On the Use and Interpretation of Certain Test Criteria for Purposes of Statistical Inference: Part II

Abstract

R. A. Fisher''s concept of likelihood is extended in this paper by defining the likelihood of the simple hypothesis that a sample, [SIGMA], has been drawn from a population, II, as [lambda] = C/C ([OMEGA]max.), where C is the chance of obtaining [SIGMA] from II, and C([OMEGA]max.) is the maximum chance of obtaining 2 from any population in the set [OMEGA]. The likelihood of the composite hypothesis that [SIGMA] has been drawn from one or the populations of the sub-set [omega] is defined as [lambda]1 = C ([omega]max.)/C ([OMEGA]max.). This leads to the x2 test of goodness of fit. In testing a function fitted to a sample we may ask 2 questions: (1) How probable is it that the observed sample would have been drawn from a specified population whose group proportions are actually those of the fitted function? In this case n[image]=k, the number of classes. (2) How probable is it that the sample would have been drawn from some one of the sub-set of populations whose law of frequency is given by the functional form? In this case n[image]=k[long dash]c, where c is the number of parameters. The correlation of x 2, the measure of deviation from the sampled population, with x i2, the measure of deviation from the fitted function, is derived, and confirmed by a sampling experiment.