Sean Meldrum wrote: >Dale, > >Thank you very much for your responses to my posts on SAS-L. I'm really >happy to get some feedback, advice and opinions on this stuff. As always, >your posts have been very helpful and thought provoking to me. > >I appreciate your comments on GEE. I am comfortable using SUDAAN, though >I'd hoped to start using GENMOD more. I didn't realize that GENMOD fits >only cumulative logit for polytomous variables, though I guess I could get >around that if I need to. I am not concerned about whether the data will >run with GEE - we have 625 subjects (clusters) with up to 3 repeated >measures each - more than 90% have all three observations. >

GEE is designed just for this type of data. Asymptotics will apply nicely here.

> >I would appreciate a copy of the bootstrap macro you mentioned, if it's not >too much trouble - I'm curious. >

I'll send that under separate cover to the list.

> >Thank you for the pitfalls of non-linear modeling that you mention. I've >found hints of the same problems in the literature - specifically troubles >getting the models to converge. Several places have suggested feeding SAS >initial parameter estimates from a conventional logistic regression, but >that doesn't seem to be a guaranteed method. I'm also quite concerned about >comments I've heard about the stability of the models, which you also echo. >Along those same lines, there is very little published advice that I've been >able to find regarding model fitting techniques for these types of models. > >But I'm still perplexed about 'population odds' and 'individual odds'. I've >always been under the impression that the parameters from a conventional >logistic regression represent individual odds - have I been wrong??? It >certainly seems that is how we present the results to people. I mean, after >we do studies we take the parameter estimates and interpret them by saying >things like "On average, if a male age 50-60 stops smoking his risk of >stroke in the next year will decrease by XX%." This sounds like an >individual odds...? Is there something about the parameter estimates from a >non-linear regression that make them uninterprettable outside of the context >of the regression themselves? >

When you fit a logistic regression model to data which are a simple random sample, then there is no difference between individual and population interpretations of odds ratios. The problem arises when you have correlated data. I will do my best with the figure below to demonstrate individual vs population average interpretations.

P(y) | 1111111112222222ppppp33 | 1111 2222pppppp333 | 111 222 ppp 333 | 11 22 pp 33 | 11 22 pp 33 | 11 22 p 33 | 1 2 pp 3 | 1 2 pp 3 | 11 22pp 33 | 11 22p 33 | 1 pp 3 | 1 p 3 | 11 pp2 33 | 11 p22 33 | 11 pp22 33 | 11 p 22 33 | 11 p 22 33 | 11 pp 22 33 | 11 ppp 22 33 | 111 ppp 222 333 | 1111ppppp2222 3333 |11111222222223333333333 ----------------------------------------------------------- X

This figure shows the cumulative probability function for three individuals (identified by values 1, 2, or 3) as they might be found from a conditional logistic regression model. The three probability plots are parallel. Given a certain cumulative probability value, say 0.10, occuring at X(1), X(2), or X(3), then the cumulative probability function at X(1)+d, X(2)+d, and X(3)+d will be the same. What causes the difference between the three curves is that each subject has a different intercept, which shift the curve left or right along the X axis. Also shown on this plot is a population average cumulative probability value as obtained, say, from a GEE. The population average model is identified by the letter p. This population average cumulative probability does not increase at the same rate as the subject specific cumulative probabilities. The population average model flattens out the curve. This implies a lower odds ratio, which is what you typically obtain when you fit a logistic regression model employing GEE as opposed to an NLMIXED or GLIMMIX. Note that by averaging over the subject- specific cumulative probabilities, you can obtain the population average cumulative probability. So, by fitting the subject-specific model, you can subsequently obtain a population average interpretation if you desire. You cannot turn this around and infer a subject- specific effect from a population average effect.

But then the question is what is the level for which you want to make some inference, individual subjects or the population? If you want to make a policy statement that modification of fat intake by such and such an amount in the population will have this certain reduction in population probability of prostate cancer, then you want to employ a population average model. If you are a doctor advising a patient, then fat modification by the same amount will have a different (larger) benefit. This is where you need to make your decision: who is going to use this information? Will individuals be counseled, or will the results be disseminated for policy making?

> >Sorry to keep pounding at the same question - I'm just feeling very dense >this week. Pretty soon I'm just going to have to make a decision and go >with it, and marginal models certainly would be a legitimate way to go. >(And they are bound to be easier for me to figure out!) But before I go >down that road I want to make sure that I'm giving the analysis the best >shot I can - and the chance to learn a new technique just whets my >appetite. > >Thank you again. I hope that others may have some more comments to add, >too. > >Sean Meldrum >University of Rochester >

--------------------------------------- Dale McLerran Fred Hutchinson Cancer Research Center mailto: dmclerra@fhcrc.org Ph: (206) 667-2926 Fax: (206) 667-5977 ---------------------------------------

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