Regression through the origin Procedures for when you know the regression function must pass through the origin (0,0)

Examples Circumference of circle = π×diameter Man hours = β1×number of items processed Distance traveled = β1×speed Blood alcohol content = β1×number drinks

No intercept model where: β1 is unknown slope parameter Xi are known constants i are unknown, independent normally distributed error terms with mean 0 and variance σ2

Example: Circumference = β1×diameter Diam Circum 6.8 21.7 10.8 33.4 5.6 18.0 1.9 6.4 2.6 8.0 4.4 12.9 9.4 29.5 16.6 51.4

No intercept model in Minitab Stat >> Regression >> Regression … Specify response and predictor. Under Options…, remove the default check mark from the “Fit the intercept” box. Note: Stat >> Regression >> Fitted line plot does not handle regression through origin.

Diam Circum DxC D_sq The regression equation is Circum = 3.11 Diam Predictor Coef SE Coef T P Noconstant Diam 3.11170 0.02011 154.73 0.000 S = 0.4876 Diam Circum DxC D_sq 6.8 21.7 147.56 46.24 10.8 33.4 360.72 116.64 5.6 18.0 100.80 31.36 1.9 6.4 12.16 3.61 2.6 8.0 20.80 6.76 4.4 12.9 56.76 19.36 9.4 29.5 277.30 88.36 16.6 51.4 853.24 275.56 ------- ------ 1829.3 587.89

S = 0.4876 Diam Circum RESI1 RESI1_sq 6.8 21.7 0.540409 0.292042 10.8 33.4 -0.206409 0.042605 5.6 18.0 0.574454 0.329998 1.9 6.4 0.487761 0.237911 2.6 8.0 -0.090432 0.008178 4.4 12.9 -0.791500 0.626472 9.4 29.5 0.249977 0.062489 16.6 51.4 -0.254296 0.064666 --------- -------- 0.50996 1.6644 is unbiased estimator of σ2.

Summary of key points Residuals don’t necessarily sum to 0 for regression through the origin. Error degrees of freedom is n-1, not n-2, since estimating only one parameter. Formulas are different for no-intercept model.

Analysis of Variance Source DF SS MS F P Regression 1 5692.4 5692.4 23941.06 0.000 Error 7 1.7 0.2 Total 8 5694.0 Diam Circum RESI1 RESI1_sq Circ_sq FITS FITS_sq 6.8 21.7 0.540409 0.292042 470.89 21.1596 447.73 10.8 33.4 -0.206409 0.042605 1115.56 33.6064 1129.39 5.6 18.0 0.574454 0.329998 324.00 17.4255 303.65 1.9 6.4 0.487761 0.237911 40.96 5.9122 34.95 2.6 8.0 -0.090432 0.008178 64.00 8.0904 65.46 4.4 12.9 -0.791500 0.626472 166.41 13.6915 187.46 9.4 29.5 0.249977 0.062489 870.25 29.2500 855.56 16.6 51.4 -0.254296 0.064666 2641.96 51.6543 2668.17 -------- ------- ------- ------- 1.6644 5694.00 5692.4

Summary of key points n total degrees of freedom, n-1 error degrees of freedom Total sum of squares is “uncorrected for the mean.” It is just sum of squared observed responses. Regression sum of squares also uncorrected for the mean. Just sum of squared fitted responses. SSTOU = SSRU + SSE

The regression equation is Circum = 3.11 Diam Predictor Coef SE Coef T P Noconstant Diam 3.11170 0.02011 154.73 0.000 S = 0.4876 ???? Many software packages, Minitab included, do not display an R2 value for regression through the origin. This is because it is possible that it is negative when you force the regression line through the origin, and therefore has no meaningful interpretation here.

Predicted Values for New Observations Diam Fit SE Fit 95.0% CI 95.0% PI 7.00 21.782 0.141 (21.449,22.115) (20.581,22.983)