Null Hypothesis for Linear Regression

Null hypothesis (\text{denoted as } H_0) is a starting assumption where there is no relationship between the independent variables and the dependent variable. In linear regression, it asserts that changes in predictor variables do not significantly influence the outcome variable.

Mathematically, the null hypothesis for each predictor in a linear regression model can be written as:

H_0: \beta_i = 0 \quad \text{for all} \quad i

Where:

• H_0 represents the null hypothesis.
• \beta_i is the coefficient for the i-th independent variable.

It suggests that any observed relationship in the sample data could be due to random chance rather than a true effect.

In this article, we will discuss the null hypothesis for linear regression, its importance, and how it is tested.

What is the Null Hypothesis in Linear Regression?

In the case of simple linear regression, the model typically follows this form:

Y = \beta_0 + \beta_1 X + \epsilon

Where:

• Y is the dependent variable.
• X is the independent variable.
• \beta_0 is the intercept.
• \beta_1​ is the slope (regression coefficient).
• \epsilon represents the error term or residuals.

The null hypothesis for linear regression involves testing whether the coefficient \beta_1 (the slope) is equal to zero. This indicates that the independent variable X has no effect on the dependent variable Y. Mathematically, the null hypothesis is expressed as:

H_0: \beta_1 = 0

The alternative hypothesis (denoted as H_A​) suggests that \beta_1 is not equal to zero, which means that there is a statistically significant relationship between X and Y:

H_A: \beta_1 \neq 0

Testing Process

To test the null hypothesis, the following steps are typically followed:

• Step 1: First, fit a linear regression model to the data, estimating the regression coefficients \beta_0​ and \beta_1​.
• Step 2: Once the model is fitted, the test statistic for the null hypothesis is calculated. This is usually done using the t-statistic for the slope coefficient \beta_1, which is computed as:

t = \frac{\hat{\beta_1}}{SE(\hat{\beta_1})}

Where:

• \hat{\beta_1}​ is the estimated value of the slope coefficient.
• SE(\hat{\beta_1}) is the standard error of the slope estimate.

• Step 3: The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, under the assumption that the null hypothesis is true. A small p-value (typically less than 0.05) indicates that the null hypothesis can be rejected in favor of the alternative hypothesis.
• Step 4: If the p-value is less than the chosen significance level (commonly \alpha = 0.05), you reject the null hypothesis. This means there is sufficient evidence to conclude that \beta_1 is significantly different from zero, and thus, the independent variable X has a significant relationship with the dependent variable Y.

Interpreting the Results

If the null hypothesis is not rejected, it means there is insufficient evidence to support the claim that the independent variable X significantly affects the dependent variable Y. This does not prove that there is no relationship; rather, it suggests that any observed relationship is likely due to chance or insufficient data.

On the other hand, if the null hypothesis is rejected, it suggests that there is a statistically significant relationship between X and Y, and the independent variable X can be considered a useful predictor for the dependent variable Y in the linear regression model.

Implementation: Hypothesis Testing for Linear Regression

We are implementing linear regression model in Python using statsmodels, which allows us to test the null hypothesis for each of the model’s coefficients. This code also computes the p-value for each coefficient, helping to assess the significance of the predictors.

We fit the ordinary least squares (OLS) model using statsmodels.api.OLS, which will provide the necessary statistics for hypothesis testing and results.summary() generates a comprehensive summary of model.

import statsmodels.api as sm
import numpy as np
import pandas as pd
from sklearn.datasets import make_regression

# Generate synthetic data
X, y = make_regression(n_samples=100, n_features=3, noise=0.1)
# Add a constant to the independent variables matrix 
X = sm.add_constant(X)

# Fit the model
model = sm.OLS(y, X)
results = model.fit()
print(results.summary())

Output:

The null hypothesis for each of these coefficients is that the coefficient is equal to zero, implying that there is no effect of the corresponding independent variable on the dependent variable y.

P-values for each coefficient:

• P>|t| for all coefficients (const, x1, x2, x3) is 0.000, which is significantly smaller than the common significance level of 0.05.
• Interpretation: For all the coefficients (except the intercept), the p-values are extremely low, which means that we reject the null hypothesis for each of the independent variables. This suggests that the independent variables x1, x2, and x3 are all statistically significant predictors of y.