# In a survival study, is interval censoring simplifiable to midtime imputation?

In epidemiological studies, it is common that event are interval censored, since an incident case (like a new diagnosis of disease) could have happened between 2 waves of data collection.

No software has default implementation for handling this. Something I wondered is if the midpoint imputation, where the date of event is fixed as the midpoint of the interval and then considered as right-censored (not much described but you can see link1 or link2), was a valid way to handle these missing event times. Since my intervals have a wide distribution, I tended to think not.

I discovered the SmoothHazard library lately, which unlike above can handle left-truncation and interval-censoring (with a fully-parametric model), but the 2 methods (interval-censoring and midpoint imputation) gave very close results.

Indeed, "parameters of the Weibull distribution" for both a and b coefficients, convergence criteria and coefficients were close to within 1%, while a Cox model gave very similar results (<30% different).

The only real difference was the log-likelihood: -18594.84 for right-censored, -20343.59 for interval-censored and -31352.55 for Cox model

Maths involved in those models are unfortunately quite out of my league for now, is there a mathematic shortcut that allows to simplify interval-censoring to midpoint imputation? Else, under which condition could I approximate the former by the latter?

Here is some code for R enthusiasts:

library(survival)
library(SmoothHazard)
library(prodlim)

#Parametric model with left-truncation and right-censoring on mid-time between questionnaires
fit = shr(formula = Hist(time=age_surv, event=event, entry=age_origin) ~ X + A1 + A2 + A3, data = db)
#Parametric model with left-truncation and interval-censoring
fit.i = shr(formula = Hist(time=list(age_surv_inf, age_surv_sup), event=event, entry=age_origin) ~ X + A1 + A2 + A3, data = db)
#Cox model with left-truncation and right-censoring on mid-time between questionnaires
fit.cox = coxph(formula = Surv(age_origin, age_surv, event) ~ X + A1 + A2 + A3, data = db)

• Unless you can provide a reference on this method (midpoint imputation), I think your question is really whether the proposed approach is valid. Commented Mar 27, 2019 at 15:38
• @AdamO Actually, my approach was "I've not been told a proper method and I'm really surprized that it gives the same results as a well described one" Commented Mar 27, 2019 at 15:45
• Be careful with "the same results": you have presented one case, but you actually should endeavor to understand the mathematics a little better here. What if the hazard is non-linear? What if the interval censoring is wide or non-uniform? What about attrition? Many opportunities to obtain biased analyses. Commented Mar 27, 2019 at 15:48
• @AdamO I think the fact that I've got such close results is too uncanny to be random. I clarified my question: it seems that it can happen that the midpoint imputation is acceptable, what would be the conditions for this ? Commented Mar 28, 2019 at 8:24

## 1 Answer

Midpoint imputation is not a valid form of imputation in any survival model. That's because the midpoint has no statistical property that is justified for survival analyses. If I know that an event occurred between time $$a$$ and $$b$$, then I highly doubt that the distribution of that time is symmetric (survival times are nearly always skewed), so $$(a+b)/2$$ is not a good estimator of the median or the mean. The ideal approach is to actually use an estimator of the hazard function to perform simulations of possible event times and pool multiply imputed datasets to obtain unbiased and efficient estimates of the effects.

If the goal, on the other hand, is to have a pragmatic imputation, then I would prefer to impute the maximum $$b$$ since, barring any informed guess as to when an event may have occurred, it is most valid to simply use all "known" data at a given time. For a subject who is seronegative for Hepatitis B at survey Wave 1 and seropositive for Hepatitis B at survey Wave 2: to the best of my knowledge they were at risk for Hepatitis for the entire ellapsed period between survey waves.

• I'm sorry, it seems that my question was awfully asked. I reworked it a bit and clarified some points. Is it clearer? Your answer is still very interesting though, how could I build such an estimator? Is there a validated way to do so? Commented Mar 28, 2019 at 7:42
• On second thought, since epidemiologic questionnaires and events are usually independant, if an event occurred between time $a$ and $b$, why wouldn't survival time be uniformly distributed between $a$ and $b$? (but it maybe would not be the same in an interventionnal study) Commented Mar 28, 2019 at 14:46
• "epidemiologic questionnaires and events are usually independant" I don't understand what you mean. To your 2nd point: here's a mathematical statistics problem for you. Suppose $Y \sim \text{Exp}(\theta)$. Show that, if $a<Y<b$, $Y$ is not distributed uniformly. Commented Mar 29, 2019 at 14:39
• In epidemiology, if you are studying cancer events in the general population, answering to a questionnaire is independant with getting cancer (unlike in an interventionnal study where time 0 may be a surgery which is related to cancer letality). Thus, age is generally defined as the timescale so the time variable is age_at_event, and data are left-truncated (because we don't follow-up since birth). I'm not sure that the survival times (ages) are skewed in this case. Could this alone explain the similarity of results? Commented Mar 29, 2019 at 15:07
• ok so you mean response likelihood doesn't depend on cancer. the questionnaire is the paper form. Commented Mar 29, 2019 at 15:41