raoscott {aod}

Test of Proportion Homogeneity using Rao and Scott'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 Rao and Scott (1993).

Usage

raoscott(formula = NULL, response = NULL, weights = NULL, 
              group = NULL, data, pooled = FALSE, deff = 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: 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 proportions.

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.

pooled

Logical indicating if a pooled design effect is estimated over the I groups.

deff

A numerical vector of I design effects.

Details

The method is based on the concepts of design effect and effective sample size.

The design effect in each group i is estimated by deff_i = vratio_i / vbin_i, where vratio_i is the variance of the ratio estimate of the probability in group i (Cochran, 1999, p. 32 and p. 66) and vbin_i is the standard binomial variance. A pooled design effect (i.e., over the I groups) is estimated if argument pooled = TRUE (see Rao and Scott, 1993, Eq. 6). Fixed design effects can be specified with the argument deff.
The deff_i are used to compute the effective sample sizes nadj_i = n_i / deff_i, the effective numbers of successes yadj_i = y_i / deff_i in each group i, and the overall effective proportion padj = \sum_{i} yadj_i / \sum_{i} deff_i. The test statistic is obtained by substituting these quantities in the usual \chi^2 statistic, yielding:

X^2 = \sum_{i}\frac{(yadj_i - nadj_i * padj)^2}{nadj_i * padj * (1 - padj)}

which is compared to a \chi^2 distribution with I - 1 degrees of freedom.

Value

An object of formal class “drs”: see drs-class for details. The slot tab provides the proportion of successes, the variances of the proportion and the design effect for each group.

Author(s)

Matthieu Lesnoff [e-mail], Renaud Lancelot [e-mail]

References

Cochran, W.G., 1999, 2nd ed. Sampling techniques. John Wiley & Sons, New York.
Rao, J.N.K., Scott, A.J., 1992. A simple method for the analysis of clustered binary data. Biometrics 48, 577-585.

See Also

chisq.test, donner, iccbin, drs-class

Examples

  data(rats)
  # deff by group
  raoscott(cbind(y, n - y) ~ group, data = rats)
  raoscott(y/n ~ group, weights = n, data = rats)
  raoscott(response = cbind(y, n - y), group = group, data = rats)
  raoscott(response = y/n, weights = n, group = group, data = rats)
  # pooled deff
  raoscott(cbind(y, n - y) ~ group, data = rats, pooled = TRUE)
  # standard test
  raoscott(cbind(y, n - y) ~ group, data = rats, deff = c(1, 1))
  data(antibio)
  raoscott(cbind(y, n - y) ~ treatment, data = antibio)