Missing values, atypical observations, and heterogeneity across latent groups are common sources of complexity in regression data. The contaminated Gaussian cluster-weighted model (CG-CWM) provides a natural framework for handling atypical observations, including outliers and leverage points, in model-based clustering. We extend the CG-CWM to data with missing-at-random (MAR) values in both the response and covariate spaces. The proposed model provides clustering in regression analysis while distinguishing typical observations, outliers, and good and bad leverage points. By treating covariates as random, the model preserves assignment dependence, allowing them to contribute directly to cluster formation. Maximum likelihood estimation is performed through an expectation-conditional maximization (ECM) algorithm that accounts for four sources of incomplete information: missing responses and covariates, unknown component memberships, and latent contamination indicators. Conditional on these indicators, the joint distribution of responses and covariates is multivariate Gaussian, yielding closed-form conditional distributions for missing values and incorporating missingness uncertainty directly into parameter updates. Thus, missing values are handled within model fitting rather than by preliminary imputation. The framework provides clustering, clusterwise regression, model-based treatment of MAR values, and detection of atypical observations. Performance is assessed through numerical studies under varying levels of contamination and missingness patterns, and a real data application.
Missing values present a common challenge in statistical modeling, so handling them properly is an important research direction. Among the various mechanisms that can generate missing values, the most common is the missing-at-random (MAR) mechanism, in which the probability of missingness depends only on observed data and not on unobserved data. This paper addresses the problem of estimating a multivariate linear regression model with multiple random covariates in the presence of MAR values in both the response and covariate spaces using a maximum likelihood (ML) framework. The proposed methodology models the joint distribution of responses and covariates through a conditional-marginal factorization of a multivariate Gaussian distribution. This formulation can be interpreted as a reparameterization of the multivariate normal distribution when the variables can be naturally partitioned into responses and covariates. Parameter estimation is performed using the expectation-maximization (EM) algorithm, which facilitates the imputation of missing values while preserving the distinct roles of responses and covariates. We extend this framework to the model-based clustering setting by considering a mixture of multivariate linear regressions with multiple random covariates. This extension enables soft clustering under incomplete data and accommodates MAR values in both the multivariate responses and covariates. Hence, it represents one of the most general model-based clustering solutions for regression data currently available in the literature. The effectiveness of the methodology is demonstrated through a simulation study, and the advantages of the proposed reparameterization are illustrated using the Automobile dataset, which contains missing values.
