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Because SSE 1n p2 is the error or residual mean square and SST 1n 12 is a constant, R2 will adj only increase when a variable is added to the model if the new variable reduces the error mean square. Note that for the multiple regression model for the pull strength data R2 0.979 (see the adj Minitab output in Table 12-4), whereas in Example 11-8 the adjusted R2 for the one-variable model is R2 0.962. Therefore, we would conclude that adding x2 die height to the model adj does result in a meaningful reduction in unexplained variability in the response.

NOTE: Adapted from Table 7.4 in Matarazzo (1972, p. 186), based on work by H. C. Lehman (1953, 1954).

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The adjusted R2 statistic essentially penalizes the analyst for adding terms to the model. It is an easy way to guard against overfitting, that is, including regressors that are not really useful. Consequently, it is very useful in comparing and evaluating competing regression models. We will use R2 for this when we discuss variable selection in regression in Section 12-6.3. adj

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The preceding sections have argued for the hypothesis of an age-related decline in fluid ability that is not due solely to a decline in speed. Whereas the evidence for that contention is quite strong, it may be more sensible to try to integrate the Botwinick and Horn explanations for the classic aging pattern rather than trying to support one and deny the other. Indeed, the close relationship between perceptual speed and cognitive decline is congruent with the finding that sensorimotor variables are extremely predictive of intellectual functioning in old age (70 103 years) (Lindenberger & Baltes, 1997). Sensory Acuity Furthermore, sensory variables (with no motor involvement) may be related to declines on all three Horn factors (Gf, Gv, and Gs) in view of compelling evidence of substantial correlations between measures of sensory processes and measures of intellectual processes (Baltes & Lindenberger, 1997; Lindenberger & Baltes, 1994, 1997). In one large-scale study, both visual and auditory acuity were substantial correlates of IQ in old age (70 103) much more so than during the earlier adult years (25 69), with fluid abilities correlating substantially with sensory functioning in both age groups (Baltes & Lindenberger,

We are frequently interested in testing hypotheses on the individual regression coefficients. Such tests would be useful in determining the potential value of each of the regressor variables in the regression model. For example, the model might be more effective with the inclusion of additional variables or perhaps with the deletion of one or more of the regressors presently in the model. Adding a variable to a regression model always causes the sum of squares for regression to increase and the error sum of squares to decrease (this is why R2 always increases when a variable is added). We must decide whether the increase in the regression sum of squares is large enough to justify using the additional variable in the model. Furthermore, adding an unimportant variable to the model can actually increase the error mean square, indicating that adding such a variable has actually made the model a poorer fit to the data (this is why R2 is adj a better measure of global model fit then the ordinary R2). The hypotheses for testing the significance of any individual regression coefficient, say , are j

H0: H1:

1997). These researchers have found that, within very old samples, more than 90% of the agerelated variance in IQ can be accounted for by differences among the elderly individuals in their sensory acuity. Although Baltes and Lindenberger found higher correlations for older than younger adult cohorts, quite substantial correlations between sensory and intellectual variables have been reported for younger adult samples as well by Salthouse and his colleagues (Salthouse, Hambrick, & McGuthrey, 1998; Salthouse, Hancock, Meinz, & Hambrick, 1996). All Wechsler Performance subtests are heavily dependent on visual acuity for interpreting the pictures and designs, and on auditory acuity for understanding the examiner s instructions. These sensory variables have generally been overlooked when trying to account for declines in Performance IQ across the entire adult life span. It is inconceivable that the dramatic declines on Gf, Gv, and Gs tasks across the life span are actually caused by declines in vision and hearing, regardless of how high the correlations are between the two. It is quite possible, however, that there is a common cause for the agerelated declines in intellectual and sensory function (Lindenberger & Baltes, 1994; Salthouse et al., 1996), a hypothesis that requires empirical research to explore what the common factor actually represents (Backman et al., 2000, p. 505).

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0 0 (12-23)

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