Master Thesis
Testing Structural Relations with General Intelligence: An Alternative to
the Method of Correlated Vectors
Jonnemei M. Colnot
University of Amsterdam
Supervisor: dr. Dylan Molenaar
August 24, 2016
To investigate if a variable (X) is related to general intelligence, Jensen (1998) developed the Method of Correlated Vectors (MCV). The MCV involves calculating the correlation between a vector of IQ subtests’ g-loadings and a vector of these tests’ correlations to X. Although it was an honest attempt to develop a method without the need of raw data and to enable meta-analysis, the MCV has substantial limitations. It lacks specificity, the vector correlations are difficult to interpret, and it has insufficient power to reject Spearman’s hypothesis. An alterna-tive to the MCV is proposed in this paper – the Method of Explicit Relations (MER)– that uses an augmented correlation matrix to fit a SEM model. The performance of this alternative method is compared to that of the MCV by means of a simulation study. Results show that the MER has a higher power, lower false positives, and in addition estimates the strength of the relationship between g and X. With that the MER proves to be a valuable alternative to Jensen’s MCV. The practical value of the new method is illustrated with a real dataset (Naglieri & Jensen, 1987).
Intelligence tests are known to show positive corre-lations between subtests. These positive correcorre-lations, known as the positive manifold (Spearman, 1904), are commonly explained with the general intelligence factor (g).
1 Method of Correlated Vectors
To determine whether an external variable, X, is re-lated to g, Jensen (1985, 1998) developed the Method of Correlated Vectors (MCV). This method intends to describe the relation between the g factor, that is ex-tracted from several tests, and variable X. Variable X is an external variable that can represent nearly all sub-ject characteristics, as long as it is not a subtest variable used to extract the g-loadings (Jensen, 1998).
The MCV is extensively used in studies explor-ing the relation between g and a wide range of ex-ternal variables (Ashton & Lee, 2005) including sex (e.g., Colom, Garc´ıa, Juan-Espinosa, & Abad, 2002), race (e.g., Rushton, 1999), educational level (Colom, Abad, Garc´ıa, & Juan-Espinosa, 2002), generation
co-hort (Must, Must, & Raudik, 2003), gray and white matter volumes (Colom, Jung, & Haier, 2006), and head size (Jensen & Johnson, 1994)
In short, the MCV determines the correlation be-tween two vectors. One vector of the n tests’ factor loadings (usually the g-factor), and the other vector con-sists of each test’s correlations with a single variable, X, that is independent of the set of tests that is used to derive the factor loadings (e.g. sex, height, ethnicity etc.). If the correlation between those vectors is signifi-cantly larger than zero, Jensen (1998) concludes that X and g are correlated. The rationale behind this method is that if g fully mediates the relationship between X and g, the vector of factor loadings and the vector of correlations with X would be perfectly collinear. If g partially mediates this relationship, the vectors would not be perfectly collinear, but a positive correlation can still be expected. Therefore, a vector correlation larger than 0 is interpreted as support for the hypothesis that X is related to g.
The MCV is most extensively used to investigate the role of g in IQ differences between African-Americans 1
and Caucasians. Between these two groups an aver-age IQ difference of roughly one standard deviation is generally found (Jensen, 1998; Lynn, 2006; Rushton & Jensen, 2005). This difference grows larger when tests have higher g-loadings. For that reason Jensen (1985) formulated the so-called Spearman’s hypothesis, that attributes the IQ differences mainly to a difference in g. Spearman’s hypothesis consists of a strong and a weak version. The strong version ascribes all variation in mean IQ differences to g. That is, the vectors con-structed in the MCV will be perfectly collinear. The weak version attributes the variation in mean IQ dif-ferences primarily to g, but it also recognizes possible contributions by lower-order factors. In terms of the MCV, the vectors will then not be perfectly collinear, but they can still be expected to show a positive corre-lation.
2 Critisism
Despite its extensive use, the Method of Correlated Vec-tors (MCV) has received substantial criticism. The lim-itations that have been proposed are both directed to-wards the general weaknesses of the MCV as well as di-rected specifically towards the MCV’s support of Spear-man’s hypothesis.
First, the MCV lacks specificity in defining the role of g in its relationship to variable X. That is, large vec-tor correlations are not necessarily indicative of a central role of g. A high correlation can even be found when a g-factor is absent in the data (Sch¨onemann, 1997; Lubke, Dolan, & Kelderman, 2001). Thus, a high correlation can indicate (many) other scenarios that describe the relation between X and IQ differences, and the MCV lacks the ability to detect this relationship.
