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)

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.

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.

Author

Matthieu Lesnoff matthieu.lesnoff@cirad.fr, Renaud Lancelot renaud.lancelot@cirad.fr

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.

Examples

Run this code

  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)

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