Question

Suppose I measure the height of the students in my class: I make a lot of measurements for each student, since each one is rather imprecise, and so I end up with the data: student : measured height, where each student appears many times.

Now I split the class into two groups: say, those, born in Winter/Spring and Summer/Autumn, and I am interested in comparing the means.

The question is how to weight my measurements for, say, Winter/Spring group: should I find the sample mean for each student, and treat the all students with equal weights? - then I somehow loose info that for some students with more measurements my sample mean is much more precise?

Or should I pool all the measurements for the group and consider each as a data point? - then it can happen, that there are a lot of measurements, but of very few students, and I get a false confidence in my estimate (say, with t-test for 2 groups).

Any thoughts? Is there a standard way to go?

Answer 1

The proper way to do this is to model the $j$-th measurement $x_{ij}$ on the $i$-th student as

$$x_{ij} = \mu_i + \delta_i + \epsilon_{ij} \quad (1 \le j \le n_i)$$

where $n_i$ is the number of measurements for student $i$. Assume that $\mu_i = \mu_{WS}$ if student $i$ was born in Winter/Spring and $\mu_{i} = \mu_{SA}$ otherwise. You want to compare $\mu_{WS}$ to $\mu_{SA}$.

You can assume that the means of the $\delta_i$ over the two groups are both zero and have to assume that the $\epsilon_{ij}$ are independent random errors with zero means and the same standard deviations.

Now use Analysis of Variance. All standard computer programs can do that for you.