## What are tetra- and polychoric correlations?

Polychoric correlations estimate the correlation between two theorized normal distributions given two ordinal variables. In psychological research, much of our data fits this definition. For example, many survey studies used with introductory psychology pools use Likert scale items. The responses to these items typically range from 1 (Strongly disagree) to 6 (Strongly agree). However, we don’t *really* think that person’s relationship to the item is actually polytomous. Instead, it’s an imperfect approximation.

Similarly, tetrachorics are special cases of polychoric crrelations when the variable of interest is dichotomous. The participant may have gotten the item either correct (i.e., 1) or incorrect (i.e., 0), but the underlying knowledge that led to the items’ response is probably a continuous distribution.

When you have polytomous rating scales but want to disattenuate the correlations to more accurately estimate the correlation betwen the latent continuous variables, one way of doing this is to use a tetrachoric or polychoric correlation coefficient.

## The problem

At the SAPA Project, the majority of our data is polytomous. We ask you the degree to which you like to go to lively parties to estimate your score on latent *extraversion*. Presently, we use `mixed.cor()`

, which calls a combination of the `tetrachoric()`

and `polychoric()`

functions in the `psych`

package (Revelle, W., 2013).

However, each time we build a new dataset from the website’s SQL server, it takes *hours*. And that’s **if** everything goes well. If there’s an error in the code or a bug in a new function, it may take hours to hit the error, wasting your day.

After a bit of profiling, it was revealed that much of our time building the SAPA dataset was used estimating the tetrachoric and polychoric correlation coefficients. When you do this for 250,000+ participants for 10,000+ variables, it takes *a long time*. So Bill and I thought about how we could speed them up and feel others may benefit from our optimization.

A serious speedup to tetrachoric and polychoric was initiated with the help of Bill Revelle. The increase in speed is roughly 1- (nc-1)^{2} / nc^{2} where nc is the number of categories. Thus, for tetrachorics where nc=2, this is a 75% reduction, whereas for polychorics of 6 item responses this is just a 30% reduction.