Exercise: Power simulation and power curves for t-test
================

## Temporal value asymmetry

“participants … were asked to imagine that they had agreed to spend 5 hr
entering data into a computer and to indicate how much money it would be
fair for them to receive. Some participants imagined that they had
completed the work 1 month previously, and others imagined that they
would complete the work 1 month in the future . . . Participants
believed that they should receive 101% more money for work they would do
1 month later ($M = \$125.04$) than for identical work that they had
done 1 month previously ($M = \$62.20$), $t(119) = 2.22$, $p = .03$,
$d = 0.41$” (Caruso, Gilbert, and Wilson 2008, 797)

### Plan a direct replication

1.  What is a plausible standard deviation? Hint: $d = (M1 − M2)/SD$
2.  What is an interesting minimal effect size (in \$, Euro, or min)?
3.  Simulate responses for 120 participants in both the *past* and the
    *future* condition, assuming normal distributions with the same
    variance. Use the standard deviation and the minimal effect size
    from 1. and 2.
4.  Parameter recovery: Repeat the simulation from 3. 2000 times to
    re-estimate the parameters ($\mu_1, \mu_2, \sigma$) from the
    simulated responses. Visualize the recovered parameters in box
    plots.  
    Hint: $SE = 2/\sqrt{n} \cdot SD$, where $n$ is the total sample
    size.

``` r
t <- t.test(x, y, mu = 0, var.equal = TRUE)
c(t$estimate, sd.pool = sqrt(n) / 2 * t$stderr)
```

5.  Power simulation: Increase the total sample size to find out the `n`
    necessary for 80% power for the t-test.
6.  Power curves:
    - Write an `R` function that takes sample size `n`, minimal effect
      `d`, standard deviation `sd`, and number of replications `nrep` as
      arguments. It should return the simulated power.
    - Use this function to simulate the power for each combination of 4
      different standard deviations and 4 sample sizes.
    - Visualize these power curves in a single plot.

### References

<div id="refs" class="references csl-bib-body hanging-indent">

<div id="ref-CarusoGilbert08" class="csl-entry">

Caruso, E. M., D. T. Gilbert, and T. D. Wilson. 2008. “A Wrinkle in
Time: Asymmetric Valuation of Past and Future Events.” *Psychological
Science* 19 (8): 796–801.
<https://doi.org/10.1111/j.1467-9280.2008.02159.x>.

</div>

</div>
