I'm trying to understand the best way for analysis whether the tightness of the correlation between to items correlates with a third variable; a toy example might be the question:
"does the connection between "money-at-hand"(MO) and "happiness"(HA) change with the growing "age"(AGE) ?
---------- where the hypothese might be that with growing age this relation weakens.
[update] Well, thinking a bit deeper says that even that "toy example" has its trickeries... Reading over this, one could think it means that both items (together) reduce its importance in older age. But when we look precisely we find that the model measures in a way that in higher age either both are seen positive or both are seen negative (but not complementary) and that in the youth it is more likely that the two aspects are seen as complementary: either much MO AND little HA or little MO and much HA and perhaps one could find a more smooth example for such type of a model... But I'll leave the initial example just as it is [/update]
If age were taken in two or a couple of groups I could think of comparing the correlation MO x HA for the young and for the old people in terms of the Pearson's r 's . But now the third variable is metric and not a nominal "factor".
I've often thought that a possibility were to go back to the case-level and determine the case-specific contribution to the correlation MO x HA. Let MO,HA and AGE be already centered, then to introduce a new variable $CMH_i = MO_i \cdot HA_i $ . Then CMH has
- high positive values when MO and HA have either both high positive or both high negative values
- high negative values when MO and HA have high values with opposite sign
- small values if both MO and HA have small absolute values.
I thought that this one can then be correlated with age. Some experiments with toy data show that it is - although the idea looks simple and convincing to me- not trivial to configure the setting (for instance center CMH ?), but possibly one can overcome this after deeper analysis, for instance doing this in a small program I come to expressions like $x_i \cdot y_i \cdot z_i$ thus crossproductswith three multiplicative factors instead of two as we are used to apply for covariance-computation.
However, before trying to step more into this there might be a "canonical" solution for this? So my question is:
Q How would I model the measure of correlation between two metric items as dependend on a third metric variable?