Multiple Linear Regression

Under Construction

This calculator is still under development. Some of the functionality may not work as expected. Please use with caution and verify results with other tools.

Calculator

1. Load Your Data

2. Select Variables & Options

Dependent Variable (Y):

Independent Variables (X):

Exclude Outliers

Standardize Variables

Learn More

Multiple Linear Regression

Definition

Multiple Linear Regression models the relationship between a dependent variable and two or more independent variables, assuming a linear relationship. It extends simple linear regression to account for multiple predictors.

Model Equation

y = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_kx_k + \epsilon

Where:

$y$ = dependent variable
$x_i$ = independent variables
$\beta_0$ = intercept
$\beta_i$ = regression coefficients
$\epsilon$ = error term

Key Formulas:

Sum of Squares:

SST = \sum(y_i - \bar{y})^2

SSR = \sum(\hat{y}_i - \bar{y})^2

SSE = \sum(y_i - \hat{y}_i)^2

Where $\hat{y}_i$ is the predicted value and $\bar{y}$ is the mean

R-squared:

R^2 = \frac{SSR}{SST} = 1 - \frac{SSE}{SST}

Adjusted R-squared:

R^2_{adj} = 1 - (1-R^2)\frac{n-1}{n-k-1}

Key Assumptions

Linearity: Linear relationship between variables

Independence: Independent residuals

Homoscedasticity: Constant variance of residuals

Normality: Normal distribution of residuals

No Multicollinearity: Independent variables not highly correlated

Practical Example

Step 1: State the Data

Housing prices model:

House	Price (K)	Sqft	Age	Bedrooms
1	300	1500	15	3
2	250	1200	20	2
3	400	2000	10	4
4	550	2400	5	4
5	317	1600	12	3
6	389	1800	8	3

Step 2: Calculate Matrix Operations

Design matrix X:

\mathbf{X} = \begin{bmatrix} 1 & 1500 & 15 & 3 \\ 1 & 1200 & 20 & 2 \\ \vdots & \vdots & \vdots & \vdots \\ 1 & 1800 & 8 & 3 \end{bmatrix}

Coefficients calculation:

\hat{\boldsymbol{\beta}} = (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{y}

Step 3: Model Results

Fitted equation:

\hat{y} = -50.2 + 0.21x_{sqft} - 3.15x_{age} + 25.3x_{bedrooms}

R² = 0.892
Adjusted R² = 0.821
F-statistic = 13.78 (p-value = 0.014)

Step 4: Interpretation

For each additional square foot, price increases by $210
Each year of age decreases price by $3,150
Each additional bedroom adds $25,300 to price
Model explains 89.2% of price variation

Model Diagnostics

Key diagnostic measures:

VIF (Variance Inflation Factor):
$VIF_j = \frac{1}{1-R^2_j}$
Residual Standard Error:
$RSE = \sqrt{\frac{SSE}{n-k-1}}$

Code Examples

library(tidyverse)
library(broom)

# Example data
data <- tibble(
  price = c(300, 250, 400, 550, 317, 389),
  sqft = c(1500, 1200, 2000, 2400, 1600, 1800),
  age = c(15, 20, 10, 5, 12, 8),
  bedrooms = c(3, 2, 4, 4, 3, 3)
)

# Fit model
model <- lm(price ~ sqft + age + bedrooms, data = data)

# Model summary
tidy(model)      # Coefficients
glance(model)    # Model statistics

par(mfrow = c(2, 2))  # Arrange plots in a 2x2 grid
plot(model)

# Predictions
new_data <- tibble(
  sqft = 1800,
  age = 10,
  bedrooms = 3
)
pred = predict(model, new_data)
print(str_glue("Predicted price: {pred}"))

Python

import pandas as pd
import numpy as np
from statsmodels.formula.api import ols
import statsmodels.api as sm

# Example data
df = pd.DataFrame({
    'price': [300, 250, 400, 550, 317, 389],
    'sqft': [1500, 1200, 2000, 2400, 1600, 1800],
    'age': [15, 20, 10, 5, 12, 8],
    'bedrooms': [3, 2, 4, 4, 3, 3]
})

# Fit the model
model = ols('price ~ sqft + age + bedrooms', data=df).fit()

# Print summary
print(model.summary())

# For just coefficients and R-squared
print("Coefficients:")
print(model.params)
print("R-squared:", model.rsquared)

# Predictions
X_new = pd.DataFrame({
    'sqft': [1800],
    'age': [10],
    'bedrooms': [3]
})
predictions = model.predict(X_new)
print("Predicted price:", predictions[0])

Multiple Linear Regression

Under Construction

Calculator

1. Load Your Data

2. Select Variables & Options

Learn More

Multiple Linear Regression

Definition

Model Equation

Key Formulas:

Key Assumptions

Practical Example

Step 1: State the Data

Step 2: Calculate Matrix Operations

Step 3: Model Results

Step 4: Interpretation

Model Diagnostics

Code Examples

Alternative Methods

Related Calculators

Simple Linear Regression

Correlation Coefficient Calculator

Help us improve