# Distribution of One and Two Sample Permutation Tests Density, distribution function and quantile function for the distribution of one and two sample permutation tests using the Shift-Algorithm by Streitberg & R\"ohmel. ```r dperm(x, scores, m, paired=NULL, tol = 0.01, fact=NULL, density=FALSE, simulate=FALSE, B=10000) pperm(q, scores, m, paired=NULL, tol = 0.01, fact=NULL, alternative=c("less", "greater", "two.sided"), pprob=FALSE, simulate=FALSE, B=10000) qperm(p, scores, m, paired=NULL, tol = 0.01, fact=NULL, simulate=FALSE, B=10000) rperm(n, scores, m) ``` ## Arguments - `x, q`: vector of quantiles. - `p`: vector of probabilities. - `scores`: arbitrary 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 rounding after multiplication with an appropriate factor. Make sure that the absolute difference between the each possible test statistic for the original scores and the rounded scores is less than `tol`. This might not be possible due to memory/time limitations, a warning is given in this case. - `fact`: real. If `fact` is given, real valued scores are mapped into integers using `fact` as factor. `tol` is ignored in this case. - `n`: number of random observations to generate. - `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. - `density`: logical. When `x` is a scalar and `density` is `TRUE`, `dperm` returns the density for all possible statistics less or equal `x` as a data frame. - `simulate`: logical. Use conditional Monte-Carlo to compute the distribution. - `B`: number of Monte-Carlo replications to be used. ## Details The exact distribution of the sum of the first `m` scores is evaluated using the Shift-Algorithm by Streitberg & 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 statistics which differ not more than `tol` from the statistics computed for the original scores. This can be done as follows. Without loss of generality let $a_i > 0$ denote real valued scores in reverse ordering and $f$ a positive factor (this is the `fact` argument). Let $R_i = f \cdot a_i - round(f \cdot a_i)$. 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 $1/f \sum_{i=1}^m f \cdot a_i$ and $1/f \sum_{i=1}^n round(f \cdot a_i)$ is given by $|\sum_{i=1}^m R_i|$. Therefore one searches for $f$ with $$ |\sum_{i=1}^m R_i| \le \sum_{i=1}^m |R_i| \le tol. $$ If $f$ induces more that 100.000 columns in the Shift-Algorithm by Streitberg & R\"ohmel, $f$ is restricted to the largest integer that does not. The second idea is to map the scores into integers by taking the integer part of $a_i N / \max(a_i)$ (Hothorn & Lausen, 2002). 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 valued 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-5 manual, page 225, equation (9.31) and equation (8.18), p. 179 (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`. ## Returns `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`. ## References Bernd Streitberg & 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 & 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 News** 1 (1), 11--12. Cyrus R. Mehta & Nitin R. Patel (2001), **StatXact-5 for Windows.** Manual, Cytel Software Cooperation, Cambridge, USA Torsten Hothorn & Berthold Lausen (2003), On the exact distribution of maximally selected rank statistics. **Computational Statistics & Data Analysis**, 43 (2), 121-137. ## Examples ```r # 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 # Shift-Algorithm by 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 the Shift-Algorithm dwexac <- dperm((n*(n+1)/2):(n^2 + n*(n+1)/2), r, n) sum(dwexac) # should be something near 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 the Shift-Algorithm sc <- cscores(c(x,y), type="Ansari") dabexac <- dperm(0:(n*(2*n+1)/2), sc, n) sum(dabexac) # 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 ```

dperm function