Objective: This paper aims to present a novel procedure for testing a set of population proportions against an ordered alternative with a control. Material and Methods: The distribution of the test statistic for the proposed test was determined theoretically and through Monte-Carlo experiments. The efficiency of the proposed test method was compared with the classical Chi-square test of homogeneity of population proportions using their empirical Type I error rates and powers at various sample sizes. Results: The new test statistic that was developed for testing a set of population proportions against an ordered alternative with a control was found to have a Chi-square distribution with non-integer values degrees of freedom v that depend on the number of population groups k being compared. Table of values of v for comparing up to 26 population groups was constructed while an expression was developed to determine v for cases where k > 26. Further results showed that the new test method is capable of detecting the superiority of a treatment, for instance a new drug type, over some of the existing ones in situations where only the qualitative data on users' preferences of all the available treatments (drug types) are available. The new test method was found to be relatively more powerful and consistent at estimating the nominal Type I error rates (α), especially at smaller sample sizes than the classical Chi-square test of homogeneity of population proportions. Conclusion: The new test method proposed here could find applications in pharmacology where a newly developed drug might be expected to be more preferred by users than some of the existing ones. This kind of test problem can equally exist in medicine, engineering and humanities in situations where only the qualitative data on users' preferences of a set of treatments or systems are available.

This study investigated the efficiency of Seemingly Unrelated Regression (SUR) estimator of Feasible Generalized Least Square (FGLS) compared to robust MM-BISQ, M-Huber, and Ordinary Least Squares (OLS) estimators when the variances of the error terms are non-constant and the distribution of the response variables is not Gaussian. The finite properties and relative performance of these other estimators to OLS were examined under four forms of heteroscedasticity of the error terms, levels of Contemporaneous Correlation (Cc) with gamma responses. The efficiency of four estimation techniques for the SUR model was examined using the Root Mean Square Error (RMSE) criterion to determine the best estimator(s) under different conditions at various sample sizes. The simulation results revealed that the SUR estimator (FGLS) showed superior performance in the small sample situations when the contemporaneous correlation ( ) is almost perfect ( =0.95) with the gamma response model while MM-BISQ was the best under low contemporaneous correlation. The relative efficiencies of MM-BISQ, M-Huber and FGLS estimators over the OLS are respectively 89%, 71%, and 14% in a small sample 30) and 49%, 32% and 1% in large sample sizes  under gamma response model. The study concluded that MM-BISQ and M-Huber estimators are the most efficient estimators for modeling systems of simultaneous equations with non-Gaussian responses under either homoscedastic or multiplicative heteroscedastic error terms irrespective of the sample size.Keywords—, Contemporaneous correlation, Feasible Generalized Least Square, Heteroscedasticity, Homoscedasticity, Seemingly unrelated Regression.