An analyst has identified 3 independent variables (X₁, X₂, X₃) which might be used to predict Y. He has computed the regression equations using all combinations of the variables and the results are summarized in the following table. Why is the R² value for the X₃ model the same as the R² value for the X₁ and X₃ model, but the Adjusted R² values differ? $\begin{array}{lcccl}\text { Independent}\\\text { Variable in the}&\text { Adjusted}\\ \hline\text { Model } & \mathrm{R}^{2} & -\mathrm{R}^{2} & \mathrm{~S}_{\mathrm{e}} & \text { Parameter Estimates } \\\hline \mathrm{X}_{1} & 0.00089 & -0.124 & 23.548 & \mathrm{~b}_{0}=93.7174, \mathrm{~b}_{1}=0.922 \\\mathrm{X}_{2} & 0.3870 & 0.3104 & 18.448 & \mathrm{~b}_{0}=57.0803, \mathrm{~b}_{2}=1.545 \\\mathrm{X}_{1} \text { and } \mathrm{X}_{2} & 0.3910 & 0.2170 & 19.654 & \mathrm{~b}_{0}=50.2927, \mathrm{~b}_{1}=1.952, \mathrm{~b}_{2}=1.554 \\\mathrm{X}_{3} & 0.8413 & 0.8214 & 9.3858 & \mathrm{~b}_{0}=31.6238, \mathrm{~b}_{3}=1.132 \\\mathrm{X}_{1} \text { and } \mathrm{X}_{3} & 0.8413 & 0.7960 & 10.033 & \mathrm{~b}_{0}=31.133, \mathrm{~b}_{1}=0.148, \mathrm{~b}_{3}=1.132 \\\mathrm{X}_{2} \text { and } \mathrm{X}_{3} & 0.9863 & 0.9824 & 2.948 & \mathrm{~b}_{0}=14.169, \mathrm{~b}_{2}=0.985, \mathrm{~b}_{3}=0.995 \\\text { All three } & 0.9871 & 0.9807 & 3.085 & \mathrm{~b}_{0}=11.113, \mathrm{~b}_{1}=0.899, \mathrm{~b}_{2}=0.990 \\&&&&\mathrm{~b}_{3}=0.993\end{array}$

Question

An analyst has identified 3 independent variables (X₁, X₂, X₃) which might be used to predict Y. He has computed the regression equations using all combinations of the variables and the results are summarized in the following table. Why is the R² value for the X₃ model the same as the R² value for the X₁ and X₃ model, but the Adjusted R² values differ?

\begin{array}{lcccl}\text { Independent}\\\text { Variable in the}&\text { Adjusted}\\ \hline\text { Model } & \mathrm{R}^{2} & -\mathrm{R}^{2} & \mathrm{~S}_{\mathrm{e}} & \text { Parameter Estimates } \\\hline \mathrm{X}_{1} & 0.00089 & -0.124 & 23.548 & \mathrm{~b}_{0}=93.7174, \mathrm{~b}_{1}=0.922 \\\mathrm{X}_{2} & 0.3870 & 0.3104 & 18.448 & \mathrm{~b}_{0}=57.0803, \mathrm{~b}_{2}=1.545 \\\mathrm{X}_{1} \text { and } \mathrm{X}_{2} & 0.3910 & 0.2170 & 19.654 & \mathrm{~b}_{0}=50.2927, \mathrm{~b}_{1}=1.952, \mathrm{~b}_{2}=1.554 \\\mathrm{X}_{3} & 0.8413 & 0.8214 & 9.3858 & \mathrm{~b}_{0}=31.6238, \mathrm{~b}_{3}=1.132 \\\mathrm{X}_{1} \text { and } \mathrm{X}_{3} & 0.8413 & 0.7960 & 10.033 & \mathrm{~b}_{0}=31.133, \mathrm{~b}_{1}=0.148, \mathrm{~b}_{3}=1.132 \\\mathrm{X}_{2} \text { and } \mathrm{X}_{3} & 0.9863 & 0.9824 & 2.948 & \mathrm{~b}_{0}=14.169, \mathrm{~b}_{2}=0.985, \mathrm{~b}_{3}=0.995 \\\text { All three } & 0.9871 & 0.9807 & 3.085 & \mathrm{~b}_{0}=11.113, \mathrm{~b}_{1}=0.899, \mathrm{~b}_{2}=0.990 \\&&&&\mathrm{~b}_{3}=0.993\end{array}

Quizplus · Accepted Answer

The Adjusted R² accounts for the number of predictors in the model. Adding X₁ to the model with X₃ does not sufficiently reduce the error sum of squares (ESS) to justify its inclusion, hence the Adjusted R² decreases.

An Analyst Has Identified 3 Independent Variables (X1, X2, X3)

An Analyst Has Identified 3 Independent Variables (X₁, X₂, X₃)