Effects of mismodelling and mismeasuring explanatory variables on tests of their association with a response variable

We consider three commonly-used statistical tests for assessing the association between an explanatory variable and a measured, binary, or survival-time, response variable, and investigate the loss in efficiency from mismodelling or mismeasuring the explanatory variable. With respect to mismodelling, we examine the consequences of using an incorrect dose metameter in a test for trend, of mismodelling a continuous explanatory variable, and of discretizing a continuous explanatory variable. We also examine the consequences of classification errors for a discrete explanatory variable and of measurement errors for a continuous explanatory variable. For all three statistical tests, the asymptotic relative efficiency (ARE) corresponding to each type of mis-specification equals the square of the correlation between the correct and fitted form of the explanatory variable. This result is evaluated numerically for the different types of mis-specification to provide insight into the selection of tests, the interpretation of results, and the design of studies where the 'correct' explanatory variable cannot be measured exactly.