Goodness of Fit of Cumulative Logit Models for Ordinal Response Categories and Nominal Explanatory variables with Two-Factor Interaction

Abstract

Power and the assessing goodness of fit of cumulative models for ordinal response data with two nominalinteraction term of explanatory variables are investigated. The magnitude of goodness-of-fit statistics, thecoefficients of determination or R2 analogs, the likelihood ratio statistic,GM, AIC (Akaike Information Criterion,Akaike, 1973),and BIC (Bayesian Information Criterion, Schwarz, 1978) are calculated. The simulations havebeen conducted for the multinomial logit models with K=3 response categories and two random explanatoryvariables X₁ and X₂ whose joint distribution of (X₁, X₂) is assumed to be multinomial with probabilities π1 π2π3and π4, corresponding to (X₁, X₂) values of (0, 0), (0,1), (1, 0), (1, 1), respectively. Three sets of (π₁, π₂, π₃,π₄ ) are studied to represent different distributional shapes, which were chosen to induce possibly strong effectssuch that β₁= log 2, β₂= log3, and β₁₂= 0.0 - 4.5 (increment 0.3), namely (X₁, X₂)~multinomial(0.10,0.35,0.45,0.10), (X₁, X₂)~ multinomial (0.50,0.30,0.10,0.10), and (X₁, X₂)~multinomial (0.25,0.25,0.25,0.25). Four sets of the three ordered category distributing corresponding with the (X₁, X₂) were againgenerated through the models under the proportions of (p₁, p₂, p₃), namely Y~multinomial(p₁, p₂, p₃):(0.05,0.20,0.75), (0.25,0.50,0.25), (0.5,0.20,0.25), and (0.33,0.33,0.33) from which it follows that the truemodel intercepts are α1 = log p₁ , α2 = log p₁ + p₂ , corresponding to the proportions of Y = 1, 2, 3respectively. Four sample sizes of 600, 800, 1,000, and 1,500 units were performed. Each condition was carriedout for 1,000 repeated simulations using the developed macro program run with the Minitab Release 11.The results under the distribution conditions of (X₁, X₂)~ multinomial (0.1,0.35,0.45,0.1) and Y ~(0.55,0.20, 0.25) show that all goodness-of-fit statistics perform better than those of the distribution conditions ofwhich Y~(0.25,0.5,0.25) and Y~(0.33,0.33,0.33) in term of the power of the tests, means and standard deviationsof goodness-of-fit statistics. These results are also similar to the condition when (X₁, X₂)~ (0.50,0.30,0.1,0.1).However, when the distribution conditions are symmetric such that (X₁, X₂)~ (0.25,0.25,0.25,0.25) and Y~ (0.33,0.33,0.33) all statistics are much generally improved the model fits. In conclusion it probably isrecommended to use large sample sizes in the analysis of ordinal categorical responses when the distributions ofvariables are asymmetric, except only when the distribution of the response categories is clearly increasing inorder. Besides this, there is also a tendency to improve the model fit by using the models with an interaction termwhen the correlated structures between the explanatory variables are evident.