If <em>A</em> and <em>B</em> correlate .80 and <em>A</em> and <em>C</em> correlate .75, the correlation between <em>B</em> and <em>C</em> is no longer free: it has to lie between .20 and 1. A .05 there is not an odd finding, it is a matrix that does not exist.
The intuition is geometric. Each standardized variable is a vector, and the correlation between two of them is the cosine of the angle they form. If A and B are close together, and A and C are too, then B and C cannot be far apart: angles add and subtract like angles.
Formally it is the requirement that the matrix be positive semi-definite, and with three variables it solves out: the third correlation must fall within r₁₂·r₁₃ ± √((1−r₁₂²)(1−r₁₃²)).
When the first two are high, that interval narrows sharply. With .90 and .90, the third is forced above .62. With .95 and .95, above .81.
Correlations computed on different samples. The most common cause and the hardest to see: under pairwise deletion each cell uses the people who have those two variables, and they are not the same people across the three cells. The matrix is then not a matrix but three loose numbers, and it has no obligation to cohere.
A flipped sign. A reverse-worded item recoded in one analysis and not in another. Flip the sign of a whole row and the triad stops closing.
A transcription error. A .05 that was .50, a 7 that was a 1. It is the first thing to rule out, which is why it is worth checking before reaching for more interesting explanations.
Because a matrix that is not positive semi-definite cannot be used: no factor solution, no multiple regression, no structural equation model. Software warns with messages almost nobody reads ("matrix is not positive definite") and much of it carries on with an automatic correction that changes the results without saying so.
In meta-analyses of correlations the problem is worse, because the correlations come from different studies with different samples and the pooled matrix can be impossible even when every entry is right on its own. There, coherence has to be checked before the matrix goes into a model.
How to present a correlation matrix properly, with its per-cell N, is in how to report a correlation in APA 7.
The positive semi-definite requirement applies to any correlation matrix, partials included, but the interval this page computes is for simple correlations among three variables. With partials the terms change.
The condition is the same, but there is no short formula: you check that all eigenvalues are zero or above. The value of the triad is that it catches most transcription errors without needing the whole matrix.
It almost always means two variables are the same thing: a total that includes the other, a recode, or a duplicated column. It is mathematically possible but in real data it is rarely what it looks like.
No. Correlation algorithms return bounded values by construction. A 1.02 in a table is a transcription error, or a number that is not a correlation.