Signed rank statistics for coherent predictions.

A generalization of Wilcoxon's signed rank test is proposed for testing a dose-response relationship with one or more outcomes. The test is useful in matched observational studies or in nonrandomized experiments that use dose-response relationships and predictions about multiple outcomes in an effort to distinguish actual treatment effects from hidden biases. A sensitivity analysis indicating whether a dose-response relationship or multiple predictions are confirmed with sufficient strength to reduce sensitivity to hidden bias is performed. Together, the test and the sensitivity analysis help to quantify the degree to which a coherent pattern of associations is present or absent, and the degree to which this strengthens or fails to strengthen evidence of cause and effect. Formal properties of tests of this kind are examined. The form of the optimal test is determined, though this test is not usable because it depends upon the values of the unknown parameters under study. Also examined are the conditions under which the proposed test resembles the optimal test, as well as the impact of various violations of those conditions on power. An example involving matched pairs exposed to varying doses of cadmium is considered in detail.