Unknown start times & idle gaps in survival analysis of log data — interval-censored model or MI?

Question

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1️⃣ What the student is really asking

Situation	What we observe	What we don’t observe	How it shows up in a survival analysis
Right‑censoring	The student never solves the problem → we only know the time of the last submission.	–	Standard right‑censoring (the event “first correct submission’’ never occurs).
Unknown start time (`x`)	We see the last submission on the previous problem (`L`) and the first submission on the current problem (`R`).	The exact moment the student actually began thinking about the current problem (`S`).	`S` lies somewhere in the interval ((L,R]). If the interval is huge (overnight break) we cannot trust it.
Idle gaps (`y`)	Two consecutive submissions on the same problem, with a long elapsed wall‑clock time.	How much of that elapsed time was spent away from the computer (idle) vs. actually working on the problem.	If the idle part is not removed, the recorded “time‑to‑event’’ is inflated and biases the hazard.

The student wants to know how to estimate the effect of a time‑varying strategy (categorical) on the hazard of solving the problem, while dealing with the two “odd’’ censoring mechanisms above.

Two ideas have been floated:

Fit an interval‑censored survival model (which can treat the start‑time interval properly) – but most software for interval‑censored data does not let you include a time‑varying covariate.
Multiple imputation (MI) of the unknown start times and idle periods, then run a usual Cox model – but the student is unsure whether MI is appropriate and how to do it correctly.

Below is a complete, step‑by‑step solution that shows why multiple imputation combined with a counting‑process Cox model (or a parametric analogue) is a sound approach, and how to implement it in practice.

2️⃣ Full worked solution

Step 0 – Notation

Symbol	Meaning
(i = 1,\dots,n)	student‑problem observations
(L_i)	timestamp of the last submission on the previous problem (lower bound for start)
(R_i)	timestamp of the first submission on the current problem (upper bound for start)
(S_i)	true (unobserved) start time of the current problem
(C_i)	censoring indicator for the event (1 = solved, 0 = never solved)
(T_i)	observed wall‑clock time from the first submission on the current problem to the first correct submission (or last submission if censored)
(G_{ij})	observed gap between submission (j) and (j+1) on the same problem
(\tilde G_{ij})	active (non‑idle) part of (G_{ij}) – what we would like to recover
(\mathbf{Z}_i(t))	vector of covariates that can change over active time (here: the strategy indicator)
(h_i(t))	hazard of solving at active time (t) for observation (i)
(M)	number of imputed data sets (typical values: 20–50)

Step 1 – Decide what is missing and what is censored

Quantity	Observed?	Type of missingness
(S_i) (true start)	No (only interval ((L_i,R_i]))	Interval‑censored latent variable – treat as missing data that we can impute because the interval is known.
(\tilde G_{ij}) (active part of a long gap)	No (only the total gap (G_{ij}) is observed)	Missing at random (MAR) if we assume the size of an idle break depends only on observed information (e.g., length of the gap, time of day).
Event time (first correct)	Partially observed – right‑censored for some students	Standard right‑censoring – keep it as‑is in the survival model.

Thus all the “problematic’’ pieces are missing rather than censored, which makes MI a natural tool.

Step 2 – Choose an imputation model for the unknown start time

A convenient and often realistic assumption is that, conditional on the observed interval, the true start time is uniform over ((L_i,R_i]).
If you have reason to believe that students tend to start soon after the previous submission (e.g., most gaps are short), you can use a truncated exponential or truncated normal distribution that places more mass near the left endpoint.

Imputation model 1 (uniform):

[ S_i^{(m)} \sim \text{Uniform}(L_i,\,R_i) \qquad\text{for } m=1,\dots,M ]

Imputation model 2 (truncated exponential):

[ S_i^{(m)} \mid L_i,R_i \sim \text{Exp}(\lambda) \text{ truncated to } (L_i,R_i] ]

Choose (\lambda) by fitting an exponential to the empirical distribution of short gaps (those you deem trustworthy).
For records flagged as “untrustworthy’’ (the Tukey‑fence outliers) you do not impute a start time; instead you treat them as completely missing and let the imputation model draw a value from the overall distribution of start‑times (the same uniform or exponential you used for the trustworthy records).

