F-Test in Statistics

The F-test is a statistical hypothesis testing used to compare the variances of two independent samples. It helps determine whether the variability in two populations is significantly different. The F-test is widely used in hypothesis testing, ANOVA (Analysis of Variance) and statistical model comparison in data science and analytics.

To properly understand the F-test, it is important to first understand the F-distribution, since the test statistic follows this distribution.

F-distribution

The F-distribution is a continuous probability distribution that arises as the ratio of two independent chi-square distributed random variables divided by their respective degrees of freedom.

It is defined by two parameters:

• df₁: degrees of freedom of the numerator
• df₂: degrees of freedom of the denominator

Formula:

F = \frac{(X_1 / df_1)}{(X_2 / df_2)}

where:

• X_1​ and X_2​ follow chi-square distributions
• df_1​ and df₂​ are their respective degrees of freedom

The F-statistic is always greater than or equal to 0 because it is a ratio of variances, and variance cannot be negative.

How the F-Test Works

The F-test compares two variances by forming their ratio. Depending on the research question, the test can be:

• Left-tailed
• Right-tailed
• Two-tailed

The F-test is applicable when:

• The populations are normally distributed
• Samples are random and independent

Hypothesis Testing Framework for F-test

For various hypothesis tests the F test formula is provided as follows:

1. Left Tailed Test

• Null Hypothesis: H_0 : \sigma_{1}^2 = \sigma_{2}^2
• Alternate Hypothesis: H_1:\sigma_{1}^2 < \sigma_{2}^2
• Decision-Making Standard: Reject H_0​ if the F statistic is less than the critical value.

2. Right Tailed Test

• Null Hypothesis: H_0:\sigma_{1}^2 = \sigma_{2}^2
• Alternate Hypothesis: H_1: \sigma_{1}^2 > \sigma_{2}^2
• Decision-Making Standard: Reject H₀​ if the F statistic is greater than the critical value.

3. Two Tailed Test

• Null Hypothesis: H_0 : \sigma_{1}^2 = \sigma_{2}^2
• Alternate Hypothesis: H_1: \sigma_{1}^2 \neq \sigma_{2}^2
• Decision-Making Standard: Reject H₀​ if the F statistic falls in the rejection region.

F Test Statistics

The F test statistic or simply the F statistic is a value that is compared with the critical value to check if the null hypothesis should be rejected or not. The F test statistic formula is given below:

For large samples: F_{calc}=\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}

For small samples: F_{calc}=\frac{s_{1}^{2}}{s_{2}^{2}}

where:

• \sigma_{1}^{2}  is the variance of the first population and \sigma_{2}^{2} is the variance of the second population.
• s_{1}^{2} is the variance of the first sample and s_{2}^{2} is the variance of the second sample.

Steps to Perform an F-Test

1. Compute variances of both samples
2. Define null and alternative hypotheses
3. Calculate the F statistic
4. Determine degrees of freedom
5. Find the critical F value using significance level α
6. Compare F statistic with critical value

Decision Rule

• If F_{calc} < F_{table} ​: Do not reject H₀
• If F_{calc} > F_{table} ​: Reject H_0​

Example

Consider the following example In this we conduct a two-tailed F-Test on the following samples: 

Step 1: Hypotheses

• H_0: \sigma_1^2 = \sigma_2^2
• H_1: \sigma_1^2 \neq \sigma_2^2

Step 2: Compute Variances

• s_1^2 = (10.47)^2 = 109.63
• s_2^2 = (8.12)^2 = 65.99

Step 3: Degrees of Freedom

• df_1 = 41 - 1 = 40
• df_2 = 21 - 1 = 20

Step 4: Critical Value

• Two-tailed test with α = 0.05:
• \alpha/2 = 0.025
• From the F-table: F_{table} = 2.287

Step 5: Decision

• 1.66 < 2.287
• Do not reject the null hypothesis. The variances of the two populations are statistically similar.

Python Implementation of F-Test

• Two samples are generated assuming normal distributions.
• Sample variances are computed using unbiased estimation (ddof=1).
• The F-statistic is calculated as the ratio of variances.
• Degrees of freedom are derived from sample sizes.
• The p-value is obtained using the F-distribution.
• Results are printed in a readable format.

import numpy as np
from scipy.stats import f

sample1 = np.random.normal(0, 10.47, 41)
sample2 = np.random.normal(0, 8.12, 21)

var1 = np.var(sample1, ddof=1)
var2 = np.var(sample2, ddof=1)

f_stat = var1 / var2

df1 = len(sample1) - 1
df2 = len(sample2) - 1

p_value = 2 * min(f.cdf(f_stat, df1, df2), 1 - f.cdf(f_stat, df1, df2))

print(f"F-statistic: {f_stat:.4f}")
print(f"P-value: {p_value:.4f}")

Output:

F-statistic: 1.2978
P-value: 0.2697

Interpretation: Since the p-value > 0.05, we fail to reject the null hypothesis, indicating that the variances are statistically similar.