Solved – Group level random effect

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.

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.

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