Page's L Test

Development

Let xij be the observed value under the jth treatment in the ith block (where j = 1,2,…,c and i = 1,2,…,n). In each of the n independent blocks, replace the c observations by their corresponding ranks such that a rank of 1 is given to the smallest observation in the block and a rank of c is given to the largest. Thus, Rij is the rank (from 1 to c) associated with the jth treatment in the ith block.

The layout for the ranked responses from a sample of either n subjects or n blocks of matched subjects over c levels of a treatment condition are shown in Table 1.

Note that under the null hypothesis of no treatment effects, each ranking within a block is equally likely, so there are c! possible rankings within a particular block and (c!)n possible arrangements of ranks over all n blocks. The sum of the ranks within each block is c(c + 1)/2, the sum of the ranks assigned to each of the c treatment levels. If the null hypothesis were perfectly true, the sum of the ranks for each of the c treatment levels would be n(c + 1)/2. On the other hand, if the alternative hypothesis is true and there is a perfect ordering in the c rankings among all n subjects, the sum of the ranks for each of the c treatment levels would respectively be n,2n,…, cn if the treatment effects were monotonically increasing, and cn,(c − 1)n,…, n if the treatment effects were monotonically decreasing.

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Table 1 Data Layout for the Page L Test
		Treatments
Block	1	2	…	c	Totals
1	R 11	R 12	…	R 1c	c(c + 1)/2
2	R 21	R 22	…	R 2c	c(c + 1)/2
.	.	.	.	.	.
.	.	.	.	.	.
.	.	.	.	.	.
n	R n1	R n2	…	R nc	c(c + 1)/2
Totals	R .1	R .2	…	R .c	nc(c + 1)/2
where
c = the number of treatment levels (i.e., columns)
n = the number of blocks (i.e., rows of subjects)
Rij = the rank assigned to the jth treatment in the ith block
R.j = the sum of the ranks for treatment level j

To develop the Page test statistic L, the ranks assigned to each of the c treatment levels are totaled over all n blocks. That is, R.j =Σni=1Rij is obtained for each of the treatments (where j = 1,2,…,c). Page's test statistic L is