Spearman's Law of Diminishing Returns (SLODR) has caused discussions since the very moment when it appeared in the work of Spearman (1927). Spearman supposed that there are higher subtest correlations in the lower regions of the g-distribution and, vice versa, lower subtest correlations correspond to higher g regions. The so-called traditional method of SLODR investigation uses the decomposition of the sample group into two subgroups, which differ in g, and then the comparison of subgroup g-variations or mean correlations. Some years ago, researchers started using SEM method with comparing subgroup factor structures. The most recent change was to use Moderated Factor Analysis, continuously changing parameters in structural models (Molenaar et al, 2010). Using linear moderators, the authors researchers aim to find in the model the loci of mean correlation change (e.g., whether those are the subtest residuals, the first-order factor specific variances, and so on). This method allows one to avoid the problems appearing due to sample division, but, as Blum & Holling (2017) wrote in their meta-analytical review, in some cases in SLODR researches, the parameters were found to change nonlinearly (at least, when age was considered as a moderator). So the loci problem becomes more complicated: not only the model parameter has to be found as the source of correlation change, but the shapes of parameter dependence on the moderating variable (for example g) as well.
We developed a simple method 'Variance running estimate' based on taking into account the dependence of individual participant’s subtest data variance on g. This function’s growth points out to the SLODR at some form, but the method allows us to distinguish between the different non-linear forms of mean correlation change. Note that some U-shaped mean correlation changes cannot be detected even with the help of CEM.
We developed the data simulation process described in (Molenaar et al, 2010). The simulated data are formed by combining several normally distributed 9- and 4-dimentional samples with different correlation structures. The low-g region features high correlations between the ‘abilities’, while the high-g region features low ones. The correlations between the ‘subtests’ forming each factor is controlled, and, in this study, it is chosen to show a weak increase. The volumes of subsamples are chosen so as to make general distributions virtually normal. The subsample occupying the middle position may have different correlations of ‘abilities’ providing possible variants of the non-linearity of mean correlation dependence on g level.
We checked whether our method, based on the dependence of participant’s subtest data variance on g, was able to distinguish between the different forms of non-linear mean correlation dependence. The deviation from linear dependence may be localized and statistically estimated using bootstrap procedures.
The application of this method to the empirical data from original computerized intellectual-ability test (almost 12000 participants) shows that its combining with ‘traditional’ and CEM methods allows us to discern some details of correlation distribution. For example, we can differ distribution asymmetry influence to the SLODR presence of two kinds: the test psychometric properties (such as seiling effect) vs the sample asymmetry.
The work was supported by Russian Foundation for Basic Research, project # 17-29-07030
