Tetrachoric

Tetrachoric correlation is a statistic used to estimate the correlation between two latent continuous variables from a pair of observed binary variables. The concept assumes that each binary variable results from thresholding an underlying normally distributed variable, and that the two latent variables follow a bivariate normal distribution. Under this model, the observed 0/1 outcomes reflect whether each latent variable exceeds its threshold.

Calculation involves a 2x2 contingency table of the binary data. The thresholds for each variable are derived from their marginal proportions using the inverse standard normal CDF. The correlation parameter, rho, that defines the bivariate normal distribution is then estimated by maximum likelihood, maximizing the probability of the observed counts for that rho and the fixed thresholds. There is no closed-form formula for rho; numerical optimization is required. The resulting estimate lies in the interval [-1, 1].

Uses and interpretation: The tetrachoric correlation is preferred over the phi coefficient when the binary measures are thought to reflect underlying continuous traits that are normally distributed, rather than being purely discrete. It is commonly used in psychometrics, item response theory, and factor analysis of binary data, as well as in meta-analytic synthesis and structural equation modeling.

Limitations: The method relies on the normality and threshold assumptions. Estimates can be unstable with small sample sizes or imbalanced margins; sparse data, boundary cells, or zero counts can cause convergence issues. Software implementations exist in statistical packages, often as part of routines for polychoric/tetrachoric correlations.