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For example, I have three variables: A, B, C.

I analysis the partial correlation between A and B while controlling C (expressed as A~B[C]), and the partial correlation between A and C while controlling B (expressed as A~C[B]).

The result shows that the correlation coefficient of A~B[C] is 0.67, p-value < 0.001, however, the p-value of A~C[B] also < 0.001, but the correlation coefficient is small, only 0.02.

Could I explain as: at a given variable C level, A is more possibly determined by B rather than C?

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Keep in mind that the p-value depends on the sample size. I guess you have a large sample size thus both values are p < 0.001. My interpretation would be that A is correlated with B and less so with C. Try to compare the partial correlation results with the normal correlation between A and B/A and C.

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is there any method which could be used to control/correct for the large sample sizes? I am currently running partial correlations on 5933 samples, even tiny correlations are incredibly significant. Personally I think we place too much emphasis on the p-value and we should use common-sense interpretation more often (but that is an aside, if I want to get published I have to succumb to the power of the p!) – rg255 Mar 6 at 10:47
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For large sizes I usually rank the tests using the test statistics since p becomes meaningless. So yes you are right, use common-sense interpretation and if your p is "too small" then you are on the safe side for publishing. – simmmons Mar 6 at 14:42
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At a given level of variable C, C cannot determine A because C cannot vary and thus cannot covary either.

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Sorry for my misleading of phase usage. Here, 'at a give level of variable C', I meant that the variable C is between a small range (comparing the whole dataset, which variable C can vary in a very large range). What I want to know is, within a specific level of variable C, whether A is determined by B or by C. – Richard Ma Jun 8 '12 at 4:57

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