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These models should now be evaluated further using residuals plots and the other techniques discussed earlier in the chapter, to see if either model is satisfactory with respect to the underlying assumptions and to determine if one of them is preferable. It turns out that the residual plots do not reveal any major problems with either model. The value of PRESS for the three-variable model is 56.0525 and for the four-variable model it is 60.3927. Since PRESS is smaller in the model with three regressors, and since it is the model with the fewest predictors, it would likely be the preferred choice. Stepwise Regression Stepwise regression is probably the most widely used variable selection technique. The procedure iteratively constructs a sequence of regression models by adding or removing variables at each step. The criterion for adding or removing a variable at any step is usually expressed in terms of a partial F-test. Let fin be the value of the F-random variable for adding a variable to the model, and let fout be the value of the F-random variable for removing a variable from the model. We must have fin fout, and usually fin fout. Stepwise regression begins by forming a one-variable model using the regressor variable that has the highest correlation with the response variable Y. This will also be the regressor producing the largest F-statistic. For example, suppose that at this step, x1 is selected. At the second step, the remaining K 1 candidate variables are examined, and the variable for which the partial F-statistic Fj SSR 1 MSE 1xj, x1 2

j0 1, 0 2

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the present generation, and in some countries these gains have been quite large. Japan joins six Western European countries (East Germany, Austria, West Germany, France, Netherlands, Belgium) in gaining more than 5 points per decade (more than a standard deviation per generation!). Of the 14 nations listed in the figure, only data from the following countries are status 1, or verified evidence of generational IQ gains: Netherlands (6.7 points), Belgium (5.8), Canada (4.6), Norway (3.2), and New Zealand (2.4). Consequently, all evidence of extreme gains in Figure 2.1 (half a standard deviation or more) is either tentative or speculative. As impressive as the three-point gain per decade for people in the United States has seemed to readers of Flynn s (1984) article, the United States has outgained only two of the nations studied by Flynn (1987) New Zealand and Great Britain and the latter country showed gains of 5.4 points per decade in a subsequent study of adults from 1942 to 1992 (Raven, Raven, & Court, 1993) that used the same test (Raven s Matrices) that was administered in Flynn s (1987) study. In addition, the United States has been outgained by four nations with verified data the Netherlands, Belgium, Norway (barely), and neighboring Canada. And, to add insult to injury, the U.S. gain of three points seems to have dwindled to 2.5 points in Flynn s (1998c) recent study of WISC-III and WAIS-III data from 1972 to 1995. Nonetheless, all comparisons among nations must remain highly speculative because the data presented in the figure are not directly comTABLE 2.4

3

(12-48)

parable. They have different statuses as scientific evidence, different tests were generally used in different countries, the samples were not comparable in age or background from one country to the next, and so forth. Gains on Verbal versus Nonverbal Tests As Flynn (1987) points out, gains on tests of fluid intelligence like Raven s Progressive Matrices have more theoretical meaning than gains on crystallized tests like Wechsler s Information subtest. Gains on the former imply true improvement in abstract problem-solving ability; gains on the latter may merely reflect greater mastery of the specific content of the items (i.e., improved achievement, not cognitive ability). As Flynn reminds us, [t]he average person today would outscore Aristotle or Archimedes on general information, but this hardly shows greater intelligence (p. 184). In fact, Flynn (1987) showed fairly consistently that gains were greater on fluid tests than on crystallized tests, indicating a true increase in problem-solving ability. For all nations studied by Flynn (1987), the median gain on the Raven and related tests was 5.9 points per decade, compared to 3.7 points per decade on verbal tests (either the Otis, tests of verbal and math ability, or a Binet-like test), as shown in Table 2.4. A value about midway between these extremes (5.2) was obtained by taking the median of the gains on tests with both verbal and nonverbal content

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