Balancing effect of sample size and amount of trials on power I ran a power simulation for a linear mixed model with an interaction of a continuous predictor and a predictor of condition (3 conditions), their intercepts and a varying intercept for subjects. Now I thought about having 1 or 3 trials per condition and ran simulations for both experiments. The estimated power did not really differ but I read that both, trial amount and sample size, affect power.
I now wonder how to balance the number of participants and the number of trials to get a well powered study. Any helpful hints?
 A: It is better to have more participants than to have more trials per participant*.
(from the point of view of the accuracy of the measurements, more participants might be more costly or difficult sampling)
See for instance the simplified example estimating the mean of a population where the participants follow a normal distribution and measurements within participants follow a normal distribution as well.
The mean of a participant $i$ be distributed as
$$\mu_i \sim N(\mu,\sigma_{p})$$
and the measurement of a particular participant
$$X_{ij} \sim N(\mu_i, \sigma_{t})$$
Let's consider that we measure $n_p$ participants and for each participant $n_t$ trials.
Then the mean of observations of participant $i$ is distributed as
$$\bar{X}_{i} = \frac{1}{n_t}\sum_{j=1}^{n_t} X_{ij} \sim N\left(\mu_i, \sqrt{\sigma_{t}^2/n_t}\right)$$
The mean of the mean of the participants is distributed as:
$$\bar{X} = \frac{1}{n_p}\sum_{i=1}^{n_p} \bar{X}_{i} \sim N\left( 0, \sqrt{\sigma_{p}^2/n_p + \sigma_{t}^2/(n_p \cdot n_t)}\right)$$

*

*Increasing the number of participants decreases both the variation $\sigma_p^2$ due to the distribution among the participants and the variation $\sigma_t^2$ due to the distribution within the participants.


*Increasing the number of trials reduces only the variation $\sigma_t^2$ due to the distribution within the participants
If the within-participant variation $\sigma_t$ is dominant, then the difference between the effects of increasing number of trials and increasing number of participants will be closer to each other.

Intuitively you can see it as sampling $n_p$ 'error' terms for each participant and $n_p \cdot n_t$ 'error' terms for each trial in each participant. Your mean is an average over these terms, and the variance will be less the more you sample, but the number of trials does not reduce the variance of the mean of the $n_p$ terms for the between participant variation.

However if there are additional costs to obtaining a participant, and obtaining more trials is cheaper, then there might be some optimum where we need more trials per participant. See: Optimize the number of individuals inside a sample
