# How to fit data with an interval-censored explanatory variable?

I have a series of human growth data that I wish to fit to a 3 parameter logarithmic growth curve:

s(i) = Beta0 + B1*T + B2*ln(t), where s is a length and t is an age.

The only problem is that this age is interval-censored - that is, the age is an estimate with a range between T(L) < U < T(R) where the true age U is some estimate between the left and right values. I'd like to do this in R, but I am struggling with the best way to model and/or 'maximize' this interval so as to return a fitted curve based on the interval values. I have thought through imputation methods involving kernel density, midpoint approximation, and various other maximizing techniques.

Any suggestions on how to handle interval-censored explanatory data when fitting data to a logarithmic (or any) model such as the one described above.

• Are T(L) and T(R) definitive limits of age or just upper and lower ends of a prediction error? Example of definitive limit of age: we want age at seroconversion, but all we know is that subject had a medical examination at age 23 and tested negative and another examination at 31 and tested positive. Age at seroconversion has to be in [23,31]. Example of prediction error interval: we estimate age as 27 +-4 based on a 95% prediction interval from another model we fit. – Cliff AB Sep 11 '17 at 4:44
• They are a prediction interval. It is an age distribution based on tooth development. Essentially, they are the upper and lower limits of that distribution, yes. I'd like to model growth based on this distribution as a whole to account for that error in age estimation. – Chris Wolfe Sep 11 '17 at 5:26
• If the intervals containing the independent variable are from a prediction model, not definitive intervals that contain the independent variable, this won't actually be considered interval censored, but rather an error in measurement problem. – Cliff AB Sep 11 '17 at 17:09
• I did not understand that distinction, thank you Cliff! This may be beyond the scope of this post, but do you know of a good way to deal with such a model in R. Thanks! – Chris Wolfe Sep 11 '17 at 17:16