# output from stepwise backward regression is as expected Code ols_step_backward_p(model) Output Stepwise Summary ------------------------------------------------------------------------- Step Variable AIC SBC SBIC R2 Adj. R2 ------------------------------------------------------------------------- 0 Full Model 736.390 756.280 586.665 0.78184 0.74305 1 alc_mod 734.407 752.308 584.276 0.78177 0.74856 2 gender 732.494 748.406 581.938 0.78142 0.75351 3 age 730.620 744.543 579.638 0.78091 0.75808 ------------------------------------------------------------------------- Final Model Output ------------------ Model Summary ------------------------------------------------------------------- R 0.884 RMSE 184.276 R-Squared 0.781 MSE 38202.426 Adj. R-Squared 0.758 Coef. Var 27.839 Pred R-Squared 0.700 AIC 730.620 MAE 137.656 SBC 744.543 ------------------------------------------------------------------- RMSE: Root Mean Square Error MSE: Mean Square Error MAE: Mean Absolute Error AIC: Akaike Information Criteria SBC: Schwarz Bayesian Criteria ANOVA ----------------------------------------------------------------------- Sum of Squares DF Mean Square F Sig. ----------------------------------------------------------------------- Regression 6535804.090 5 1307160.818 34.217 0.0000 Residual 1833716.447 48 38202.426 Total 8369520.537 53 ----------------------------------------------------------------------- Parameter Estimates ------------------------------------------------------------------------------------------------ model Beta Std. Error Std. Beta t Sig lower upper ------------------------------------------------------------------------------------------------ (Intercept) -1178.330 208.682 -5.647 0.000 -1597.914 -758.746 bcs 59.864 23.060 0.241 2.596 0.012 13.498 106.230 pindex 8.924 1.808 0.380 4.935 0.000 5.288 12.559 enzyme_test 9.748 1.656 0.521 5.887 0.000 6.419 13.077 liver_test 58.064 40.144 0.156 1.446 0.155 -22.652 138.779 alc_heavy 317.848 71.634 0.314 4.437 0.000 173.818 461.878 ------------------------------------------------------------------------------------------------ --- Code ols_step_backward_p(model, progress = TRUE) Output Backward Elimination Method --------------------------- Candidate Terms: 1. bcs 2. pindex 3. enzyme_test 4. liver_test 5. age 6. gender 7. alc_mod 8. alc_heavy Variables Removed: => alc_mod => gender => age No more variables to be removed. Stepwise Summary ------------------------------------------------------------------------- Step Variable AIC SBC SBIC R2 Adj. R2 ------------------------------------------------------------------------- 0 Full Model 736.390 756.280 586.665 0.78184 0.74305 1 alc_mod 734.407 752.308 584.276 0.78177 0.74856 2 gender 732.494 748.406 581.938 0.78142 0.75351 3 age 730.620 744.543 579.638 0.78091 0.75808 ------------------------------------------------------------------------- Final Model Output ------------------ Model Summary ------------------------------------------------------------------- R 0.884 RMSE 184.276 R-Squared 0.781 MSE 38202.426 Adj. R-Squared 0.758 Coef. Var 27.839 Pred R-Squared 0.700 AIC 730.620 MAE 137.656 SBC 744.543 ------------------------------------------------------------------- RMSE: Root Mean Square Error MSE: Mean Square Error MAE: Mean Absolute Error AIC: Akaike Information Criteria SBC: Schwarz Bayesian Criteria ANOVA ----------------------------------------------------------------------- Sum of Squares DF Mean Square F Sig. ----------------------------------------------------------------------- Regression 6535804.090 5 1307160.818 34.217 0.0000 Residual 1833716.447 48 38202.426 Total 8369520.537 53 ----------------------------------------------------------------------- Parameter Estimates ------------------------------------------------------------------------------------------------ model