raoscott: Test of Proportion Homogeneity using Rao and Scott's... In aod: Analysis of Overdispersed Data

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

 ```1 2``` ```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:

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.

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.

`chisq.test`, `donner`, `iccbin`, `drs-class`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ``` 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) ```