We know that the first-year parents who consented were usually better off economically, and hence their children, if left untreated, were more likely to get polio than those whose parents didn't consent. The treatment lowered the risk of polio almost to the level of the children of non-consenting parents. In particular, they differ in the likelihood of getting polio. This study was controlling for the confounding variables age and sex. This is the wrong conclusion to draw from the evidence.
Many people choose to stop smoking because they are getting sick.
Thus, stopping smoking was probably the effect, not the cause, of health problems. Again, Simpson's Paradox is at work here. The overall success rate is higher for Hope Hospital, but Mercy is more successful for each individual type of operation. It may seem like Hope's surgeons are better overall, but accounting for the ease of the operation shows that Mercy's doctors have a higher success rate. It is generally wise to focus on the record for your type of surgery, which leads to a decision to choose Mercy.