Second, the correlations that are calculated in the MCV are difficult to interpret (Dolan & Hamaker, 2001). Vector correlations significantly greater than zero are considered evidence of a central role of g (Jensen, 1998). However, significant correlations are found even when the effect size of the relation between X and g is small. As Jensen (1998) pointed out himself, the vector correlation essentially shows whether the ex-ternal variable is fully or partially mediated through g. This does not depend on the strength of the relation-ship. Interpreting significant correlations when effect sizes are small can result in a large number of false pos-itives.
A final limitation also holds when the MCV is ap-plied to test Spearman’s hypothesis. The MCV lacks the power to reject Spearman’s hypothesis when the contribution of g to the group differences diminishes (Lubke et al., 2001; Dolan & Lubke, 2001). The MCV still considers these (weak) correlations support
for Spearman’s hypothesis, despite that the mean dif-ference is not primarily due to g.
Following these limitations, the MCV has difficulties distinguishing models that describe the relationship be-tween g and X. In addition, the MCV has no guidelines to determine when a correlation is substantial enough to support Spearman’s strong hypothesis over the weak hypothesis, nor does it have means of goodness of fit testing to compare competing models. Ideally, explor-ing the relationship between g and X (e.g. sex, eth-nicity) and testing competing models (e.g. Spearman’s hypothesis and alternatives) is done by fitting a hierar-chical second-order factor model (Jensen, 1998) with an external variable X to the raw data. A second-order fac-tor model can explicitly compare competing models and hypotheses through the use of fit measures. Whereas the vector correlation is the MCV’s only way support for Spearman’s hypothesis. However, fitting such a model is not always possible, as it requires raw data obtained from a single study.
Given the criticism discussed above, question arises why the Method of Correlated Vectors (MCV) is still used to such a large extend. One reason is that the MCV is a way to use (non-raw) data from different studies to explore the role of g in relation to an ex-ternal variable, and is therefore commonly applied in meta-analytic settings where raw data is not available (te Nijenhuis, van den Hoek, & Armstrong, 2015; Arm-strong, Woodley, & Lynn, 2014; Flynn, te Nijenhuis, & Metzen, 2014; Peach, Lyerly, & Reeve, 2014; te Nijen-huis & van der Flier, 2013; Woodley & Fernandes, 2014). That is, researchers rely on data from multiple sources; g-loadings are acquired from IQ test manuals, while the external group variables have a different source.
Considering the limitations of the MCV we propose an alternative method with the same benefits as the MCV, but without its limitations.
3 Alternative Method
The proposed alternative method, Method of Explicit Relations (MER), does not require raw data and yet allows for model comparison. In short, the MER aug-ments an expected correlation matrix by adding a vector V . This correlation matrix is then used to fit a SEM model.
Vector V contains the correlations between subtest scores and variable X. Similar to the MCV, the subtest scores for both groups of X are acquired from litera-ture. These scores are used to calculate the correlations between subtest scores and X to form vector V . 2
Figure 1: Schematic full MER model
An expected correlation matrix can be computed with the vector of g-loadings (V g). This vector V g can be acquired from previous studies, through a principal component analysis, or as stated in test manuals. These ways to obtain g-loadings are also seen in the MCV.
The expected correlation matrix V is calculated with the following equation:
V = V gV gt+ diag(1 − V g2) (1) The correlation matrix is augmented by adding vec-tor V and is then used to fit a SEM model to test several models that potentially describe the relation between g and variable X. A full SEM model of the MER is shown in figure 1. Here X is a categorical variable representing group membership and all paths from subtest and g to X are estimated.
We can, for example, fit a model representing the strong version of Spearman’s hypothesis, where g (second-order factor) is the only source of group differ-ences, and compare this to the fit of a full MER model where the group differences are also due to first-order factors like in figure 1. Comparing these two models
could give more insight into the role of g in these group differences.
Contrary to the MCV, the MER is an explicit sta-tistical model and therefore enables the evaluation of model fit. We can establish how well our model fits with fit indices (i.e. RMSEA, AIC, BIC) from the SEM modeling framework and compare this fit to other mod-els. The possibility of model comparison contributes to a higher sensitivity and power of the MER. An explicit model, like the MER, also takes all variance into ac-count. That is, the variance of g and first-order factors, where the MCV only takes the variance of g into ac-count. This limits confounding factors in the MER that could increase the number of false positives.
