aod (version 1.3.1)

raoscott: 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.

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.

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.

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

# NOT RUN {
  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)
  
# }