Correlation of a calculated variable This might be a very basic question, but I've failed in finding an explanation on the web ...
How is it possible that a variable (Var1) that significantly correlates with two different variables (Var2 and Var3), would not correlated also with a fourth variable (Var4) which is the result of dividing the later two variables (Var4 = Var2/Var3)?
For example:
I have an image in which one of the properties is "Quality".
I use this image to obtain two measurements: $M_1$ and $M_2$. From $M_1$ and $M_2$ I calculate $F_1$ (which is my endpoint) by dividing $M_1$ by $M_2$ ($F_1 = M_1/M_2$).
Although I get a significant correlation between "Quality" to both $M_1$ and $M_2$ (Spearman's -0.625, P<0.0001 and -0.636, P<0.0001 respectively), the correlation between $F_1$ and "Quality" is non-significant (-0.095, p=0.565).
It should also be noted that $M_1$ and $M_2$ strongly correlate with each other (0.692, P<0.0001).   
 A: Well, the most simple (and extreme) example of this would be if all those variables are the same.  If $X_1 = X_2 = X_3$ then we have $X_2/X_3 = 1$, which is constant, so that:
$$\begin{equation} \begin{aligned}
\mathbb{Corr}(X_1,X_2) &= 1, \\[6pt]
\mathbb{Corr}(X_1,X_3) &= 1, \\[6pt]
\mathbb{Corr}(X_1,X_2/X_3) &= 0. \\[6pt]
\end{aligned} \end{equation}$$
As you can see, in this extreme case the value $X_1$ has perfect correlation with $X_2$ and $X_3$, but zero correlation with their ratio $X_2/X_3$.  So this kind of phenomenon is certainly possible in the mathematical sense described here.  In this case you can see that the correlation with the individual variables is "cancelled out" in the correlation with the ratio variable.
Practically speaking, in your particular analysis I notice that $\widehat{\mathbb{Corr}}(X_1,X_2) \approx \widehat{\mathbb{Corr}}(X_1,X_3)$ and you also find no significant evidence of correlation with the ratio of these two latter variables.  It is possible that the correlation in the individual variables is "cancelling out" in the ratio in a manner roughly analogous to the above extreme example.  You can probably see exactly what is going on by generating a scatterplot matrix of your data, including the ratio variable.
