# Systematic vs. statistical error

I'm currently working with some measurements of the following form:

• Three test subjects, three trials per test subject, three measurement points per trial:

• At each measurement point within each trial, I have time measurements and velocity measurements, each with associated +/- errors. I then compute velocity from these measurements and propagate my error. Next, I take the average of the three measurement points per subject subject, propagating my error, to find the average velocity per trial. Finally, I average the three trials, leaving me with a value and propagated error per test subject. I also calculate the standard error of the mean on this average.

• Now I have two error values, one +/i and one standard error. I'd like to do more calculations with these average velocity measurements. Is there a generally accepted method to determine which error I should use in these future calculations?

Please let me know if there are any questions regarding my experimental setup. In a nutshell, I'd like to know a way to compare systematic versus statistical error and determine which to propagate. Many thanks for all of your help!

Cheers.

To me, this looks like a balanced, nested design with 3 levels of random effect: subject, trial, measurement point.

I'm not clear to me on what happens at the measurement point level. It looks as if you get several measurements there as well. Perhaps you could clarify.

What sort of error structure do you think you have? Are the errors normally distributed? If you average over the velocity measurements at the end, can you assume (or transform) to normality?

To me, it looks as if your model is something like this:

$$Y_{ijk}=\mu + \alpha_i+\beta_{ij}+\gamma_{ijk}$$

where alpha, beta and gamma are error terms attributed to subject, trial and measurement point.

if so, then fitting a general linear model with random effects will give you an estimate for the variance at each stage. You have no fixed effects apart from the overall mean. If you want to predict the response of a random subject, your best estimate will be that overall mean. Confidence intervals will be based on the sum of the estimated variances at each stage.

$$\sqrt{\hat\sigma_{\alpha}^2+\hat\sigma_{\beta}^2+\hat\sigma_{\gamma}^2}$$