Testing additivity in two-way classifications with no replications: the locally best invariant test

Row x column interaction is frequently assumed to be negligible in two-way classifications having one observation per cell. Absence of interaction allows the researcher to estimate experimental error and to proceed with making inferences about row and column effects. If additivity is suspect, it is conventional to test it against a structured alternative. If the structured alternative missspecifies the existing nonadditivity, then the power of the test is low, even if the magnitude of the existing nonadditivity is large. The locally best invariant (LBI) test of additivity is less subject to model misspecification because a particular structural alternative need not be hypothesized. This paper illustrates the LBI test of additivity and compares its power to that of the Johnson-Graybill likelihood ratio (LR) test. The LBI test performs as well as the LR test under a Johnson-Graybill alternative and performs better than the LR test under more general alternatives.