The determinant of a correlation matrix becomes zero or near zero when some of the variables are perfectly correlated or highly correlated with each other. However, while computing the reliability statistics of a measure of around 79 items (on a sample of 188 participants using SPSS) with none of the pairs having very high correlation except a few with r = .7, I encountered a message from the SPSS the determinant of the matrix is near zero.
I also tried colinearity diagnostics option of the regression module of the SPSS. The tolerance and VIF also appears to be less problemetic. However, the condition number (the ratio of the sqrt of the highest eigenvalue with the lowest eigenvalue) was very high (above 130) suggesting problem of multicolinearity.
While surfing the internet I came to know that when there are numerous variables having low variance one may encounter such problem even if the variables are not highly correlated or perfectly predicted by the remaining variables. This case applies to my data inasmuch as the SD of most of the items were below 1 with a few having SD in between 1 to 1.5.
Further, when I computed the reliability of the sub-scales of this measure, I never received the message that determinant is near zero. Similarly, I tried various combinations of on only 49 items without this message. But as soon as I add any one item making the total number of items 50, I receive the message that the determinant is near zero.
Please suggest what may be the reason(s) and what are the potential solutions.
@ Jan Boehnke, Thanks a lot for your response and initiating the discussion with raising some fundamental issues that may be potentially related to the problem that I am facing.
I agree with you that if some items are highly correlated then it may be because of similarity of the psychological content and thus some may be dropped. I would like to invite your view about what value of inter-item correlation should be considered high suggesting considerable overlap of the content. In my case the highest correlation observed is around .7 and thus reflecting an overlap of around 49 % variance and the remaining 51% variance (including both specific and error) may be considered unique.
Here I would like to share the fact that I am evaluating the psychometric properties of the Hindi version of already validated measure and thus at this stage I am not thinking of dropping the items. I am waiting for more data so that I can run a CFA and then take the final decision about retaining/rejecting items.
Further, I would also like to share that the problem of near zero determinant remains even I dropped those items that were highly correlated.
To felicitate the discussion, I would like to share that I tried to compute alpha coefficient of other measures having larger number of items with lower variance and encountered the same message that the determinant is near zero. For instance, the reliability analysis of SCL-90-R was associated with same problem. I guess that the problem is somehow related with the larger number of items with lower variance. But still I am not able to learn how to cope up with this problem. If this problem will be present one cannot compute the SMC (squared multiple correlation)- an index of item reliability.
@Professor Pandey. If one wants to compute the reliability of the testscore using Cronbach's alpha reliability coefficient, one needs only the covariance matrix and the number of items.
http://en.wikipedia.org/wiki/Cronbach's_alpha
I do not see what the correlation matrix has to do with it. Why standardize the item scores?
Om Shanti Shanti Shanti!
@ H. Kelderman. Hari Om Tat Sat
Thanks for your comments. Actually I am interested in getting the SMC (squared multiple correlation) of each item as an index of item reliability. And for this a near zero determinant will create problem particularly in inversion of the matrix. In view of this, I have requested the esteemed members to help me in this regard.
Om Shanti
You may want to use regularized regression analysis to obtain the statistic in this case. See section 3.4 of Hastie et al.: http://www-stat.stanford.edu/~tibs/ElemStatLearn/
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