In addition, the MER is able to capture the strength of the relationship between g and X. As noted before, the MCV only indicates if a full or if a partial mediation is present, but the strength of this relationship cannot be determined.
This paper compares the performance of the pro-posed alternative method – the MER – to the MCV by means of a simulation study. The simulation is set up to generate data with and without mediation through g and though first-order factors. Their performance is then assessed in terms of the methods’ specificity, false positive rate and power to distinguish different under-lying models.
4 Design of the study
The simulation study is set up to establish whether the MER can better identify the relationship between g and variable X (e.g. sex, ethnicity) than the MCV. Spear-man’s hypothesis is used as a basis for the underlying models. Specifically, data is generated with an exter-nal variable X that represents group membership. The group difference in IQ is manipulated in 4 ways: a differ-ence fully mediated through g, a full mediation through a first-order factor, a partial mediation through both g and a first-order factor, and no relationship between g and X (and thus no group differences in the IQ sub-tests). These data generating models are represented schematically in figure 2 and explained more extensively below.
Figure 2: Model simulation and predicted outcomes according to underlying models. Bold variables represent the location of the group difference. Bold arrows represent the predicted significant paths. Note that model 4 is not depicted, as no differences will be simulated and thus µgroup is expected to show no significant paths.
4.1 Data Generating Models
The data are simulated in R (R Core Team, 2015). All four models will be based on a two-group second-order factor model with 13 subtests, 3 first-second-order fac-tors and a g factor. More specifically they are based on the WISC-R factor model described in Dolan (2000), and Jensen and Reynolds (1982). The data are gen-erated with two different factor structures. A factor structure with cross-loadings, as one generally finds in test-manuals and research, and one with a simple struc-ture. The simple structure will ensure the first-order effects to be more isolated and easier to detect.
Data provided in Dolan (2000) are used to set up the data generating models. Factor loadings and factor scores are standardized over the two groups.
The basic matrix representation of the data generating models is:
y = υ + Λη + ε (2)
where y is a column vector of observed scores on n sub-tests. η denotes the q x 1 vector of q factor scores. As we use the data from Dolan (2000), n is 13 and q is 3. υ represents the n x 1 vector containing the intercepts of the subtests, and ε is the vector of residual scores of the n subtests. The n x q matrix Λ contains the factor loadings of the n subtests on the q first-order factors.
η is modeled in a linear regression on g, where g is the score on the general intelligence factor (Molenaar, Dolan, Wicherts, & van der Maas, 2010; Dolan & Hamaker, 2001).
η = Γ g + ω (3)
where ω is a q x 1 vector of first-order residuals. The vector Γ contains second-order factor loadings, which can also be interpreted as regression coefficients or the regression of the first-order factors (η) on the general intelligence factor (g).
The model can now be represented as:
y = υ + Λ(Γ g+ω) + ε (4)
The covariance matrix of y will then be Σ = Λ[Γ σ2
gΓt+ Σω]Λt+ Σω (5) where σ2
g is the variance of general intelligence and Σω is the p x p covariance matrix of residuals. The expected mean can then be denoted by:
µy= υ + Λ(Γ µg+ µω) (6) where µgdenotes the mean score on general intelligence and µω represents the first-order residual means. The models will generate data for two different groups. To asses the mean group difference, the mean of group 1 (µg1and µω1) is used as a reference group and is set to zero. The model of means for group 1 and group 2 are respectively
µy1= υ (7)
µy2= υ + Γi∆µg2+ ∆µω2 (8) where ∆µg2 represents the mean difference between group 1 and group 2 (∆µg2 = µg2− µg1) and ∆µω2 is the q x 1 vector of first-order residual factor means differences between group 1 and group 2.
Note that we assume measurement invariance across groups, as found by Dolan (2000). In addition, we as-sume invariance of σg2and σω2. That is, the differences between groups are solely due to differences in ∆µω2 an
The vector correlation of the MCV can be represented with:
cor(ΛΓ ), (Γ ∆µg2+ ∆µω2)
(9) The basic data generating model is adjusted to rep-resent several scenarios. The following data generating models are used for data simulation and to test the per-formance of the MER and MCV:
Model 1 (’g-only’) has a group difference that is fully mediated through g. Here IQ subtest scores are simu-lated with a mean group difference on g (∆µg2) which represents Spearman’s strong hypothesis.
