It modifies the Pearson chi-squared test to incorporate a suspected ordering in the effects of the k categories of the second variable.
The hypothesis of no association (the null hypothesis) can be expressed as: Assuming this holds, then, using iterated expectation, The variance can be computed by decomposition, yielding and as a large sample approximation, The weights ti can be chosen such that the trend test becomes locally most powerful for detecting particular types of associations.
Thus, the suspected trend in effects is not an assumption that must hold in order for the test results to be meaningful.
For example, consider the following data, in which the genotype frequencies vary linearly in the cases and are constant in the controls: In genetics applications, the weights are selected according to the suspected mode of inheritance.
Thus, we obtain a stronger significance level if the weights corresponding to additive (codominant) inheritance are used.