Tetrachoric correlations infer a latent Pearson correlation from a two x two table of frequencies with the assumption of bivariate normality. The estimation procedure is two stage ML. Cell frequencies for each pair of items are found. In the case of tetrachorics, cells with zero counts are replaced with .5 as a correction for continuity (correct=TRUE).
The data typically will be a raw data matrix of responses to a questionnaire scored either true/false (tetrachoric) or with a limited number of responses (polychoric). In both cases, the marginal frequencies are converted to normal theory thresholds and the resulting table for each item pair is converted to the (inferred) latent Pearson correlation that would produce the observed cell frequencies with the observed marginals. (See draw.tetra and draw.cor for illustrations.)
This is a computationally intensive function which can be speeded up considerably by using multiple cores and using the parallel package. The number of cores to use when doing polychoric or tetrachoric may be specified using the options command. The greatest step up in speed is going from 1 cores to 2. This is about a 50% savings. Going to 4 cores seems to have about at 66% savings, and 8 a 75% savings. The number of parallel processes defaults to 2 but can be modified by using the options command: options("mc.cores"=4) will set the number of cores to 4.
tetrachoric and polychoric can find non-symmetric correlation matrices of set of x variables (coluumns) and y variable (rows). This is useful if extending a solution from a base set of items to a different set. See fa.extension for an application of this.
The tetrachoric correlation is used in a variety of contexts, one important one being in Item Response Theory (IRT) analyses of test scores, a second in the conversion of comorbity statistics to correlation coefficients. It is in this second context that examples of the sensitivity of the coefficient to the cell frequencies becomes apparent:
Consider the test data set from Kirk (1973) who reports the effectiveness of a ML algorithm for the tetrachoric correlation (see examples).
Examples include the lsat6 and lsat7 data sets in the bock data.
The polychoric function forms matrices of polychoric correlations by an local function (polyc) and will also report the tau values for each alternative. Earlier versions used John Fox's polychor function which has now been replaced by the polyc function.
For finding one polychoric correlation from a table, see the Olsson example (below).
polychoric replaces poly.mat and is recommended. poly.mat was an alternative wrapper to the polycor function.
biserial and polyserial correlations are the inferred latent correlations equivalent to the observed point-biserial and point-polyserial correlations (which are themselves just Pearson correlations).
The polyserial function is meant to work with matrix or dataframe input and treats missing data by finding the pairwise Pearson r corrected by the overall (all observed cases) probability of response frequency. This is particularly useful for SAPA procedures (https://www.sapa-project.org/) (Revelle et al. 2010, 2016, 2020) with large amounts of missing data and no complete cases. See also the International Cognitive Ability Resource (https://www.icar-project.org/) for simiilar data.
Ability tests and personality test matrices will typically have a cleaner structure when using tetrachoric or polychoric correlations than when using the normal Pearson correlation. However, if either alpha or omega is used to find the reliability, this will be an overestimate of the squared correlation of a latent variable with the observed variable.
A biserial correlation (not to be confused with the point-biserial correlation which is just a Pearson correlation) is the latent correlation between x and y where y is continuous and x is dichotomous but assumed to represent an (unobserved) continuous normal variable. Let p = probability of x level 1, and q = 1 - p. Let zp = the normal ordinate of the z score associated with p. Then, \(rbi = r s* \sqrt(pq)/zp \).
As nicely discussed by MacCallum et al, 2002, if artificially dichotomizing at the mean, the point biserial will be .8 of the biserial (actually .798). Similarly, the phi coefficient (a Pearson on dichotmous data will be 2[arcsin(rho)/pi) of the real value (rho).
The 'ad hoc' polyserial correlation, rps is just \(r = r * sqrt(n-1)/n) \sigma y /\sum(zpi) \) where zpi are the ordinates of the normal curve at the normal equivalent of the cut point boundaries between the item responses. (Olsson, 1982)
All of these were inspired by (and adapted from) John Fox's polychor package which should be used for precise ML estimates of the correlations. See, in particular, the hetcor function in the polychor package. The results from polychoric match the polychor answers to at least 5 decimals when using correct=FALSE, and global = FALSE.
Particularly for tetrachoric correlations from sets of data with missing data, the matrix will sometimes not be positive definite. Various smoothing alternatives are possible, the one done here is to do an eigen value decomposition of the correlation matrix, set all negative eigen values to 10 * .Machine$double.eps, normalize the positive eigen values to sum to the number of variables, and then reconstitute the correlation matrix. A warning is issued when this is done.
For very small data sets, the correction for continuity for the polychoric correlations can lead to difficulties, particularly if using the global=FALSE option, or if doing just one correlation at a time. Setting a smaller correction value (i.e., correct =.1) seems to help.
John Uebersax (2015) makes the interesting point that both polychoric and tetrachoric correlations should be called latent correlations or latent continuous correlations because of the way they are found and not tetrachoric or polychoric which is the way they were found in the past. That is, what is the correlation between two latent variables that when artificially broken into two (tetrachoric) or more (polychoric) values produces the n x n table of observed frequencies.
For combinations of continous, categorical, and dichotomous variables, see mixed.cor.
If using data with a variable number of response alternatives, it is necessary to use the global=FALSE option in polychoric. (This is set automatically if this condition is detected).
For relatively small samples with dichotomous data if some cells are empty, or if the resampled matrices are not positive semi-definite, warnings are issued. this leads to serious problems if using multi.cores (the default if using a Mac). The solution seems to be to not use multi.cores (e.g., options(mc.cores =1)
mixedCor which calls polychoric for the N = 4,000 with 145 items of the spi data set, on a Mac book Pro with a 2.4 GHz 8-core Intel i9, 1 core took 130 seconds, 2 cores, 68 seconds, 4 37, 8 22 and 16 22.8. Just finding the polychoric correlations (spi[11:145]) for 4 cores took 34 and for 8 cores, 22 seconds.
As the number of categories increases, the computation time does as well. For more than about 8 categories, the difference between a normal Pearson correlation and the polychoric is not very large*. However, the number of categories (num.cat) may be set to large values if desired. (Added for version 2.0.6).
*A recent article by Foldnes and Gronneberg suggests using polychoric is appropriate for categorical data even if the number of categories is large. This particularly the case for very non-normal distributions of the category frequencies.