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

= an optional data frame providing the variables in formula input

= 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.

=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}!… n_{1k_1}!}\times…\times \frac{N_M!}{n_{M1}!… n_{Mk_M}!}

of the final distribution vector, were N_i = n_{i1}+…+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.

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.

= 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.

"Kruskal-Wallis", "van der Waerden scores", or

"normal scores"

number of blocks of samples being compared

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

vector of length M of total sample sizes involved in each of the M comparisons of k.i samples, respectively

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

list of M matrices giving the qn results from qn.test, applied to the samples in each of the M blocks

2 (or 3) vector containing the observed QN.comb, asymptotic P-value, (simulated or exact P-value).

logical indicator, warning = TRUE when at least one n.ij < 5.

simulated or enumerated null distribution of the QN.comb statistic. It is NULL when dist = FALSE or when method = "asymptotic".

The method used.

The number of simulations used for each block of samples.