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Having an issue wrapping my head around a problem today. I have 4 vectors: $A, B, X$ and $Y$. $A$ is linearly correlated to $X$ with an $R^2$ of ~0.93, and $B$ is linearly correlated to $Y$ with an $R^2$ of ~0.91. However, the ratios ($\frac{A}{B}$) and ($\frac{X}{Y}$) show almost 0 correlation.

The funky part (in my opinion) is that even if I use the regression equations to forecast $X$ and $Y$ from $A$ and $B$, the ratio of ($\frac{A}{B}$) is still uncorrelated to the forecasted ratio of ($\frac{X}{Y}$).

I've done some preliminary reading on Spurious correlation, but it did not set off any lightbulbs for me. Any insight is very much appreciated.

Note: Sample data added below

A   B   X   Y   A/B X/Y
1   0.212918    4.347578    -17.4018    4.696633    -0.24984
2   2.201479    4.587501    -9.96324    0.90848 -0.46044
3   3.424614    5.435605    -10.3609    0.876011    -0.52462
4   2.623962    6.153195    -10.6694    1.524413    -0.57671
5   4.548159    8.802416    -8.24004    1.099346    -1.06825
6   6.04264 8.172989    -2.88426    0.992943    -2.83365
7   8.100475    10.57117    1.131157    0.864147    9.345455
8   6.161973    10.20388    -5.07726    1.298286    -2.00972
9   10.15207    11.16542    4.943855    0.886519    2.258444
10  10.23762    12.08092    6.442936    0.976789    1.875065
11  11.31451    13.79457    8.014128    0.972203    1.721282
12  11.23338    15.60182    8.206886    1.068245    1.901065
13  12.81461    16.48746    8.299397    1.014467    1.986585
14  15.94658    16.90622    16.45668    0.877931    1.027317
15  14.63955    18.13013    14.93188    1.024622    1.214189
16  17.96807    19.57305    20.4805 0.890468    0.955692
17  16.02187    20.2595 16.72331    1.06105 1.211453
18  17.7038 20.45644    20.36031    1.016731    1.004721
19  17.36019    22.08522    20.12868    1.094458    1.097202
20  20.78462    22.06984    27.09491    0.96225 0.814538
21  22.37151    23.69444    27.46959    0.938694    0.86257
22  22.61773    25.19785    28.55082    0.972688    0.882562
23  24.23659    26.044  33.86888    0.948978    0.768965
24  24.70908    27.85312    33.03254    0.971303    0.843202
25  24.37781    27.46584    33.7228 1.025523    0.814459

correlations of A to X and B to Y

correlation of (A/B) and (X/Y)

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  • $\begingroup$ are B and Y correlated ? $\endgroup$
    – IrishStat
    Dec 20, 2016 at 20:23
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    $\begingroup$ Suppose $Y=X$ with just a tiny bit of independent random noise added. Then $Y$ and $X$ will be strongly positively correlated. Let $X$ have a narrow range of large positive values. What do you think the ratios $Y/X$ will behave like? This little thought experiment might help you reason through the situation you describe. $\endgroup$
    – whuber
    Dec 20, 2016 at 20:25
  • $\begingroup$ Would Y/X then have the same "tiny bit of independent random noise" for any value X? It basically then comes out to modeling the noise at any value X, which (if normal distribution assumptions are met) should be uncorrelated? $\endgroup$
    – JVal
    Dec 20, 2016 at 20:31
  • $\begingroup$ What is the ratio of two vectors? I understand the ratio of two random variables but not the ratio of two random vectors. $\endgroup$ Dec 20, 2016 at 20:58
  • $\begingroup$ Some ratio distrubutions do not admit a covariance relationship so standard tools built on least squares will show a random covariance relationship. I work with this problem quite a bit in the real world. Is this an abstraction or real data for a real problem? $\endgroup$ Dec 20, 2016 at 21:17

1 Answer 1

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The covariance of two ratios will vary as a function of both the numerator and denominator of both ratios in terms of the covariance as well as the expected values for all of X, Y, A, and B. Basically, you did not account for the within and between correlation between the numerator and denominator. Taking a ratio of two factors very rarely suffices to create a proper standardization, so other methods should be used, like log transformations.

One can use the delta method to derive the actual estimate of the covariance of the two values.

Without showing the long winded algebra, I basically get:

$Cov(A/B, X/Y) \approx \frac{Cov(A, X)}{BX} - \frac{A Cov(Y,X)}{B^2X} - \frac{X Cov(A, Y)}{B Y^2} + \frac{A X Cov(B, Y)}{B^2 Y^2}$

Now your assumptions boil down to $Cov(A,X) > 0$ and $Cov(B, Y) > 0$. So basically it is very possible to obtain approximately 0 covariance in the ratios. You just need to "contrive" of values that set the above display to 0. Take $B$ very very large, or $X$ and $Y$ very very large, you can also set covariances of $X$ and $Y$ and/or $A$ and $Y$ positive offset the covariance of the other terms.

Basically it's just a balancing act. Use some other parameters and explore a simulation setting which gives you a non-zero correlation.

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  • $\begingroup$ Possibly stupid question but how is $BX$ defined here? If $B$ and $X$ are vectors, are we doing their pointwise product? Or the inner product? If we are doing their pointwise product the dimensions don't match up. $\endgroup$ Jul 20, 2022 at 18:17

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