Test for study design few subjects many paired observations per subject

Question

I'm having difficulties in contructing a proper statistical approach to analyzing the data described below.

We have 3 subjects which we each measured on 20 different days. On each day we measured a certain variable Y in two conditions: A and B. These measurements are thus paired. The effect of specific day is not of interest (different days for each subject and there is also no order effect) though there is variation in Y between days.

We are interested in whether the value of Y differs between condition A and B.

Using all observations and performing a paired t test of condition A and B would violate the assumption that observations are independent, since 20 pairs are from subject 1, 20 pairs from subject 2 and 20 pairs from subject 3.

A one sample t test on the difference score (A-B) is also not appropriate for the same reason as above.

Aggregating the data and performing a paired t test on the aggregated data is not an option since than we are only left with 6 values (3 subjects x 2 conditions).

Performing a multi level model with a random effect of subject is possible, however then the paired nature is disregarded.

Performing an 2 by 3 ANOVA with factors Condition and Subject is also not appropriate given that we have paired data (condition A and B are measured on the same day).

I would have a preference to perform a test on the condition difference score (A-B) but do there exist one sample t tests in which we could take into account the non-independence (i.e., since it are multiple measurements of 3 subjects)?

Answer 1

You could try a mixed model on the difference scores $D=A-B$. Let $$ D_{ij}=\mu + \tau_i + \epsilon_{ij} $$ where $i=1,2,3$ indexes persons and $j=1,2,\dotsc, 20$ days within person. $\tau_i$ is a random effect modeled as normal with zero mean, $\epsilon$ the usual error term.

In R this might be

library(lme4)
mod0 <- lme(D ~ 1 + (1 | day), data=your_data_frame)

and then you can make a confidence interval for the intercept.