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type I error rates) in several different runs to see if this substantially affects the choice of the final model. EXAMPLE 12-14 Table 12-18 gives the Minitab stepwise regression output for the wine quality data. Minitab uses fixed values of for entering and removing variables. The default level is 0.15 for both decisions. The output in Table 12-18 uses the default value. Notice that the variables were entered in the order Flavor (step 1), Oakiness (step 2), and Aroma (step 3) and that no variables were removed. No other variable could be entered, so the algorithm terminated. This is the three-variable model found by all possible regressions that results in a minimum value of Cp. Forward Selection The forward selection procedure is a variation of stepwise regression and is based on the principle that regressors should be added to the model one at a time until there are no remaining candidate regressors that produce a significant increase in the regression sum of squares. That is, variables are added one at a time as long as their partial F-value exceeds fin. Forward selection is a simplification of stepwise regression that omits the partial F-test for deleting variables from the model that have been added at previous steps. This is a potential weakness of forward selection; that is, the procedure does not explore the effect that adding a regressor at the current step has on regressor variables added at earlier steps. Notice that if we were to apply forward selection to the wine quality data, we would obtain exactly the same results as we did with stepwise regression in Example 12-14, since stepwise regression terminated without deleting a variable.

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Table 12-19 Minitab Backward Elimination Output for the Wine Quality Data Stepwise Regression: Quality versus Clarity, Aroma, . . . Backward elimination. Alpha-to-Remove: 0.1 Response is Quality on 5 predictors, with N = 38 Step Constant Clarity T-Value P-Value Aroma T-Value P-Value Body T-Value P-Value Flavor T-Value P-Value Oakiness T-Value P-Value S R-Sq R-Sq(adj) C p 1 3.997 2.3 1.35 0.187 0.48 1.77 0.086 0.27 0.82 0.418 1.17 3.84 0.001 0.68 2.52 0.017 1.16 72.06 67.69 6.0 1.26 4.52 0.000 0.66 2.46 0.019 1.16 71.47 68.01 4.7 1.20 4.36 0.000 0.60 2.28 0.029 1.16 70.38 67.76 3.9 2 4.986 1.8 1.12 0.269 0.53 2.00 0.054 0.58 2.21 0.034 3 6.467

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6); that statement applies as well to Matrix Reasoning How well a person solves problems in an abstract setting versus a social setting is, therefore, of considerable clinical value regarding inferences about a person s personality, intellectual potential in quite different circumstances, and neuropsychological integrity The best validation of the Abstract Thought Social Comprehension differential comes from Browning and Quinlan s (1985) study of the Wechsler profiles of 91 adolescent and young adult psychiatric patients (45 females, 46 males) ranging in age from 13 to 30 years (mean = 19) These patients were administered Loevinger s Sentence Completion Test (SCT) for measuring ego development (Loevinger & Wessler, 1970), along with either the WISC-R (N = 16), WAIS (N = 46), or the WAIS-R (N = 29).

Backward Elimination The backward elimination algorithm begins with all K candidate regressors in the model. Then the regressor with the smallest partial F-statistic is deleted if this F-statistic is insignificant, that is, if f fout. Next, the model with K 1 regressors is fit, and the next regressor for potential elimination is found. The algorithm terminates when no further regressor can be deleted. Table 12-19 shows the Minitab output for backward elimination applied to the wine quality data. The value for removing a variable is 0.10. Notice that this procedure removes Body at step 1 and then Clarity at step 2, terminating with the three-variable model found previously. Some Comments on Final Model Selection We have illustrated several different approaches to the selection of variables in multiple linear regression. The final model obtained from any model-building procedure should be subjected to the usual adequacy checks, such as residual analysis, lack-of-fit testing, and examination of the effects of influential points. The analyst may also consider augmenting the original set of candidate variables with cross-products, polynomial terms, or other transformations of the original variables that might improve the model. A major criticism of variable selection methods such as stepwise regression is that the analyst may conclude there is one best regression equation. Generally, this is not the case, because several equally good regression models can

The SCT is meant to measure an aspect of personality functioning akin to social intelligence: The exercise of social judgment and the capacity for anticipation and planning may be compared to Loevinger s description of ego development as a.

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