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)

 A: 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.
