Converting between exposure-response parameters

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Converting between exposure-response parameters helps in using different kinds of input data for exposure-response modelling. Especially, sometimes relateive risks (RR) are given, while sometimes odds ratios (OR) are used. These can be transformed from one to another if enough background data is available.

Scope

How to transform different exposure-response parameters from one to another and use them in a coherent way? Especially, we are interested in relative risk, odds ratio, and unit risk.

Definition

The definition of relative risk (or risk ratio, RR) is

RR = PX1/PX0,

where P is the incidence of disease (or alternatively the probability of the disease during a fixed time period), X1 is the exposure in the group of interest, and X0 is the exposure in the control group (or background group).

Using the same terms, odds ratio (OR) is

OR = ( PX1/(1-PX1) ) / ( (PX0/(1-PX0) ).

We can solve the P_X1 (incidence in the exposed group) from both equations:

PX1 = RR*PX0 and

PX1 = OR*PX0/( 1-PX0+OR*PX0 ) and therefore

RR = OR/( 1-PX0+OR*PX0 ).

Thus, we can see that OR can be translated to RR (and vice versa) only if the background incidence is known. If P_X0 is small, OR is close to RR, but the difference may be substantial if P_X0 and OR are large.

We can also look at these parameters using the regression coefficient ß obtained from an epidemiological study. The interpretation and usage of ß depends on the statistical model used.

If a linear model is used, then it is assumed that

ln(PX) = α + ß*X

for each exposure level X. Then,

RR = exp(ß*(X1-X0)).

If a logistic model is used (which is typical in case-control studies where the background parameter α cannot be estimated), then it is assumed that

ln(PX/( 1-PX) ) = α + ß*X.

Then,

OR = exp(ß*(X1-X0)).

It must be noted that the parameters ß are NOT the same in linear and logistic approaches. The linear and logistic betas can be converted by the following formula

ßRR = ßOR - ln(PX0exp(ßOR*(X1-X0))-PX0+1) / (X1-X0)