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Can a percent difference between groups be extrapolated to individuals? For example, if a group receiving a drug has 10% more survivors than the control group, is an individual that receives the drug 10% more likely to survive?

gung - Reinstate Monica
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mowglis_diaper
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Note: I am interpreting your question to be, "an individual that receives the drug 10% more likely to survive than they would have been if they had not received the drug". That is, I interpret it to be about the causal effect of the drug.

In general, no, what is true for groups cannot be assumed to hold for the constituent members of those groups. The most basic issue is that groups can be endogeneous. For instance, one particular case of endogeneity is what's called the ecological fallacy. Another relevant entry point into this topic would look at the study of causal inference with observational data, specifically Rubin's propensity score methods. Within that framework it is common to distinguish between the average treatment effect (ATE), the average treatment effect on the treated (ATT) and the average treatment effect for the untreated (ATU). What's important here is that ${\rm ATT}\ne{\rm ATU}$, in general (cf., Why is Average Treatment Effect different from Average Treatment effect on the Treated?).

It's true that both examples above are primarily concerned with observational data, whereas you may be specifically asking about an experimental context. Nonetheless, both of these assume that causal forces can be heterogeneous. For an example that falls within an experimental setting, consider longitudinal studies where there are multiple measurements per patient. Mixed effects models are typically used for such data, and a standard question is whether to include only random intercepts for the patients, or both random intercepts and random slopes. The latter is often preferred, and it can commonly be seen that the random slopes were appropriate. Note that this can only be seen in a study when there are multiple measurements over time. Nonetheless, this shows again that causal forces can be heterogeneous.

Once you are willing to accept that causal forces can be heterogeneous, it follows as a matter of logic that the difference between groups need not be the effect on every member. On the other hand, with randomly assigned or appropriately matched groups (such that all other variables are constant), if you knew a-priori that the effect was perfectly homogeneous, you could assume that the group difference was the same as the effect on every individual.

gung - Reinstate Monica
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  • Isn't there also a "yes" version of this answer? It would have to interpret the question narrowly in the sense of "if we randomly select an individual from the control group and randomly and independently select an individual from the treatment group, is the chance of observing a survivor 10% greater for the treated individual?" And then, in the case where these groups were selected randomly from a population of interest, can we not extend this conclusion from the experimental subjects to that population? – whuber Jun 11 '18 at 15:40
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    Hmmm, that's a defensible interpretation, @whuber, but it's not what I took the question to be. I understood it to be, "an individual that receives the drug 10% more likely to survive *than they would have been if they had not received the drug*". Even under this reading, there is a possible yes discussed in the last sentence. – gung - Reinstate Monica Jun 11 '18 at 15:52