TetCorr 2.1 (Enzmann, 07/2005)

Version 2.0 of TetCorr replaced a former version that was based
on a routine published by Kirk (1973). In case of extreme asymetric
marginal distributions the former version failed to calculate tetra-
choric correlation coefficients or produced incorrect values.

Version 2.0 did compute wrong estimates if one cell frequency was
zero. This has been corrected in version 2.1: The correlation will
be -1 and a warning message will be issued.

Adapting a routine published Brown (1977) and slightly modified by
Uebersax (2000), TetCorr computes a matrix of tetrachoric correlation
coefficients of up to 50 variables and (dependent on the memory
available) up to 8,000 cases. The input file needs raw data coded '0'
and '1' in free format with one row of data per case, the columns
separated by blanks.

Compare the file WMT.DAT for an example of an input file with data of
501 cases and 24 variables.

If you want to analyse data with more than 50 variables you have to
adapt the source code TETCORR.PAS to your needs (set the constant VMAX
to a different value) and compile it as real mode (= DOS) with Borland
Pascal (7.0). If the memory available is not sufficient you may also
compile the source code with a value of the constant VMAX less than
50. The absolute number of cases, however, is 15,000.

----------------------------------------------------------------------

If you are working with SPSS the following steps can be used for a
factor analysis of dichotomous items based on tetrachoric correlation
coefficients. It is assumed that you want to analyse 24 items (WMT01
to WMT24) stored consecutively in your working file.

1) If the values are not coded (0) 'no/false' and (1) 'yes/true' but
   differently, first recode the variables accordingly. For example,
   assume that the values of your variables are coded (1) 'no/false'
   and (2) 'yes/true'. In that case:

RECODE WMT01 to WMT24 (1=0) (2=1).

2) All values of all cases must be valid. Thus, for listwise deletion
   of missing values use:

COUNT WMTMiss = WMT01 to WMT24 (missing).
SELECT IF WMTMiss=0.

3) Write the raw data to a text file (for example WMT.DAT) and note
   the number of cases and the min/max values of the subsequent
   DESCRIPTIVES procedure (all values must be in the range 0 and 1):

WRITE OUTFILE='WMT.DAT'
     /WMT01b to WMT24b (24(x,F1.0)).
DESCRIPTIVES WMT01 to WMT24

4) Use the program TETCORR.EXE to produce a matrix of tetrachoric
   correlations, specify (for example) WMT.DAT as input file and
   WMT.COR as output file. Check that the number of cases and number
   of variables reported by TETCORR.EXE are the same as in the
   previous DESCRIPTIVES procedure.

5) Use SPSS to read the output of TETCORR.EXE (in this example
   WMT.COR) with the matrix of tetrachoric correlations:

MATRIX DATA variables = wmt01 to wmt24
      /FILE = 'wmt.cor'
      /FORMAT FREE LOWER
      /CONTENTS=COR.

6) Now you can use the procedure FACTOR (in this example a
   principal components analysis with three "factors" and
   oblimin rotation, see /EXTRACTION, /CRITERIA, and /ROTATION)
   for a factor analysis of the items WMT01 to WMT24 based on
   tetrachoric correlation coefficients. With /ANALYSIS you
   can specify a subset of variables, if necessary:

FACTOR
  /MATRIX = IN(cor=*)
  /ANALYSIS wmt01 to wmt04
            wmt06 wmt08 wmt09 wmt12 wmt15
            wmt17 to wmt18
            wmt20 wmt21 wmt23 wmt24
  /FORMAT SORT
  /PRINT initial ROTATION
  /CRITERIA FACTORS(3) ITERATE(50)
  /EXTRACTION=PC
  /CRITERIA ITERATE(100)
  /ROTATION oblimin .

  Be aware that in case of extreme asymetric marginal distributions
  or in case of very low cell frequencies it is possible that the
  correlation matrix is not positive definite. Sometimes only one
  variable is responsibe but it is not always easy to identify this
  variable. You should look for variables that are extremely easy
  or difficult, and you should look for pairs of variables of which
  the standard error of the correlation is extraordinary large (see
  r_tetra.sps).

----------------------------------------------------------------------

References:

Brown, M.B. (1977). Algorithm AS 116: The tetrachoric correlation
   and its standard error. Applied Statistics, 26, 343-351.
Kirk, D.B. (1973). On the numerical approximation of the bivariate
   normal (tetrachoric) correlation coefficient. Psychometrika, 38,
   259-268.
Uebersax, J.S. (2000). The tetrachoric and polychoric correlation
   coefficients. Internet:
   http://ourworld.compuserve.com/homepages/jsuebersax/tetra.htm