aod (version 1.3.3)

donner: Test of Proportion Homogeneity using Donner's Adjustment

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 \(\chi^2\) statistic proposed by Donner (1989).

Usage

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

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.

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.

Author

Matthieu Lesnoff , Renaud Lancelot

Details

The \(\chi^2\) statistic is adjusted with the correction factor \(C_i\) computed in each group \(i\). The test statistic is given by: $$X^2 = \sum_{i}\frac{(y_i - n_i * p)^2}{C_i * n_i * p * (1 - p)}$$ where \(C_i = 1 + (nA_i - 1) * \rho\), \(nA_i\) 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 \(\chi^2\) distribution with \(I - 1\) degrees of freedom. Fixed correction factors can be specified with the argument C.

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.

See Also

chisq.test, raoscott, drs-class

Examples

  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)