A Monte Carlo Evaluation of the Tetrachoric Correlation Coefficient
The tetrachoric correlation coefficient (r(tet)), computed from a phi coefficient, approximates what the bivariate normal correlation would have been had the dichotomous variables been analyzed in their continuous form with underlying normal distributions. Although often used by early researchers to adjust phi when marginal distributions had extreme proportions, r(tet), more commonly, has been regarded with suspicion. The purpose of these Monte Carlo simulations was to investigate the inaccuracy of r(tet). More specifically, the bias and standard error of r(tet) was examined for dichotomized scores computed from bivariate normal and lognormal continuous raw scores when the proportions of 0s and 1s ranged from somewhat to extremely skewed. Findings indicated that rtet tended to estimate what Pearson's r may have been if symmetry in the marginal distributions of continuous raw scores had been induced by transformation and that in general (a) r(tet) exhibited little bias with standard errors slightly less than double those for Pearson's r with proportions of 1s between .3 and .7, (b) r(tet) performed well with proportions more extreme than .3 and .7 provided there were no empty cells, (c) r(tet) resulted in less bias and smaller standard errors with larger sample sizes, and (d) the .5 adjustment further reduced bias and the standard error of r(tet).
Educational and Psychological Measurement
Greer, T. F.,
Dunlap, W. P.,
Beatty, G. O.
(2003). A Monte Carlo Evaluation of the Tetrachoric Correlation Coefficient. Educational and Psychological Measurement, 63(6), 931-950.
Available at: https://aquila.usm.edu/fac_pubs/3129