Max R. P. Grossmann

Page's L test is a little-known nonparametric test for testing for a trend. It was originally described in this paper (shorter version here).

There is already this R implementation, but it has somewhat odd behavior and gives warnings without explaining how to address them. Here is an alternative implementation. Let X be a matrix of outcomes from a “within” experiment in wide format. That is, each column represents one treatment; each row represents one experimental subject.

In the following, I will use the example from Page's paper. My code (licensed under CC0) for Page's L test will automatically transform the data in X into ranks:

X <- matrix(c(2, 1, 3, 4,
              1, 3, 4, 2,
              1, 3, 2, 4,
              1, 4, 2, 3,
              3, 1, 2, 4,
              1, 2, 4, 3), ncol = 4, byrow = T)

page.l <- function (X) {
    n <- ncol(X)
    m <- nrow(X)

    X_rank <- t(apply(X, 1, rank))

    L <- sum(colSums(X_rank) * seq(1, n))
    stat <- ((12*L - 3*m*n*(n+1)^2)^2)/(m*n^2*(n^2-1)*(n+1))
    p <- 1-pchisq(stat, 1)

    list(L = L, stat = stat, p.value = p)
}

print(page.l(X))

The result is:

$L
[1] 168

$stat
[1] 6.48

$p.value
[1] 0.0109095

Indeed, the p-values are calculated from a χ² approximation described in Page's original papers. The p.value returned by the code above reflects the two-tailed test; as Page wrote, if “a one-sided test is desired, as will almost always be the case, the probability discovered [from the χ² distribution] should be halved” (emphasis in original). The exact critical value for his example (one-tailed test, m = 6, n = 4) is 167 at α = 0.01; from the code above, we would estimate a p-value of the one-sided test of p ≈ 0.0055.

Perhaps I will one day reconstruct Page's original computer programs that he used to calculate the exact critical values. It must have been very time-consuming to run these in the 1950s.

The size of Page's L test

In my opinion, Page's L test typically performs very well; as is often expected of nonparametric tests—not entirely correctly—it is well-powered. But, as is also well known, one could construct a test with power 1 by always rejecting the H₀. Let us thus investigate the (empirical) size of Page's L test. Is the test solid? We will simulate data consistent with the null hypothesis of no trend and see how the p-values are distributed.

I choose to simulate the case of 100 subjects and 5 treatments. The outcome is N(0,1) and there is no trend. Feel free to change these parameters and report back. Here is the code:

library(pbapply) # for pbreplicate
set.seed(622213478)

gen.data <- function () {
    matrix(rnorm(5*100, 0, 1), ncol = 5)
}

p.sim <- pbreplicate(10000, page.l(gen.data())$p.value)

ks.test(p.sim, punif) # rejects at 5% level

mean(p.sim < 0.01) # fine
mean(p.sim < 0.05) # fine
mean(p.sim < 0.1) # fine

summary(p.sim) # also fine

plot(ecdf(p.sim)) # looks fine, too
abline(0, 1, col = "red")

This shows that a Kolmogorov-Smirnov test rejects the null hypothesis of the p-values being uniformly distributed on [0,1]. (Under the H₀, p-values are U(0,1).) Visual inspection and an investigation using summary() reveal, however, that this is likely due to slightly irregular behavior at high values for α. Overall, Page's L test seems to be well sized. At usual values for α, almost exactly the fraction of tests that should be (falsely) rejected is rejected. At unusual values for α, it may get a little wobbly!

In my opinion, Page's L test is slightly conservative if X has many ties. But this requires further investigation.