Beta Std. Error Std. Beta t Sig lower upper ------------------------------------------------------------------------------------------------ (Intercept) -1178.330 208.682 -5.647 0.000 -1597.914 -758.746 bcs 59.864 23.060 0.241 2.596 0.012 13.498 106.230 pindex 8.924 1.808 0.380 4.935 0.000 5.288 12.559 enzyme_test 9.748 1.656 0.521 5.887 0.000 6.419 13.077 liver_test 58.064 40.144 0.156 1.446 0.155 -22.652 138.779 alc_heavy 317.848 71.634 0.314 4.437 0.000 173.818 461.878 ------------------------------------------------------------------------------------------------ --- Code ols_step_backward_p(model, details = TRUE) Output Backward Elimination Method --------------------------- Candidate Terms: 1. bcs 2. pindex 3. enzyme_test 4. liver_test 5. age 6. gender 7. alc_mod 8. alc_heavy Step => 0 Model => y ~ bcs + pindex + enzyme_test + liver_test + age + gender + alc_mod + alc_heavy R2 => 0.782 Initiating stepwise selection... Step => 1 Removed => alc_mod Model => y ~ bcs + pindex + enzyme_test + liver_test + age + gender + alc_heavy R2 => 0.78177 Step => 2 Removed => gender Model => y ~ bcs + pindex + enzyme_test + liver_test + age + alc_heavy R2 => 0.78142 Step => 3 Removed => age Model => y ~ bcs + pindex + enzyme_test + liver_test + alc_heavy R2 => 0.78091 No more variables to be removed. Variables Removed: => alc_mod => gender => age Stepwise Summary ------------------------------------------------------------------------- Step Variable AIC SBC SBIC R2 Adj. R2 ------------------------------------------------------------------------- 0 Full Model 736.390 756.280 586.665 0.78184 0.74305 1 alc_mod 734.407 752.308 584.276 0.78177 0.74856 2 gender 732.494 748.406 581.938 0.78142 0.75351 3 age 730.620 744.543 579.638 0.78091 0.75808 ------------------------------------------------------------------------- Final Model Output ------------------ Model Summary ------------------------------------------------------------------- R 0.884 RMSE 184.276 R-Squared 0.781 MSE 38202.426 Adj. R-Squared 0.758 Coef. Var 27.839 Pred R-Squared 0.700 AIC 730.620 MAE 137.656 SBC 744.543 ------------------------------------------------------------------- RMSE: Root Mean Square Error MSE: Mean Square Error MAE: Mean Absolute Error AIC: Akaike Information Criteria SBC: Schwarz Bayesian Criteria ANOVA ----------------------------------------------------------------------- Sum of Squares DF Mean Square F Sig. ----------------------------------------------------------------------- Regression 6535804.090 5 1307160.818 34.217 0.0000 Residual 1833716.447 48 38202.426 Total 8369520.537 53 ----------------------------------------------------------------------- Parameter Estimates ------------------------------------------------------------------------------------------------ model Beta Std. Error Std. Beta t Sig lower upper ------------------------------------------------------------------------------------------------ (Intercept) -1178.330 208.682 -5.647 0.000 -1597.914 -758.746 bcs 59.864 23.060 0.241 2.596 0.012 13.498 106.230 pindex 8.924 1.808 0.380 4.935 0.000 5.288 12.559 enzyme_test 9.748 1.656 0.521 5.887 0.000 6.419 13.077 liver_test 58.064 40.144 0.156 1.446 0.155 -22.652 138.779 alc_heavy 317.848 71.634 0.314 4.437 0.000 173.818 461.878 ------------------------------------------------------------------------------------------------ # output from stepwise backward hierarchical regression Code ols_step_backward_p(model, hierarchical = TRUE) Output Stepwise Summary ------------------------------------------------------------------------- Step Variable AIC SBC SBIC R2 Adj. R2 ------------------------------------------------------------------------- 0 Full Model 736.390 756.280 586.665 0.78184 0.74305 1 alc_mod 734.407 752.308 584.276 0.78177 