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I have this question below , and I am unable to understand what is internal consistency, can anyone please tell the concept , I have read its wiki page but I couldn't understand how to solve a numerical using that theory , also I couldn't find anything on this platform about this concept .

Examine , after deducing necessary results , if the following set of correlation coefficients are internally consistent. $ r_{12} =0.62 , r_{13}= 0.55 , r_{23} = 0.42 $

kjetil b halvorsen
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simran
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  • I think you'll need to add the self-study tag since this appears to be a homework problem. What have you tried so far and what are your thoughts so far so we can guide you figuring this out on your own? – StatsStudent Jul 20 '22 at 10:04
  • The term "internal consistency" arises mostly in the domain of reliability and validity of tests/questionnaires. What are yor tests? – ttnphns Jul 20 '22 at 10:27
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    Probably these are bivariate correlations and maybe you should read that first: pg 206-207 https://www.researchgate.net/publication/280839401_Internal_consistency_Do_we_really_know_what_it_is_and_how_to_assess_it – M. Chris Jul 20 '22 at 10:41
  • @statsstudent , I have added a self study tag , actually I am not clear with concept of internally consistent so I needed to know what does it actually mean to start my work – simran Jul 20 '22 at 11:47

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If you assign arbitrarily the three correlations between three variables you may end up with a correlation matrix that is not positive definite. The assignment is consistent if the correlation matrix is positive definite. In your case, it is. Check the eigenvalues.

user4422
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  • Hey , so I calculated the eigen values , and they are 2.06 , 0.58 , 0.34 , hence when the eigen values are positive we say that it is internally consistent or can we simply calculate the determinant of the correlation matrix to say it is positive definite? – simran Jul 20 '22 at 11:46
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    Checking the eigenvalues is a good conceptual description but a poor practical one, because it involves relatively too much calculation. See the duplicates for answers to all your questions. In general, Sylvester's Criterion is a practical way to address the trivariate case. – whuber Jul 20 '22 at 12:55
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    As suggested by whuber, checking one determinant is not enough. You need to check 3 determinants (the leading principal minors). As an alternative, you have the criterion: a symmetric matrix is positive definite if and only if all its eigenvalues are positive. – user4422 Jul 20 '22 at 13:55
  • @user4422 , you said in this question it is internally consistent but one of the principle minor is negative ( 3rd one) can you tell me why so ? Actually i have exam in coming week and this question have always been a 10 marker , can you please tell the fixed criteria to check ? – simran Jul 29 '22 at 09:53