Editing Power (statistics) (section)

=== Simulation solution ===

Alternatively we can use a [[Monte Carlo simulation]] method that works more generally.<ref>{{cite conference|url=https://support.sas.com/resources/papers/proceedings/proceedings/sugi24/Posters/p236-24.pdf|last=Graebner|first=Robert W.|title=Study design with SAS: Estimating power with Monte Carlo methods|year=1999|conference=SUGI 24}}</ref> Once again, we return to the assumption of the distribution of <math>D_n</math> and the definition of <math>T_n</math>. Suppose we have fixed values of the sample size, variability and effect size, and wish to compute power. We can adopt this process:

1. Generate a large number of sets of <math>D_n</math> according to the null hypothesis, <math>N(0, \sigma_D)</math>

2. Compute the resulting test statistic <math>T_n</math> for each set.

3. Compute the <math>(1-\alpha)</math>th quantile of the simulated <math>T_n</math> and use that as an estimate of <math>t_\alpha</math>.

4. Now generate a large number of sets of <math>D_n</math> according to the alternative hypothesis, <math>N(\theta, \sigma_D)</math>, and compute the corresponding test statistics again.

5. Look at the proportion of these simulated alternative <math>T_n</math> that are above the <math>t_\alpha</math> calculated in step 3 and so are rejected. This is the power.

This can be done with a variety of software packages. Using this methodology with the values before, setting the sample size to 25 leads to an estimated power of around 0.78. The small discrepancy with the previous section is due mainly to inaccuracies with the normal approximation.