I study labor and delivery as an epidemiologist. It is well established that a large fetus has a higher risk of causing maternal birth trauma. But a large baby is also likely to be delivered by cesarean section due to difficult labor. Those delivered by cesarean section will not have any risk for maternal birth trauma and, therefore, excluded from an analysis using trauma as the outcome. Thus, informative censoring is a real issue to estimate an unbiased relative risk, particularly when the cesarean rate is high. Are there any methods to adjust for such a bias?
When I initially wrote the comments below I had assumed that the Heckman estimator could be used for dichotomous outcomes, but the second paper I cite says there is no direct analog. Hopefully someone can point to different and more applicable resources. I still leave my initial comment up as I still feel those papers are helpful. I'm not sure how acceptable it would be viewed to use OLS (as oppossed to logistic regression) simply so you can incorporate the Heckman correction estimate.
The work of James Heckman would be applicable to your problem, especially if you have an instrument with which you can estimate the probability of being chosen for a C-section independent of trauma risk.
Sample Selection Bias as a Specification Error by: James J. Heckman Econometrica, Vol. 47, No. 1. (1979), pp. 153-161.
Also as an intro into the logic of the Heckman selection estimator intended for a largely non-technical audience, I enjoy this paper
Is the Magic Still There? The Use of the Heckman Two-Step Correction for Selection Bias in Criminology by: Shawn Bushway, Brian Johnson, Lee Slocum Journal of Quantitative Criminology, Vol. 23, No. 2. (1 June 2007), pp. 151-178.
As another option here, how about using a multinomial logistic regression (with the outcomes being trauma, [non-elective] caesarean, and no trauma?)
I'm not entirely sure if this approach will fully address the issue of bias, but one would get some measures of association about the associations between e.g. fetus size and having a non-elective caesarean section, and could see these side by side with the associations between the same exposure and trauma, and at the least make some qualitative comparisons about magnitudes of effect.
[This question was bumped by an answer edit, with an inactive original questioner, so I'm really just floating another idea here (plus I'm interested in this kind of question)]