qn.test.combined: Combined Rank Score k-Sample Tests

In kSamples: K-Sample Rank Tests and their Combinations

Combined Rank Score k-Sample Tests

Description

This function combines several independent rank score 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.

Usage

qn.test.combined(..., data = NULL, test = c("KW", "vdW", "NS"),
	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 have same 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
test	= c("KW", "vdW", "NS"), where "KW" uses scores 1:N (Kruskal-Wallis test) "vdW" uses van der Waerden scores, qnorm( (1:N) / (N+1) ) "NS" uses normal scores, i.e., expected values of standard normal order statistics, invoking function normOrder of package SuppDists (>=1.1-9.4) For the above scores N changes from block to block and represents the total pooled sample size N_i for block i.
method	=c("asymptotic","simulated","exact"), where "asymptotic" uses only an asymptotic chi-square approximation for the P-value. The adequacy of asymptotic P-values for use with moderate sample sizes may be checked with method = "simulated". "simulated" uses simulation to get Nsim simulated QN statistics for each block of samples, adding them component wise across blocks to get Nsim combined values, and compares these with the observed combined value to get the estimated P-value. "exact" uses full enumeration of the test statistic value for all sample splits of the pooled samples within each block. The test statistic vectors for each block 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. This "addition" is done one block at a time. 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, were N_i = n_{i1}+\ldots+n_{ik_i} is the sample size of the pooled samples for the i-th block. Otherwise, it reverts to the simulation method using the provided Nsim.
dist	FALSE (default) or TRUE. If TRUE, the simulated or fully enumerated convolution vector null.dist is returned for the QN test statistic. Otherwise, NULL is returned.
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.

Details

The rank score QN criterion QN_i for the i-th block of k_i samples, 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). See qn.test for the definition of the QN criterion and the exact calculation of its null distribution.

The combined QN criterion QN_{\rm comb} = QN_1 + \ldots + QN_M is used to simultaneously test whether the samples in block i come from the same continuous distribution function F_i(x). However, the unspecified common distribution function F_i(x) may change from block to block.

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

The asymptotic approximating chi-square distribution has f = (k_1-1)+\ldots+(k_M-1) degrees of freedom.

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 the asymptotic, simulated or exact P-values are viewed conditionally, given the tie patterns within each block. Under such randomization any conclusions are valid only with respect to the blocks of subjects that were randomly allocated. In case of ties the average rank scores are used across tied observations within each block.

Value

A list of class kSamples with components

test.name	"Kruskal-Wallis", "van der Waerden scores", or "normal scores"
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	vector of length M of total sample sizes involved in each of the M comparisons of k_i samples, respectively
n.ties	vector giving the number of ties in each the M comparison blocks
qn.list	list of M matrices giving the qn results from qn.test, applied to the samples in each of the M blocks
qn.c	2 (or 3) vector containing the observed QN_{\rm comb}, asymptotic P-value, (simulated or exact P-value).
warning	logical indicator, warning = TRUE when at least one n_{ij} < 5.
null.dist	simulated or enumerated null distribution of the QN_{\rm comb} statistic. It is NULL when dist = FALSE or when method = "asymptotic".
method	The method used.
Nsim	The number of simulations used for each block of samples.

Note

These tests are useful in analyzing treatment effects of shift nature in randomized (incomplete) block experiments.

References

Lehmann, E.L. (2006), Nonparametric, Statistical Methods Based on Ranks, Springer Verlag, New York. Ch. 6, Sec. 5D.

See Also

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

kSamples documentation built on Oct. 8, 2023, 1:07 a.m.