Re: st: Additive interactions with Stata

[Constantine Daskalakis <C_Daskalakis@mail.jci.tju.edu>]

At 12:26 PM 1/23/2004, Andrea Baccarelli wrote:
> My question is: Does anybody used Stata to test for additive interactions?
>
> There are different ways to test for additive interactions:
> 1.
> The simplier is to calculate the Sinergy Index (SI). SI is the ratio 
> Observed/Expected, where Observed is the OR for the joint effect of A and 
> B (OR_AB) and Expected is obtained from the main effects of A and B as 
> follows: (OR_A+OR_B-1). If SI is significantly different from 1, then we 
> have an additive interaction. I believe it could be possible to go this 
> way with Stata, by using "lincom" after my usual logistic model (i.e., xi: 
> logit depvar i.A*i.B, or). SI, in  the end,  is just a combination of ORs. 
> Has anybody ever tried this?
> SI, however, has been criticised and other methods are often prefered
Yes, under the logistic regression model, "no additive interaction" implies 
that

OR_A + OR_B - 1 = OR_AB   (the sum of the main effects minus 1 equals the 
joint effect)

However, the synergy index you cite (one of many proposed) is actually a 
*non-linear* combination of odds ratios, because it is a *ratio* of odds 
ratios. But even if you used some other index that is indeed a linear 
combination of ORs, such as

RERI = OR_AB - OR_A - OR_B + 1

you would still not be able to use -lincom- or -test-, because these 
require a linear combination of beta coefficients, *not* odds ratios 
(logistic regression is linear on the logit, or log-odds scale).

There is a way to use the multivariate delta method to obtain the variance 
for such non-linear quantities (and hence conduct tests). You might look 
into -testnl- for that. But this approach performs very poorly in small to 
moderate samples. In your particular situation, the actual type-I error for 
a nominal .05-level test can be as high as .25, and the coverage of the 
confidence intervals is just awful.

A likelihood ratio test performs much better, but then you have to fit both
(i) the full logistic regression model -- the usual unconstrained model 
with main effects for A, B, as well as their A*B interaction term; *and*
(ii) the constrained model -- one that assumes no additive interaction, ie, 
one where exp(b_AB) is constrained to be equal to exp(b_A)+exp(b_B)-1.

The first is no problem. But the second one is a non-standard non-linear 
logistic regression model and cannot be easily fit with standard software. 
So, unless you write your own code (which I have done in SAS but not 
Stata), I think you're stuck.


> 2. Additive relative risk model (as implemented in Epicure)
> Is there a way to do this in Stata?
You can do that with the -glm- command that is very flexible in Stata.
This is the *true* additive model and should be the preferred approach.

(1) and (2) are not one and the same. The broad difference is that

(1) considers additivity of A and B in the context of an otherwise 
multiplicative model (ie, the world is multiplicative, but maybe A and B 
are additive)

while

(2) assumes additivity of A and B within an overall additive model.

Greenland has a paper discussing the nuances of these issues.


> I hope this was clear enough. Any suggestion/trick/experience on the issue 
> would be most welcomed.
>
> Thanks,
>
> Andrea
>
> Andrea Baccarelli, MD, MPH, PhD
> Genetic Epidemiology Branch
> Division of Cancer Epidemiology and Genetics
> National Cancer Institute, NIH, DHHS
> 6120 Executive Blvd. - EPS 7110
> Bethesda, MD 20892-7236
> Tel. (301)-496-5786
> Fax (301)-402-4489
> baccarea@mail.nih.gov


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________________________________________________________________

Constantine Daskalakis, ScD
Assistant Professor,
Biostatistics Section, Thomas Jefferson University,
211 S. 9th St. #602, Philadelphia, PA 19107
   Tel: 215-955-5695
   Fax: 215-503-3804
   Email: c_daskalakis@mail.jci.tju.edu
   Webpage: http://www.jefferson.edu/medicine/pharmacology/bio/  
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