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We propose and demonstrate a new two-stage maximum likelihood estimator for parameters of a social relations structural equation model (SR-SEM) using estimated summary statistics (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi mathvariant="sans-serif">Σ</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula>) as data, as well as uncertainty about <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi mathvariant="sans-serif">Σ</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula> to obtain robust inferential statistics. The SR-SEM is a generalization of a traditional SEM for round-robin data, which have a dyadic network structure (i.e., each group member responds to or interacts with each other member). Our two-stage estimator is developed using similar logic as previous two-stage estimators for SEM, developed for application to multilevel data and multiple imputations of missing data. We demonstrate out estimator on a publicly available data set from a 2018 publication about social mimicry. We employ Markov chain Monte Carlo estimation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi mathvariant="sans-serif">Σ</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula> in Stage 1, implemented using the R package rstan. In Stage 2, the posterior mean estimates of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi mathvariant="sans-serif">Σ</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula> are used as input data to estimate SEM parameters with the R package lavaan. The posterior covariance matrix of estimated <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi mathvariant="sans-serif">Σ</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula> is also calculated so that lavaan can use it to calculate robust standard errors and test statistics. Results are compared to full-information maximum likelihood (FIML) estimation of SR-SEM parameters using the R package srm. We discuss how differences between estimators highlight the need for future research to establish best practices under realistic conditions (e.g., how to specify empirical Bayes priors in Stage 1), as well as extensions that would make 2-stage estimation particularly advantageous over single-stage FIML.