This is my sequel to that story:
The patient, still frozen mid-flinch, opens one eye.
“Wait. Wait wait wait. Don’t stick me yet. I think I actually understand this now.”
The doctor lowers the needle again, visibly annoyed but curious despite herself.
“The odds ratio is the same in both subgroups because it’s not being distorted by anything,” the patient says slowly, working it out loud. “It’s the outcome that’s heterogeneous, not the treatment effect. My gene doesn’t change how the drug works. It changes how sick I already was to begin with.”
“Go on,” says the doctor, arms crossed.
“The overall OR of 2 is a mixture. It’s contaminated by the mixing proportions, eighty percent of the trial has no variant, twenty percent has it. Mix two very different baseline risks together and you get a number that describes nobody. The subgroup OR of 4, that’s the real, portable parameter. It doesn’t care how many people in the room have the variant. It applies to me, specifically, regardless of which group I fall into. That’s not a paradox, that’s the whole point of conditioning on a prognostic variable that isn’t a confounder.”
“I’m listening,” says the doctor, though she is smiling now.
“So there’s no interaction on the odds scale. Same OR, four and four, in both subgroups. But…” the patient’s eyes narrow, “…let me see those percentages again.”
He grabs the tables back.
“Without the variant: eighty-five percent recovery on placebo, ninety-six percent on treatment. Risk difference, eleven points. With the variant: nine percent on placebo, twenty-nine percent on treatment. Risk difference, twenty points. Those aren’t the same. Are you telling me now that there IS an interaction, just on a different scale?”
“That was going to be my next question for you,” says the doctor.
The patient sits with it. “No. No, I don’t think so. I think that’s not a second finding, it’s the same finding wearing a different coat. Probability is stuck between zero and one. My no-variant self is already sitting up near ninety-six percent, there’s barely any room left for the drug to push me any higher, the ceiling eats the effect. My variant self is way down at nine percent, there’s tons of room, so the same underlying multiplicative boost turns into a much bigger jump in raw percentage points. The risk difference isn’t disagreeing with the odds ratio. It’s just what a constant odds ratio looks like once you force it through a bounded scale at two very different starting points.”
The doctor raises an eyebrow. “So which one do you trust?”
“The odds ratio is the model,” the patient says, sounding more certain now. “It’s unconstrained, it goes from zero to infinity, and once you have it plus my baseline risk, you can generate the exact risk difference for me or anyone else without losing any information. The risk difference is just that model’s output, translated into a number I can feel in my gut. It’s not competing evidence. It’s arithmetic downstream of the real thing. If I’d looked at the risk differences first and concluded there was some special interaction going on between my genes and the drug, I’d have been fooled by the ruler, not the reading.”
“So,” says the doctor, needle hovering again, “no confounding, no interaction, one consistent multiplicative effect, and a genetic variant that’s telling you about your prognosis, not about how the drug works on you specifically.”
“Correct.”
“And you have the variant.”
“Correct.”
“Which means your real expected jump is from nine percent to twenty-nine percent recovery, not the eleven-point bump the no-variant group gets.”
“Also correct.”
“So,” the doctor says, raising the needle one final time, “shot or no shot?”
The patient rolls up his sleeve without flinching.
“Odds ratio of four, applies to me regardless of anyone else in the room. Yeah. Give me the shot.”