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.

#### Best Answer

## 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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