GIFSQ Tools · Module 16 — Statistics, Experimental Design & Research · Tool 312 of 500
Fit y = b₀ + b₁x₁ + b₂x₂ to two predictors, with R², adjusted R² and the overall F test — for responses that depend on two factors at once.
Try sample values — fills the form with the worked example below.
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Coefficient b₁
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Coefficient b₂
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R²
Equation: Coefficients solve the normal equations (X′X)b = X′y by Gaussian elimination. R² = SSR ÷ SST; adjusted R² = 1 − (1 − R²)(n − 1) ÷ (n − 3). F = (SSR ÷ 2) ÷ (SSE ÷ (n − 3)).
Method: Ordinary least-squares multiple regression with two predictors (standard textbook method; Draper and Smith, Applied Regression Analysis). The F p-value comes from the F distribution itself (incomplete beta function).
Assumptions: the response is linear in the predictors, errors are independent with constant variance and approximately normal; the predictors are not collinear.
Limitations: with few observations the fit is unstable — at least 5 observations per predictor is a common minimum. Correlated predictors make individual coefficients hard to interpret even when R² is high.
Rounding: coefficients to 4 decimal places; R² to 4 decimal places; F to 4 decimal places; intermediates unrounded.
This is a calculation aid, not a laboratory report — confirm against your accredited method before citing a value.
Method: Ordinary least-squares multiple regression with two predictors (standard textbook method; Draper and Smith, Applied Regression Analysis). The F p-value comes from the F distribution itself (incomplete beta function).
Assumptions: the response is linear in the predictors, errors are independent with constant variance and approximately normal; the predictors are not collinear.
Limitations: with few observations the fit is unstable — at least 5 observations per predictor is a common minimum. Correlated predictors make individual coefficients hard to interpret even when R² is high.
Rounding: coefficients to 4 decimal places; R² to 4 decimal places; F to 4 decimal places; intermediates unrounded.
This is a calculation aid, not a laboratory report — confirm against your accredited method before citing a value.
Worked example: x₁ 1–6, x₂ 2,1,4,3,6,5, y 3,5,6,8,11,12: b₀ = 0.9375, b₁ = 1.7708, b₂ = 0.1042, R² = 0.9824, F = 83.6538 (p 0.0023).
Enter values above to see results.
Check the predictors are not near-duplicates of each other before trusting the separate coefficients. Tool 312 of 500 — GIFSQ
