عنوان مقاله [English]
Regression coefficient estimation of multiple responses is an important problem which has been previously studied. In this paper, a heuristic algorithm has been proposed to estimate regression coefficients of the relationship between control and correlated binary response variables. In this paper, a log-linear model has been used for analyzing experiments with more than one categorical response variable. The considered model in this study is called the saturated log-linear model, because the responses (2 responses) of this research are
cross correlated. To estimate the parameters of the logistic regression model for dependent responses, a heuristic iterative nonlinear method is used to maximize the number of concordance.
The proposed heuristic approach is a development of the parameter estimation method (Yeh et al. 2009) that is presented for the univariate binary logistic regression model, which is then applied to estimate the parameters of the log-linear model.
The proposed approach uses the concept of concordance. Concordance means that the joint probability of the occurrence of dependent responses in each treatment is more than other probabilities in the same treatment. Although much research has been undertaken on issues of single and multiple continuous responses and single categorical response problems, this study presents a new approach for simultaneous estimation of the parameters of the log-linear model with correlated categorical responses. Hence, it can be useful in real experimental cases. To indicate the efficiency of the heuristic method, the proposed approach has been compared to existing approaches in some hypothetical examples with simulated data and different sizes (seven, ten and fifteen
treatments). Thus, initially, the parameter values of dependent responses were estimated, and then, the model parameters for each response variable were calculated separately. After estimating parameters, the joint probability values were obtained for both cases of dependent and independent response problems.
It should be noted that joint probability values, in the case of independence between variables, are equal to the product of the individual probabilities of two independent responses.
Because the number of concordance in the proposed heuristic method is greater than in the second case (independence between variables), the proposed heuristic method represents a good performance compared to the estimated coefficients of individual variables, on the basis of concordance measurement. Also, the proposed approach has been analyzed for a real case study from the literature, and with the existing method of (Yeh et al. 2009), and analyses show the efficiency of the proposed approach.