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.
Stephen W. Lagakos (1988) studied this question.