A Note on Decomposing the Difference of Two Ratios

Consider a ratio statistic (e.g. the mean) built from observations assigned into classes. An example would be losses=L, claim counts=C, and expos tres=E each aggregated by rating class with the applicable statistic being either case severity=L/C or case frequency=C/E. The note discusses comparing two observed values for such a statistic. The difference is expressed as a sum of two components. One component measures the change due to the change in class mix. The other measures the change "holding the class mix constant ". It is shown that a T-test on each component can assess whether it represents a nonzero difference. A simple numeric example is presented and an Appendix provides a SAS routine to perform the calculations.