# Within-subject dataset: Fixed vs. random effects in SPSS

I have two classes of 25 students. At the beginning of the semester, they took a personality test that measured a given variable. Let's call it "competence".

Every week, each student from a particular class (class #) was paired randomly with another student from the same class and they performed a task. Then, they each (source) gave their partner (target) feedback by email, and rated the quality of each other's feedback (feedback quality).

I would like to see if "competence" has an effect on "feedback quality" while controlling for the fact that data is within-subject and that I have two classes. My dataset looks like this:

Class  I  Source Name  I  Competence I Target Name I Feedback Quality
1    I     AHR       I     6       I    HVD      I       5
2    I     TGH       I     3       I    POL      I       2
1    I     HVD       I     4       I    AHR      I       7


I am using SPSS and I understand what I am trying to do is a linear mixed model. My understanding is that competence is a "random" effect while "class" and "target" are fixed effects. Is this correct?

It's useful for me to write out the model. I think you're interested in something like the following:

$$y_i=\alpha+\beta x_i + \delta_i + \phi_j +\epsilon_{ti}$$

where feedback quality of individual $i$ is $y_i$, their competence is $x_i$, and $\phi_j$ is the fixed or random effect of the person receiving the feedback and $\delta_i$ is the fixed or random effect of the person giving the feedback. $\epsilon_{ti}$ is an idiosyncratic error.

The terms fixed effects and random effects mean different thing to different people. From an econometric perspective, the question is simply whether $\delta_i$ or $\phi_j$ are correlated with $x_i$. $\delta_i$ likely is, while $\phi_j$ is not due to random assignment. From that perspective than $\delta_i$ is a fixed effect and $\phi_i$ is a random effect.

The problem though is that the coefficient $\beta$ is not identified if $\delta_i$ is a fixed effect. You can't disentangle someone's fixed effect from the unchanging measure of competency.

Which gets at a larger problem that you'll never be able to control for all characteristics that might explain feedback quality. There will always be some un-observed time-constant or time-changing characteristics that are correlated with competency and feedback quality.

Whether class is fixed or not depends on how students are sorted into classes.