estimating start point of a production of y" -ie findx for y=0 given y values for x>0

I hope I can explain the problem clearly and its not an obvious solution. the application area is say an event (in the body) at say day=0 starts the production of, say, a biomarker the concentration of which increases. Essentially the problem is to estimate what day zero is for a particular individual 0 and we have a number if individuals. The timing of the event is always unknown and can only be inferred from the increasing biomarker values. So given measures on various days (always day>0) from various people of the biomarker maybe some multiple measures from the same person I can fit a linear regression ( I know a mixed model with patient effects might be better with multiple measures). Then for an individual I might have say 2 or 3 biomarker values taken on various days. Is there some established method - (inverse regression ? with regression parameters supplied as a prior ? neither my strong suit ) where I can estimate when that persons day zero is eg if have the parameters (slope etc) of a linear regression (estimated on numeric people) and, say, values of 13, 32 and 100 on March 17, 19 and 23 can I estimate on what day the biomarker production started ? ie on what day the biomarker values was zero for this person ? March 8 ? March 13 ?

many many thanks in advance for reading this long and tricky description - and any suggestions (and hope I've complied with the board rubric)

By the way I've been trying to see if there is anything on estimating infection data using serial bacterial counts (or measures of antibodies that react to infection like CD4) - I guess the difference is when an infection is identified it is usually treated, one wouldn't usually spend days measuring bacterial or viral levels rising in an untreated patient.

many thanks for your help
Re: estimating start point of a production of y" -ie findx for y=0 given y values for


using one (mixed) model including the data from all patients is somewhat tricky, since day zero for each person is unknown, thus we do not know "where to start" with our time dependent variable. Which is actually the forcus of your question.

I would set up a regression model for each person, fit this to the data (introducing the data as a numerical variable with day = 0 for the fist existing measurement). Using the parameters from the regression model, you subsequently can set y=0 and find the corresponding x (which is a negative value telling you how many days before the first measurement the production of the marker started).

E.g. using bootstrap you can even calculate confidence intervals for the real "day zero"


Omega Contributor
Re: estimating start point of a production of y" -ie findx for y=0 given y values for

So you are trying to extrapolate outside of your data space. Mmercker'sapproach is reasonable. Though repeatsd measure (multilevel) is important to address (potential autocorrelation structure between measures). I would plot your data to examine the relationship between time and value to understand the shape. If it is linear, a constant slope, the approach should be a good approximate. Are there any patients or information in the literature where patients report their initial exposure and you know the incubation time, to try to validate your estimate.

Your second comment reminds me of procalcitonin values. They have markers for bacterial infection.
Re: estimating start point of a production of y" -ie findx for y=0 given y values for

many thanks, mmercker and hlsmith, lots to think about

I probably wasn't clear, mmercker (or maybe I hadn't decided this at this point) but I would initially do a simulation (with various assumptions) so I would know which days post start each person has measurement on for that)

then I'd use the estimates from the mixed model on new patients where I'd just know the separation in days between measures not the number of days from the start (I'd use just the fixed estimates not the random effects as they'd be unknown for new patients) - I'd invert in some way to get the day where the outcome is predicted to be zero aka the start

many thanks again for all the useful ideas