Mike and Mike, Thanks for your stimulating responses on this topic. I think I agree with practically everything you both have contributed but I think I need to expand more on the reasons why it seems to me that traditional time to event analysis has not really described the full story.
I accept (Mike C) that the Cox model assumes only that the undescribed hazard is proportional and not the same in both groups (for simplicity I will make remarks as if we are studying a clinical trial with a placebo and active drug group). Do you know of any realistic examples where the hazard might be proportional but not equal? I would think that in any randomized trial setting that the randomization would allow the assumption of equality. It seems to me that there are three situations were standard textbooks and reports in the medical literature fail to deal fully with reality. I have had Collett's book with me for 18 months now and have read it from cover to cover. It is the best source I have found for a parametric hazard model perspective but it only hints at these issues. The first situation can be investigated with the semiparametric approach but I havent seen reports of what I would like to see. What interests me is the time course of the relative risk. Typical reports of clinical trial provide hazard ratios (e.g. WHI 2003). The HR is the average relative risk over the duration of the trial. Or sometimes trial will report relative risk at one or two times -- typically towards the end of the trial e.g. 3 y and 5 y after starting treatment. But they dont try to describe the time course of the relative risk as a continuous variable e.g. if you look at the Scandinavian Simvastatin Survival Study (1994) you will see that the survival curves are essentially the same in placebo and treated groups for about a year and only after that do they diverge. I have never heard any quantitative discussion of this phenomenon although it seems it is quite important for understanding the time course of benefit. The obvious explanatory factor in this model would relate to the time course (continuous time varying) of cholesterol (or other lipids) but this has not been attempted as far as I am aware and may not even be possible with standard survival packages (see my third discussion point below). The second situation relates to the merits of a parametric approach. Consider the 4S trial. The Kaplan-Meier plot shows only that the placebo and simvastatin treated groups differ in their mortality but it doesnt tell us about the underlying hazard of death from cardiovascular disease. Imagine that the hazard for death is greatest after initial diagnosis because the patients are younger and still working and living a more stressful lifestyle. It could happen that as time passes and the patients (in both groups) get older and develop a more relaxed life style that the underlying hazard of death from CV disease decreases and becomes negligible (I doubt if this is true but I am using this as a thought experiment). The KM plot we see in the Lancet does not reveal easily to us the underlying hazard time course. With a parametric model for the baseline hazard (i.e. the hazard common to both groups over time) we could discover how the hazard changed with time. In the hypothetical ex ample I have given one might conclude that treatment with a statin after say 5 years confers no additional benefit because the hazard of CV disease has become negligible. The third situation may apply to both parametric and semiparametric situations. The hazard of fracture in the WHI trial (WHI 2003) is related to the time course of bmd changes due to hormone treatment (Garnett 2006). In addition to the drug induced changes one needs also to consider the changes due to natural disease progression. The bmd change due to disease progression is a continuous variable. Therefore one should try to use it as a continuous time varying factor in the hazard function. Collett gives some hints for time varying covariates in a semiparametric situation. But the way I have seen this described for SAS or Splus it is an approximate solution that depends on a piece wise solution with discrete values of the explanatory variable as it changes with time. NONMEM can solve this problem because it explicitly defines the hazard function and integrates it continuously over time. I dont know how one can get SAS or Splus to do this. If you know how then I'd love to hear from you with code and data showing how to do it. Best wishes, Nick 1. Scandinavian Simvastatin Survival Study Group. Randomised trial of cholesterol lowering in 4444 patients with coronary heart disease: the Scandinavian Simvastatin Survival Study (4S). Lancet. 1994 November 19;344:1383-89. 2. Cauley JA, Robbins J, Chen Z, Cummings SR, Jackson RD, LaCroix AZ, et al. Effects of Estrogen Plus Progestin on Risk of Fracture and Bone Mineral Density: The Women's Health Initiative Randomized Trial. JAMA. 2003 October 1, 2003;290(13):1729-38. 3. Garnett C, Holford NHG. Bone mineral density progression linked to dropout and time-to-fracture: application to postmenopausal women taking hormone replacement therapy. 5th International Symposium on Measurement and Kinetics of In Vivo Drug Effects; 2006 April 26-29; Noordwijkerhout, the Netherlands; 2006. "Smith, Mike K" wrote: > > Nick, > > I would argue that parametric survival models are dependent on the > "structural model" (Weibull, Exponential, Gompertz etc.) that you choose > for the hazard function and so suffer the same issues as standard PK > model building where the choice of covariates, error structures etc. > depend on the correct choice of hazard model. The choice of model is > still an assumption... > > On the other hand my understanding of Proportional Hazards models is > that we don't necessarily care what the parametric form of the hazard > is. We assume that the hazards changes proportionately with changes in > the covariates (hence the name). Treatment, dose or exposure variable > could be a covariate and although it is usually added in a linear form > it doesn't have to. In many cases the form or "shape" of the hazard > function itself is a bit of a "nuisance variable" and what we want to > know is the influencing factors on survival rates. In this case the > proportional hazards model does just fine. > > I'm hoping that your last paragraph was written at least partly > tongue-in-cheek... I would argue that if the range of parametric hazard > models you may have tried do not capture features in your data then you > may want to examine proportional hazards models. There's a fairly huge > statistical literature on these topics (and I have to confess I'm not an > expert by any means!). > > A good reference book is by D. Collett: "Modelling Survival Data in > Medical Research", Chapman & Hall / CRC Press. 