Metric or Method to examine the correlation of a ratio Let $X$, $Y$, $W$, and $Z$ be random variables where $X$ and $Y$ are different metrics of the same data point and $W$ and $Z$ are also different metrics of the same baseline datapoint. The assumption is $X$ and $Y$ are linearly correlated as are $W$ and $Z$.
Given an instance of $\frac{X}{W}$ I want to know or predict $\frac{Y}{X}$. In particular, I want to know when $\frac{X}{W}$ is high, can I also assume $\frac{Y}{Z}$ is high. This simple metric is not enough. Even though the numerator and denominator are correlated the ratios are not, and this makes sense because of the cross correlations.
Even if I try predicting the numerator and denominator, the predictions are not correlated to the real values. The only place where the predictions do well are when the ratio $\frac{X}{W}$ is close to $1$. This is not very helpful since I want to understand when $\frac{X}{W}$ is large. I want to know if an experiment $X$ performs much better in the baseline under metric $1$ ($W$) it should also perform better than the baseline for metric $2$.
So I have two questions

*

*Is there a better way to measure "Better than the baseline"? It seems ratios are not correlated and prediction is only good then it performs "equal" to the baseline. What are other ways to measure when something does dramatically better than the baseline?


*If the ratio is the correct measurement, Is there a better way to predict $\frac{Y}{Z}$ from $X$ and $W$?. Note $\frac{Y}{Z}$ and $\frac{X}{Z}$ can range from 0 to $\infty$

You state: "...The assumption is $X$ and $Y$ are linearly correlated as are $W$ and $Z$."
In other words, there exist $\alpha,\beta\in\mathbb R$ such that:
$$
\begin{align}
Y &= \alpha X\\
Z &= \beta W.
\end{align}
$$
You ask "...In particular, I want to know when $\frac{X}{W}$ is high, can I also assume $\frac{Y}{Z}$ is high."
But from those equations, we have (assuming $Z\ne 0$):
$$
\frac{Y}{Z} = \frac{\alpha}{\beta}\frac{X}{W}.
$$
Thus, $\frac{Y}{Z}$ is proportional to the ratio $\frac{X}{W}$. And if $\frac{\alpha}{\beta} > 0$ then, indeed, if $\frac{X}{W}$ increases, so does $\frac{Y}{Z}$.
