Assessing the correlations between psychological variabless, such as abilities and improvements, is one essential goal of psychological science. However, psychological variables are usually only available to the researcher as estimated parameters in mathematical and statistical models. The parameters are often estimated from small samples of observations for each research participant, which results in uncertainty (aka sampling error) about the participant-specific parameters. Ignoring the resulting uncertainty can lead to suboptimal inferences, such as asserting findings with too much confidence. Hierarchical models alleviate this problem by accounting for each parameter’s uncertainty at the person- and average levels. However, common maximum likelihood estimation methods can have difficulties converging and finding appropriate values for parameters that describe the person-level parameters’ spread and correlation. In this post, I discuss how Bayesian hierarchical models solve this problem, and advocate their use in estimating psychological variables and their correlations.

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