shuffle {permute}

Unrestricted and restricted permutations

Description

Unrestricted and restricted permutation designs for time series, line transects, spatial grids and blocking factors.

Usage

shuffle(n, control = how())

permute(i, n, control)

Arguments

n

numeric; the length of the returned vector of permuted values. Usually the number of observations under consideration. May also be any object that nobs knows about; see nobs-methods.

control

a list of control values describing properties of the permutation design, as returned by a call to how.

i

integer; row of control$all.perms to return.

Details

shuffle can generate permutations for a wide range of restricted permutation schemes. A small selection of the available combinations of options is provided in the Examples section below.

permute is a higher level utility function for use in a loop within a function implementing a permutation test. The main purpose of permute is to return the correct permutation in each iteration of the loop, either a random permutation from the current design or the next permutation from control$all.perms if it is not NULL and control$complete is TRUE.

Value

For shuffle a vector of length n containing a permutation of the observations 1, ..., n using the permutation scheme described by argument control.

For permute the ith permutation from the set of all permutations, or a random permutation from the design.

Author(s)

Gavin Simpson

References

shuffle() is modelled after the permutation schemes of Canoco 3.1 (ter Braak, 1990); see also Besag & Clifford (1989).

Besag, J. and Clifford, P. (1989) Generalized Monte Carlo significance tests. Biometrika 76; 633–642.

ter Braak, C. J. F. (1990). Update notes: CANOCO version 3.1. Wageningen: Agricultural Mathematics Group. (UR).

See Also

check, a utility function for checking permutation scheme described by how.

Examples

set.seed(1234)

## unrestricted permutations
shuffle(20)

## observations represent a time series of line transect
CTRL <- how(within = Within(type = "series"))
shuffle(20, control = CTRL)

## observations represent a time series of line transect
## but with mirroring allowed
CTRL <- how(within = Within(type = "series", mirror = TRUE))
shuffle(20, control = CTRL)

## observations represent a spatial grid, 5rx4c
nr <- 5
nc <- 4
CTRL <- how(within = Within(type = "grid", ncol = nc, nrow = nr))
perms <- shuffle(20, control = CTRL)
## view the permutation as a grid
matrix(matrix(1:20, nrow = nr, ncol = nc)[perms],
       ncol = nc, nrow = nr)

## random permutations in presence of strata
plots <- Plots(strata = gl(4, 5))
CTRL <- how(plots = plots, within = Within(type = "free"))
shuffle(20, CTRL)
## as above but same random permutation within strata
CTRL <- how(plots = plots, within = Within(type = "free",
            constant = TRUE))
shuffle(20, CTRL)

## time series within each level of block
CTRL <- how(plots = plots, within = Within(type = "series"))
shuffle(20, CTRL)
## as above, but  with same permutation for each level
CTRL <- how(plots = plots, within = Within(type = "series",
            constant = TRUE))
shuffle(20, CTRL)

## spatial grids within each level of block, 4 x (5r x 5c)
nr <- 5
nc <- 5
nb <- 4 ## number of blocks
plots <- Plots(gl(nb, 25))
CTRL <- how(plots = plots,
            within = Within(type = "grid", ncol = nc, nrow = nr))
shuffle(100, CTRL)
## as above, but with same permutation for each level
CTRL <- how(plots = plots,
            within = Within(type = "grid", ncol = nc, nrow = nr,
                            constant = TRUE))
shuffle(100, CTRL)

## permuting levels of plots instead of observations
CTRL <- how(plots = Plots(gl(4, 5), type = "free"),
            within = Within(type = "none"))
shuffle(20, CTRL)
## permuting levels of plots instead of observations
## but plots represent a time series
CTRL <- how(plots = Plots(gl(4, 5), type = "series"),
            within = Within(type = "none"))
shuffle(20, CTRL)

## permuting levels of plots but plots represent a time series
## free permutation within plots
CTRL <- how(plots = Plots(gl(4, 5), type = "series"),
            within = Within(type = "free"))
shuffle(20, CTRL)

## permuting within blocks
grp <- gl(2, 10) # 2 groups of 10 samples each
CTRL <- how(blocks = grp)
shuffle(length(grp), control = CTRL)

## Simple function using permute() to assess significance
## of a t.test  
pt.test <- function(x, group, control) {
    ## function to calculate t
    t.statistic <- function(x, y) {
        m <- length(x)
        n <- length(y)
        ## means and variances, but for speed
        xbar <- mean(x)
        ybar <- mean(y)
        xvar <- var(x)
        yvar <- var(y)
        pooled <- sqrt(((m-1)*xvar + (n-1)*yvar) / (m+n-2))
        (xbar - ybar) / (pooled * sqrt(1/m + 1/n))
    }
    ## check the control object
    #control <- check(x, control)$control ## FIXME
    ## number of observations
    Nobs <- nobs(x)
    ## group names
    lev <- names(table(group))
    ## vector to hold results, +1 because of observed t
    t.permu <- numeric(length = control$nperm) + 1
    ## calculate observed t
    t.permu[1] <- t.statistic(x[group == lev[1]], x[group == lev[2]])
    ## generate randomisation distribution of t
    for(i in seq_along(t.permu)) {
        ## return a permutation
        want <- permute(i, Nobs, control)
        ## calculate permuted t
        t.permu[i+1] <- t.statistic(x[want][group == lev[1]],
                                    x[want][group == lev[2]])
    }
    ## pval from permutation test
    pval <- sum(abs(t.permu) >= abs(t.permu[1])) / (control$nperm + 1)
    ## return value
    return(list(t.stat = t.permu[1], pval = pval))
}

## generate some data with slightly different means
set.seed(1234)
gr1 <- rnorm(20, mean = 9)
gr2 <- rnorm(20, mean = 10)
dat <- c(gr1, gr2)
## grouping variable
grp <- gl(2, 20, labels = paste("Group", 1:2))
## create the permutation design
control <- how(nperm = 999, within = Within(type = "free"))
## perform permutation t test
perm.val <- pt.test(dat, grp, control)
perm.val

## compare perm.val with the p-value from t.test()
t.test(dat ~ grp, var.equal = TRUE)