Factor models are widely applied to the analysis of multivariate data across disparate fields of research. However, modern scientific data are often incomplete, and estimating a factor model from partially observed data can be very challenging. In this work, we show that if the data are structurally incomplete, the factor model likelihood function can be decomposed into a product of likelihood functions for multiple factor models relative to different observed data subsets. If these factor models are linked together by common parameters, we can obtain complete maximum likelihood estimates of the full factor model parameters. We call this modeling framework Linked Factor Analysis (LINFA). LINFA can be used for covariance matrix completion, dependence estimation, dimension reduction, and data completion. We compute the maximum likelihood estimator through an efficient Expectation-Maximization algorithm, accelerated by a novel Group Vertex Tessellation algorithm. We establish the conditions for the consistency and asymptotic normality of the estimator. We design confidence regions, hypothesis tests, bootstrap algorithms, and methods for selecting the number of factors. Finally, we illustrate the application of LINFA in an extensive simulation study and in the analysis of neuroscience data.

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