Wechsler Arithmetic test: Measure of Gq or Gf?

What does the Wechsler Arithmetic test measure? Why has it's interpretation been so variable over the decades? Why is it now classified (as per CHC theory) as a mixed measure of Gsm (Short-term Memory - Working memory; MW) and Gf (Fluid Reasoning - Quantitative reasoning; RQ) in the latest Essentials of Cross-Battery Assessment book (Flanagan, Ortiz and Alfonso, 2007)? [Click here if you need more information on CHC theory and the major abilities, definitions, and abbreviations]

While preparing for my recent presentation at the Third National School Psychology Neuropsychology Conference, I consulted the 2nd edition of the Essentials of Cross-Battery book. I noticed on page 310 that, in contrast to prior cross-battery classifications of the Arithmetic test as a primary measure of Gq (Quantitative Knowledge-Math Achievement; A3) and Gf (Fluid Reasoning-Quantitative Reasoning; RQ) [Note - I was involved in these prior classifications as a coauthor of the first cross-battery book (ITDR: McGrew and Flanagan, 1998) and the Wechsler-specific spin-off cross-battery book (Flanagan, McGrew & Ortiz, 2000), it had now changed to Gsm and Gf.

First, a historical note. In the ITDR and Wechsler cross-battery books the primary Gq classification was based on a series of CHC/Gf-Gc designed cross-battery (joint) factor studies. The secondary Gf classification was a logical content analysis based classification, for which no available CHC/Gf-Gc cross-battery factor analysis supported the classification.

A review of page 310 in the 2nd Edition of the Essentials of CB indicates that the Gsm and Gf classifications are based on "factor analysis from the WISC-IV technical and interpretive manual (Psychological Corporation, 2003) and from the results of factor analysis reprted in Keith, Fine, Taub, Reynolds, and Kranzler (2006)." My problem with this change is that these supporting analyses are all within-battery (WISC-IV only) confirmatory factor analysis studies (CFA), and thus do not include the complete range of CHC indicators in the analysis, especially other Gq markers. If you want to see a prior post I made about my disagreement with the Keith et al. Arithmetic classification, click here. Personal communication with one of the XBA Essentials books authors indicates that the Gf classification is also intended to reflect content validity evidence.

Why do I take issue with the use of within-battery CFA to make CHC test classifications? If you want the long story you can read about the strengths and limitations of within- and cross-battery CFA studies in the first two XBA books listed above. If you want a short-course on the issue, you can check out an on-line PPT show I just uploaded that conceptually illustrates the limitations of within-battery CFA studies....and...more importantly, the advantage of CHC-designed cross-battery (joint) CFA studies classifying tests as per CHC theory.

To date I know of 9 different Wechsler/Woodcock-Johnson CHC/Gf-Gc designed cross-battery studies. All 9 studies include other markers of Gq (math achievement tests). In ALL of these analysis the Wechsler Arithmetic test has a large a significant loading on Gq (average loading in the mid .70's)....none on Gf. In one study (Phelps et al. 2005) there is a small secondary loading on Gs. In none of these cross-battery studies does Arithmetic display a significant Gsm factor loading. You can view a summary of these CHC-designed cross-battery findings, as well as the Keith et al. (2006) model upon which the current (and I believe inaccurate) Arithmetic Gf classification is based (due to a loading of .79 on a Gf factor), at the following link.

My two cents. I believe the best available analysis argue for the Wechsler Arithmetic test being interpreted primarily as a measure of Gq (Math Achievment-A3). I believe practitioners should not interpret this test as a primary or strong measure of Gf (Fluid Reasoning - Quantitative Reasoning; RQ). This does not mean that RQ is probably not involved. What the data indicate that if RQ is involved, the amount of variance is trivial and dwarfed by Gq-A3.

Conflict of interest disclosure - I'm a coauthor of the WJ III and thus have a financial interest in a competitor to the Wechsler batteries. I no longer receive any royalties from the two respective cross-battery books I coauthored.