Multiple linear regression calculator
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Multiple Linear Regression Calculator
This document provides an overview of the multiple linear regression calculation process, including the calculation of coefficients, the R-squared value, and the statistical significance of each feature.
Theory and Equations
Multiple Linear Regression Model
In multiple linear regression, we aim to model the relationship between a target variable and multiple predictor variables . The model can be represented as:
where:
- is the intercept.
- (for ) are the coefficients of the predictor variables.
- is the error term.
The coefficients are estimated such that the difference between the observed values and the predicted values is minimized. The Ordinary Least Squares (OLS) method is typically used to find these estimates.
R-squared ()
The R-squared value is a measure of how well the regression model explains the variability of the target variable. It is given by:
where:
- are the observed values.
- are the predicted values from the model.
- is the mean of the observed values.
Statistical Significance (p-values)
To assess the statistical significance of each coefficient, we compute the p-values. The p-value for each feature indicates whether the corresponding coefficient is significantly different from zero. Lower p-values suggest that the feature has a significant impact on the target variable.
The p-value for a term is calculated using hypothesis testing. For each feature , the null hypothesis is that . The p-value helps in deciding whether to reject this null hypothesis.
Example
Let's walk through the steps to calculate the coefficients and p-values in a multiple linear regression model.
1. Organize the Data
Suppose we have the following dataset with three independent variables (features and one dependent variable (target):
- Features Matrix :
- Target Vector :
2. Add the Intercept
To account for the intercept, add a column of ones to the features matrix :
3. Fit the Model
Fit the multiple linear regression model using the least squares method to estimate the coefficients:
Where:
- are the estimated coefficients,
- is the transpose of ,
- is the inverse of ,
- is the target vector.
4. Calculate Coefficients and P-values
After fitting the model, you obtain:
- Intercept (): 40.214
- Coefficient for Feature 1 (): -0.0029711
- Coefficient for Feature 2 (): -0.063915
- Coefficient for Feature 3 (): -0.31671
P-values for each coefficient test the null hypothesis that the coefficient is zero:
- Intercept p-value: 0.35605
- Feature 1 p-value: 0.78453
- Feature 2 p-value: 0.52916
- Feature 3 p-value: 0.82945
5. Calculate
The coefficient of determination measures the proportion of variance in the target variable that is explained by the features:
Where:
- RSS is the residual sum of squares,
- TSS is the total sum of squares.
For this example, , indicating that 69% of the variance in is explained by the model.
Conclusion
The model provides an value of 0.9837, showing a good fit. However, most coefficients (including the intercept) have high p-values, suggesting that they are not statistically significant. This means the model explains the variance well, but the individual contributions of the features may not be substantial.