0.74856 2 gender 732.494 748.406 581.938 0.78142 0.75351 3 age 730.620 744.543 579.638 0.78091 0.75808 ------------------------------------------------------------------------- Final Model Output ------------------ Model Summary ------------------------------------------------------------------- R 0.884 RMSE 184.276 R-Squared 0.781 MSE 38202.426 Adj. R-Squared 0.758 Coef. Var 27.839 Pred R-Squared 0.700 AIC 730.620 MAE 137.656 SBC 744.543 ------------------------------------------------------------------- RMSE: Root Mean Square Error MSE: Mean Square Error MAE: Mean Absolute Error AIC: Akaike Information Criteria SBC: Schwarz Bayesian Criteria ANOVA ----------------------------------------------------------------------- Sum of Squares DF Mean Square F Sig. ----------------------------------------------------------------------- Regression 6535804.090 5 1307160.818 34.217 0.0000 Residual 1833716.447 48 38202.426 Total 8369520.537 53 ----------------------------------------------------------------------- Parameter Estimates ------------------------------------------------------------------------------------------------ model Beta Std. Error Std. Beta t Sig lower upper ------------------------------------------------------------------------------------------------ (Intercept) -1178.330 208.682 -5.647 0.000 -1597.914 -758.746 bcs 59.864 23.060 0.241 2.596 0.012 13.498 106.230 alc_heavy 317.848 71.634 0.314 4.437 0.000 173.818 461.878 pindex 8.924 1.808 0.380 4.935 0.000 5.288 12.559 enzyme_test 9.748 1.656 0.521 5.887 0.000 6.419 13.077 liver_test 58.064 40.144 0.156 1.446 0.155 -22.652 138.779 ------------------------------------------------------------------------------------------------ --- Code ols_step_backward_p(model, hierarchical = TRUE, progress = TRUE) Output Backward Elimination Method --------------------------- Candidate Terms: 1. bcs 2. alc_heavy 3. pindex 4. enzyme_test 5. liver_test 6. age 7. gender 8. alc_mod Variables Removed: => alc_mod => gender => age No more variables to be removed. Stepwise Summary ------------------------------------------------------------------------- Step Variable AIC SBC SBIC R2 Adj. R2 ------------------------------------------------------------------------- 0 Full Model 736.390 756.280 586.665 0.78184 0.74305 1 alc_mod 734.407 752.308 584.276 0.78177 0.74856 2 gender 732.494 748.406 581.938 0.78142 0.75351 3 age 730.620 744.543 579.638 0.78091 0.75808 ------------------------------------------------------------------------- Final Model Output ------------------ Model Summary ------------------------------------------------------------------- R 0.884 RMSE 184.276 R-Squared 0.781 MSE 38202.426 Adj. R-Squared 0.758 Coef. Var 27.839 Pred R-Squared 0.700 AIC 730.620 MAE 137.656 SBC 744.543 ------------------------------------------------------------------- RMSE: Root Mean Square Error MSE: Mean Square Error MAE: Mean Absolute Error AIC: Akaike Information Criteria SBC: Schwarz Bayesian Criteria ANOVA ----------------------------------------------------------------------- Sum of Squares DF Mean Square F Sig. ----------------------------------------------------------------------- Regression 6535804.090 5 1307160.818 34.217 0.0000 Residual 1833716.447 48 38202.426 Total 8369520.537 53 ----------------------------------------------------------------------- Parameter Estimates ------------------------------------------------------------------------------------------------ model Beta Std. Error Std. Beta t Sig lower upper ------------------------------------------------------------------------------------------------ (Intercept) -1178.330 208.682 -5.647 0.000 -1597.914 -758.746 bcs 59.864 23.060 0.241 2.596 0.012 13.498 106.230 alc_heavy 317.848 71.634 0.314 4.437 0.000 173.818 461.878 pindex 8.924 1.808 0.380 4.935 