Chance in Guttman scaling

In their recent paper (1977, Social Science Research, 6, 188–196), Kenny and Rubin offer an interpretation of “chance” in Guttman scaling. Their approach is based on the matrix of probabilities p_ij, giving the chance that the ith respondent passes the jth item. It is shown that their interpretation of chance in fact corresponds to a stochastic model of a perfect unidimensional scale. An alternative condition for chance based on the matrix of p_ij's is proposed. This is then proved to be equivalent to the usual interpretation.