exactRankTests (version 0.7-5)

dperm: Distribution of One and Two Sample Permutation Tests

Description

Density, distribution function and quantile function for the distribution of one and two sample permutation tests using the Shift-Algorithm by Streitberg and R"ohmel.

Usage

dperm(x, scores, m, paired=NULL, tol = 0.01, fact=NULL)
pperm(q, scores, m, paired=NULL, tol = 0.01, fact=NULL,
      alternative=c("less", "greater", "two.sided"), pprob=FALSE)
qperm(p, scores, m, paired=NULL, tol = 0.01, fact=NULL)
rperm(n, scores, m)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

scores

ranks, midranks or real valued scores of the observations of the x (first m elements) and y sample.

m

sample size of the x sample. If m = length(x) scores of paired observations are assumed.

paired

logical. Indicates if paired observations are used. Needed to discriminate between a paired problem and the distribution of the total sum of the scores (which has mass 1 at the point sum(scores))

tol

real. real valued scores are mapped into integers by multiplication. Make sure that the absolute difference between the "true" quantile and the approximated quantile is less than tol. This might not be possible due to memory/time limitations.

fact

real. If fact is given, real valued scores are mapped into integers using fact as factor. tol is ignored.

n

number of observations.

alternative

character indicating whether the probability $P(T \le q)$ (less), $P(T \ge q)$ (greater) or a two sided p-value (two.sided) should be computed in pperm.

pprob

logical. Indicates if the probability $P(T = q)$ should be computed additionally.

Value

• dperm gives the density, pperm gives the distribution function and qperm gives the quantile function. If pprob is true, pperm returns a list with elements
• PVALUE
  the probability specified by alternative.
• PPROB
  the probability $P(T = q)$.
• rperm is a wrapper to sample.

Details

The exact distribution of the sum of the first m scores is evaluated using the Shift-Algorithm by Streitberg and R"ohmel under the hypothesis of exchangeability (or, equivalent, the hypothesis that all permutations of the scores are equally likely). The algorithm is able to deal with tied scores, so the conditional distribution can be evaluated.

The algorithm is defined for positive integer valued scores only. There are two ways dealing with real valued scores. First, one can try to find integer valued scores that lead to quantiles which differ not more than tol from the quantiles induced by the original scores. This can be done as follows.

Without loss of generality let $a_i > 0$ denote real valued scores and $f$ a positive factor. Let $R_i = a_i - round(f \cdot a_i)$. Then

$$\sum_{i=1}^m f \cdot a_i = \sum_{i=1}^m round(f \cdot a_i) - R_i.$$

Clearly, the maximum difference between $\sum_{i=1}^m f \cdot a_i$ and $\sum_{i=1}^n round(f \cdot a_i)$ is given by $|\sum_{i=1}^m R_{(i)}|$ or $|\sum_{i=m+1}^N R_{(i)}|$, respectively. Therefore one searches for $f$ with

$$\max(|\sum_{i=1}^m R_{(i)}|, |\sum_{i=m+1}^N R_{(i)}|)/f \le tol.$$

If $f$ induces more that 20.000 columns in the Streitberg-R"ohmel Shift-Algorithm, $f$ is restricted to the largest integer that does not.

The second idea is to map the scores itself into integers by taking the integer part of $a_i N / \max(a_i)$. This induces additional ties, but the shape of the scores is very similar. That means we do not try to approximate something but use a different test (with integer scores), serving for the same purpose (due to a similar shape of the scores). However, this has to be done prior to calling pperm (see the examples).

Exact two-sided p-values are computed as suggested in the StatXact-4 manual, page 209, equation (9.32) and equation (8.19), p. 165 (paired case). In detail: For the paired case the two-sided p-value is just twice the one-sided one. For the independent sample case the two sided p-value is defined as $$p_2 = P( |T - E(T)| \ge | q - E(T) |)$$ where $q$ is the quantile passed to pperm.

References

Bernd Streitberg and Joachim R"ohmel (1986). Exact Distributions for Permutations and Rank Tests: An Introduction to Some Recently Published Algorithms. Statistical Software Newsletter 12(1), 10--17.

Bernd Streitberg and Joachim R"ohmel (1987). Exakte Verteilungen f"ur Rang- und Randomisierungstests im allgemeinen $c$-Stichprobenfall. EDV in Medizin und Biologie 18(1), 12--19.

