I don’t want you to get the impression from this example that aggregating the data is always wrong or that partitioning the data is always right. It depends on the process that generated the data. In the Monty Hall paradox, we saw that changing the rules of the game also changed the conclusion. The same principle works here. I’ll use a different story to demonstrate when pooling the data would be appropriate. Even though the data will be precisely the same, the role of the “lurking third variable” will differ and so will the conclusion.
So the question is whether we need to aggregate and take the average, or we need to take the average of the averages. In the case of a confounder, it must be a weighted average of the freq. OF the different values the confounder can take.
Other important point here is that the rules of the game fundamentally changes the way we need to interpret the data.
