# Logistic Regression as an adjunct to Survival Analysis

As I understand it, survival analysis was created for situations where, if we followed everyone indefinitely, everyone reaches the "event" (death). Let's say that this isn't one of those situations - not everyone will get to the "event", but time still increases the likelihood that they'll get to the "event".

In this situation, is there a better tool to use? Specifically, I was thinking about using logistic regression where the dependent variable is the "event" (yes/no). But I want to include time in the model. I have the unique entry date of each participant and the date when the study stopped collecting data on everyone, and from that I can make a variable for the time that they were in the study. Is there a reason why I can't just include this variable as one of my many independent variables when constructing my logistic regression model? Would that not account for the influence of time (which increases the likelihood of the event, but does not guarantee it by any means)? Is there something I'm missing? It's a solution that seems really obvious but I couldn't find anything on it - so I'm assuming there's a good reason why no one does this. It seems to good to be true because it actually allows me to use the entire data set.

Keep in mind that this is still a situation where I will use survival analysis for the short-term analysis, but I'd like to complement it with a "long-view" using something else that doesn't cause me to lose 60% of my data.

• The first sentence is simply wrong. Survival analysis is designed to handle varying intervals of followup and the possibility of incomplete follow-up (which technically is referred to as 'censoring'.)
– DWin
Dec 31, 2015 at 6:30
• @Dwin I would disagree with that statement. Survival analysis is for handling time-to-event data, which often contains censoring, but is not necessarily. The OP's statement is a little inaccurate: many parametric survival models assume that all events will occur in a finite amount of time, but not all (i.e. cure-rate models) Dec 31, 2015 at 18:36
• Although it is true that survival analysis may be appropriate when complete follow-up is available (which is not the situation described here) it is more generally considered more appropriate than logistic regression when the followup time is variable. It sounds as though this question may be implying that a) the risk is varying (decreasing) over time, and /or b) the censoring process is highly informative. Cox models can produce biased results in either of these situations.
– DWin
Jan 1, 2016 at 0:41

Even if $P(T < t) \rightarrow 1$ as $t \rightarrow \infty$ does not hold does not mean that survival analysis methods cannot be used.

It is true that most survival parametric models (exponential, weibull, etc.) enforce this limit. However, cure rate models are method of handling the case when the event will not necessarily occur. These can be seen as a two stage model. First, $Z \sim Bern(p)$, where $Z$ is an indicator that the subject is "cured" (i.e. will never have the event). So if $Z = 1$, then $T$ (the event time) = $\infty$. If $Z = 0$, then $T \sim F(\theta)$, where $F(\theta)$ is some standard survival model, presumably with the property that $P(T < t) \rightarrow 1$ as $t \rightarrow \infty$.

Alternatively, many non-parametric and semi-parametric models, such as Kaplan Meier curves and the Cox-PH models, do not make the assumption that $P(T < t) \rightarrow 1$ as $t \rightarrow \infty$. However, they do not explicitly model the hierarchical structure that cure rate models do, i.e. you don't end up with an explicit cure rate, which may be an very interesting parameter to you.

• Thanks for your answer, I'm looking into cure rate models as we speak. But I'm still mainly interested in why I can't use logistic regression for the purpose that I laid out above. Appreciate any thoughts you might have on the matter. Dec 29, 2015 at 18:23
• @MatthewBowen You certainly could do that. But you'll need to do a lot of thinking of the interaction between time and the other variables. This will also make your model a little more difficult to build and interpret. Dec 29, 2015 at 18:26
• @MatthewBowen actually your example is a perfect case of a simple effect in survival analysis that would be complicated in a logistic regression setting. If you don't include an interaction with sex and time, let's suppose that $\beta_{time} = 0.1$ and $\beta_{male} = 0.5$. Then the difference of male to female is that males are exactly equivalent to being a 5 year older female. There's plenty of reasons why this would be a bad model (such as predicting a positive probability of a negative 1 year old male experiencing an event). Dec 29, 2015 at 18:51
• As my example demonstrates, you will also note that it will be difficult to enforce $P(T < 0) = 0$. Dec 29, 2015 at 19:01
• If every subject who hasn't failed has been followed the same period of time $t_{max}$ you can do a binary logistic model to predict the probability that $T > t_{max}$ and get a meaningful result. But you will have less precision and power than with a time-to-event analysis. Dec 29, 2015 at 20:51

The situation you describe might be handled well with either Poisson regression or with Cox PH regression. (Although there are some hints in your question that the constant hazards and/or the uninformative censoring assumptions may be violated.) In Poisson regression the time aspect can be entered as a weighting variable; typically one would add an offset term where the 'offset' was the log(time). This would implicitly model the risks associated with the other elements as being constant since the link function is also the logarithm of the sum of predictors. In Cox PH modeling the time variable serves to determine which cases are in the risk set at the time of each of the events. You can examine the validity of the implicit assumptions of constant risk (for Poisson regression) or constant relative risk (for Cox models). Alternative approaches using parametric models may impose further (or quite different) assumptions, which if met may allow more statistically powerful tests.

Including time as a weighting factor in a logistic regression model has been proposed in the past (before Cox models were widely accessible). That approach performed fairly well, but it might be seen by reviewers of submissions to academic journals as been a method from the "last century".