ad.test.combined {kSamples}

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 block come from a common unspecified distribution, while the common distributions may vary from block to block. Both versions of the Anderson-Darling test statistic are provided.

Usage

ad.test.combined(..., data = NULL,
	method = c("asymptotic", "simulated", "exact"),
	dist = FALSE, Nsim = 10000)

Arguments

...

Either a sequence of several lists, say L_1, \ldots, L_M (M > 1) where list L_i contains k_i > 1 sample vectors of respective sizes n_{i1}, \ldots, n_{ik_i}, where n_{ij} > 4 is recommended for reasonable asymptotic P-value calculation. N_i=n_{i1}+\ldots+n_{ik_i} is the pooled sample size for block i,

or a list of such lists,

or a formula, like y ~ g | b, where y is a numeric response vector, g is a factor with levels indicating different treatments and b is a factor indicating different blocks; y, g, b are or equal length. y is split separately for each block level into separate samples according to the g levels. The same g level may occur in different blocks. The variable names may correspond to variables in an optionally supplied data frame via the data = argument,

data

= an optional data frame providing the variables in formula input

method

= c("asymptotic","simulated","exact"), where

"asymptotic" uses only an asymptotic P-value approximation, reasonable for P in [0.00001, .99999], linearly extrapolated via \log(P/(1-P)) outside that range. See ad.pval for details. The adequacy of the asymptotic P-value calculation may be checked using pp.kSamples.

"simulated" uses simulation to get Nsim simulated AD statistics for each block of samples, adding them across blocks component wise to get Nsim combined values. These are compared with the observed combined value to obtain the estimated P-value.

"exact" uses full enumeration of the test statistic values for all sample splits of the pooled samples within each block. The test statistic vectors for the first 2 blocks are added (each component against each component, as in the R outer(x,y, "+") command) to get the convolution enumeration for the combined test statistic. The resulting vector is convoluted against the next block vector in the same fashion, and so on. It is possible only for small problems, and is attempted only when Nsim is at least the (conservatively maximal) length

\frac{N_1!}{n_{11}!\ldots n_{1k_1}!}\times\ldots\times \frac{N_M!}{n_{M1}!\ldots n_{Mk_M}!}

of the final distribution vector. Otherwise, it reverts to the simulation method using the provided Nsim.

dist

FALSE (default) or TRUE. If TRUE, the simulated or fully enumerated convolution vectors null.dist1 and null.dist2 are returned for the respective test statistic versions. Otherwise, NULL is returned for each.

Nsim

= 10000 (default), number of simulation splits to use within each block of samples. It is only used when method = "simulated" or when method = "exact" reverts to method = "simulated", as previously explained. Simulations are independent across blocks, using Nsim for each block. Nsim is limited by 1e7.

Details

If AD_i is the Anderson-Darling criterion for the i-th block of k_i samples, its standardized test statistic is T_i = (AD_i - \mu_i)/\sigma_i, with \mu_i and \sigma_i representing mean and standard deviation of AD_i. This statistic is used to test the hypothesis that the samples in the i-th block all come from the same but unspecified continuous distribution function F_i(x).

The combined Anderson-Darling criterion is AD_{comb}=AD_1 + \ldots + AD_M and T_{comb} = (AD_{comb} - \mu_c)/\sigma_c is the standardized form, where \mu_c=\mu_1+\ldots+\mu_M and \sigma_c = \sqrt{\sigma_1^2 +\ldots+\sigma_M^2} represent the mean and standard deviation of AD_{comb}. The statistic T_{comb} is used to simultaneously test whether the samples in each block come from the same continuous distribution function F_i(x), i=1,\ldots,M. The unspecified common distribution function F_i(x) may change from block to block. According to the reference article, two versions of the test statistic and its corresponding combinations are provided.

The k_i for each block of k_i independent samples may change from block to block.

NA values are removed and the user is alerted with the total NA count. It is up to the user to judge whether the removal of NA's is appropriate.

The continuity assumption can be dispensed with if we deal with independent random samples, or if randomization was used in allocating subjects to samples or treatments, independently from block to block, and if we view the simulated or exact P-values conditionally, given the tie patterns within each block. Of course, under such randomization any conclusions are valid only with respect to the blocks of subjects that were randomly allocated. The asymptotic P-value calculation assumes distribution continuity. No adjustment for lack thereof is known at this point. The same comment holds for the means and standard deviations of respective statistics.

Value

A list of class kSamples with components

test.name

= "Anderson-Darling"

M

number of blocks of samples being compared

n.samples

list of M vectors, each vector giving the sample sizes for each block of samples being compared

nt

= (N_1,\ldots,N_M)

n.ties

vector giving the number of ties in each the M comparison blocks

ad.list

list of M matrices giving the ad results for ad.test applied to the samples in each of the M blocks

mu

vector of means of the AD statistic for the M blocks

sig

vector of standard deviations of the AD statistic for the M blocks

ad.c

2 x 3 (2 x 4) matrix containing AD_{comb}, T_{comb}, asymptotic P-value, (simulated or exact P-value), for each version of the combined test statistic, version 1 in row 1 and version 2 in row 2

mu.c

mean of AD_{comb}

sig.c

standard deviation of AD_{comb}

warning

logical indicator, warning = TRUE when at least one n_{ij} < 5

null.dist1

simulated or enumerated null distribution of version 1 of AD_{comb}

null.dist2

simulated or enumerated null distribution of version 2 of AD_{comb}

method

the method used.

Nsim

the number of simulations used for each block of samples.

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.

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

ad.test, ad.pval

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) )
# and a corresponding data frame datx1x2
x1x2 <- c(unlist(x1),unlist(x2))
gx1x2 <- as.factor(c(rep(1,5),rep(2,6),rep(3,5),rep(1,6),rep(2,5)))
bx1x2 <- as.factor(c(rep(1,16),rep(2,11)))
datx1x2 <- data.frame(A = x1x2, G = gx1x2, B = bx1x2)

## Run ad.test.combined.
set.seed(2627)
ad.test.combined(x1, x2, method = "simulated", Nsim = 1000) 
# or with same seed
# ad.test.combined(list(x1, x2), method = "simulated", Nsim = 1000)
# ad.test.combined(A~G|B,data=datx1x2,method="simulated",Nsim=1000)