CHC Theory, Cognitive Assessment, Tutorial

Factor Analysis of the WISC-IV Integrated with a Schmid-Leiman Transformation

On the IAPCHC listserv, the question of what the WISC-IV Integrated Spatial Span measures came up recently. I obtained permission from Pearson to use Tables 5.1, 10.1, and 10.2 to construct a correlation matrix of the subtests in the WISC-IV Integrated. I removed the process scores that have to do with time bonuses. I also removed subtests that are the sum of two, more basic subtests (e.g., Digit Span is the sum of Digits Forward and Digits Backward). This was necessary because the correlation matrix is not positive definite if the parts and the sum are included. A matrix that is not positive definite is impossible to factor analyze.

I used the correlation matrix to extract 5 principal factors with promax rotation. I played around with other factor extractions but a variety of concerns led me to settle on 5 factors. I used the Cattell-Horn-Carroll names for the factors (Gc, Gv, Gsm, Gq, and Gs). I then applied a Schmid-Leiman transformation to the analysis so that it would be parallel to Carroll’s analyses.

Any loading over 0.10 was highlighted. Click the picture to see a larger version.

Note the absence of Gf. I think that Picture Concepts is not a strong enough Gf measure to team up with Matrix Reasoning for it to emerge. Gf often does not emerge anyway because it is so correlated with g.

I used the reliability estimates (averaged across all ages) to estimate how much of the variance in each subtest is error, specific, and shared by g and the 5 smaller factors.


12 thoughts on “Factor Analysis of the WISC-IV Integrated with a Schmid-Leiman Transformation

  1. Very interesting analysis. Could the absence of Gf also be a consequence of the difficulties of EFA rotations (e.g promax) to cope with measures of complexity greater than one (factor indicators have
    loadings on more than one factor, typical rotations seek simple structure) ??


    • You are suggesting that the analysis may not be up to the job. This may be true. However, I really think that any analysis would have a hard time generating a well-defined Gf factor with the WISC-IV. If a extract 9 factors, I can get a very weak Gf factor. My guess is that much of the specific variance in Matrix Reasoning would become a good Gf factor if there were another good measure of Gf in the WISC-IV.

      • I also agree with your point. I really like EFA because, as Carroll wrote many times, it let the data speak for itself. Nevertheless, there are some structures that are problematic to recover with EFA/rotations. In bifactor models (were all factor indicators loads on two factors, g and a group factor) the g factor tends to get “spread” on other factors because of EFA/rotations to simple structure. In quite a few CFA studies, Gf tends to be statistically indistinguishable from g (correlation = 1). So maybe Gf tends to vanish from EFA for the same reasons (?)


  2. Brad says:

    “g” is useless, get rid of it, it does nothing for identification or intervention. The notion that IQ reflects ability/potential/intelligence is one of the biggest myths there is in psychology. It renders your findings meaningless.

    • You might be right. I don’t know why abilities correlate. Let’s call the sum total effect of all those unknown reasons “g” (or something else, if you like). I have never maintained that g is an ability. By the way, neither did Spearman.

      • Alexandre says:

        Abilities correlate because there’s a single nervous system that underlies them. To say it is useless is simply ignorant. G is a biological property of the brain.

        To say it doesn’t exist is absurd and simply ignores what the empirical data is hinting at.

      • Hi Alexandre,
        I am sympathetic to the “single nervous system” hypothesis as a partial explanation but there seems to be more to the story. Why do some abilities correlate with each other more than they do with others? Why are abilities heritable? Why do g loadings differ systematically? Why are there non-g factors? Why don’t abilities correlate with other traits that are equally rooted in brain functioning (e.g., neuroticism)?

  3. Alexandre says:

    G itself has a high heritability rating in adulthood (.8 or so); just because it isn’t 1.0 should not be taken as a question against the validity of g as an expression of one’s biology. (Jensen, among others, makes the argument for g being a biological property of the brain.) Furthermore, the brain is the only place in the human organism in which all genes are expressed (unlike in other organs and cells thereof). Skill factors do not necessarily correlate with other skills, and I do not see why the should. Skill factors, in any case, are nature’s way – one might say – of securing future evolutionary paths, which may or may not develop into more general abilities further down the road.
    To directly answer to your last question, neuroticism is a personality feature. Some personality features do correlate with g, like openness to experience (one of the Big Five), but the correlation isn’t surprising since openness to experience is also the most heritable of personality traits among those in the FFM. So, some traits do in fact correlate with brain functioning – just not all of them.

    It is naturally a part of the project of psychometrics to determine why certain factors intercorrelate the way they do. I do not think it is a good idea to deny the primacy of g simply because certain aspects do not apparently correlate with it (those being few in number, though). That would be to throw out a lot of baby with the bath water and to ignore a lot of good science.

    (One thing I should mention, though, is that we don’t have just a neocortex, but a limbic system and a cerebellum. These and other factors most definitely must be factored in a full-fledged analysis to adequately answer all of your (and my) why questions.)

    You may find this paper interesting as it relates certain aspects of intelligence testing with structural (network) properties of the brain: .

    Anyway I’ve said enough. But let me say you’re doing a great job here. I really enjoy your tutorials and look forward to more!

  4. Hello, I’d like to know how do you use Schmid Leiman Transformation? What software do you use? Sorry, I am not familiar with programming, yet I have to do the transformation for my graduation thesis.

    I really need your help 🙂

  5. Thank you for your splendid website. I’m learning a lot from it.

    If you have a raw data and script for calculating g-loadings from scratch, could you please upload it so that relative newbies in psychometrics can get a hang of this procedure.

    • Thanks!

      First install the psych package in R.

      Suppose that your raw data is in a data frame called d. The code for the Schmid-Leiman transformation for a 4-factor model would be:


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