Regression Coefficient Calculator
Compute regression coefficient using validated scientific equations. See step-by-step derivations, unit analysis, and reference values.
Formula
b1 = Sxy / Sxx, b0 = mean(Y) - b1 * mean(X)
Where b1 is the slope, b0 is the intercept, Sxy = Sum((Xi - meanX)(Yi - meanY)) is the sum of cross-deviations, and Sxx = Sum((Xi - meanX)^2) is the sum of squared deviations of X. The predicted value is Y-hat = b0 + b1*X. R-squared = (Sxy)^2 / (Sxx * Syy) measures the proportion of variance explained.
Worked Examples
Example 1: Drug Dose-Response Relationship
Problem: A study measures drug concentration (X: 0, 5, 10, 15, 20 mg/L) and cell viability (Y: 98, 85, 72, 55, 40 %). Find the regression equation.
Solution: n=5, Mean X=10, Mean Y=70\nSxx = 250, Sxy = -725, Syy = 2210\nb1 = -725/250 = -2.90\nb0 = 70 - (-2.90)(10) = 99.0\nEquation: Y = 99.0 - 2.90X\nR-squared = (-725)^2 / (250 * 2210) = 0.951
Result: Y = 99.0 - 2.90X | Each mg/L increase reduces viability by 2.9% | R-squared = 0.951
Example 2: Body Weight and Metabolic Rate
Problem: Predict metabolic rate from body weight. Data: Weight (kg: 50, 60, 70, 80, 90, 100) and Rate (kcal/day: 1250, 1420, 1580, 1700, 1850, 2000).
Solution: n=6, Mean X=75, Mean Y=1633.3\nSxx = 1750, Sxy = 26250, Syy = 399333.3\nb1 = 26250/1750 = 15.0\nb0 = 1633.3 - 15.0(75) = 508.3\nEquation: Y = 508.3 + 15.0X\nR-squared = (26250)^2 / (1750 * 399333.3) = 0.986
Result: Y = 508.3 + 15.0X | Each kg adds 15 kcal/day to metabolic rate | R-squared = 0.986
Frequently Asked Questions
What is linear regression and what are regression coefficients?
Linear regression models the relationship between a dependent variable (Y) and one or more independent variables (X) by fitting a straight line: Y = b0 + b1*X. The regression coefficients are b0 (intercept, the predicted Y when X=0) and b1 (slope, the change in Y for each unit increase in X). In biostatistics, regression is used to model dose-response relationships, predict outcomes from biomarkers, and adjust for confounding variables. The coefficients are estimated using the ordinary least squares method, which minimizes the sum of squared residuals between observed and predicted values.
How do I interpret the slope coefficient (b1)?
The slope b1 represents the average change in Y for every one-unit increase in X, holding all else constant. If b1 = 2.05, then for each unit increase in X, Y increases by 2.05 units on average. A positive slope indicates a positive relationship, negative indicates inverse. The p-value for the slope tests whether b1 is significantly different from zero. In biological contexts, for example, if X is drug dosage (mg) and Y is blood pressure reduction (mmHg), b1 = -0.5 means each additional mg reduces blood pressure by 0.5 mmHg on average.
What does R-squared tell me about my regression model?
R-squared (coefficient of determination) measures the proportion of variance in Y explained by the regression model. R-squared = 0.85 means 85% of the variability in Y is accounted for by the linear relationship with X. Adjusted R-squared corrects for sample size and is more appropriate for comparing models. However, a high R-squared does not mean the model is correct. Always examine residual plots for patterns that suggest non-linearity. In biological data, R-squared values of 0.7+ are generally considered good, but this varies by field.
What are the assumptions of linear regression?
Linear regression assumes: (1) Linearity - the relationship between X and Y is linear. (2) Independence - observations are independent. (3) Homoscedasticity - the variance of residuals is constant across all X values. (4) Normality - residuals are approximately normally distributed. (5) No significant outliers or influential points. Violations can lead to biased estimates, incorrect standard errors, and invalid p-values. Check these by examining residual plots: residuals vs fitted values for linearity and homoscedasticity, Q-Q plots for normality, and Cook's distance for influential points.
When should I use simple vs multiple regression?
Simple linear regression uses one predictor variable and is appropriate when you want to understand the relationship between two variables in isolation. Multiple regression uses two or more predictors and is necessary when multiple factors influence the outcome or when you need to control for confounders. In biostatistics, multiple regression is almost always preferred because biological outcomes rarely depend on a single factor. For example, predicting patient recovery time might require age, severity, treatment type, and comorbidities. Start with simple regression for exploratory analysis, then build multiple regression models for more accurate predictions.
What is regression analysis and when should I use it?
Regression models the relationship between a dependent variable and one or more independent variables. Linear regression fits a straight line (y = mx + b). Use it to predict outcomes, identify which variables matter most, and quantify relationships. R-squared tells you what percentage of variation is explained by the model.