# Multilevel alternative to Paired t-test

I have a problem where we have N patients, and we measure some property of tumoral lesions for each patient, where the number of lesions varies on each patient (patient A could have 1 lesion, and patient B could have 5).

For each lesion we measure the property using two different methods, so from this point of view we could use a paired t-Test, but, I would like to account on possible differences between patients.

Is there any alternative test to the paired t-test that could account for differences in each patient?

Thanks,

• What do you mean by accounting for possible differences between patients? It's not immediately clear to me that you're wanting to do something that t-tests don't already do. Do you mean that you want to know whether there's a relationship between the number of lesions and the measurement? Jul 5, 2016 at 8:45
• I mean, in patient 1, maybe I measure 5 lesions with method A and B, and there is a decrease in the measurement, but it is due to the patient. And patient 1 counts 5 time more than patient 2, which only have one lesion and with method A and be the measurement increases. Jul 5, 2016 at 8:53
• Are the measurements using the two different measures taken at different time points? Or are the different lesions measured at different times but both with the same method? Without knowing about the design of your study it's really difficult to suggest an appropriate test. Jul 5, 2016 at 8:57
• The measures are the same measure but using two different methods. Jul 5, 2016 at 9:11
• Okay, so what do you mean in your earlier comment when you say "decrease in measurement"? In order to show a decrease in something you would need to have two measurements, one before the decrease and one after. It sounds like you have two measurements, but they're both taken at the same time, hence no way of observing a decrease. Do you mean something else? I'd stress that you're much more likely to get an answer if you edit the question to make a lot of these details (e.g., the design of the data, your hypothesis, etc.) explicit Jul 5, 2016 at 9:14

## 1 Answer

Following on from discussion in the comments, my suggestion would be that, in general, for a case like this mixed-effects regression would be the appropriate approach to take.

Your model would contain two random effects (unless there's others not mentioned): patient, and lesion which would be nested under patient. There would be at least one fixed effect: the method used for measurement.

You would likely want to test both random intercepts and random slops for both patient and lesion.

Now the catch, it sounds like you may have some patients who only have one lesion. For these patients you couldn't estimate distinct random effects for both patient and lesion (because you couldn't tease apart the variance associated with each). Assuming such a model would converge, I'm not sure whether this would have a consequence for the estimates associated with the fixed effect (i.e., assessing whether there's a difference overall difference between methods). Someone else may know.

The alternatives here would be to remove the random effect for lesion (the between lesion variance within a patient would no longer be being accounted for) or to retain the random effect but remove the nesting (between lesion variance would still be accounted for, but the fact that a pair of lesions within a single person may be more similar than a pair of lesions between two people would not). I'm not sure whether either way is better than the other...

Some reading:

https://en.wikipedia.org/wiki/Mixed_model

https://cran.r-project.org/web/packages/lme4/index.html

@book{gelman2007data,
address =  {New York, NY},
author =   {A. Gelman and J. Hill},
publisher = {Cambridge University Press},
title =    {Data analysis using regression and multilevel/hierarchical models},
year =     2007
}