[leereeves]

I see what you mean, there's a clear gap in the fitted (regression) lines, suggesting the trend of dementia with age is different in the two groups.

But I wonder if that's just a statistical artifact. The overall trend looks the same in both groups if you ignore a couple points (ages) on either side of introducing the vaccine.

A single line appears to fit all the points well, except two points on either side of the divide.

[yayachiken]

The normalization of the age variable still looks problematic, as it requires that there is a linear dependency between age and incidence.

Both the scatter plot and also the probability graph seem to be curving upwards w.r.t age, i.e. the incidence of dementia cases seems to accelerate with age. Which at least makes a lot of intuitive sense, and would forbid just plotting in a regression line.

Having a more or less circumstantial point to arbitrarily cut your regression line in half also just begs for introducing Simpson's Paradox.