A random effect autologistic regression model with application to the characterization of multiple microstructure samples

The microstructure of a material can strongly influence its properties such as strength, hardness, wear resistance, etc., which in turn play an important role in the quality of products produced from these materials. Existing studies on a material's microstructure have mainly focused on the characteristics of a single microstructure sample and the variation between different microstructure samples is ignored. In this article, we propose a novel random effect autologistic regression model that can be used to characterize the variation in microstructures between different samples for two-phase materials that consist of two distinct parts with different chemical structures. The proposed model differs from the classic autologistic regression model in that we consider the unit-to-unit variability among the microstructure samples, which is characterized by the random effect parameters. To estimate the model parameters given a set of microstructure samples, we first derive a likelihood function, based on which a maximum likelihood estimation method is developed. However, maximizing the likelihood function of the proposed model is generally difficult as it has a complex form. To overcome this challenge, we further develop a stochastic approximation expectation maximization algorithm to estimate the model parameters. A simulation study is conducted to verify the proposed methodology. A real-world example of a dual-phase high strength steel is used to illustrate the developed methods.