A less important motivation/​mechanism is probabilities/​ratios (instead of odds) are bounded above by one. For rare events ‘doubling the probability’ versus ‘doubling the odds’ get basically the same answer, but not so for more common events. Loosely, flipping a coin three times ‘trebles’ my risk of observing it landing tails, but the probability isn’t 1.5. (cf).
E.g.
Sibling abuse rates are something like 20% (or 80% depending on your definition). And is the most frequent form of household abuse. This means by adopting a child you are adding something like an additional 60% chance of your other child going through at least some level of abuse (and I would estimate something like a 15% chance of serious abuse). [my emphasis]
If you used the 80% definition instead of 20%, then the ‘4x’ risk factor implied by 60% additional chance (with 20% base rate) would give instead an additional 240% chance.
[(Of interest, 20% to 38% absolute likelihood would correspond to an odds ratio of ~2.5, in the ballpark of 3-4x risk factors discussed before. So maybe extrapolating extreme event ratios to less-extreme event ratios can do okay if you keep them in odds form. The underlying story might have something to do with logistic distributions closely resemble normal distributions (save at the tails), so thinking about shifting a normal distribution across the x axis so (non-linearly) more or less of it lies over a threshold loosely resembles adding increments to log-odds (equivalent to multiplying odds by a constant multiple) giving (non-linear) changes when traversing a logistic CDF.
But it still breaks down when extrapolating very large ORs from very rare events. Perhaps the underlying story here may have something to do with higher kurtosis : ‘>2SD events’ are only (I think) ~5X more likely than >3SD events for logistic distributions, versus ~20X in normal distribution land. So large shifts in likelihood of rare(r) events would imply large logistic-land shifts (which dramatically change the whole distribution, e.g. an OR of 10 makes evens --> >90%) much more modest in normal-land (e.g. moving up an SD gives OR>10 for previously 3SD events, but ~2 for previously ‘above average’ ones)]
Yep, I should have definitely kept the probabilities in log-form, just to be less confusing. It wouldn’t have made a huge difference to the outcome, but it seems better practice than the thing that I did.
A less important motivation/​mechanism is probabilities/​ratios (instead of odds) are bounded above by one. For rare events ‘doubling the probability’ versus ‘doubling the odds’ get basically the same answer, but not so for more common events. Loosely, flipping a coin three times ‘trebles’ my risk of observing it landing tails, but the probability isn’t 1.5. (cf).
E.g.
If you used the 80% definition instead of 20%, then the ‘4x’ risk factor implied by 60% additional chance (with 20% base rate) would give instead an additional 240% chance.
[(Of interest, 20% to 38% absolute likelihood would correspond to an odds ratio of ~2.5, in the ballpark of 3-4x risk factors discussed before. So maybe extrapolating extreme event ratios to less-extreme event ratios can do okay if you keep them in odds form. The underlying story might have something to do with logistic distributions closely resemble normal distributions (save at the tails), so thinking about shifting a normal distribution across the x axis so (non-linearly) more or less of it lies over a threshold loosely resembles adding increments to log-odds (equivalent to multiplying odds by a constant multiple) giving (non-linear) changes when traversing a logistic CDF.
But it still breaks down when extrapolating very large ORs from very rare events. Perhaps the underlying story here may have something to do with higher kurtosis : ‘>2SD events’ are only (I think) ~5X more likely than >3SD events for logistic distributions, versus ~20X in normal distribution land. So large shifts in likelihood of rare(r) events would imply large logistic-land shifts (which dramatically change the whole distribution, e.g. an OR of 10 makes evens --> >90%) much more modest in normal-land (e.g. moving up an SD gives OR>10 for previously 3SD events, but ~2 for previously ‘above average’ ones)]
Yep, I should have definitely kept the probabilities in log-form, just to be less confusing. It wouldn’t have made a huge difference to the outcome, but it seems better practice than the thing that I did.