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The present study uses doubly latent models to estimate the effect of average mathematics achievement at the class level on students' subsequent mathematics achievement (the "Peer Spillover Effect") and mathematics self-concept (the "Big-Fish-Little-Pond-Effect; BFLPE"), controlling for individual differences in prior mathematics achievement. Our data, consisting of 13-year-old students from Canada, the USA, and New Zealand, come from a unique cross-national database with a longitudinal design at the student level: the Second International Mathematics Study (SIMS80). This historical survey was administered by IEA in the 1980s and highly influenced the development of educational policies in the following decades. We replicate a widely cited study based on SIMS80, interrogating the validity of its findings of a positive peer spillover effect. When we adjust for measurement error, using doubly latent models, we observe that originally positive peer spillover effects become less positive or disappear altogether. On the contrary, negative BFLPEs become more negative and remain statistically significant throughout. Our study is the only cross-national study to have evaluated both the BFLPE and the peer spillover effect with controls for a true measure of prior achievement -- and the only study to test the peer spillover effect cross-nationally using doubly latent models. Our findings question the empirical results of past and current research evaluating school- and class-level compositional effects based on sub-optimal models that fail to control for measurement error.