st: RE: risk ratio

[jhilbe@aol.com]

In response to the StataLister asking about creating a synthetic binary 
response model that
can be used to estimate a relative risk ratio:

I have an article coming out in the next Stata Journal that details how 
to create synthetic models for a wide  variety
of discrete response regression models. For your problem though, I 
think that the best approach is to create a synthetic
binary logistic model with a single predictor - as you specified. Then 
model the otherwise logistic data as
Poisson with a robust variance estimator. And the coefficient must be 
exponentiated. It can be interpreted as a relative
risk ratio.

Below is code to create a simple binary logistic model. Then model as 
mentioned above. You asked for a
continuous pseudo-random variate, so I generated it from a normal 
distribution. I normally like to use pseudo-random
uniform variates rather normal variates when creating these types of 
models, but it usually makes little difference.
Recall that without a seed the model results will differ each time run. 
If you want the same results, pick a seed. I used my birthday.

I hope that this is what you were looking for.

Joseph Hilbe

*  intercept = 2;  Beta for X1=0.75
clear
set obs 50000
set seed 1230
gen x1 = invnorm(runiform())
gen xb = 2 + 0.75*x1
gen exb = 1/(1+exp(-xb))
gen by = rbinomial(1, exb)
glm by x1, nolog fam(bin 1)
glm by x1, nolog fam(poi) eform robust


. glm by x1, nolog fam(bin 1)
Generalized linear models                          No. of obs      = 
50000
Optimization     : ML                              Residual df      = 
49998
                                                 Scale parameter = 1
Deviance         =  37672.75548                    (1/df) Deviance = 
.7534852
Pearson          =  49970.46961                    (1/df) Pearson  = 
.9994494
Variance function: V(u) = u*(1-u)                  [Bernoulli]
Link function    : g(u) = ln(u/(1-u))              [Logit]
                                                  AIC             = 
.7535351
Log likelihood   = -18836.37774                    BIC             = 
-503294.5
-------------------------------------------------------------------------
-----
           |                 OIM
         by |      Coef.   Std. Err.      z    P>|z|     [95% Conf.  
Interval]
-------------+-----------------------------------------------------------
-----
         x1 |   .7534291   .0143134    52.64   0.000     .7253754  
.7814828
      _cons |   1.993125   .0149177   133.61   0.000     1.963887   
2.022363
-------------------------------------------------------------------------
-----


. glm by x1, nolog fam(poi) eform robust
Generalized linear models                          No. of obs      
=50000
Optimization     : ML                              Residual df     = 
49998
                                                 Scale parameter = 1
Deviance         =  12673.60491                    (1/df) Deviance = 
.2534822
Pearson          =   7059.65518                    (1/df) Pearson  = 
.1411988
Variance function: V(u) = u                        [Poisson]
Link function    : g(u) = ln(u)                    [Log]
                                                  AIC             = 
1.970592
Log pseudolikelihood = -49262.80246                BIC             = 
-528293.7
-------------------------------------------------------------------------
-----
           |               Robust
         by |        IRR   Std. Err.      z    P>|z|     [95% Conf. 
Interval]
-------------+-----------------------------------------------------------
-----
         x1 |   1.104476   .0021613    50.78   0.000     1.100248 
1.10872
-------------------------------------------------------------------------
-----
.
Tomas Lind wrote:
Does anyone know how to generate fake data for a dichotomous outcome 
(0, 1)
that is dependent on a continuous exposure variable in an 
epidemiological
relative risk context. I know how to use the logit transformation but in
that case exposure is proportional to log(ods) and not to risk.


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