Nonparametric Statistics
Page’s L Trend Test Formula for Ordered Treatments in Blocked Ranks
Detect monotonic dose-response or learning-curve trends across blocked repeated-measures ranks using Page’s (1963) weighted rank sum L = SUM(j * R_j) and directional Z_L score.
See how Page’s directional L statistic captures order that omnibus Friedman ignores
When k repeated-measures conditions follow a known a priori sequence (such as Dose 1 < Dose 2 < Dose 3 < Dose 4 or Trial 1..4), Friedman’s Q squares deviations without regard to column order — shuffling columns reverses a trend while leaving Q unchanged. Page (1963) weights each treatment’s rank sum R_j by its hypothesized order index j = 1..k: L = 1*R1 + 2*R2 + ... + k*Rk.
Verify by hand for N = 6 blocks and k = 4 ordered dosages whose within-row rank sums (entered in B2:E2) are R1 = 8, R2 = 13, R3 = 17, R4 = 22 (checksum = 60 = 6*4*5/2): L = 1*8 + 2*13 + 3*17 + 4*22 = 8 + 26 + 51 + 88 = 173! Under H0, exact mean mu_L = N * k * (k + 1)^2 / 4 = 6 * 4 * 25 / 4 = 150 and exact variance var_L = N * k^2 * (k + 1) * (k^2 - 1) / 144 = 7,200 / 144 = 50, giving Z_Page = (173 - 150) / SQRT(50) = 3.2527.
| Page Trend Metric (N = 6, k = 4) | Exact Formula | Sample Output |
|---|---|---|
| Page Weighted Rank Sum L | SUMPRODUCT({1,2,3,4}, R1:R4) | 173 |
| Null Expectation mu_L | N * k * (k + 1)^2 / 4 | 150 |
| Null Variance var_L | N * k^2 * (k + 1) * (k^2 - 1) / 144 | 50 (SD = 7.0711) |
| Standardized Directional Z_Page | (L - mu_L) / SQRT(var_L) | +3.2527 (Reversing order -> -3.2527) |
Page’s L Statistic with Rank-Sum Integrity Check Formula
Validate that the k column rank sums equal N * k * (k + 1) / 2 before reporting L so mis-ranked rows or unadjusted missing values are caught immediately.
=LET(n_blk, B5, k_ord, COLUMNS(B3:E3), chk_sum, n_blk * k_ord * (k_ord + 1) / 2, IF(ABS(SUM(B3:E3) - chk_sum) > 0.001, "Rank Sum Mismatch", SUMPRODUCT(B2:E2, B3:E3)))Standardized Directional Z_Page Formula
Test directional sensitivity by reversing the column rank sums to 22, 17, 13, 8: omnibus Friedman Q stays at 10.6000, whereas L drops from 173 to 127 and Z_Page flips sign from +3.2527 to -3.2527.
=LET(n_blk, B5, k_ord, COLUMNS(B3:E3), chk_sum, n_blk * k_ord * (k_ord + 1) / 2, IF(OR(n_blk < 2, k_ord < 3, ABS(SUM(B3:E3) - chk_sum) > 0.001), "Invalid Rank Table", LET(l_val, SUMPRODUCT(B2:E2, B3:E3), mu_l, n_blk * k_ord * ((k_ord + 1)^2) / 4, var_l, (n_blk * (k_ord^2) * (k_ord + 1) * (k_ord^2 - 1)) / 144, ROUND((l_val - mu_l) / SQRT(var_l), 4))))Try the example
Use the original values or change them to test your understanding.