The Inverse Variance Weighted Least Squares Simple Regression as Generalized Cochran–Armitage Test for Trend

Trend tests such as the Cochran-Armitage test for trend, the Jonckheere-Terpstra test, and Cuzick’s test require specific data format, limiting their applicability. This article shows that the inverse variance weighted least squares simple regression model works as the generalized Cochran-Armitage test for trend, which is useful when the requirements for the Cochran-Armitage test for trend, the Jonckheere-Terpstra test, and Cuzick’s test are not met. We propose naming this test the ‘inverse variance trend test’ or the ‘generalized Cochran-Armitage test’ for clarity. We assume that k ordered groups have interval-scaled dependent variables y, where the distances between values are consistent, along with their associated variance v. The groups are assigned group score x, typically ranging from 1 to k, though not necessarily. The weight w is defined as the inverse of the variance v for each group. It is assumed that the error terms in the regression analysis follow a normal distribution, and the sample size should be sufficiently large for the central limit theorem to apply. Z^2 = [Σ_i w_i (y_i – y_avg)(x_i – x_avg)]^2 /[Σ_i w_i (x_i – x_avg)^2)] ~ χ^2 (df = 1). The null hypothesis is that 𝑦 is independent from 𝑥, or the slope = 0. We demonstrate the broad applicability of this test across various datasets by applying it to five medical case examples.