There are several packages in R that will find tetrachoric and polychoric correlations. John Fox's polycor function does this very well. The tetrachoric and polychoric functions in my psych package will do this as well.

In terms of a formula, there is not one, for it is found iteratively. That is, given the observed cut points of x and y (the marginal frequencies), what is the underlying bivariate normal Pearson correlation that best approximates the observed cell frequencies.

For small samples, this is a very unstable estimate. Even for large samples, if there are some empty cells, the estimates are very noisy.

Hello, you can use the package polycor in r. The pdf doucumentation is:

http://cran.r-project.org/web/packages/polycor/polycor.pdf

The formula for tetrachoric correlation is

cos(180/(√AD/BC)+1)

Where in the 2×2 table A, B, C,D represent

(AD/BC)=odds ratio

Stata says the tetrachoric correlation is the ratio of (alpha -1)/(alpha +1), where alpha is (AD/BC)^(pi/4) and where A,D,B,C have the same '2x2 table frequency of events' meaning as above. That is n_00, n_11, n_01, n_10.

But pretty hard to see how the cosine function above is going to give the same results.

Cosine function above is in "S. K. MANGAL. Statistics in psychology and education. second edition. 2010."

Page 223

I am interested in the same thing. I found an article which seems to give it (I haven't read it carefully yet):

http://www.scribd.com/doc/67834562/Ekstro%CC%88m-2010-On-the-relation-between-the-polychoric-correlation-coefficient-and-Spearman-s-rank-correlation-coefficient

The polychor function in the polycor package of R seems to work OK but the hetcor function to calculate a matrix of polychoric correlations causes problems because first you have to convert a matrix of numbers into a matrix of factors. I managed to do that but it continues to send error messages.

If you are doing the same thing, please let me know how you are getting on.

I stand corrected, Dr Mahmoodi is entirely correct. I guess I was a bit too biased in favor of Stata's formulas!

A bit more reading convinced me that both the Stata formula and the cosine formula are good approximations to finding , where is the correlation between standard bivariate normals that are assumed to give rise to the observed distribution of events.

That is, if you assume that positive events happen in one dichotomous variable with frequency B+D when some standard normal score exceeds some critical value, and if you similarly assume that positive events happen in the other dichotomous variable with frequency C+D when the other standard normal score exceeds some critical value, then positive events happen in both variables with frequency D.

With this bivariate normal model, D is then the volume under the standard bivariate normal surface from those respective critical values for each variable through infinity.

Some fancy geometry then yields either approximation.

Trust this helps.

Ertugrul:

Is that in SAS? If so I read:

"There are no SAS procedures specifically designed to create a polychoric correlation matrix, and there are no SAS procedures specifically designed to calculate a polyserial correlation or correlation matrix."

at

http://support.sas.com/resources/papers/proceedings10/256-2010.pdf

Similarly, SPSS, which my university uses, doesn't calculate it (although it does calculate sham statistics like Cronbach's alpha).

The problem is not to calculate a single polychoric correlation but a matrix of them.

John, you can use SAS for tetrachoric and polychoric correlation as follow:

Proc freq data=datename;

tables x*y/plcore;

Marzieh:

Yes, but my university doesn't have it. Thanks anyway.

Amir:

Yes, I had read the article you pointed out. I managed to convert a matrix of numbers into factors then used hetcor(). With some matrices there is no problem, e.g:

V1 V2 V3 V4

1 1 3 1 5

2 4 2 1 3

3 3 3 1 2

4 1 2 3 3

5 4 3 1 5

But with this one:

V1 V2 V3 V4 V5 V6 V7

1 1 3 1 5 4 3 2

2 2 1 3 4 2 1 3

3 1 2 1 3 3 1 2

4 5 4 2 1 2 1 3

5 3 2 1 4 3 1 5

I get this error:

Error in optim(0, f, control = control, hessian = TRUE, method = "BFGS") :

non-finite finite-difference value [1]

I am not an expert in computer languages and am interested as a sociologist user so any help you can give will be appreciated.

John

Amir:

Wouldn't the singularity of the matrices also affect Pearson correlations? There is no problem with these. I tried investigating individual polychoric correlations between the variables - some are fine but others, eg. between V5 and V6, produce:

[1] 0.9995721

Warning messages:

1: In optimise(f, interval = c(-1, 1)) :

NA/Inf replaced by maximum positive value

2: In optimise(f, interval = c(-1, 1)) :

NA/Inf replaced by maximum positive value

I am new to R so I don't know what it's telling me. It gives me the rho but then tells me it had to mess with the data. I suppose I could do the 5040 calculations necessary to find out where there are problems but I don't think I will.

John

Amir,

I see the point. The problem is that with 7 variables I have to inspect (7x7 -7)/2 =21 correlations which is not too bad. Unfortunately, the data I have is for 36 variables. That's 630 correlations. In addition, it's not just one variable which is highly correlated with another(V5 and V6), I get the same problem with V4 and V6. If I start eliminating them I will end up with very few variables.

The other point is that perfect correlation is desirable from the point of view of the survey instrument I am using. I don't want to eliminate the highly correlated variables - it defeats the purpose.

John

Hello Ertugrul:

Thanks for the link. I've been checking it out with a matrix of 23 respondents and 36 Likert type items. With a reduced number of items, say 5 or 6, it works okay but anything bigger and it fails to converge, as Amir predicted. However, according to Ekstrom (http://escholarship.org/uc/item/7j01t5sf ) Spearman's rho and the polychoric correlation, in practice, converge - well, he explains it better. I am quite happy to use Spearman if the resulting matrix is not far from the polychoric one.

I've discovered a paper which seems to explain a lot of the problems:

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3103762/pdf/nihms267587.pdf

Most books only give a simplified method for estimating a tetrachoric correlation. The more precise method is apparently exceedingly complex. I believe that the SPSS interface to R will let you calculate a polychoric correlation through R. I would not be surprised if other packages in R did not calculate a tetrachoric or polychoric correlation.

There are several packages in R that will find tetrachoric and polychoric correlations. John Fox's polycor function does this very well. The tetrachoric and polychoric functions in my psych package will do this as well.

In terms of a formula, there is not one, for it is found iteratively. That is, given the observed cut points of x and y (the marginal frequencies), what is the underlying bivariate normal Pearson correlation that best approximates the observed cell frequencies.

For small samples, this is a very unstable estimate. Even for large samples, if there are some empty cells, the estimates are very noisy.

For Multiple Choice Questions that has been dichotomously scored.