Fwd: st: Convert Odds Ratios to Risk Ratios after clogit?

[Steve Samuels <sjsamuels@gmail.com>]

I just looked up the Zhang and Yu (1998) publication (full references
please!) and see my formula is equivalent. My point is that  displaying p* and the risk difference d* = p* - p is more informative than the RR,
especially when the range of p is large.

I still recommend the full predictive mixed-effect model, if the
data permit, over conditional logistic regression. One place where
the data do not permit is a matched case-control study, but you haven't
mentioned this as your design.
 
Steve

Reference: Zhang, J., & Kai, F. Y. (1998). What's the relative risk?: A method of correcting the odds ratio in cohort studies of common outcomes. Jama, 280(19), 1690-1691.


In publications, I've used a simpler approach to interpret odds ratios:
Take a range of probabilities p for the outcome and determine what 
estimated probabilities p* would result from applying the estimated OR
to the p's. The following steps follow from the definition of the odds ratio


OR = (p*/(1-p*))/(p/(1-p)) = odds*/odds where 
odds* = p*/(1-p*)  
odds  = p/(1-p)

odds* = OR x odds   = a*/b*, say

p* = a*/(a* + b*)  = OR*p/(1-p +OR*p)

So: if OR = 2 and  p = 0.3 
odds = 3/7  odds* = 2 x 3/7 = 6/7, 
p* = 6/(6+7) = 6/13 = 0.46

Note that the same formula can be applied to endpoints of the CI for the
odds ratio  Unlike the relative risk conversion, this approach works
whether the baseline probabilities are high or low.

Steve
sjsamuels@gmail.com


You'd get much more interpretable results from fitting a sufficiently rich
mixed-effects model with -melogit- or -meqrlogit-. By "sufficiently rich", I mean
one that considers interactions, including differential heterogeneity in
subgroups.

These commands produce predictions on the probability scale. You can
then use -margins- to plot the results, which might lead to simple
interpretations (i.e. of linear differences in probabilities). As a
bonus, you can have group-level variables as predictors.


Steve
sjsamuels@gmail.com


> On Mar 13, 2014, at 10:50 AM, Marcel Raab <marcel.raab@wzb.eu> wrote:
> 
> Christian,
> Being a social scientist I am not used to the terminology in epidemiology. After I checked the Stata Manual entry on -clogit- I would say that I have k1i:k2i matching with k1i >= 1 and i denoting that matching can change from group to group. My groups/strata are families consisting of a varying number of children (2-10). For most groups I have only one case but in roughly 25% of the groups I have multiple positive outcomes.
> 
> I was asking for risk ratios because our Information and Communication Department is struggling with the Odds Ratio interpretation. They (and also I) would prefer a more accessible interpretation of our multivariate results.
> 
> Therefore, I was first trying to work with some kind of predicted probabilities. But I had the impression that these are not correct in my case. The pc1-option of predict calculates the probability of a positive outcome conditional on one positive outcome within group but I have multiple positive outcomes in a lot of groups. And the pu0-option calculates the probability of a positive outcome, assuming that the fixed effect is zero which according to a Statalist-post of Maarten Buis is "a rather weird hybrid between average marginal effects and marginal effects at average values of explanatory variables" (http://www.stata.com/statalist/archive/2012-03/msg01167.html). Finally I turned to the OR to RR conversion which also seems to be problematic if I understand you correctly (unmatched matched case-control)..
> 
> Marcel
> 
> 
> 
Am 13.03.2014 13:57, schrieb Christian Bautista:
> Marcel,
> 
> The formula given by Zhang and Yu is for odds ratios from unmatched
> case-control studies but I see that you're using "clogit" which is the
> standard method for matched case-control studies. What kind of matching
> you have used? frequency-matching or incidence density-matching?
> 
> /Christian
> 
>> Date: Thu, 13 Mar 2014 13:48:58 +0100
>> From: marcel.raab@wzb.eu
>> To: statalist@hsphsun2.harvard.edu
>> Subject: st: Convert Odds Ratios to Risk Ratios after clogit?
>> 
>> Dear Statalisters,
>> 
>> although I am aware of the criticism that has been raised against
>> converting Odds Ratios to Risk Ratios I was wondering if the formula
>> proposed by Zhang and Yu (1998) can also be used in the context of a
>> fixed effects model. As the -oddsrisk- module does not work for this
>> purpose I was trying to apply the formula manually
>> 
>> RR = OR / ((1 - pu) + (pu * OR))
>> (pu = incidence rate of the unexposed group)
>> 
>> Here is my example:
>> 
>> . webuse union, clear
>> . clogit union age grade not_smsa, group(idcode) or
>> . sum union if not_smsa == 0 & e(sample) // mean is pu(?)
>> . display exp(_b[not_smsa]) / ((1 - r(mean)) + (exp(_b[not_smsa]) *
>> r(mean)))
>> 
>> In the example the OR = .9673623 and the RR = .97958943.
>> 
>> I read about the conversion of ORs to RRs only recently and I am
>> definitely not an expert in the field of non linear models. Hence, I
>> would be very glad if anyone could help me with this issue.
>> Is it appropriate to convert ORs in RRs in a clogit context like
>> suggested above?
>> Is there an alternative/superior method to do it? And finally, if it is
>> possible what would be the best way to obtain confidence intervals for
>> the RRs?
>> 
>> Thanks for your consideration,
>> Marcel
>> 
>> --
>> Reference:
>> J. Zhang and K. Yu, 1998. What's the Relative Risk, JAMA, Vol 280, No
>> 19, pp 1690-1691.
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