20th Aug 2001
Many people asked for a summary of replies to my QUERY, some most enthiastically. I am grateful to all those below who took the time to consider the query and to reply. Thank you.

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11th Aug 2001
"QUERY: when one has performed multivariate logistic regression, is there a way of taking the odds ratio estimated for any one co-variate (i.e. adjusted for the others), and calculating an adjusted relative risk ? This is particularly of interest when rates are known not to be 'rare': events might be as high as frequent as 30%. It is not possible to calculate odds directly, if one wishes to include adjustment for confounders. This becomes important when people are comfortable with interpreting 'relative risk' and, understandably, less than comfortable with interpreting 'odds ratios'. I believe this is a general concern.

I've explored use of Cochrane-Mantel-Haentzel in this respect. It gave a different value for the odds ratio derived within Proc Genmod (SAS), presumably because it does not look at individual cases. Anyway, my QUERY is, "Can this be done using the output from a logistic regression." From the latter, I've tried back-calculating to probabilities, but the intercept has a 95% Confidence Interval of (77.9, 278.9): not the way to go.

Davies and Crombie give a way of calculating RR from the OR (I read this as unadjusted RR and OR), in Davies H.T.O., Crombie I.K. (1998) "When can odds ratios mislead ?" BMJ 316:989-991. I am interested in the adjusted case.

If there is interest in this (i.e. if it is not obvious and others request a summary) I'll be happy to summarise to the list.

Thank you,

Martin Holt
Medical Statistician
Southern Derbyshire Acute Hospitals NHS Trust"
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I am so glad you asked this question. Recently, a colleague of mine wished to estimate the relative risk following multivariate logistic regression. In all the literature she found on similar studies, odds ratios were reported and she has decided to report these also. So we would be very interested to hear of any responses or possible solutions you receive to your query.

Eileen Holmes.
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Martin,
I recently faced a similar situation for a nested case control study. However, mine was a conditional logistic regression case and having used the special "discrete" form of the COX model (see SAS documentation), I more or less, left the estimated odd ratios as "relative risks". I simply quoted as reference, a similar application and interpretation in "'Statistical methods in cancer research, Vol 1- The analysis of case-control studies' by Breslow and Day. To follow their reasoning, you may wish to start from page 253.

I'll be interested in a summary of the suggestions from the list.

Regards
Victor
Victor A Kiri MSc PhD CStat
Principal  Statistician Supporting Epidemiology
Department of Statistics and Programming
GlaxoSmithkline Research and Development
GlaxoSmithline
Tel:  +44 (0)208 9662936
Fax: +44 (0)208 9662475
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Dear Dr Holt *****ed: I am a Mr, in fact**********ed

This reference might be of interest:

What's the Relative Risk?
A Method of Correcting the Odds Ratio in Cohort Studies of Common Outcomes
Jun Zhang; Kai F. Yu
JAMA. 1998;280:1690-1691

(However I should say that there has been some correspondence in JAMA criticising this procedure for producing CIs that are too narrow).

Morven Leese

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Dear Martin

Is there any problem doing Poisson regression with appropriate offset. That will give you RR.  If the incidence is rare then log linked, bin distribution should give the same results as Poisson and approximated to RR.

Thanks and like to see the summary

Lukman Thalib

Lukman
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Hi Martin
Have you looked at James Lee's 1994 article "Odds Ratio or Relative Risk for Cross-Sectional data", Int J Epi Vol 23, 1, 201-203?
If you assume constant risk period, you can use Cox model to estimate relative risk for cross-sectional data (with all the adjustments).

I hope this helps


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Martin
Regression with binomial errors (as in logistic) but with a log rather than logit link gives relative risks rather than odds ratios.  This procedure does not seem as robust as logistic regression and failure to converge is fairly common (which is why people went down the logistic route presumably).    However the procedure "binreg" in Stata appears to have been well put together and works well.  This is worth checking out rather than spending ages in the dubious pursuit of working backwards from odds ratios - OK in simple models I expect but getting hellishly fiddly for more complicated models.
Andy


Andy Sloggett
Lecturer in Medical Demography
MSc Course Organiser
London School of Hygiene & Tropical Medicine
Tel: 020 7299 4628
Fax: 020 7299 4637

