Survival Analysis – Discrete Time Survival Analysis or Cox Regression?

cox-modeldiscrete datadiscrete timesurvival

In an RCT, I want to find out whether the treatment (treatment vs control) has an effect on the uptake of aftercare (yes/no + time). I have five measurement points, which are not equidistant (i.e., baseline, 3 weeks, 6 weeks, 12 weeks and 24 weeks after randomization). There are some questions regarding the difference between Cox regression and discrete-time survival analysis here, but I still wonder whether discrete-time survival analysis would be better in my case. As I read, there are two advantages of discrete-time over Cox regression: 1) handling of ties and 2) no proportionality of hazards. Regarding 1): the handling of ties should be no problem in interval-censored Cox regression. Regarding 2): as I normally would assume PH, but the time periods of my five measurement points are not equal in length, discrete-time survival analysis would be favored. Is that true?

EDIT: I am only interested in the first uptake of aftercare (or, conversely, no uptake at all until end of measurement period), so I don't care about whether aftercare was continued or not after uptake at other measurement timepoints, so the outcome can't change over time)

Best Answer

With only 5 time points and at most one event per individual, a discrete-time model would be the most natural. Interval-censored Cox regression is possible, but that's probably better reserved for situations where the time intervals differ among individuals. When the time intervals are the same for everyone, using binomial regression with a complementary log-log link provides a time-grouped proportional hazards model that's what you would get from an interval-censored Cox model. See this page. Other links in the binomial regression are possible, but wouldn't have the same proportional hazards interpretation.

The different lengths of the time periods don't really matter, unless you explicitly model time as other than categorical in the binomial discrete-time survival model. A Cox model per se doesn't directly evaluate event times at all. It only uses the order of events in time. The survival curves you can generate from a Cox model simply re-express the ordered events in terms of the times at which they occurred.

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Solved – Basic questions about discrete time survival analysis

Assume $K$ is the largest value of $k$ (i.e. the largest month/period observed in your data).

Here is the hazard function with a fully discrete parametrization of time, and with a vector of parameters $\mathbf{B}$ a vector of conditioning variables $\mathbf{X}$: $h_{j,k} = \frac{e^{\alpha_{k} + \mathbf{BX}}}{1 + e^{\alpha_{k} + \mathbf{BX}}}$. The hazard function may also be built around alternative parameterizations of time (e.g. include $k$ or functions of it as a variable in the model), or around a hybrid of both.

The baseline logit hazard function describes the probability of event occurrence in time $k$, conditional upon having survived to time $k$. Adding predictors ($\mathbf{X}$) to the model further constrains this conditionality.
No, logistic regression estimates (e.g. $\hat{\alpha}_{1}$, $\dots$, $\hat{\alpha}_{K}$, $\mathbf{\hat{B}}$) are not the hazard functions themselves. The logistic regression models: logit$(h_{j,k}) = \alpha_{k} + \mathbf{BX}$, and you need to perform the anti-logit transform in (1) above to get the hazard estimates.
Yes. Although I would notate it $\hat{S}_{j,q} = \prod_{i=1}^{q}{(1-h_{j,i})}$. The survival function is the probability of not experiencing the event by time $k$, and of course may also be conditioned on $\mathbf{X}$.
This is a subtle question, not sure I have answers. I do have questions, though. :) The sample size at each time period decreases over time due to right-censoring and due to event occurrence: would you account for this in your calculation of mean survival time? How? What do you mean by "the population?" What population are the individuals recruited to your study generalizing to? Or do you mean some statistical "super-population" concept? Inference is a big challenge in these models, because we estimate $\beta$s and their standard errors, but need to do delta-method back-flips to get standard errors for $\hat{h}_{j,k}$, and (from my own work) deriving valid standard errors for $\hat{S}_{j,k}$ works only on paper (I can't get correct CI coverages for $\hat{S}_{j,k}$ in conditional models).
You can use Kaplan-Meier-like step-function graphs, and you can also use straight up line graphs (i.e. connect the dots between time periods with a line). You should use the latter case only when the concept of "discrete time" itself admits the possibility of subdivided periods. You can also plot/communicate estimates of cumulative incidence (which is $1 - S_{j,k}$... at least epidemiologists will often define "cumulative incidence" this way, the term is used differently in competing risks models. The term uptake may also be used here.).

Solved – discrete time survival analysis

You can do this with static_glm function in the dynamichazard package I have made. The model you get is exactly like the multiperiod logit model used in

Shumway, T. (2001). Forecasting bankruptcy more accurately: A simple hazard model. The Journal of Business, 74(1), 101-124.

This is common method used in the litterature. The R code for your data would be

fit <- dynamichazard::static_glm(
  formula = Surv(tstart, tstop, EVENT) ~ x1 + x2 + x3 + x4 + x5,
  data = the_data_frame_you_used, # you have to change this
  max_T = 12,                     # the last time you observe
  by = 1)                         # bin into period of one year

You will though first have to transform you data into the start-stop setup. This is easily done with the tmerge function from the survival package. You can see the Using Time Dependent Covariates and Time Dependent Coefficients in the Cox Model vignette in the survival package for example on how to use the tmerge function.

Of course, you can use any other survival method which supports time-varying covariates once you have your data.frame in the stat-stop format. There is a long list of option in R. E.g., see the Survival Analysis view.

An issue though is that you companies (likely) do not default at TIME but somewhere between Time -1 and TIME. I.e. you are dealing with interval censoring which you may want to account for if you chose with the survival model you use.

You question is related to this one. Particularly, you can include TIME as random effect kinda like answer here as follows

require(lme4)
ans <- glmer(
  EVENT ~  x1 + x2 + x3 + x4 + x5 + (1|TIME), 
  data = your_intial_data_frame, # data.frame as you posted it 
  family = binomial)

Update to OP's further questions

Could you please let me know if it is possible to use "cloglog" for your both methods?

You cannot get an interval censored model (i.e., cloglog link function) with the static_glm. However, you can use the get_survival_case_weights_and_data function in the same package as I show in the Comparing methods for time varying logistic models vignette and then use whatever classifier you want like glm with a cloglog link function.

Is it allowed to use your suggestions If some companies enter the study in time 4, some others in time 7 and etc.?

This is called delayed entry. It should not be problem in a discrete time default model if your time scale is the calendar date/year.

Really, I want to predict bankruptcy using survival analysis so my covariates should be lagged for example 1 year lag.

Yes, you need to lag your covariates.

As I tried logistic regression in Python - sklearn, the solver "sag" had a better performance. Is it allowed to use this solver in your suggestions? Thanks a lot.

Seems like "sag" is a penalized logistic model. It should not be problem if you set up your data correctly.

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Related Solutions

Solved – Basic questions about discrete time survival analysis

Solved – discrete time survival analysis

Update to OP's further questions

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