Step 3 – Impute the idle portions of long gaps

Identify “long’’ gaps – e.g., any (G_{ij}) that exceeds the 75th percentile plus 1.5 × IQR of all gaps within the same problem (the same Tukey‑fence rule used for start‑time intervals).
Model the active part of a gap as a fraction of the observed gap. A simple model:

[ \tilde G_{ij}^{(m)} = \rho_{ij}\, G_{ij},\qquad \rho_{ij} \sim \text{Beta}(\alpha,\beta) ]

Estimate (\alpha,\beta) from the short gaps (those you trust to be almost entirely active). The Beta distribution is bounded on ([0,1]) and can capture the typical proportion of “working’’ time inside a gap.

Draw (\rho_{ij}^{(m)}) for each long gap, then compute (\tilde G_{ij}^{(m)}).

If a gap is deemed “untrustworthy’’ (e.g., overnight), you can set (\tilde G_{ij}^{(m)} = 0) (i.e., treat the whole gap as idle) or let the Beta draw be near zero; the exact rule is a modeling choice, but it must be applied consistently across imputations.

Step 4 – Build the active event time for each imputed data set

For each imputed dataset (m):

Compute the total active time spent before the first correct submission (or censoring).

[ \text{ActiveTime}i^{(m)} = \underbrace{(T_i - (R_i - S_i^{(m)}))}{\text{wall‑clock time after first submission}}
- \sum_{j\in\mathcal{L}i} \bigl(G{ij} - \tilde G_{ij}^{(m)}\bigr) ]
where (\mathcal{L}_i) is the set of long gaps for observation (i).
In words: start from the observed wall‑clock interval, subtract the unknown part of the start interval (the time before the first submission), then subtract the idle parts of long gaps.
Create a counting‑process representation (the “start‑stop’’ format required for a Cox model with time‑varying covariates):
- Entry time = 0 (the moment the student actually begins working on the problem, i.e. after we have removed the unknown pre‑start interval).
- Exit time = (\text{ActiveTime}_i^{(m)}).
- Event indicator = (C_i) (1 if solved, 0 if right‑censored).
If a student changes strategy at active time (t^\star), split the row at (t^\star) and assign the appropriate strategy label to each interval. Most software (e.g., survival::coxph with Surv(start, stop, event)) will handle this automatically.

Step 5 – Fit the survival model on each imputed data set

Model of interest (Cox proportional‑hazards with a time‑varying strategy):

[ h_i(t) = h_0(t)\,\exp\bigl{\beta_{\text{strategy}}(t)\, \text{Strategy}_i(t) + \boldsymbol\gamma^\top \mathbf{X}_i \bigr}, ]

where (\mathbf{X}_i) may contain baseline covariates (e.g., prior knowledge, problem difficulty).

In R:

library(survival)
fit_m <- coxph(Surv(start, stop, event) ~ strategy + X1 + X2,
               data = imp_data_m, robust = FALSE)

Repeat for all (M) imputed data sets, storing the estimate (\hat\beta^{(m)}) and its variance (\hat V^{(m)}).

Step 6 – Combine the (M) results with Rubin’s Rules

Let

(\bar\beta = \frac{1}{M}\sum_{m=1}^M \hat\beta^{(m)}) – pooled point estimate.
(\bar V = \frac{1}{M}\sum_{m=1}^M \hat V^{(m)}) – within‑imputation variance.
(B = \frac{1}{M-1}\sum_{m=1}^M (\hat\beta^{(m)}-\bar\beta)^2) – between‑imputation variance.

Total variance

[ T = \bar V + \left(1+\frac{1}{M}\right) B . ]

The pooled standard

Original question: Unknown start times & idle gaps in survival analysis of log data — interval-censored model or MI? on Cross Validated (Stats Stack Exchange), licensed CC BY-SA.