2003. > http://www.amazon.co.uk/Modelling-Survival-Medical-Research-Statistical/ > dp/1584883251/ref=pd_bowtega_3/026-9223339-1525240?ie=UTF8&s=books&qid=1 > 183564051&sr=1-3 > > Mike > Mike Cole wrote: > > Nick > > I've come in at the end of this email exchange but felt I had to 'defend' the > extremely widely used method of Cox regression and the proportional hazards > model. There are several advantages which you don't spell out in you email > and a couple of inaccuracies as well. > > 1. There is both a non-parametric version of the proportional hazards model > (the widely used Cox regression model) and a parametric version which assumes > a parametric form for the survival times but still retains the > proportionality assumption. > > 2. The hazard function in the proportional hazards model is NOT assumed to be > the same for each treatment group they are assumed to be proportional, > pedantic maybe but needed to be spelt out for clarity. The proportionality > assumption is often valid for survival data. There are extensions to the Cox > model which allows different hazard functions between subgroups or strata. > > 3. The choice of survival time distribution is often difficult to justify > with parametric models. That said, when this is possible the parametric model > allows a greater degree of interpretation and provides more precise parameter > estimates. > > 4. Whereas a parametric survival model is limited by the flexibility of the > chosen survival distribution (and corresponding shape of the hazard function) > the semi-parametric Cox method estimates this in a non-parametric way and so > is extremely flexible. > > 5. "So it depends what you want -- if you just want to collect P values then > use the semi parametric method. But if you want to understand the biology of > the disease and the effects of drug treatments you need to seriously consider > the parametric method." I would suggest that Professor Sir David Cox the > originator of this method would have a few choice words to say about this > remark :-) By the way he has written over 300 papers or books and the > original paper has now been cited over 22,000 times. > > Finally (and I might be opening myself up to a torrent of emails here, but > why would you want to analysis survival data in NONMEM when this is covered > so comprehensively in other software packages and many scripts are available > to use in R/Splus?? > > Mike > > ____________________________________________ > > Michael Cole, CStat FSS > > Statistician > Northern Institute for Cancer Research, > Paul O'Gorman Building, > Medical School, > University of Newcastle Upon Tyne, > Framlington Place, > Newcastle Upon Tyne, > NE2 4HH > > Email: [EMAIL PROTECTED] > ____________________________________________ > > > > -----Original Message----- > From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On Behalf Of Nick Holford > Sent: 03 July 2007 22:59 > To: nmusers@globomaxnm.com > Subject: Re: [NMusers] COX Proportional Hazard Model with Time > DependentCovariate > > Jeff, > > Thanks for highlighting the time to event analysis terminology issue. I think > nmusers need to pay particular attention to a major difference between two > classes of methods. > > 1. The Cox proportional hazards model is a semiparametric method that is used > to describe the difference between treatments. It assumes the underlying > hazard for both treatments is the same. > > 2. Parametric methods (e.g. using the Weibull distribution) try to describe > the undelying hazard for each treatment and do not require the assumption > that the underlying hazard is the same. > > The semiparametric method is somewhat similar to doing a bioequivalence > analysis with NCA. It can tell you about the difference between the two > formulations under the assumption that the clearance is the same but it > doesnt tell you the underlying PK parameters (clearance, volume, absorption > rate constant etc) and cannot make predictions of the time course of > concentration. The parametric time to event method describes the full hazard > function but is dependent on assuming a particular model -- just like > assuming a specific compartmental model and input function in compartmental > PK. > > As nmusers will appreciate, one can learn and understand much more from a > compartmental model than one can from doing a bioequivalence analysis. The > parametric approach does not require the restrictive assumption that the > underlying hazard is the same for both treatments (which is analogous to > having to assume clearance is the same for a bioequivalence analysis). > > So it depends what you want -- if you just want to collect P values then use > the semiparametric method. But if you want to understand the biology of the > disease and the effects of drug treatments you need to seriously consider the > parametric method. > > Nick > > [EMAIL PROTECTED] wrote: > > > > Liang - There are some examples of NONMEM code in the following link. I > > have used this in the past as a good starting point for specifying > > time-dependent hazard models. > > > > http://anesthesia.stanford.edu/pkpd/NONMEM%20Repository/ > > > > A note on nomenclature, I always felt a bit confused about these models > > till I realized the level of ambiguity in the literature. The following > > words are often used in an apparent mosaic fashion to describe different > > analyses that are actually quite similar. > > Survival analysis > > Failure analysis > > Event modeling > > Hazard regression > > Cox proportional hazards model > > Cox model > > Proportional hazards model. > > Weibull (...or insert your favorite function here...) proportional hazards > > model > > Parametric proportional hazards models > > Semi-parametric proportional hazards models > > Cox regression > > Poisson regression, etc... > > > > For more check out: > > http://en.wikipedia.org/wiki/Proportional_hazards_models > > > > Regards, Jeff > > > > > > "Nick Holford" <[EMAIL PROTECTED]> > > > > > > Sent by: [EMAIL PROTECTED] > > > > Åj¬72Z·ÃG{»%÷âÿ± > -- > Nick Holford, Dept Pharmacology & Clinical Pharmacology > University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand > email:[EMAIL PROTECTED] tel:+64(9)373-7599x86730 fax:373-7556 > http://www.health.auckland.ac.nz/pharmacology/staff/nholford/ -- Nick Holford, Dept Pharmacology & Clinical Pharmacology University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand email:[EMAIL PROTECTED] tel:+64(9)373-7599x86730 fax:373-7556 http://www.health.auckland.ac.nz/pharmacology/staff/nholford/
