# Linear regression: multiple points or averages?

I have the following set-up: A material property that is a function of temperature is measured by 3 laboratories. Each laboratory performs two measurements each at 4 different temperatures, i.e. 8 results per laboratory. I observe that for a given temperature, the difference between the averages of each lab are larger than the differences of the two results of each lab. My goal is to have the most reliable (least squares) linear regression of the property vs. temperature.

The question is: Should I perform the regression on the individual points (3 labs x 4 temperatures x 2 points per temperature = 24 points) or on the average per lab (3 labs x 4 temperatures = 12 points)? As the design is balanced, this does not have any effect on the regression parameters, but does influence the confidence interval of the regression line, with the confidence interval for the average being larger (even without applying the t-factor).

I looked at the residuals and in both versions the residuals follow a normal distribution (normal probability plot of all residuals). There is also no sign of heteroscedasticity. My tendency would go to the average per lab, as the within-lab standard deviation seems to be smaller than the between-lab standard deviation. Correct?

• I’m no expert on multilevel/mixed/nested modeling, but wouldn’t that apply here better than OLS regression? Aug 4, 2021 at 10:46

When averaging the measurements you make your dataset smaller. Initially, it was pretty small (24 samples) but here you half its size. This is why confidence intervals get larger. You are correct that with a single measurement there is less noise in the data, so this would decrease the variability of the estimates, but at the same time you make the dataset smaller, hence you observe the reverse effect.

As a side note, averaging the measurements is not the best idea. First, you are discarding valid data. Second, you are cheating by making its variability smaller than in reality (so the parameter estimates would be overly optimistic). Better to use all data, or at least something like an error-in-variables model.