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I want to answer the research question which determinants are associated with long-term survival after a myocardial infarction (MI) in a prospective patient cohort study.

More precisely: I want to compare patients who have more than 10 years healthy survival after a MI compared to patients who have a new vascular event or have died in less than 10 years after the MI. What determinants are associated with >10 years healthy surival? What characterizes these patients?

I have tried to find which model is best to use, and in similar articles I have seen two different approaches: either a multivariate cox regression model or a multivariable logistic regression model.

Which method is best? Or if there is no 'best' model, but it is dependent of what exactly you are trying to answer - what is the difference between the results from the two models?

I hope the question is clear. I found similar questions on StackExchange, but the examples in those questions were so vastly different from mine that I found I could not answer my question. Thanks!

Edit:
I am now wondering if this is even the right approach at all. Like I said, the question I am trying to answer is whether patients who have a long (i.e. longer than 15 years) healthy follow-up after myocardial infarction have different characteristics than patients who did not achieve long healthy follow-up after myocardial infarction.

I do not want to use a regular Cox model, as I think this does not really answer my question (it will say, e.g. that smoking is a risk factor for mortality over time, which is really not that informative as this is generally known information). What I really want to know is what characterizes those 'healthy survivors'. Though the two questions are really close, they are not precisely the same.
The solution I have thought of for now is this: Make groups based on follow-up time and compare the determinants between those groups (e.g. a group with 0-5 year follow-up, 5-10, 10-15, 15+ years follow-up). However, I am not sure what model I can use to find out what independent determinants are for different groups compared to one another.
Alternatively, I would like to do a sort of 'reversed' Cox, where I look at the people who have the longest follow-up time first and working back, compared to a normal Cox model which is based on events throughout the follow-up time. But I have NO idea whatsoever if there is any type of model that would allow for this.

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    $\begingroup$ Interesting question. If the moment of death is not relevant for your research (dying after 1 year is as bad as dying after 11 years), you might opt for logistic regression. However, this is quite a statement. IMHO I think a patient would be rather interested in taking into account the expected time still alive, as I am going to die some day anyways. A cox model would do this. ... $\endgroup$ – IWS Jul 11 '17 at 12:41
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    $\begingroup$ cont'd ... Additionally, using the extended cox regression options you could also model time varying variables and effects. These would have you account for other 'death-causing' events which occur during follow-up and determinant effects which change over time. In other words, you are probably able to get closer to a more realistic 'model' when using cox regression. Do note, that interpretation of hazard ratios (the effect size in cox models) can be difficult when all these options are used. $\endgroup$ – IWS Jul 11 '17 at 12:44
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    $\begingroup$ Just to add to @IWS excellent comments if you have people lost to follow-up before 10 years (other than from death) you need Cox as you have censoring. $\endgroup$ – mdewey Jul 11 '17 at 12:48
  • $\begingroup$ @mdewey how could I forget (one of) the most important reason(s) to do survival analysis: taking into account censoring! You are absolutely right. $\endgroup$ – IWS Jul 11 '17 at 12:49
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    $\begingroup$ It is abit like the difference between (1) what predicts who can cimb to the summit of Mount Everest versus (2) what predicts how far up Mount Everest you can climb. They may be different I suppose. Incidentally if you want to ping @IWS or me you need to do that. You cannot ping more than one of us in the same comment though. $\endgroup$ – mdewey Jul 11 '17 at 13:34
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Logistic regression models dichotomic (e.g. 0/1) variables, while Cox regression models survival times.

This means that logistic regression focuses on wheather a patient died within 10 years or not while Cox regression takes into account exact survival time (e.g. 11.5 years).

A simple example. Imagine you have 100 men and 100 women in your sample. All men died after 9 years and 360-364 days and all women died after 10 years and 1-5 days.

Logistic regression will tell you that sex is significant determinant of 10-years-survival (because all women survived and all men did not). Cox regression will (most probably) tell you that sex is not significant determinant (since women lived only a few days longer than men).

Note also, that in Cox regression you have determinants of survival while in logistic regression you have determinants od X-years-survival and you have to choose X.

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