Statistical Power Analysis in Python

In this chapter we focus specifically on statistical power. We will use the NHANES dataset, so let’s first set that up.


import numpy as np
import pandas as pd

np.random.seed(12345) 

from nhanes.load import load_NHANES_data

nhanes_data = load_NHANES_data() 
adult_nhanes_data = nhanes_data.query('AgeInYearsAtScreening > 18')
adult_nhanes_data = adult_nhanes_data.dropna(subset=['WeightKg']).rename(columns={'WeightKg': 'Weight'})

Power analysis

We can compute a power analysis using functions from the statsmodels.stats.power package. Let’s focus on the power for an independent samples t-test in order to determine a difference in the mean between two groups. Let’s say that we think than an effect size of Cohen’s d=0.5 is realistic for the study in question (based on previous research) and would be of scientific interest. We wish to have 80% power to find the effect if it exists. We can compute the sample size needed for adequate power using the TTestIndPower() function:


import scipy.stats
import statsmodels.stats.power as smp
import matplotlib.pyplot as plt

power_analysis = smp.TTestIndPower()
sample_size = power_analysis.solve_power(effect_size=0.5, power=0.8, alpha=0.05)
sample_size
63.765611775409525

Thus, about 64 participants would be needed in each group in order to test the hypothesis with adequate power.

Power curves

We can also create plots that can show us how the power to find an effect varies as a function of effect size and sample size, at the alpha specified in the power analysis. We will use the plot_power() function. The x-axis is defined by the dep_var argument, while sample sizes (nobs) and effect sizes (effect_size) are provided as arrays.

#+
effect_sizes = np.array([0.2, 0.5, 0.8])
sample_sizes = np.array(range(10, 500, 10))

plt.style.use('seaborn')
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
fig = power_analysis.plot_power(
    dep_var='nobs', nobs=sample_sizes,  
    effect_size=effect_sizes, alpha=0.05, ax=ax, 
    title='Power of Independent Samples t-test\n$\\alpha = 0.05$')

#-
_images/09-StatisticalPower_5_0.png

Simulating statistical power

We can also simulate data to see whether the power analysis actually gives the right answer. We will sample data for two groups, with a difference of 0.5 standard deviations between their underlying distributions and a sample size based on power analysis, and we will then look at how often we reject the null hypothesis.

#+
num_runs = 5000
effectSize = 0.5

# perform power analysis to get sample size
power_analysis = smp.TTestIndPower()
sampleSize = power_analysis.solve_power(
    effect_size=effectSize, power=0.8, alpha=0.05)

# round up from estimated sample size
sampleSize = np.int(np.ceil(sampleSize))

# create a function that will generate samples and test for
# a difference between groups using a two-sample t-test


def get_t_result(sampleSize, effectSize):
    """
    perform a ttest on random data of n=sampSize
    """
    
    group1 = np.random.normal(loc=0.0, scale=1.0, size=sampleSize)
    group2 = np.random.normal(loc=effectSize, scale=1.0, size=sampleSize)
    ttresult = scipy.stats.ttest_ind(group1, group2)
    return(ttresult.pvalue)


# create input data frame for output
power_sim_results = pd.DataFrame({'p_value': np.zeros(num_runs)})

for run in range(num_runs):
    power_sim_results.loc[run, 'p_value'] = get_t_result(sampleSize, effectSize)


p_reject = np.mean(power_sim_results['p_value'] < 0.05)
p_reject
#-
0.7994

This should return a number very close to 0.8.