Exponential decay model with repeated measurements

[JoakimE]

Hi,

I am working with a study where we expect an exponential decay relationship between increasing dose and a pharmacodynamic response variable of interest. An initial idea was therefore to model this using an exponential regression model on the form

y = ab^x

where b is the parameter of interest to estimate (x is the dose). This could be done fairly easily, since it transforms to linear model on log-scale. However, there are also repeated measurements per patient, since the same patient is measured at different dose levels. To avoid a potential problem of correlated data points in the model, a proposal is to

1) fit the model separately for each patient and

2) Retrieve all estimates of b and

3) Perform a t-test of the values of b to investigate if mean of b is below 1 (proving a dose-dependent response in Y).

However, while this seems to solve the problem of correlated data points, I am doubtful if it is the most efficient/powerful approach to analyse the data. Some patients have very few datapoints (as few as 3), so it does not seem very robust to me to analyse all of them separately. I am contemplating whether the current model could be extended instead with a mixed effect for patient, something like this:

y = apb^x

where p is the random effect for patient, allowing observations to be correlated within the same individual.

Would this be a viable alternative? Which approach is preferable? Do I need to extend the modelling approach even further allowing for a random coefficient model?

Any input and ideas would be helpful.

Kind Regards,

JoakimE

[StatDave · Accepted solution]

A repeated measures analysis can be done using a log-linked generalized linear model as you suggest either with PROC MIXED (if Y is normal) or PROC GLIMMIX. Another approach is a Generalized Estimating Equations model using PROC GEE which doesn't involve introducing any random effects. It allows for unequal numbers of response per subject. For example, the following fits the model log(mean)=a+bx, where mean is the mean of Y and assuming Y is distributed normal and a common correlation (TYPE=EXCH) among the repeated measures within patients. See other examples in the PROC GEE documentation.

proc gee;
class patient;
model y=x / dist=normal link=log;
repeated subject=patient / type=exch;
run;

[JoakimE]

Thanks for sharing your ideas StatDave. I believe GEE would be a good option for us (that I had not thought of).