drs {aods3}

Test of Proportion Homogeneity between Groups using Donner's and Rao-Scott's Adjustments

Description

The function tests the homogeneity of probabilities between J groups (H_0: \mu_1 = \mu_2 = ... = \mu_J) from clustered binomial data {(n_1, m_1), (n_2, m_2), ..., (n_N, m_N)}, where n_i is the size of cluster i, m_i the number of “successes” (proportions are y = m/n), and N the number of clusters. The function uses adjusted chi-squared statistics, with either the correction proposed by proposed by Donner (1989) or the correction proposed by Rao and Scott (1993).

Usage

  drs(formula, data, type = c("d", "rs"), C = NULL, pooled = FALSE)
  
  ## S3 method for class 'drs'
print(x, ...)
  

Arguments

formula

An formula where the left-hand side is a matrix of the form cbind(m, n-m) (the modelled proportion is m / n). The right-hand side must specify a single grouping variable.

type

A character string: either “d” for the Donner's test and “rs” for the Rao and Scott's test.

data

A data frame containing n, m) and the grouping variable.

C

An optional vector of a priori J cluster correction factors used for the Donner's test or design effects factors used for the Rao-Scott's test. If C is set no NULL (default), it is calculated internally (see details).

pooled

Logical indicating if a pooled design effect is estimated over the J groups for the Rao-Scott's test (see details). Default to FALSE.

x

An object of class “drf”.

...

Further arguments to be passed to print.

Details

Donner's test
The chi-squared statistic is adjusted with the correction factor C_j computed in each group j. The test statistic is given by:

X^2 = \sum_{j} ( (m_j - n_j * \mu)^2 / (C_j * n_j * \mu * (1 - \mu)) )

where \mu = \sum_{j} (m_j) / \sum_{j} (n_j) and C_j = 1 + (n_{A,j} - 1) * \rho. n_{A,j} 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 J - 1 degrees of freedom. Fixed correction factors C_j can be specified with the argument C.

Rao ans Scott's test
The method uses design effects and “effective” sample sizes. The design effect C_j in each group j is estimated by C_j = v_{ratio,j} / v_{bin,j}, where v_{ratio,j} is the variance of the ratio estimate of the probability in group i (Cochran, 1999, p. 32 and p. 66) and v_{bin,j} is the standard binomial variance. The C_j are used to compute the effective sample sizes n_{adj,j} = n_j / C_j, the effective numbers of successes m_{adj,j} = m_j / C_j in each group j, and the overall effective proportion mu_adj = \sum_{j} m_{adj,j} / \sum_{j} C_j. The test statistic is obtained by substituting these quantities in the usual chi-squared statistic, yielding:

X^2 = \sum_{j} ( (m_{adj,j} - n_{adj,j} * muadj)^2 / (n_{adj,j} * muadj * (1 - muadj)) )

which is compared to a chi-squared distribution with J - 1 degrees of freedom.
A pooled design effect over the J groups is estimated if argument pooled = TRUE (see Rao and Scott, 1993, Eq. 6). Fixed design effects C_j can be specified with the argument C.

Value

An object of class drs, printed with print.drs.

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

Examples

data(dja)
# Donner
drs(formula = cbind(m, n - m) ~ group, data = dja, type = "d")
# Rao and Scott
drs(formula = cbind(m, n - m) ~ group, type = "rs", data = dja)
drs(formula = cbind(m, n - m) ~ group, type = "rs", data = dja, pooled = TRUE)
# standard chi2 test
drs(formula = cbind(m, n - m) ~ group, data = dja, type = "d", C = c(1:1))
drs(formula = cbind(m, n - m) ~ group, data = dja, type = "rs", C = c(1:1))