Editing G factor (psychometrics) (section)

==Factor structure of cognitive abilities==
[[Image:SpearmanFactors.svg|right|thumb|200px|An illustration of Spearman's two-factor intelligence theory. Each small oval is a hypothetical mental test. The blue areas correspond to test-specific variance (''s''), while the purple areas represent the variance attributed to ''g''.]]

[[Factor analysis]] is a family of mathematical techniques that can be used to represent correlations between intelligence tests in terms of a smaller number of variables known as factors. The purpose is to simplify the correlation matrix by using hypothetical underlying factors to explain the patterns in it. When all correlations in a matrix are positive, as they are in the case of IQ, factor analysis will yield a general factor common to all tests. The general factor of IQ tests is referred to as the ''g'' factor, and it typically accounts for 40 to 50 percent of the variance in IQ test batteries.<ref>Mackintosh 2011, 44–45</ref> The presence of correlations between many widely varying cognitive tests has often been taken as evidence for the existence of ''g'', but McFarland (2012) showed that such correlations do not provide any more or less support for the existence of ''g'' than for the existence of multiple factors of intelligence.<ref>{{Cite journal |last=McFarland |first=Dennis J. |date=2012 |title=A single g factor is not necessary to simulate positive correlations between cognitive tests |journal=Journal of Clinical and Experimental Neuropsychology |volume=34 |issue=4 |pages=378–384 |doi=10.1080/13803395.2011.645018 |issn=1744-411X |pmid=22260190|s2cid=4694545 |quote=The fact that diverse cognitive tests tend to be positively correlated has been taken as evidence for a single general ability or "g" factor...the presence of a positive manifold in the correlations between diverse cognitive tests does not provide differential support for either single factor or multiple factor models of general abilities.}}</ref>

Charles Spearman developed factor analysis in order to study correlations between tests. Initially, he developed a model of intelligence in which variations in all intelligence test scores are explained by only two kinds of variables: first, factors that are specific to each test (denoted ''s''); and second, a ''g'' factor that accounts for the positive correlations across tests. This is known as Spearman's two-factor theory. Later research based on more diverse test batteries than those used by Spearman demonstrated that ''g'' alone could not account for all correlations between tests. Specifically, it was found that even after controlling for ''g'', some tests were still correlated with each other. This led to the postulation of ''group factors'' that represent variance that groups of tests with similar task demands (e.g., verbal, spatial, or numerical) have in common in addition to the shared ''g'' variance.<ref>Jensen 1998, 18, 31–32</ref>

[[Image:Carroll three stratum model of human Intelligence.png|right|thumb|500px|An illustration of [[John B. Carroll]]'s [[three stratum theory]], an influential contemporary model of cognitive abilities. The broad abilities recognized by the model are fluid intelligence (Gf), crystallized intelligence (Gc), general memory and learning (Gy), broad visual perception (Gv), broad auditory perception (Gu), broad retrieval ability (Gr), broad cognitive speediness (Gs), and processing speed (Gt). Carroll regarded the broad abilities as different "flavors" of ''g''.]]

Through [[Factor analysis#Rotation methods|factor rotation]], it is, in principle, possible to produce an infinite number of different factor solutions that are mathematically equivalent in their ability to account for the intercorrelations among cognitive tests. These include solutions that do not contain a ''g'' factor. Thus factor analysis alone cannot establish what the underlying structure of intelligence is. In choosing between different factor solutions, researchers have to examine the results of factor analysis together with other information about the structure of cognitive abilities.<ref name="carroll1995">Carroll 1995</ref>

There are many psychologically relevant reasons for preferring factor solutions that contain a ''g'' factor. These include the existence of the positive manifold, the fact that certain kinds of tests (generally the more complex ones) have consistently larger ''g'' loadings, the substantial invariance of ''g'' factors across different test batteries, the impossibility of constructing test batteries that do not yield a ''g'' factor, and the widespread practical validity of ''g'' as a predictor of individual outcomes. The ''g'' factor, together with group factors, best represents the empirically established fact that, on average, overall ability differences ''between'' individuals are greater than differences among abilities ''within'' individuals, while a factor solution with orthogonal factors without ''g'' obscures this fact. Moreover, ''g'' appears to be the most heritable component of intelligence.<ref name="jensen1982">Jensen 1982</ref> Research utilizing the techniques of [[confirmatory factor analysis]] has also provided support for the existence of ''g''.<ref name="carroll1995"/>

A ''g'' factor can be computed from a correlation matrix of test results using several different methods. These include exploratory factor analysis, [[principal components analysis]] (PCA), and confirmatory factor analysis. Different factor-extraction methods produce highly consistent results, although PCA has sometimes been found to produce inflated estimates of the influence of ''g'' on test scores.<ref name="floyd2009"/><ref>Jensen 1998, 73</ref>

There is a broad contemporary consensus that cognitive variance between people can be conceptualized at three hierarchical levels, distinguished by their degree of generality. At the lowest, least general level there are many narrow first-order factors; at a higher level, there are a relatively small number – somewhere between five and ten – of broad (i.e., more general) second-order factors (or group factors); and at the apex, there is a single third-order factor, ''g'', the general factor common to all tests.<ref name="deary2012">Deary 2012</ref><ref>Mackintosh 2011, 57</ref><ref>Jensen 1998, 46</ref> The ''g'' factor usually accounts for the majority of the total common factor variance of IQ test batteries.<ref>Carroll 1997. The total common factor variance consists of the variance due to the ''g'' factor and the group factors considered together. The variance not accounted for by the common factors, referred to as ''uniqueness'', comprises subtest-specific variance and measurement error.</ref> Contemporary hierarchical models of intelligence include the [[three stratum theory]] and the [[Cattell–Horn–Carroll theory]].<ref name="d&k">Davidson & Kemp 2011</ref>