# Appropriate "semi-paired" t-test

From what I understand, the paired t-test is used when measuring the same "thing" in two different scenarios (e.g. one student's pre-test vs. post-test scores). However, is there a test that takes into consideration the variation in the measurment? Take the data set below. The objective is to determine if the two instruments produce similar results (Data 1 vs. Data 2). The testing is destructive so a direct paired t-test is not possible. However, each batch of material (15 in total) is tested in triplicate on each instrument. There is batch to batch variation, so I feel like doing a non-paired t-test loses that information. What is the best way to determine if the instruments are statistically indistinguishable? • A multilevel, or mixed, model is designed to solve this problem. Aug 8, 2018 at 19:37
• Thanks! I am not familiar with a this model. Can you point me to where I can learn about it? Aug 8, 2018 at 20:25
• I think I got it. I assume it is a multilevel regression, and if the coefficient for "instrument" is statistically significant, then the instrument is a factor that affects the measurement? Aug 8, 2018 at 20:29

A model can be written as $$Y_{ijt} = \mu + b_i + \beta_t + \epsilon_{ijt}$$ where $$i$$ is batches, $$j$$ is material within batch, and $$t$$ (A or B) is instrument. $$b_i$$ is a random batch effect, $$\beta_t$$ is treatment effect. In R this can be done as (not tested)
library(lme4)

This model is assuming constant variance of the error term $$\epsilon_{ijt}$$. This is a variance component model, see What is a variance component model?