qn.test: Rank Score k-Sample Tests
In kSamples: K-Sample Rank Tests and their Combinations
qn.test R Documentation
Rank Score k-Sample Tests
Description
This function uses the QN criterion (Kruskal-Wallis, van der Waerden scores, normal scores) to test
the hypothesis that k independent samples arise
from a common unspecified distribution.
Usage
qn.test(..., data = NULL, test = c("KW", "vdW", "NS"),
method = c("asymptotic", "simulated", "exact"),
dist = FALSE, Nsim = 10000)
Arguments
...
Either several sample vectors, say
x_1, \ldots, x_k,
with x_i containing n_i sample values.
n_i > 4 is recommended for reasonable asymptotic
P-value calculation. The pooled sample size is denoted
by N=n_1+\ldots+n_k,
or a list of such sample vectors,
or a formula y ~ g, where y contains the pooled sample values
and g (same length as y) is a factor with levels identifying
the samples to which the elements of y belong.
data
= an optional data frame providing the variables in formula y ~ g.
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 standard normal order statistics,
invoking function normOrder of package SuppDists (>=1.1-9.4)
method
= c("asymptotic","simulated","exact"), where
"asymptotic" uses only an asymptotic chi-square approximation
with k-1 degrees of freedom to approximate the P-value.
This calculation is always done.
"simulated" uses Nsim simulated QN statistics based on random
splits of the pooled samples into samples of sizes
n_1, \ldots, n_k, to estimate the P-value.
"exact" uses full enumeration of all sample splits with resulting
QN statistics to obtain the exact P-value.
It is used only when Nsim is at least as large as the number
ncomb = \frac{N!}{n_1!\ldots n_k!}
of
full enumerations. Otherwise, method
reverts to "simulated" using the given Nsim. It also reverts
to "simulated" when ncomb > 1e8 and dist = TRUE.
dist
FALSE (default) or TRUE. If TRUE, the
simulated or fully enumerated null distribution vector null.dist
is returned for the QN test statistic. Otherwise, NULL
is returned. When dist = TRUE then Nsim <- min(Nsim, 1e8),
to limit object size.
Nsim
= 10000 (default), number of simulation sample splits to use.
It is only used when method = "simulated",
or when method = "exact" reverts to method =
"simulated", as previously explained.
Details
The QN criterion based on rank scores v_1,\ldots,v_N is
QN=\frac{1}{s_v^2}\left(\sum_{i=1}^k \frac{(S_{iN}-n_i \bar{v}_{N})^2}{n_i}\right)
where S_{iN} is the sum of rank scores for the i-th sample and
\bar{v}_N and
s_v^2 are sample mean and sample variance (denominator N-1)
of all scores.
The statistic QN is used to test the hypothesis that the samples all come
from the same but unspecified continuous distribution function F(x).
QN is always adjusted for ties by averaging the scores of tied observations.
Conditions for the asymptotic approximation (chi-square with k-1 degrees of freedom)
can be found in Lehmann, E.L. (2006), Appendix Corollary 10, or in
Hajek, Sidak, and Sen (1999), Ch. 6, problems 13 and 14.
For small sample sizes exact null distribution
calculations are possible (with or without ties), based on a recursively extended
version of Algorithm C (Chase's sequence) in Knuth (2011), which allows the
enumeration of all possible splits of the pooled data into samples of
sizes of n_1, \ldots, n_k, as appropriate under treatment randomization. This
is done in C, as is the simulation.
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 from any common distribution,
or if randomization was used in allocating
subjects to samples or treatments, and if
the asymptotic, simulated or exact P-values are viewed conditionally, given the tie pattern
in the pooled sample. Under such randomization any conclusions
are valid only with respect to the subjects that were randomly allocated
to their respective treatment samples.
Value
A list of class kSamples with components
test.name
"Kruskal-Wallis", "van der Waerden scores", or
"normal scores"
k
number of samples being compared
ns
vector (n_1,\ldots,n_k) of the k sample sizes
N
size of the pooled samples = n_1+\ldots+n_k
n.ties
number of ties in the pooled sample
qn
2 (or 3) vector containing the observed QN, its asymptotic P-value,
(its simulated or exact P-value)
warning
logical indicator, warning = TRUE when at least one
n_i < 5
null.dist
simulated or enumerated null distribution
of the test statistic. It is NULL when dist = FALSE or when
method = "asymptotic".
method
the method used.
Nsim
the number of simulations used.
warning
method = "exact" should only be used with caution.
Computation time is proportional to the number of enumerations.
Experiment with system.time and trial values for
Nsim to get a sense of the required computing time.
In most cases
dist = TRUE should not be used, i.e.,
when the returned distribution objects
become too large for R's work space.
References
Hajek, J., Sidak, Z., and Sen, P.K. (1999), Theory of Rank Tests (Second Edition), Academic Press.
Knuth, D.E. (2011), The Art of Computer Programming, Volume 4A
Combinatorial Algorithms Part 1, Addison-Wesley
Kruskal, W.H. (1952), A Nonparametric Test for the Several Sample Problem,
The Annals of Mathematical Statistics,
Vol 23, No. 4, 525-540
Kruskal, W.H. and Wallis, W.A. (1952), Use of Ranks in One-Criterion Variance Analysis,
Journal of the American Statistical Association,
Vol 47, No. 260, 583–621.
Lehmann, E.L. (2006),
Nonparametrics, Statistical Methods Based on Ranks, Revised First Edition,
Springer Verlag.
