Nonlinearities in Intelligence

Most of our everyday measurements are linear measurements. A linear measurement is one in which a constant interval means the same thing at any point on the scale. For instance, adding one inch to a six-foot board produces the same change in length that adding one inch to a five-foot board does. We are so familiar with linear measurements that we often assume that the properties of linear measurements apply to any characteristic that is described by numbers. That is not so, and the erroneous assumption can be particularly confusing when we deal with intelligence.

In psychometric theories intelligence is calculated by determining a person's standard score on an IQ test. The standard score is the deviation of a person's absolute score of a test from the mean test score of a reference population, divided by the standard deviation (a measurement of the variability of scores in the reference population):

where x_i is the ith person's score in absolute units (usually the number of correct answers on a test) and ? and σ are, respectively, the population mean and standard deviation. If this equation were applied strictly, a person of exactly average intelligence would have a score of zero, and people with below-average intelligence would have negative scores. Since the ideas of zero and negative intelligence do not seem reasonable, it is conventional to report IQ scores by rescaling standard scores, using the equation

IQ = 15z + 100

This gives the person of average intelligence a score of 100. This equation is simply a scaling convention; the real definition is contained in the first equation, which makes the standard deviation the unit of scoring. Herrnstein and Murray refer to the standard deviation as "like an inch," but it is not. The standard deviation is determined not by the absolute values of the scores in a population, but rather by the extent to which one score is likely to be different from another. In addition, the zero point of the IQ scale (IQ = 100) is determined by the population mean, not by a definition of "average intelligence" in terms of intellectual performance. Therefore the IQ score of an individual is a relative score, compared to the mean and variability in the reference population, rather than an absolute measure of mental competence. If we measured height the way that we measured IQ, a six-foot, six-inch man would have a standard score of somewhat greater than 2, in the North American male population. The same person would have a standard score of about 0 if the reference population were professional basketball players.

The distinction between the relative and absolute definitions of intelligence becomes important when we consider the relation between IQ, defined by standard scores, and various dependent measures, such as school achievement and workplace performance. Suppose a psychometrician records the job performance and intelligence-test scores of a group of workers. The relationship would be expressed by this equation, where B is the regression coefficient, or the rate at which job performance changes as IQ changes:

job performance = average job performance + B * IQ

B is calculated to make predictions as accurate as they can be. The actual degree of accuracy is measured by the correlation coefficient , which varies from 0 (no accuracy at all) to 1 (perfect prediction). Determining the regression and correlation coefficients from a given set of data is straightforward. The problem comes when an extrapolation is made to new situations, where some data points lie outside the range of IQ units observed in the original study. An example might be extrapolating the grade-IQ relationship observed in high-school students to grade-IQ relations among college students. Such extrapolations implicitly assume that IQ scores are linear measures of the intellectual traits that they are supposed to measure. This is not true. Suppose that a person in his 20s suffered a brain injury or infection that reduced his IQ score by 20 points. (Such things are possible.) If he were a medical or law school student with an original IQ of 140, he would probably still complete his coursework, though perhaps with not quite so high a class rank as before. If the person were a blue-collar worker with an original IQ of 80 he would, at IQ 60, have a substantial risk of homelessness, poverty and a number of other serious social problems.

The issue of nonlinearity applies to the very definition of intelligence, and in particular to the question of whether there is one type of intelligence or several. Suppose that general intelligence is equally important at all levels of mental competence. In this case the results of a factor-analytic study of test scores, based on data from people with high levels of intelligence, should be similar to the results of a study based on data from people of lower absolute levels of intelligence. Historically there have been suggestions that this is not so. The general-intelligence model was first developed by Charles Spearman (1904, 1927), based on analysis of test results from English schoolchildren. In 1938 L. L. Thurstone challenged Spearman's conclusion because he found very little evidence for general intelligence in a sample of University of Chicago undergraduates. It was observed at the time that the discrepancy might have arisen because Spearman and Thurstone had taken data from people of widely different intellectual levels, which would be evidence that intelligence changes qualitatively as the level of mental competence changes. However, the results were not definitive because Spearman and Thurstone had used different tests.

An important study by Douglas Detterman and Mark Daniel (1989) showed that the relations between subtests do change as the level of scores changes. Among other things, Detterman and Daniel examined correlations between subtests of the WAIS and found higher correlations between subtest scores for people with below-average IQ than for people with above-average IQ. David Waller and Derek Chung and I found the same thing when we analyzed the ASVAB scores that Herrnstein and Murray used in The Bell Curve to determine the relation between IQ and various indicators of social adjustment. It appears that general intelligence may not be an accurate statement, but general lack of intelligence is!

The conclusion that the relation between different indices of mental competence depends on the general level of competence is not consistent with psychometric approaches, but it is consistent with the cognitive-psychology approach. Recall that the cognitive-psychology approach assumes that mental competence is produced by a cascade of progressively more refined abilities, moving from information processing to problem-solving techniques to knowledge possession. It follows that problems at the information-processing level will be general, whereas potentials established at higher levels will be specific. In fact, Detterman and Daniel did find that the relation between information-processing measures and intelligence-test performance is higher at low levels of intelligence. Similar observations have been made by scientists who have studied very high-level performance, in fields ranging from physics to literature. A certain amount of intelligence seems to be needed to gain entry to an intellectually demanding field, but beyond that point success is determined by the effort put into the job, social support, and just sheer experience. (See Ericsson, Krampe and Tesch-Romer (1993) on expertise, Simonton (1984) on creativity, and Gardner (1993) for some interesting biographical data.)

In economic terms it appears that the IQ score measures something with decreasing marginal value. It is important to have enough of it, but having lots and lots does not buy you that much. My regrets to Mensa, but that is the way things are. Nonlinearity becomes important when we ask a key question raised by Herrnstein and Murray: What is the relation between intelligence and workplace performance?