Torsten Hothorn (2001). On Exact Rank Tests in R. R News 1(1), 11--12. Cyrus R. Mehta and Nitin R. Patel (1998). StatXact-4 for Windows. Manual, Cytel Software Cooperation, Cambridge, USA

Examples

# exact one-sided p-value of the Wilcoxon test for a tied sample

x <- c(0.5, 0.5, 0.6, 0.6, 0.7, 0.8, 0.9)
y <- c(0.5, 1.0, 1.2, 1.2, 1.4, 1.5, 1.9, 2.0)
r <- cscores(c(x,y), type="Wilcoxon")
pperm(sum(r[seq(along=x)]), r, 7)

# Compare the exact algorithm as implemented in ctest and the
# Streitberg-Roehmel for untied samples
 
# Wilcoxon:

n <- 10
x <- rnorm(n, 2)
y <- rnorm(n, 3)
r <- cscores(c(x,y), type="Wilcoxon")

# exact distribution using Streitberg-Roehmel

dwexac <- dperm((n*(n+1)/2):(n^2 + n*(n+1)/2), r, n)
su <- sum(dwexac)           # should be something near 1 :-)
su
if (su != 1) stop("sum(dwexac) not equal 1")

# exact distribution using dwilcox

dw <- dwilcox(0:(n^2), n, n)

# compare the two distributions:

plot(dw, dwexac, main="Wilcoxon", xlab="dwilcox", ylab="dperm")      
# should give a "perfect" line

# Wilcoxon signed rank test

n <- 10
x <- rnorm(n, 5)
y <- rnorm(n, 5)
r <- cscores(abs(x - y), type="Wilcoxon")
pperm(sum(r[x - y > 0]), r, length(r))
wilcox.test(x,y, paired=TRUE, alternative="less")
psignrank(sum(r[x - y > 0]), length(r))

# Ansari-Bradley

n <- 10
x <- rnorm(n, 2, 1)
y <- rnorm(n, 2, 2)

# exact distribution using Streitberg-Roehmel

sc <- cscores(c(x,y), type="Ansari")
dabexac <- dperm(0:(n*(2*n+1)/2), sc, n)
sum(dabexac)
tr <- which(dabexac > 0)

# exact distribution using dansari (wrapper to ansari.c in ctest)

dab <- dansari(0:(n*(2*n+1)/2), n, n)

# compare the two distributions:

plot(dab[tr], dabexac[tr], main="Ansari", xlab="dansari", ylab="dperm")

# real scores are allowed (but only result in an approximation)
# e.g. v.d. Waerden test

n <- 10
x <- rnorm(n)
y <- rnorm(n)
scores <- cscores(c(x,y), type="NormalQuantile")
X <- sum(scores[seq(along=x)])  # <- v.d. Waerden normal quantile statistic

# critical value, two-sided test

abs(qperm(0.025, scores, length(x)))

# p-values

p1 <- pperm(X, scores, length(x), alternative="two.sided")

# generate integer valued scores with the same shape as normal quantile
# scores, this no longer v.d.Waerden, but something very similar

scores <- cscores(c(x,y), type="NormalQuantile", int=TRUE)

X <- sum(scores[seq(along=x)])
p2 <- pperm(X, scores, length(x), alternative="two.sided")

# compare p1 and p2

p1 - p2

# the blood pressure example from StatXact manual, page 221:

treat <- c(94, 108, 110, 90)
contr <- c(80, 94, 85, 90, 90, 90, 108, 94, 78, 105, 88)

# compute the v.d. Waerden test and compare the results to StatXact-4 for
# Windows:

sc <- cscores(c(contr, treat), type="NormalQuantile")
X <- sum(sc[seq(along=contr)])
round(pperm(X, sc, 11), 4) 	# == 0.0462 (StatXact)
round(pperm(X, sc, 11, alternative="two.sided"), 4) 	# == 0.0799 (StatXact)

# the alternative method returns:

sc <- cscores(c(contr, treat), type="NormalQuantile", int=TRUE)
X <- sum(sc[seq(along=contr)])

round(pperm(X, sc, 11), 4)      # compare to 0.0462 
round(pperm(X, sc, 11, alternative="two.sided"), 4)     # compare to 0.0799