The relationship between factorial composition of test items and measures of test reliability

For continuous distributions associated with dichotomous item scores, the proportion of common-factor variance in the test,H², may be expressed as a function of intercorrelations among items.H² is somewhat larger than the coefficienta except when the items have only one common factor and its loadings are restricted in value. The dichotomous item scores themselves are shown not to have a factor structure, precluding direct interpretation of the Kuder-Richardson coefficient,r_K-R, in terms of factorial properties. The value ofr_K-R is equal to that of a coefficient of equivalence,H², when the mean item variance associated with common factors equals the mean interitem covariance. An empirical study with synthetic test data from populations of varying factorial structure showed that the four parameters mentioned may be adequately estimated from dichotomous data.This study was supported in part by an Air Force project (Contract Number AF18(600-170), monitored by the Crew Research Laboratory, Air Force Personnel and Training Research Center, Randolph Air Force Base, Randolph Field, Texas. Permission is granted for reproduction, translation, publication, use and disposal in whole and in part by or for the United States Government. Further support was given by the Northwestern University Graduate School. The computational assistance of Mr. Norman Miller is acknowledged. Professor Meyer Dwass provided mathematical advice both directly and indirectly relevant to the paper.