R: Test of Proportion Homogeneity using Rao and Scott's...

raoscott {aod} | R Documentation

## 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-squared 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(yadj_i) / sum(deff_i). The test statistic is obtained by substituting these quantities in the usual chi-squared statistic, yielding:

X^2 = sum( (yadj_i - nadj_i * padj)^2 / (nadj_i * padj * (1 - padj)) )

which is compared to a chi-squared 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 matthieu.lesnoff@cirad.fr, Renaud Lancelot renaud.lancelot@cirad.fr

### 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)

[Package aod version 1.3 Index]