Statistical & Econometric
Cochran-Armitage Trend Test (Z_CA & χ²_trend) for Dose-Response Proportions
Test monotonic dose-response trends across ordered 2×k clinical trial cohorts using the Cochran-Armitage statistic Z_CA = T / SQRT(Var(T)) and 1-df χ²_trend = Z_CA^2.
Why standard 2×k Pearson χ² throws away your dose ordering—and loses statistical power!
An omnibus 2×k Pearson chi-square test treats Placebo (0 mg), Low Dose (10 mg), and High Dose (20 mg) as unordered nominal labels—wasting degrees of freedom (k - 1 df) and missing monotonic dose-response trends! William G. Cochran (1954) and Peter Armitage (1955) concentrated the linear trend onto a single 1-degree-of-freedom statistic using dose scores x_i (0, 1, 2), responders r_i, group sizes n_i, and pooled proportion p̄ = R / N: numerator T = SUMPRODUCT(x, r) - p̄ * SUMPRODUCT(x, n) and exact hypergeometric-null variance estimate Var(T) = p̄ * (1 - p̄) * (SUMPRODUCT(n, x^2) - (SUMPRODUCT(n, x)^2) / N)!
In a 3-arm Phase II trial (n = 50 per arm; responders 4/50 = 8% on Placebo, 11/50 = 22% on Low Dose, 21/50 = 42% on High Dose), R = 36, N = 150, p̄ = 0.24, T = 17.0, Var(T) = 18.24, Z_CA = +3.980, and 1-df χ²_trend = 15.844 (far exceeding the p = 0.001 critical threshold 10.828)!
| Dose Arm (Score x_i) | Responders r_i / Total n_i | Response Rate (%) | Cochran-Armitage Trend Output |
|---|---|---|---|
| Placebo (x = 0) | 4 / 50 | 8.0% | Pooled Rate p̄ = 36 / 150 = 24.0% |
| Low Dose (x = 1) | 11 / 50 | 22.0% | Trend Z_CA = +3.980 (p < 0.0001) |
| High Dose (x = 2) | 21 / 50 | 42.0% | 1-df χ²_trend = 15.844 (> 3.841) |
Cochran-Armitage Z_CA Trend Statistic
Computes T / SQRT(Var(T)) across dose scores B2:B4, responder counts C2:C4, total sample sizes D2:D4, and squared dose helper column E2:E4 (=B2^2).
=IF(OR(SUM(C2:C4) <= 0, SUM(C2:C4) >= SUM(D2:D4)), "Zero Variance Cohort", ROUND((SUMPRODUCT(B2:B4, C2:C4) - (SUM(C2:C4) / SUM(D2:D4)) * SUMPRODUCT(B2:B4, D2:D4)) / SQRT((SUM(C2:C4) / SUM(D2:D4)) * (1 - SUM(C2:C4) / SUM(D2:D4)) * (SUMPRODUCT(D2:D4, E2:E4) - ((SUMPRODUCT(B2:B4, D2:D4) ^ 2) / SUM(D2:D4)))), 3))1-df Chi-Square Linear Trend Statistic χ²_trend = Z_CA^2
Squares Z_CA for direct comparison against 1-df χ² critical cutoffs (3.841 at α = 0.05; 6.635 at α = 0.01).
=IF(OR(SUM(C2:C4) <= 0, SUM(C2:C4) >= SUM(D2:D4)), "Zero Variance Cohort", ROUND(((SUMPRODUCT(B2:B4, C2:C4) - (SUM(C2:C4) / SUM(D2:D4)) * SUMPRODUCT(B2:B4, D2:D4)) ^ 2) / ((SUM(C2:C4) / SUM(D2:D4)) * (1 - SUM(C2:C4) / SUM(D2:D4)) * (SUMPRODUCT(D2:D4, E2:E4) - ((SUMPRODUCT(B2:B4, D2:D4) ^ 2) / SUM(D2:D4)))), 3))Try the example
Use the original values or change them to test your understanding.