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I've not yet had the chance to explore the references quoted, but so far it seems to me that the consensus is, "So, 'tis Tepperary you want to get to ? Well, I wouldn't start from here !". I.e. trying to obtain relative risks directly from the ORs output from logistic regression, as I was trying to do, is not the best way to go. Alternatively, as it has been done before, a suggested way forward is to 'publish ORs as relative risks and be damned !'. I would not chose to do this where frequencies are 'high' (Davies and Crombie - ref. in original query above - suggest that > 20% is too high, or not 'rare' enough).
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The following was an aside in which Philip McShane questioned the premises of my QUERY:
----- Original Message -----
From: "Philip McShane" <[log in to unmask]>
To: <[log in to unmask]>
Sent: Tuesday, August 14, 2001 4:06 PM
Subject: Re: QUERY: adjusted relative risk from ORs


Dear Martin

You write "This becomes important when people are comfortable with interpreting 'relative risk' and, understandably, less than comfortable with interpreting 'odds ratios'. I believe this is a general concern."


Is there any evidence for this? Are people 'comfortable with relative risks'? Are they less confortable with odds- ratios? People generally are quite familiar with odds (check with your local bookie if you don't believe me).


Is there not a suggestion (certainly there was in the reference you cite) that odds- ratios are somehow less real than relative risks or absolute differences in risk? I don't accept this; they are different but related and none is more real than the other (whatever 'real' is).

Regards


Phil

Phil McShane
Nuffield dept of Surgery
John Radcliffe Hosp
Headington
Oxford
OX3 9DU

01865 220042

8 Manor Way
Kidlington
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Dear Phil,

Point taken, and I tend to agree. I think the key to this is how odds ratios are interpreted by the statistician and by those writing paper, where there can be over-simplification: 'males were 3 times as likely than females.....' being written when the odds ratio is 3.00 as if it is OK to interpret the odds ratio as a relative risk. As I said in my email, it seemed to me that
in my application this was almost certainly not the case, if one were to take the more absolute approach called for in the electronic letters, etc, and not just say, "Well, it's so big it doesn't really matter just what absolute value it has."

I was speaking from (rather limited) experience of consulting with and writing papers with Doctors. Also, I believe most people more readily understand that the 'chances' of throwing a six with a fair die are 1 in 6 than that the odds are 1/5. With two dice, the 'chances' of throwing a total of six are 5 in 36, while the odds are 5/31. And then there is the next step: the risk of throwing a total of six with two dice relative to that of throwing a six with one die are 5/6. I think this is more intuitively understood than the corresponding odds ratio of 25/31 ? Perhaps not a very good example: it's the thought that counts ! (I hope I've got the maths right.)

This probably speaks more to a need for me to develop further, yet it has proven to be a stumbling block when communicating with those who would prefer their papers to be 'instantly' intelligible, without statistical fog. Ultimately this is a matter of opinion, driven by one's 'customer base': on how well one can explain odds ratios to be intuitively understood and on
whether the absolute value for a realtive risk is required if it is to be interpreted in this way.

Certainly, this was not intended to be a comment on whether one is more 'real' than the other; only which is more user-friendly than the other. That the article I referenced, and others such as those listed therein as references have been published shows that there is room for discussion in this area. The accompanying electronic letters support this and demonstrate
that care is needed with either approach.

Thank you for your comments. I hope the above explains where I was coming from, and I welcome any further thoughts. I do feel a little as if this is walking in the Twilight Zone, not having been able to find much written about it. Certainly, I have had a large number of emails asking enthusiastically for a summary, and will include this in the summary (after
a week has gone by), unless you would prefer me not to ?

Another angle on this is that 'reverting' to relative risks so as not to use odds ratios might act to dumb down the statistics, when maybe it is better to educate 'customers' to better understand odds ratios. I would go along with that; my only comment is, that if the first author does not comfortably understand a statistical concept, he/she is more likely to expect that his peers may not also understand it, depending on the target journal, and it is that that creates the dumbing down process. One has to tailor the statistics to one's audience. Disappointing, but a practical reality.

Best Wishes,

Martin

Martin Holt
Medical Statistician
Southern Derbyshire Acute Hospitals Trust