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I'm getting the sense that the same subject is tested twice, once at baseline and once after their assigned treatment. Is that true?

You are looking for an analog for ANCOVA with a baseline covariate for a censored response and censored baseline.  The problem is that linear regression models (I know proportional hazard models are not exactly linear but they are similar) is that the x's are known and have a defined error distribution for the response, usually normal or binary.   You could treat the baseline as categorical and divide into dummy variables where each dummy has the category of "more than censor point" as a reference.  This is what they do in tree models and some GAM models.    But some people are opposed to discretization. 

Another approach is to bring the baseline to the left side of the equation and model 'time to event' of both endpoints, introducing an assumed correlation.  There is an extensive literature on recurrence, however it usually deals with consecutive events, not events with distinct start times.  An area where these type of problems do come up is in lab work with upper and lower detection limits which act as censoring points.  There are articles on 'bivariate normal survival models' that have applications in comparing assays. 

David,

To answer your specific questions,

1.       I don't know of a specific theoretical commandment to incorporate censoring at baseline into the model, but my instincts tell me I ought to try, especially if there is a fair amount of censoring at baseline.

2.       How I might add it (if I were doing so) would be to replace BASELINE in your model with two variables, perhaps called BaseTime and BaseCens. BaseCens would be the censoring indicator for whether the baseline task-completion time was censored at the specified maximum time. BaseTime would be equal to the specified maximum time when censoring took place, or equal to the observed baseline task-completion time when censoring did not take place. I'm guessing that BaseTime would already be equal to BASELINE in your particular case, in which case all we really would be doing is adding BaseCens as a second covariate to your model. 

a.       For purposes of clarity, my modification of your model would be TIME*Censor(1) = TRT BaseTime BaseCens.

But that's just one idea. There may be better ideas than that one. Georgette Asherman made reference to lab-work problems where the analytes have upper and lower detection limits which act as censoring points. If she knows of literature articles in which the censored analytes are used as predictor variables in either a regression model or a classification model, some of those might feature better ideas.