Regression Analysis

Linear regression is a statistical method for modeling the relationship between one or more predictor variables (independent variables) and an outcome variable (dependent variable). It assumes a linear relationship where changes in the predictors are associated with proportional changes in the outcome. **Types of Linear Regression:** - **Simple Linear Regression**: Models the relationship between one X variable and Y using the equation Y = a + bX, where a is the intercept and b is the slope. - **Multiple Linear Regression**: Models multiple predictors: Y = b₀ + b₁X₁ + b₂X₂ + ... + bₙXₙ. Each coefficient represents the effect of that predictor while holding others constant. - **Polynomial Regression**: Extends simple regression to model curved relationships using powers of X: Y = b₀ + b₁X + b₂X² + b₃X³ + ... The model estimates coefficients by minimizing the sum of squared residuals (ordinary least squares method). The **R-squared value** measures how well the model explains variance in Y, ranging from 0 (no explanation) to 1 (perfect fit). Regression is widely used for prediction, trend analysis, and understanding variable relationships in fields like economics, biology, engineering, and social sciences.

This calculator supports three regression types with flexible data input options: **For Simple and Polynomial Regression:** 1. **Manual Data Entry**: Enter your X and Y data directly in the text boxes. Separate values with commas, spaces, or line breaks. This is the quickest method for small datasets. 2. **CSV Upload**: Upload a CSV file and select two columns (X and Y variables) from the sidebar. Useful for larger datasets. **For Multiple Regression:** - CSV upload is required. Select 2 or more columns from your CSV file. The last selected column becomes the outcome variable (Y), while preceding columns are predictor variables (X₁, X₂, etc.). **Configuration Options:** - **Regression Type**: Choose Simple (one predictor), Multiple (multiple predictors), or Polynomial (curved relationship). - **Confidence Level**: Select 90%, 95%, or 99% for coefficient confidence intervals. - **Polynomial Degree**: For polynomial regression, set the degree (2 for quadratic, 3 for cubic, etc.). Higher degrees fit more complex curves but risk overfitting. Click "Run Regression Analysis" to compute results. The calculator provides comprehensive output including regression equation, R², coefficient significance tests, and diagnostic plots to validate assumptions.

The calculator provides four tabs of detailed results: **Summary Tab:** - **R-squared (R²)**: Proportion of variance explained (0-1). Higher values indicate better fit. - **Adjusted R²**: R² adjusted for number of predictors. Preferred for multiple regression. - **F-statistic & p-value**: Tests overall model significance. p < 0.05 suggests the model is statistically significant. - **RMSE, MAE, MSE**: Error metrics measuring prediction accuracy. Lower values indicate better fit. **Coefficients Tab:** Shows each coefficient (intercept, slope, or multiple β values) with: - **Coefficient estimate**: The effect size. - **Standard error**: Uncertainty in the estimate. - **t-value**: Test statistic (coefficient / standard error). - **p-value**: Significance test. p < 0.05 suggests the coefficient is statistically significant. - **Confidence interval**: Range of plausible coefficient values. **Diagnostics Tab:** - Sample size and degrees of freedom - Residual standard error - Model quality metrics **Visualizations Tab:** - **Scatter plot**: Shows data points, regression line, and confidence band. - **Residual plot**: Check for patterns. Random scatter indicates good fit. - **Q-Q plot**: Tests normality assumption. Points should follow diagonal line. Use these results to interpret relationships, make predictions, and validate that regression assumptions are satisfied.

Linear regression is one of the most widely used statistical techniques across many fields: **1. Prediction and Forecasting:** - **Sales forecasting**: Predict future sales based on advertising spend, seasonality, and market trends. - **Stock price prediction**: Model stock prices using economic indicators and company financials. - **Demand forecasting**: Estimate product demand based on price, promotions, and competitor activity. **2. Trend Analysis:** - **Economic trends**: Analyze GDP growth, inflation rates, or unemployment trends over time. - **Climate studies**: Model temperature changes, sea level rise, or precipitation patterns. - **Population growth**: Project population trends based on historical data. **3. Research and Causality:** - **Medical research**: Study the effect of treatment dosage on patient outcomes while controlling for age, gender, and other factors. - **Education**: Analyze how study hours, teaching methods, and class size affect exam scores. - **Psychology**: Examine relationships between personality traits, behaviors, and mental health outcomes. **4. Quality Control:** - **Manufacturing**: Model product quality based on process parameters like temperature, pressure, and material composition. - **Engineering**: Analyze stress-strain relationships or material properties. **5. Business Analytics:** - **Pricing optimization**: Determine optimal pricing based on demand elasticity. - **Customer lifetime value**: Predict customer value based on demographics and behavior. - **Risk assessment**: Model credit risk, insurance claims, or default probability. Regression is particularly valuable because it provides both prediction capabilities and interpretable coefficients that quantify the strength and direction of relationships between variables.