Sunday, November 1, 2009

Methodology: Logistic Regression and Relative Risk - Part 2

A friend of mine read my blog on Logistic Regression and relative Risk and asked a few questions I'll try to answer here.

In that blog I gave the following example:
Suppose that we have a group of students of which some are classified as ADD. Of 80 boys, 13 were classified as ADD and 67 were not. Of 100 girls, 6 were classified as ADD and 94 were not. The odds of a boy being classified as ADD (as the logistic regression output would report) is 13/67 = .194; the odds of a girl being so classified is 6/94 = .064.

I also provided information on how to calculate estimate relative risk:
Estimated Relative Risk = Odds Ratio / ((1-Pr) + (Pr * Odds Ratio))
where Pr is the proportion of non-treated persons that exhibit the outcome of interest.

My friend asked about the following scenarios:

Can the estimated relative risk ever be 0?

Yes. This will only happen when Odds Ratio is 0, or in this case, when the odds of a boy being classified as ADD is 0 (as is the probability) meaning no boys were classified as ADD.

Can the relative risk ever be 1?

Yes. This will only happen when Odds Ratio is 1, or in this case, when the odds of a boy being classified as ADD is 1 (the probability is 100%) meaning all boys were classified as ADD.

Can the relative risk ever be negative?

Thankfully not.

Can the relative risk ever be between 0 and 1?

Yes. Let's look at the case of the relative risk for a girl being classified as ADD. It is:
Odds Ratio / ((1-Pr) + (Pr * Odds Ratio)) , where Odds Ratio = .064/.194 = 0.3298969
and Pr = 13/80 = .1625

Estimated Relative Risk = .3229 / ((1-.1625) + (.1625*.3229)) = .3229 / (.8375 + .0536) =
.3229/.8911 = .3624

Thus girls are .3624 times as likely as a boys of being classified as ADD.

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