Linear regression calculator

How to use it

Paste the predictor values (X) and the response values (Y) in the same order. Optionally enter an x at which to predict: you get the fitted value, a confidence interval for the mean response and a prediction interval for a single new observation.

Formulas

• Slope b₁ = S_xy / S_xx, intercept b₀ = ȳ − b₁x̄, with S_xx = Σ(x − x̄)², S_xy = Σ(x − x̄)(y − ȳ).
• Residual SE s = √(SSE / (n − 2)); SE(b₁) = s / √S_xx; SE(b₀) = s √(1/n + x̄²/S_xx); t = estimate / SE with n − 2 df.
• R² = 1 − SSE / SST; adjusted R² = 1 − (1 − R²)(n − 1)/(n − 2); F = MSR / MSE on 1 and n − 2 df.
• At x₀: mean response ŷ ± t s √(1/n + (x₀ − x̄)²/S_xx); prediction ŷ ± t s √(1 + 1/n + (x₀ − x̄)²/S_xx).
• Leverage h_i = 1/n + (x_i − x̄)²/S_xx; studentized residual e_i / (s √(1 − h_i)).

Assumptions and limits

• A straight-line relation, independent errors with constant variance; normal errors for the tests and intervals. Check the residual table for curvature, outliers (|studentized| > 2) and high leverage (> 4/n).
• Predictions outside the observed range of X are extrapolation and are flagged.
• One predictor only. Up to 100,000 pairs; the residual table shows the first 1,000.

Worked example

The default data are NIST’s Norris ozone-monitor calibration set (36 pairs). The fit is ŷ = −0.262323073774 + 1.00211681802 x, R² = 0.999993745884, residual SE 0.884796 on 34 df — the NIST certified values to at least 10 significant digits. At x = 500 the prediction interval is 498.97 to 502.62.

References

• NIST StRD linear regression: Norris dataset and certified values
• NIST/SEMATECH e-Handbook of Statistical Methods: least squares
• R documentation: lm (fitting linear models)
• R documentation: predict.lm (confidence and prediction intervals)