See Also
qn.test.combined
Examples
u1 <- c(1.0066, -0.9587, 0.3462, -0.2653, -1.3872)
u2 <- c(0.1005, 0.2252, 0.4810, 0.6992, 1.9289)
u3 <- c(-0.7019, -0.4083, -0.9936, -0.5439, -0.3921)
yy <- c(u1, u2, u3)
gy <- as.factor(c(rep(1,5), rep(2,5), rep(3,5)))
set.seed(2627)
qn.test(u1, u2, u3, test="KW", method = "simulated",
dist = FALSE, Nsim = 1000)
# or with same seed
# qn.test(list(u1, u2, u3),test = "KW", method = "simulated",
# dist = FALSE, Nsim = 1000)
# or with same seed
# qn.test(yy ~ gy, test = "KW", method = "simulated",
# dist = FALSE, Nsim = 1000)
kSamples documentation built on Oct. 8, 2023, 1:07 a.m.
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| qn.test | R Documentation |
Rank Score k-Sample Tests
Description
This function uses the QN criterion (Kruskal-Wallis, van der Waerden scores, normal scores) to test
the hypothesis that k independent samples arise
from a common unspecified distribution.
Usage
qn.test(..., data = NULL, test = c("KW", "vdW", "NS"),
method = c("asymptotic", "simulated", "exact"),
dist = FALSE, Nsim = 10000)
Arguments
... |
Either several sample vectors, say
or a list of such sample vectors, or a formula y ~ g, where y contains the pooled sample values and g (same length as y) is a factor with levels identifying the samples to which the elements of y belong. |
data |
= an optional data frame providing the variables in formula y ~ g. |
test |
=
|
method |
=
of
full enumerations. Otherwise, |
dist |
|
Nsim |
|
Details
The QN criterion based on rank scores v_1,\ldots,v_N is
QN=\frac{1}{s_v^2}\left(\sum_{i=1}^k \frac{(S_{iN}-n_i \bar{v}_{N})^2}{n_i}\right)
where S_{iN} is the sum of rank scores for the i-th sample and
\bar{v}_N and
s_v^2 are sample mean and sample variance (denominator N-1)
of all scores.
The statistic QN is used to test the hypothesis that the samples all come
from the same but unspecified continuous distribution function F(x).
QN is always adjusted for ties by averaging the scores of tied observations.
Conditions for the asymptotic approximation (chi-square with k-1 degrees of freedom)
can be found in Lehmann, E.L. (2006), Appendix Corollary 10, or in
Hajek, Sidak, and Sen (1999), Ch. 6, problems 13 and 14.
For small sample sizes exact null distribution
calculations are possible (with or without ties), based on a recursively extended
version of Algorithm C (Chase's sequence) in Knuth (2011), which allows the
enumeration of all possible splits of the pooled data into samples of
sizes of n_1, \ldots, n_k, as appropriate under treatment randomization. This
is done in C, as is the simulation.
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 from any common distribution,
or if randomization was used in allocating
subjects to samples or treatments, and if
the asymptotic, simulated or exact P-values are viewed conditionally, given the tie pattern
in the pooled sample. Under such randomization any conclusions
are valid only with respect to the subjects that were randomly allocated
to their respective treatment samples.
Value
A list of class kSamples with components
test.name |
|
k |
number of samples being compared |
ns |
vector |
N |
size of the pooled samples |
n.ties |
number of ties in the pooled sample |
qn |
2 (or 3) vector containing the observed |
warning |
logical indicator, |
null.dist |
simulated or enumerated null distribution
of the test statistic. It is |
method |
the |
Nsim |
the number of simulations used. |
warning
method = "exact" should only be used with caution.
Computation time is proportional to the number of enumerations.
Experiment with system.time and trial values for
Nsim to get a sense of the required computing time.
In most cases
dist = TRUE should not be used, i.e.,
when the returned distribution objects
become too large for R's work space.
References
Hajek, J., Sidak, Z., and Sen, P.K. (1999), Theory of Rank Tests (Second Edition), Academic Press.
Knuth, D.E. (2011), The Art of Computer Programming, Volume 4A Combinatorial Algorithms Part 1, Addison-Wesley
Kruskal, W.H. (1952), A Nonparametric Test for the Several Sample Problem, The Annals of Mathematical Statistics, Vol 23, No. 4, 525-540
Kruskal, W.H. and Wallis, W.A. (1952), Use of Ranks in One-Criterion Variance Analysis, Journal of the American Statistical Association, Vol 47, No. 260, 583–621.
Lehmann, E.L. (2006), Nonparametrics, Statistical Methods Based on Ranks, Revised First Edition, Springer Verlag.
See Also
qn.test.combined
Examples
u1 <- c(1.0066, -0.9587, 0.3462, -0.2653, -1.3872)
u2 <- c(0.1005, 0.2252, 0.4810, 0.6992, 1.9289)
u3 <- c(-0.7019, -0.4083, -0.9936, -0.5439, -0.3921)
yy <- c(u1, u2, u3)
gy <- as.factor(c(rep(1,5), rep(2,5), rep(3,5)))
set.seed(2627)
qn.test(u1, u2, u3, test="KW", method = "simulated",
dist = FALSE, Nsim = 1000)
# or with same seed
# qn.test(list(u1, u2, u3),test = "KW", method = "simulated",
# dist = FALSE, Nsim = 1000)
# or with same seed
# qn.test(yy ~ gy, test = "KW", method = "simulated",
# dist = FALSE, Nsim = 1000)
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