adk.combined.test {adk}R Documentation

Combined Anderson-Darling K-Sample Tests

Description

This function combines several independent Anderson-Darling k-sample tests into one overall test of the hypothesis that the independent samples within each group come from a common unspecified distribution, while the common distributions may vary from group to group. The k for each group of k independent samples may also change from group to group. All samples from all groups are independent and are assumed to come from continuous distributions. Also provided is a version that adjusts for a moderate number of ties (due to rounding).

Usage

adk.combined.test(...)

Arguments

... Either several lists, say L.1, ... , L.M, where list L.i contains K.i sample vectors of respective sizes n.i[1], ..., n.i[K.i], where n.i[j] > 4 is recommended

or a list of such lists.

Details

If AD.i is the Anderson-Darling criterion for the i-th group of K.i samples, its standardized test statistic is T.i = (AD.i - mu.i)/sig.i, with mu.i and sig.i representing mean and standard deviation of AD.i. This test statistic is used to test the hypothesis that the samples in the i-th group all come from the same but unspecified continuous distribution function F(x).

The combined Anderson-Darling criterion is AD.c=AD.1 + ... + AD.M and T.combined = (AD.c - mu.c)/sig.c is the standardized test statistic, where mu.c and sig.c represent the mean and standard deviation of AD.c. This test statistic is used to simultaneously test whether the samples in each group come from the same continuous distribution function F(x), However, the unspecified common distribution function F(x) may change from group to group.

Value

A list with components

M number of groups of samples being compared
n.samples list of vectors, each vector giving the sample sizes for each group of samples being compared
nt vector of total sample sizes involved in each of the k.data comparisons
n.ties vector giving the number of ties in each the k.data comparison group
adk.i (2*k.data) * 3 matrix containing the T.kN, P-value, and extrapolation for the M individual Anderson-Darling tests, not adjusted for ties and adjusted for ties
mu vector of means of the k.data AD statistics
sig vector of standard deviations of the k.data AD statistics
adk.c 2*3 matrix containing T.combined, P-value, and extrapolation for the combined test not adjusted for ties and adjusted for ties
mu.c mean of the combined AD statistic
sig.c standard deviation of the combined AD statistic
warning logical indicator, warning = TRUE when at least one of the sample sizes is < 5.

Note

This test is useful in analyzing treatment effects in randomized (incomplete) block experiments and in examining performance equivalence of several laboratories when presented with different test materials for comparison.

Author(s)

Fritz Scholz

References

Scholz F.W. and Stephens M.A. (1987), K-sample Anderson-Darling Tests, Journal of the American Statistical Association, Vol 82, No. 399, 918–924.

See Also

See also adk.test

Examples

## Create two lists of sample vectors.
x1 <- list( c(1, 3, 2, 5, 7), c(2, 8, 1, 6, 9, 4), c(12, 5, 7, 9, 11) )
x2 <- list( c(51, 43, 31, 53, 21, 75), c(23, 45, 61, 17, 60) )

## Run adk.combined.test.
adk.combined.out <- adk.combined.test(x1,x2)  # or out <- adk.combined.test(list(x1,x2))

## Examine list objects in adk.combined.out.
names(adk.combined.out)

## Extract matrix components adk.i and adk.c
adk.combined.out$adk.i
adk.combined.out$adk.c

## Fully formatted output 
adk.combined.out

[Package adk version 1.0 Index]