Correlation of Ratios substantially lower compared to the numerator alone

Question

I have four variables. The first two (V1, V2) are correlated .98. The third and fourth variables are correlated (.99). When I correlate the ratios V1/V3 and V2/V4 the correlation drops to .59. Any thoughts why? V1-V3 and V2-V4 are correlated about .98. Coudl that have anything to do with it? And how?

Answer 1

Since correlation of the pairs $(V_1, V_2)$ and $(V_3, V_4)$ both are high (about $0.98$) then the regression of one on the other must be almost linear. But still, the constant terms could be very far from zero! and that will be seen in the ratios. Algebraically, we have (very approximately) that $$ V_1 = k_1 + \beta_1 V_2 $$ $$ V_3 = k_2 + \beta_2 V_4 $$ (where we have left out error terms). If we have scaled the variables such that $\beta_1 = \beta_2 = \beta$ then we get, approximately, $$ V_1-V_3 = k_1-k_2 +\beta(V_2 - V_4) $$ and the correlation should not depend on the value of $k_1-k_2$. But if you take the ratio, $$ \frac{V_1}{V_3}= \frac{k_1+\beta V_2}{k_2+\beta V_4}. $$ Now, the dependence and thus the correlation will clearly depend on the constants $k_1, k_2$. Just make some simulated example data and then some plots and you will see!

Answer 2

Indeed, the correlation of the ratios can be pretty much 0:

set.seed(19291010)

x1 <- rnorm(100)
x2 <- x1 + rnorm(100, 0, .1)
x3 <- x1 + rnorm(100, 0, .15)
x4 <- x1 + rnorm(100, 0, .1)
rat1 <- x1/x2
rat2 <- x3/x4

cor(x1,x2) #.99
cor(x1,x3) #.99
cor(x1,x4) #.99
cor(x2,x3) #.98
cor(x2,x4) #.99
cor(rat1, rat2)  #-0.09

Essentially, you are correlating the left over noise.

@whuber makes the good point that ratios should only be taken on variables that are all possible. This does not change the results much:

set.seed(19291010)

x1 <- rnorm(100,10,.5)
x2 <- x1 + rnorm(100, 10, .1)
x3 <- x1 + rnorm(100, 10, .15)
x4 <- x1 + rnorm(100, 10, .1)
rat1 <- x1/x2
rat2 <- x3/x4

cor(x1,x2) #.98
cor(x1,x3) #.94
cor(x1,x4) #.98
cor(x2,x3) #.94
cor(x2,x4) #.99
cor(rat1, rat2)  #0.01