Observed/Expected rate correlates with Expected rate - A Flaw? I am trying to finish my thesis, in which I investigated the association between different hospital characteristics and (numerous)patient outcomes, using multivariable regression models. One of the dependent variables was a ratio of the expected to the observed rate of infections.
Turns out that the expected rate of infections is the best predictor of the O/E ratio. The lower the expected rate, the higher the O/E ratio.
Somehow I believe this result indicates a systematic flaw and that this O/E ratio should not be used to assess the quality of care, but I am not entirely sure if this is sound. Maybe it is a natural behavior of an O/E ratio to be correlated with E? I would deeply appreciate your thoughts on this. 
PS: I am working with the entire dataset that has been used to calculate the O/E ratios, but on a higher level of aggregation, so I can not reproduce the standardization... 
 A: You are right to suspect that the relationship here may be spurious. I give below the general formula for correlation between two ratios then some comments on your particular case.
Suppose we have four variables $A, B, C,$ and $D$
and we form the two ratios $\frac{A}{C}$ and
$\frac{B}{D}$ then letting the coefficient
of variation of each be $v$ and the correlations between them
be $r$ with obvious subscripts we have
that the correlation between $\frac{A}{C}$
and $\frac{B}{D}$ is
$$
\frac{r_{AB}v_Av_B - r_{AD}v_Av_D - r_{BC}v_Bv_C + r_{CD}c_Cv_D}{
\sqrt{v_A^2 + v_C^2 - 2 r_{AC}v_Av_C}\sqrt{v_B^2 + c_D^2 - 2r_{BD}v_Bv_D}}
$$
In your case you can say that $A$ and $D$ are both the expected, $C$ is a constant 1 and $B$ is the observed. The terms involving $C$ then vanish as it has coefficient of variation zero thus simplifying things.
However I believe this is not going to be too helpful to you in practice as


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From your question it seems unlikely you have all the necessary information


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The real issue is why, having in a sense adjusted for expected by dividing
by it, you wish to make a further adjustment.



You might also want to consider modelling the log of the ratio so as to make it symmetrical.
Source
The formula above comes from Q McNemar Psychological Statistics 1962 third edition, Wiley. He does not give an attribution for it so assumed it was well known.
