `cbind(y, n-y)`, where the modelled probability is `y/n`, or a vector of proportions to be modelled (`y/n`). In both cases, the right-hand side must specify a single grouping variable. When the left-hand side of the formula is a vector of proportions, the argument `weight` must be used to indicate the denominators of the proportions. `response` An optional argument indicating either a matrix of the form `cbind(y, n-y)`, where the modelled probability is `y/n`, or a vector of proportions to be modelled (`y/n`). `weights` An optional argument used when the left-hand side of `formula` or `response` is a vector of proportions: `weight` is the denominator of the proportion. `group` An optional argument only used when `response` is used. In this case, this argument is a factor indicating a grouping variable. `data` A data frame containing the response (`n` and `y`) and the grouping variable. `C` If not NULL, a numerical vector of I cluster correction factors.

## Details

The chi-squared statistic is adjusted with the correction factor C_i computed in each group i. The test statistic is given by:

X^2 = sum( (y_i - n_i * p)^2 / (C_i * n_i * p * (1 - p)) )

where C_i = 1 + (nA_i - 1) * ρ, nA_i is a scalar depending on the cluster sizes, and rho is the ANOVA estimate of the intra-cluster correlation, assumed common across groups (see Donner, 1989 or Donner et al., 1994). The statistic is compared to a chi-squared distribution with I - 1 degrees of freedom. Fixed correction factors can be specified with the argument `C`.

## Value

An object of formal class “drs”: see `drs-class` for details. The slot `tab` provides the proportion of successes and the correction factor for each group.

## Author(s)

Matthieu Lesnoff [email protected], Renaud Lancelot [email protected]

## References

Donner, A., 1989. Statistical methods in ophthalmology: an adjusted chi-squared approach. Biometrics 45, 605-611.
Donner, A., 1993. The comparison of proportions in the presence of litter effects. Prev. Vet. Med. 18, 17-26.
Donner, A., Eliasziw, M., Klar, N., 1994. A comparison of methods for testing homogeneity of proportions in teratologic studies. Stat. Med. 13, 1253-1264.

`chisq.test`, `raoscott`, `drs-class`
 ``` 1 2 3 4 5 6 7 8 9 10``` ``` data(rats) donner(formula = cbind(y, n - y) ~ group, data = rats) donner(formula = y/n ~ group, weights = n, data = rats) donner(response = cbind(y, n - y), group = group, data = rats) donner(response = y/n, weights = n, group = group, data = rats) # standard test donner(cbind(y, n - y) ~ group, data = rats, C = c(1, 1)) data(antibio) donner(cbind(y, n - y) ~ treatment, data = antibio) ```