توصيفگر ها :
طرح هاي بهينه موضعي , طرح اشباع شده , مدل خطي تعميم يافته , پاسخ چندمتغيره , ماتريس همبستگي
چكيده فارسي :
تحقيقات زيادي در زمينه يافتن طرحهاي بهينه در مدلهاي خطي تعميميافته صورت گرفته است. در اين مدلها، برآورد ماكسيمم درستنمايي پارامترها بهطور مجانبي نااُريب است و ماتريس واريانس كوواريانس آن را با وارون ماتريس اطلاع تقريب ميزنند. از ماتريس اطلاع براي بهدست آوردن طرحهاي بهينه براي مدلهاي مختلف از جمله مدلهاي خطي تعميميافته چند متغيره )MGLM( استفاده ميشود كه ماتريس اطلاع نيز با استفاده از روش معادلات برآورديابي تعميميافته
بهدست ميآيد. براي مدلهاي خطي تعميميافته چندمتغيره ماتريس اطلاع به پارامترهاي نامعلوم بستگي دارد كه باعث ميشود مسأله
بهينه كردن آزمايش نيز به تعيين پارامترهاي نامعلوم وابسته گردد. يكي از راهحلهاي غلبه بر اين مسأله استفاده از طرحهاي بهينه موضعي است. در اينجا براي بهدست آوردن طرحهاي بهينه موضعي از معيارهاي A-بهينگي و D-بهينگي استفاده ميشود.
چكيده انگليسي :
Abstract:
Generalized linear models (GLMs) have been used quite effectively in many areas of application in medicine, eco- nomics, social sciences, quality control, to name just a few. The generalized linear model was developed by Nelder and Wedderburn (1972). It is viewed as a generalization of the ordinary linear regression which allows continu- ous or discrete observations from one-parameter exponential family distributions to be combined with explanatory variables (factors) via proper link functions. Therefore, wide applications can be addressed by GLMs such as so- cial and educational sciences, clinical trials, insurance, industry. In particular; logistic and probit models are used for binary observations whereas Poisson models and gamma models are used for count and nonnegative continuous observations, respectively.
Methods of likelihood are utilized to obtain the estimates of the model parameters. The precision of these maximum likelihood estimates (MLEs) is measured by their variance-covaraince matrix
One important consideration in the analysis of GLMs is the choice of design, that is, the determination of the settings of the control variables that yield an adequate predicted response having desirable properties. Unfortunately, the choice of design for GLMs depends on the unknown parameters of the fitted model. This design dependence problem was the subject of a recent.
These are many research articles for finding the optimal designs under generalized linear models.
In these models, the maximum likelihood estimators for parameters are asymptoticolly unbiased, and their variance- covariance of the matrics are approximated by the inverce of the information matrix.
The information matrix is obtained by generalized estimating equations which is used to find the optimal designs under different models especially for multivariate generalized linear models (MGLM).
Since the information for MGLM depends on the unknown parameters of the model, the optimality problem of designs are also related to these parameters.
A solution for this problem is to apply locally optimal designs. Here we obtain the locally optimal designs under A- and D-optimality criteria.
Optimal designs for multivariate generalized linear models are investigated. The components of the multivariate response might be combined with linear predictors via distinct link functions. We found that the locally optimal design for the univariate generalized linear models remains the same in the multivariate structure.
It turns out that under certain assumptions the optimality problem can be reduced to the marginal models. More precisely, a locally optimal saturated design for the univariate generalized linear models remains optimal for the mul- tivariate structure in the set of all saturated designs. Moreover, the general equivalence theorem provides a necessary and sufficient condition under which the saturated design is locally D-optimal in the set of all designs.
This thesis is based on the following paper:
Idais, O., Locally optimal designs for multivariate generalized linear models, Journal of Multivariate Analysis, (2020) 1-6.
