To assess validity of a low-intensity measure of fitness (X) in a population of older adults as a proxy measure for the original, high-intensity measure (Y), we used ordinary least square regression with the new, potential proxy measure (X) as the sole explanatory variable for Y. A perfect proxy measure would be unbiased (i.e., result in a regression line with a y-intercept of zero and a slope of one) with no error (variance equal to zero). We evaluated the properties of potential biases of proxy measures. A two degree-of-freedom approach using a contrast matrix in the setting of simple linear ordinary least squares regression was compared to a one degree-of-freedom paired t test alternative approach. We found that substantial improvements in power could be gained through use of the two degree-of-freedom approach in many settings, while scenarios where no linear bias was present there could be modest gains from the paired t test approach. In general, the advantages of the two degree-of-freedom approach outweighed the benefits of the one degree-of-freedom approach. Using the two degree-of-freedom approach, we assessed the data from our motivating example and found that the low-intensity fitness measure was biased, and thus was not a good proxy for the original, high-intensity measure of fitness in older adults.
Keywords: bias; linear contrast; ordinary least squares regression; paired t test.