Does measurement level detemine what statistics are valid?
Measurement theory cannot determine some single statistical method or model as appropriate for data at a specific level of measurement. But measurement theory does in fact show that some some statistical methods are inappropriate for certain levels of measurement if we want to make meaningful inferences about the attribute being measured. If we want to make statistical inferences regarding an attribute based on a scale of measurement, the statistical method must yield invariant or equivariant results under the permissible transformations for that scale of measurement. If this invariance or equivariance does not hold, then the statistical inferences apply only to the measurements, not to the attribute that was measured. If we record the temperature in degrees Fahrenheit in Cary, NC, at various times, we can compute statistics such as the mean, standard deviation, and coefficient of variation. Since Fahrenheit is an interval scale, only statistics that are invariant or equivariant under
