*Bounty: 50*

*Bounty: 50*

I am planning a nested experimental design, where 1000 participants respond to one policy proposal coming from a 2^3 design (125 participants for each condition A x B x C).

Yet, the policy proposal can come from one of 25 policy categories. But I am not particularly interested in their specific effects, I just want to be able to make claims across these categories.

Therfore I plan to model them as random effects in a mixed model (Each policy category would receive 40 partcipants).

Ideally, the model would be:

```
lmer(response ~ A*B*C + (1 + A*B*C|category))
```

Would such a model come even close to being adequately powered? Or does the inclusion of the 25 policy categories in the experimental design increase the number of experimental conditions to 200, therefore requiring an extremely large amount of participants?

Is there a benefit, when it comes to statistical power, to model a categorical treatment variable as random?

## Edit:

Some info about likely parameter values:

Y = values from 1-7 (let’s assume it’s metric and gaussian)

Ymean = 4

BetaA = 0.0

BetaB = -0.4

BetaC = 0.4

BetaAxB = 0.2

BetaAxC = -0.2

BetaBxC = -0.1

BetaAxBxC = -0.1

Residual Standard Deviation = 1

Within-category standard deviation = 0.5

alpha = 0.05