Multiple linear regression calculator

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How to use Multiple Regression Calculator?

Usage:
1. Type Xi and Y values.
2. Click Calculate button
3. Results are generated automatically.



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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 YY and multiple predictor variables X1,X2,,XnX_1, X_2, \ldots, X_n. The model can be represented as:

Y=β0+β1X1+β2X2++βnXn+ϵY = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots + \beta_n X_n + \epsilon

where:

  • β0\beta_0 is the intercept.
  • βi\beta_i (for i=1,2,,ni = 1, 2, \ldots, n) are the coefficients of the predictor variables.
  • ϵ\epsilon is the error term.

The coefficients βi\beta_i 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 (R2R^2)

The R-squared value is a measure of how well the regression model explains the variability of the target variable. It is given by:

R2=1(YiY^i)2(YiYˉ)2R^2 = 1 - \frac{\sum (Y_i - \hat{Y}_i)^2}{\sum (Y_i - \bar{Y})^2}

where:

  • YiY_i are the observed values.
  • Y^i\hat{Y}_i are the predicted values from the model.
  • Yˉ\bar{Y} 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 XiX_i, the null hypothesis is that βi=0\beta_i = 0. 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):

Feature 1Feature 2Feature 3y35041301218369316511.5153436150111834331501216344914010.517\begin{array}{cccc} \text{Feature 1} & \text{Feature 2} & \text{Feature 3} & y \\ \hline 3504 & 130 & 12 & 18 \\ 3693 & 165 & 11.5 & 15 \\ 3436 & 150 & 11 & 18 \\ 3433 & 150 & 12 & 16 \\ 3449 & 140 & 10.5 & 17 \\ \end{array}

  • Features Matrix X X :

X=[350413012369316511.5343615011343315012344914010.5]X = \begin{bmatrix} 3504 & 130 & 12 \\ 3693 & 165 & 11.5 \\ 3436 & 150 & 11 \\ 3433 & 150 & 12 \\ 3449 & 140 & 10.5 \end{bmatrix}

  • Target Vector y y :

y=[1815181617]y = \begin{bmatrix} 18 \\ 15 \\ 18 \\ 16 \\ 17 \end{bmatrix}

2. Add the Intercept

To account for the intercept, add a column of ones to the features matrix X X :

X=[13504130121369316511.5134361501113433150121344914010.5]X' = \begin{bmatrix} 1 & 3504 & 130 & 12 \\ 1 & 3693 & 165 & 11.5 \\ 1 & 3436 & 150 & 11 \\ 1 & 3433 & 150 & 12 \\ 1 & 3449 & 140 & 10.5 \end{bmatrix}

3. Fit the Model

Fit the multiple linear regression model using the least squares method to estimate the coefficients:

β^=(XTX)1XTy\hat{\beta} = (X'^T X')^{-1} X'^T y

Where:

  • β^ \hat{\beta} are the estimated coefficients,
  • XT X'^T is the transpose of X X' ,
  • (XTX)1 (X'^T X')^{-1} is the inverse of XTX X'^T X' ,
  • y y is the target vector.

4. Calculate Coefficients and P-values

After fitting the model, you obtain:

  • Intercept (β0\beta_0): 40.214
  • Coefficient for Feature 1 (β1\beta_1): -0.0029711
  • Coefficient for Feature 2 (β2\beta_2): -0.063915
  • Coefficient for Feature 3 (β3\beta_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 R2R^2

The coefficient of determination R2R^2 measures the proportion of variance in the target variable that is explained by the features:

R2=1RSSTSSR^2 = 1 - \frac{\text{RSS}}{\text{TSS}}

Where:

  • RSS is the residual sum of squares,
  • TSS is the total sum of squares.

For this example, R2=0.69R^2 = 0.69, indicating that 69% of the variance in yy is explained by the model.

Conclusion

The model provides an R2 R^2 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.