discrete time survival analysis

Question

I would greatly appreciate if you could let me know how to do discrete time survival analysis with time varying covariates. Some part of my data set is as follows:

ID TIME EVENT   x1   x2   x3   x4   x5 
1    1    0    1.28 0.02 0.87 1.22 0.06 
1    2    0    1.27 0.01 0.82 1.00 -0.01 
1    3    0    1.05 -0.06 0.92 0.73 0.02 
1    4    0    1.11 -0.02 0.86 0.81 0.08 
1    5    1    1.22 -0.06 0.89 0.48 0.01 
2    1    0    1.06 0.11 0.81 0.84 0.20 
2    2    0    1.06 0.08 0.88 0.69 0.14 
2    3    0    0.97 0.08 0.91 0.81 0.17 
2    4    0    1.06 0.13 0.82 0.88 0.23 
2    5    0    1.12 0.15 0.76 1.08 0.28 
2    6    0    1.60 0.26 0.55 1.31 0.37 
2    7    0    1.58 0.26 0.56 1.16 0.35 
2    8    0    1.54 0.24 0.59 1.08 0.33 
2    9    0    1.72 0.22 0.55 0.84 0.29 
2    10   0    1.72 0.21 0.53 0.79 0.29 
2    11   0    1.63 0.19 0.55 0.73 0.27 
2    12   0    2.17 0.32 0.44 0.95 0.43 
3    1    0    0.87 -0.03 0.79 0.61 0.00 
3    2    1   0.83 -0.14 0.95 0.57 -0.02 

My data set is related to companies' bankruptcy. My covariates are some financial ratios which are computed at the end of each year. Besides, the issue that a company is gone bankrupt or not, is also determined at the end of each year after financial statements is prepared.

Which method should be used?: Non-parametric method (logit, cloglog),Semi-parametric method (cox) or Parametric method (exponential, loglogistic, lognormal, weibull and gamma). Should the model be estimated using fixed-effects, random-effects, mixed-effects or pooled regression?

Some R codes are also provided here.

Answer 1

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.