Measurement equivariance requires the same inference for two equivalent data points: x, measurements expressed in one scale, and y, exactly the same measurements expressed in a different scale. Formal invariance, in the end, leads to a relationship between the inferences at two different data points in the same measurement scale. Suppose an experimenter wishes to estimate 6, the mean boiling point of water, based on a single observation X, the boiling point measured in degrees Celsius. Because of the altitude and impurities in the water he decides to use the estimate T(x) = .5x + .5(100). If the measurement scale is changed to degrees Fahrenheit, the experimenter would use T*(y) = .5y + .5(212) to estimate the mean boiling point expressed in degrees Fahrenheit.
a. The familiar relation between degrees Celsius and degrees Fahrenheit would lead us to convert Fahrenheit to Celsius using the transformation 5/9 (T*(y) - 32). Show that this procedure is measurement equivariant in that the same answer will be obtained for the same data; that is, 5/9 (T*(y) - 32) = T(x).
b. Formal invariance would require that T(x) = T*(x) for all x. Show that the estimators we have defined above do not satisfy this. So they are not equivariant in the sense of the Equivariance Principle.