Power of a test

Foundations of Inference in Python

Paul Savala

Assistant Professor of Mathematics

What determines power?

Sample size	Effect size	Alpha

Simulating weight loss data

from scipy.stats import norm


# Control group: Mean 0 pounds, std 1 pound

control = norm.rvs(loc=0,

                   scale=1,

                   size=100)


# Treatment group: Mean -2 pounds, std 1 pound

treatment = norm.rvs(loc=-2, 
                     scale=1, 
                     size=100)

T-test

$H_0$: No difference in weight loss (wrong)

$H_a$: Treatment group loses weight (correct)

Conclusion: Correctly reject $H_0$ in favor of $H_a$.

from scipy.stats import ttest_ind


# Conduct a t-test
alpha = 0.05
t_test = ttest_ind(treatment, control, 
                   alternative='less')


# Check the significance
print(t_test.pvalue < alpha)

TRUE

Small sample size

control = norm.rvs(loc=0, scale=1, size=5)
treatment = norm.rvs(loc=-2, scale=1, size=5)


# Conduct a t-test
tt = ttest_ind(treatment, control, 
               alternative='less')


print(tt.pvalue < 0.05)

FALSE

Conclusion: Fail to reject H0 (wrong)

Small effect size

# Weight loss = 0.2 pounds, sample size = 100

treatment = norm.rvs(loc=-0.2, scale=1, size=100)


# Conduct a t-test
t_test = ttest_ind(treatment, control, 
                   alternative='less')


print(t_test.pvalue < 0.05)

FALSE

Small effect? Big sample!

A small group of people standing next to a large group of people.

Defining the power of a test

If there is a significant effect, will our test detect it?

A magnifying glass showing germs on a cartoon hand.

Power of a test: If the alternative hypothesis (Ha) is true, how likely is our test to reject the null (H0) given the data we've supplied it?

Calculate power before collecting a sample

Calculating power

from statsmodels.stats import power
# Power function
tt_power = power.TTestIndPower()


# Calculate power
pwr = tt_power.power(effect_size=0.2,

                     nobs1=100,

                     alpha=0.05)

print(pwr)

0.291

Low chance of detection!

Solving for power

nobs1 = TTestIndPower().solve_power(effect_size=-0.2, 
                                    nobs1=None, # Solve for
                                    alpha=0.05,
                                    power=0.8)


print(nobs1)

13735.26

Let's practice!

Foundations of Inference in Python