A study of Coleman's linear model for attributes

Conclusion

Coleman's linear model for attributes parallels the classical regression model for real variables. The aim is to obtain, from the data, a set of numbers which serve as measures of the effect of a set of characteristics on an attitude. The probability of the attitude is assumed to be a linear function of the parameters.

We have distinguished the case where the distribution of the population into the different classes defined by the characteristics is random (model A) from the case where this distribution is given (model B). We have shown that, for large samples, these two models have very similar properties.

Coleman's method of estimation, a least-squares method, leads to very simple solutions, and this even in the general case. It has, however, certain shortcomings: it uses only proportions, and not the magnitude of the classes, and does not lead to a test of goodness-of-fit. We know, however, the asymptotic distribution of the estimates from which we get confidence regions and tests for definite values of the parameters.

We propose two other methods, the maximum likelihood and the minimum X ²₁ , which both lead to unbiased estimates with asymptotic minimum variances, and to tests of hypotheses of the X ² type. The min. X ²₁ is particularly well suited in a linear context.

Section II is devoted to the case where all variables are dichotomies, and contains a numerical example.

Section III is devoted to the general treatment of the model and demonstrates its properties from basic results in probability and statistic theory, which are recalled in the Appendix.