0.000 5.288 12.559 enzyme_test 9.748 1.656 0.521 5.887 0.000 6.419 13.077 liver_test 58.064 40.144 0.156 1.446 0.155 -22.652 138.779 ------------------------------------------------------------------------------------------------ --- Code ols_step_backward_p(model, hierarchical = TRUE, details = TRUE) Output Backward Elimination Method --------------------------- Candidate Terms: 1. bcs 2. alc_heavy 3. pindex 4. enzyme_test 5. liver_test 6. age 7. gender 8. alc_mod Step => 0 Model => y ~ 1 R2 => 0.782 Initiating stepwise selection... Significance Table ----------------------- Predictor Pr(>|t|) ----------------------- bcs 0.01426 alc_heavy 0.00047 pindex 2e-05 enzyme_test 0.00000 liver_test 0.26811 age 0.72123 gender 0.78727 alc_mod 0.90601 ----------------------- Step => 1 Removed => alc_mod Model => y ~ bcs + alc_heavy + pindex + enzyme_test + liver_test + age + gender R2 => 0.78177 Significance Table ----------------------- Predictor Pr(>|t|) ----------------------- bcs 0.01331 alc_heavy 9e-05 pindex 1e-05 enzyme_test 0.00000 liver_test 0.25860 age 0.72755 gender 0.78613 ----------------------- Step => 2 Removed => gender Model => y ~ bcs + alc_heavy + pindex + enzyme_test + liver_test + age R2 => 0.78142 Significance Table ----------------------- Predictor Pr(>|t|) ----------------------- bcs 0.01285 alc_heavy 8e-05 pindex 1e-05 enzyme_test 0.00000 liver_test 0.20745 age 0.74164 ----------------------- Step => 3 Removed => age Model => y ~ bcs + alc_heavy + pindex + enzyme_test + liver_test R2 => 0.78091 Significance Table ----------------------- Predictor Pr(>|t|) ----------------------- bcs 0.01248 alc_heavy 5e-05 pindex 1e-05 enzyme_test 0.00000 liver_test 0.15457 ----------------------- No more variables to be removed. Variables Removed: => alc_mod => gender => age Stepwise Summary ------------------------------------------------------------------------- Step Variable AIC SBC SBIC R2 Adj. R2 ------------------------------------------------------------------------- 0 Full Model 736.390 756.280 586.665 0.78184 0.74305 1 alc_mod 734.407 752.308 584.276 0.78177 0.74856 2 gender 732.494 748.406 581.938 0.78142 0.75351 3 age 730.620 744.543 579.638 0.78091 0.75808 ------------------------------------------------------------------------- Final Model Output ------------------ Model Summary ------------------------------------------------------------------- R 0.884 RMSE 184.276 R-Squared 0.781 MSE 38202.426 Adj. R-Squared 0.758 Coef. Var 27.839 Pred R-Squared 0.700 AIC 730.620 MAE 137.656 SBC 744.543 ------------------------------------------------------------------- RMSE: Root Mean Square Error MSE: Mean Square Error MAE: Mean Absolute Error AIC: Akaike Information Criteria SBC: Schwarz Bayesian Criteria ANOVA ----------------------------------------------------------------------- Sum of Squares DF Mean Square F Sig. ----------------------------------------------------------------------- Regression 6535804.090 5 1307160.818 34.217 0.0000 Residual 1833716.447 48 38202.426 Total 8369520.537 53 ----------------------------------------------------------------------- Parameter Estimates ------------------------------------------------------------------------------------------------ model Beta Std. Error Std. Beta t Sig lower upper ------------------------------------------------------------------------------------------------ (Intercept) -1178.330 208.682 -5.647 0.000 -1597.914 -758.746 bcs 59.864 23.060 0.241 2.596 0.012 13.498 106.230 alc_heavy 317.848 71.634 0.314 4.437 0.000 173.818 461.878 pindex 8.924 1.808 0.380 4.935 0.000 5.288 12.559 enzyme_test 9.748 1.656 0.521 5.887 0.000 6.419 13.077 liver_test 58.064 40.144 0.156 1.446 0.155 -22.652 138.779 ------------------------------------------------------------------------------------------------