Power Analysis
Researchers can also use hierarch to determine the appropriate sample size for a future experiment. hierarch.power provides a class, DataSimulator, to assist in power analyses. DataSimulator is initialized with a list specifying the probability distributions generating the data and an optional random_state for reproducibility.
In this case, consider an experiment similar to the one above - two treatment conditions, but the sample size at each level of hierarchy is yet to be determined. First, you must posit a data-generating process for the analysis.
Suppose you assume that the column 1 values are normally distributed with mean 0 and variance 1. From past experience, you believe that the column 2 values follow a right-tailed distribution, so you choose to model it as a lognormal distribution with a scale parameter of 0.75. Finally, you decide that you want to achieve 80% power for a mean difference equal to one standard deviation. You calculate that the summed standard deviation of the two distributions you specified is 1.525 and input that as a parameter, as well.
from hierarch.power import DataSimulator
parameters = [[0, 1.525], #difference in means due to treatment
[stats.norm, 0, 1], #column 1 distribution - stats.norm(loc=0, scale=1)
[stats.lognorm, 0.75]] #column 2 distribution - stats.lognorm(s = 0.75)
sim = DataSimulator(parameters, random_state=1)
Next, you choose a experimental design to simulate. Perhaps, like above, you decide to start with three samples per treatment condition and three measurements within each sample. Calling the .fit() function will ready the DataSimulator to produce randomly-generated data according to this experimental scheme.
import scipy.stats as stats
hierarchy = [2, #treatments
3, #samples
3] #within-sample measurements
sim.fit(hierarchy)
By calling the .generate() function, DataSimulator uses the prespecified
parameters to generate a simulated dataset. ::
print(sim.generate())
0 |
1 |
2 |
3 |
|---|---|---|---|
1 |
1 |
1 |
1.014087 |
1 |
1 |
2 |
1.891843 |
1 |
1 |
3 |
1.660049 |
1 |
2 |
1 |
2.068442 |
1 |
2 |
2 |
1.843164 |
1 |
2 |
3 |
2.328488 |
1 |
3 |
1 |
0.906038 |
1 |
3 |
2 |
1.215424 |
1 |
3 |
3 |
1.027005 |
2 |
1 |
1 |
1.788798 |
2 |
1 |
2 |
1.252083 |
2 |
1 |
3 |
1.024889 |
2 |
2 |
1 |
2.986665 |
2 |
2 |
2 |
3.254925 |
2 |
2 |
3 |
3.436481 |
2 |
3 |
1 |
2.784636 |
2 |
3 |
2 |
4.610765 |
2 |
3 |
3 |
4.099078 |
You can use this to set up a simple power analysis. The following code performs a hierarchical permutation test with 50,000 total permutations (though this is overkill in the 2, 3, 3 case) on each of 100 simulated datasets and prints the fraction of them that return a significant result, assuming a p-value cutoff of 0.05.
from hierarch.stats import hypothesis_test
pvalues = []
loops = 100
for i in range(loops):
data = sim.generate()
pvalues.append(hypothesis_test(data, 0, bootstraps=500, permutations=100))
print(np.less(pvalues, 0.05).sum() / loops)
#out: 0.29
The targeted power is 0.8, so you can fit the DataSimulator with a larger sample size. You can run the following code block with different sample sizes until you determine the column 1 sample size that achieves at least 80% power.
sim.fit([2,10,3])
pvalues = []
loops = 100
for i in range(loops):
data = sim.generate()
pvalues.append(hypothesis_test(data, 0, bootstraps=500, permutations=100))
print(np.less(pvalues, 0.05).sum() / loops)
#out: 0.81
You note, however, that increasing the number of column 1 samples is much more laborious than increasing the number of column 2 samples. For example, perhaps the column 1 samples represent mice, while column 2 represents multiple measurements of some feature from each mouse’s cells. You have posited that the slight majority of your observed variance comes from the column 2 samples - indeed, in biological samples, within-sample variance can be equal to or greater than between-sample variance. After all, that is why we make multiple measurements within the same biological sample! Given that this is a reasonable assumption, perhaps 80% power can be achieved with an experimental design that makes more column 2 measurements.
sim.fit([2,8,30])
pvalues = []
loops = 100
for i in range(loops):
data = sim.generate()
pvalues.append(hypothesis_test(data, 0, bootstraps=500, permutations=100))
print(np.less(pvalues, 0.05).sum() / loops)
#out: 0.84
Of course, adding column 2 samples has a much more limited influence on power compared to adding column 1 samples - with infinite column 2 samples, the standard error for the difference of means is still dependent on the variance of the column 1 data-generating process. This is illustrated with an excessive example of 300 column 2 samples per column 1 sample, which shows no improvement in power over using only 30 column 2 samples.
sim.fit([2,8,300])
pvalues = []
loops = 100
for i in range(loops):
data = sim.generate()
pvalues.append(hypothesis_test(data, 0, bootstraps=500, permutations=100))
print(np.less(pvalues, 0.05).sum() / loops)
#out: 0.83
On the other hand, adding only four column 1 samples to each treatment group (rather than 270 to each column 1 sample) brings the power to 97%.
Finally, to ensure that hierarchical permutation is valid for the posited data-generating process, you can do another power analysis under the null hypothesis - that there is no difference between groups. To compensate for Monte Carlo error, you should increase the number of loops - at 100 loops, the error for an event that happens 5% probability is +/- 2%, but at 1000 loops, it is only +/- 0.7%.
parameters = [[0, 0], #no difference in means because we are sampling under the null hypothesis
[stats.norm, 0, 1], #column 1 probability distribution
[stats.lognorm, 0.75]] #column 2 probability distribution
sim = ha.power.DataSimulator(parameters, random_state=1)
sim.fit([2,12,30])
pvalues = []
loops = 1000
for i in range(loops):
data = sim.generate()
pvalues.append(hypothesis_test(data, 0, bootstraps=500, permutations=100))
print(np.less(pvalues, 0.05).sum() / loops)
#out: 0.05
Hierarchical permutation experiences no size distortion for this experimental design and is therefore a valid test.
Note: because these power calculations are subject to Monte Carlo error, so you should consider upping the number of loops if the precise value for power is of extreme importance. In nonclinical settings, however, small-scale power analyses are sufficient and can be a valuable guide for choosing the sample size for your study.