Group level random effect

Question

In a lab experiment Participants took part in four different treatment. At the beginning of each session they were randomly assigned to groups of 4 and played repeated public goods game for 30 periods.

So far I have collapsed the data to obtain group averages, so that observation in a period are independent. This data was used to estimate a model with group-level random effects. Since in some groups there were participants whose incentives were different than those of other members of their group, I would like to test whether difference in incentives lead to disparities in behaviour.

Would it be OK for me to split the group average into two observations:

• With the decisions made by the "special" individuals
• with the average of 3 others

The same random effect would be assigned to both.

Answer 1

Short answer & R Formula

I believe the R code you're looking for is something along the lines of the following, from the lme4 package.

myModel <- lmer(y ~ incentive + (1 | treatGroup/player), data = df)

Basically you're saying that there's a random effect for each player, who is nested inside of a treatment group with a random effect. In addition, there is a fixed effect for incentive.

Longer answer / explanation

First of all, I personally find the terms "fixed effect" and "random effect" to be confusing and used somewhat inconsistently. Andrew Gelman (Prof. at Columbia) has a great blog post on the topic.

Model formulation

So in lieu of using those terms, I'll try to write out the (basic) formulation based on what I understand from your question. In particular, it sounds like we have:

• 16 players (indexed by $i$)
• 30 observations per player (indexed by $j$)
• 4 possible groups (indexed by $k$)
• A binary "incentive" treatment $M$, where $m_{i(k)} = 1$ if the $i$-th player receives the incentive

Let $Y_{ij(k)}$ be the $j$-th observation for the $i$-th player, who is placed into the $k$-th group.

With these in mind, our regression model is as follows:

$$Y_{ij(k)} \sim N(\mu_{i(k)}, \sigma^2)$$

$$\mu_{i(k)} = \alpha_i + \alpha_k + \beta \cdot m_{i(k)}$$

Explanation of terms / quick notes

We have a few relevant terms:

• $\alpha_i$: The 'baseline' for this player
• $\alpha_k$: the effect of being in the $k$-th group
• $\beta$: The impact of the incentive treatement

Some quick notes:

• Using (1 | treatGroup / player) describes a nesting between levels (e.g., each player is in only one treatment group).

• To your point in the comments, 30 observations for each player are assumed to be drawn from a common distribution - Our goal is to estimate the impact of (a) incentives and (b) treatment group on the mean of that distribution.

• This answer assumes no interaction between the incentive and the treatment group.

Additional reading

• I used this to refresh my memory on lmer syntax since I'm more accustomed to using rstan (link).

• Skimming this tutorial (I haven't read the whole thing), it looks like a good resource, with an intuitive overview for the beginner!

• Also see this CV post on lmer formula syntax

If you really want to get into this stuff, then Gelman & Hill (2006) is an excellent book.