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.
qn.test.combined(…, data = NULL, test = c("KW", "vdW", "NS"),
method = c("asymptotic", "simulated", "exact"),
dist = FALSE, Nsim = 10000)
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.
= 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
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
$$\frac{N_1!}{n_{11}!\ldots n_{1k_1}!}\times\ldots\times \frac{N_M!}{n_{M1}!\ldots n_{Mk_M}!}$$
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
.
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.
A list of class kSamples
with components
"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 \(QN_{\rm comb}\), 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
\(QN_{\rm comb}\) statistic.
It is NULL
when dist = FALSE
or when method = "asymptotic"
.
The method
used.
The number of simulations used for each block of samples.
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.
Lehmann, E.L. (2006), Nonparametric, Statistical Methods Based on Ranks, Springer Verlag, New York. Ch. 6, Sec. 5D.
# NOT RUN {
## 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)
# }
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