Re: Ordinal logistic regression and the relative risk
- From: "m00es" <m00es@xxxxxxxxx>
- Date: 30 Nov 2006 07:53:52 -0800
Jack-of-all-trades wrote:
In binary and ordinal logistic regression we obtain odds ratios as
measures of effect. Because the odds ratio is only a good measure of
the relative risk when the outcome is rare, often it is desirable to
convert the odds ratio to a relative risk. In binary logistic
regression, the relative risk can be approximated by:
RR = OR / [(1-P0) + (P0*OR)]
Where,
RR = relative risk
OR = odds ratio from logistic regression
P0 = probability of outcome in the unexposed group
However, in ordinal logistic regression (let's assume proportional odds
ordinal regression), what the probability of the outcome in the
unexposed group would be is not evident. Furthermore, I have no idea
whether this approximation is even valid for ordinal outcomes.
Does anyone know how to estimate the relative risk from a proportional
odds ordinal logistic regression coefficient?
Here is a thought. Why not estimate the probabilities directly from the
model and then form whatever ratios you are interested in? For example,
in binary logistic regression with a single independent variable, the
predicted probability for x = 1 is:
exp( a + b*1) / ( 1 + exp( a + b*1) )
and for x = 2, it is:
exp( a + b*2) / ( 1 + exp( a + b*2) ), where a and b are the estimates
of the intercept and slope from the fitted model. Then just take the
ratio of these two and you get the estimated relative risk when x = 1
vs. when x = 2.
In a proportional odds model, you can calculate the predicted
probability P(Y <= j), where j denotes the response category. From
this, you can calculate P(Y = j) = P(Y <= j) - P(Y <= j - 1). Now you
can do this for the various j, get the predicted probabilities, and
form ratios to get estimated risk ratios. You could do this for various
values of whatever independent variables are in the model. I hope this
makes sense.
m00es
.
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