Veeranun Pongsapukdee and Sujin Sukgumphaphan

pp. 805 - 811

Abstract

     Power assessing the goodness of fits of three cumulative link models for ordinal response data with interaction term of explanatory variables is investigated. The three link models are the cumulative logit link models, the cumulative probit link models, and the cumulative complementary log-log link models. The simulations have been conducted for the models with
three response categories, K=3, and two explanatory variables, namely X₁
~Ber(0.5) and X₂~N(0,1). Data were simulated under the sample sizes of 600, 800, 1000, and 1200. Given the set of parameters, of α₁ = log(
log ( ), p
p p
 

1 2
2
3
 log ( ), p p
p



1   log , 2 2   log , 3 and 12   0.0 4.5  (increment 0.3) from which it follows that the true model parameters are corresponding to the proportions, 1
p ,
2
p , and 3
p of the response categories, Y = 1, 2, 3 in the sample data, respectively. The
response’s outcomes , j = 1,..., n, j
y are then computed through the correctly specified models under each sample size.
Each condition was carried out for 1,000 repeated simulations using the developed macro program.
The results show that the cumulative logit link models generally improve the model fits when the sample sizes and
12 are increased; however, results from the power using the score statistic, under the alternative model with indicators, are
probably stable and the power plots remain around the diagonal line. The cumulative probit link models, show that all the
power plots improve the model fits as the power approach 1 quite rapidly for every statistic as the 12 and sample sizes are
large. The last results, from the cumulative complementary log-log link models, show that every power plot is approaching 1
slowly. Therefore, overall the results reveal in general that the cumulative probit link models give the best power of the tests
for every test statistic. The likelihood ratio statistic and the Wald statistic perform better than the score statistic does.
Although, the score statistic gives the minimum power among the three test statistics, it still performs quite stably for every
link function and sample size. Thus, this statistic may not have high power, but it still has significantly high power against
the minimal null hypothesis. 

Keywords

• cumulative logit models
• interaction terms
• goodness of fits
• link functions