Model 2 (’first-order only’) has a group difference that is fully mediated through a first-order factor. It will include IQ subtest scores with group differences due to first-order factor (e.g. a verbal factor). Thus, (∆µω2) deviates from zero, and in turn causes a mean difference in IQ subtest scores. All subtests that load on this verbal factor will show a group difference.
Model 3 (’g and first-order’) has a group difference that is partially mediated through g and partially through a first-order factor. It will generate IQ subtest scores with a group difference on both g (∆µg2) and the ver-bal factor (∆µω2).
Model 4 (’no difference’) assumes no group differ-ence and thus produces IQ subtest scores that are not different between groups.
All models will use an effect size of group differences based on Naglieri and Jensen (1987) and Dolan (2000).
4.2 Fitted Models
The above described generated data is then analyzed with the MER and the MCV. The MER analyses are done with the use of the Lavaan package (Rosseel, 2012). The g-loadings used in the MER models are deter-mined in two ways. The true g-loadings as provided in the literature (Dolan, 2000) and g-loadings as extracted with a principal component analysis (PCA). The follow-ing MER models are fit to the data.
1. full-model : estimating paths from X to g and all subtests. See model 1 in figure 2
2. g-model : estimating only the path between X and g. See model 2 in figure 2.
3. subtest-model : estimating paths only from X to all subtests. See model 3 in figure 2.
4. null-model : not estimating paths with X, but X is represented in the model.
5. full - f1 : full-model without estimating paths from X to the subtests loading on the first (first-order) factor.
6. full - f2 : full-model without estimating paths from X to the subtests loading on the second (first-order) factor.
7. full - f3 : full-model without estimating paths from X to the subtests loading on the third (first-order) factor.
8. subtest - f1 : subtest-model without estimating paths from X to the subtests loading on the first (first-order) factor.
9. subtest - f2 : subtest-model without estimating paths from X to the subtests loading on the sec-ond (first-order) factor.
10. subtest - f3 : subtest-model without estimating paths from X to the subtests loading on the third (first-order) factor.
4.3 Role of g
To determine the role of g with the Method of Corre-lated Vectors (MCV) a Pearson product moment corre-lation between the vector of g-loadings and the vector of group differences is calculated. For every underlying (data generating) model, the correlation and its signifi-cance is determined on every iteration. The normality of the g-loadings and the vector of differences is checked with Shapiro Wilk’s test of normality. If normality is rejected Spearman’s correlation is calculated instead of Pearson’s correlation (Jensen, 1998).
The Method of Explicit Relations (MER) determines the regression path from X to g on every iteration. The significance of this path is determined by fitting the full-model to the data. The full-full-model estimates paths from the external group variable (X; race) to g and to all 13 subtests. Competing models are also fitted and evalu-ated on the several fit criteria (i.e. RMSEA, CFI, AIC, BIC).
4.4 Power and false positives
The power and the number of false positives are de-termined for the MCV and MER. For the MCV the vector correlation is evaluated and determined if its sig-nificance matches with the underlying (data generating) model. For the MER the significance of the path from g to X (race) is analyzed and evaluated on its corre-spondence to the data generating model (and thus the true path). The regression paths from X (race) to sub-test are also evaluated with the MER. These analyses are done for the full-model, g-model, subtest-model, and the null-model.
In addition, a log-likelihood ratio test compares seven of the above described fitted MER models. Their power to detect mediation through g and mediation through first-order factors is evaluated. Here the focus is partic-ularly on detecting the true underlying model when g is not the main source of group differences.
The general expectation is that the Method of Ex-plicit Relations (MER) will perform better than the Method of Correlated Vectors (MCV); it will find the true underlying model of group differences where the MCV is inconclusive. More concretely, the MER will have less false positives and more power than the MCV. See figure 2 for a schematic representation of the pre-dicted outcomes.
4.5 Real data
Finally, the MER is applied to a real dataset to at-tempt to find the role of g in group differences. Naglieri and Jensen (1987) describe the scores on the WISC-R (Wechsler, 1974) and the K-ABC (Kaufman & Kauf-man, 1983) of 86 African-American and 86 Caucasian children. The correlations of the subtests with variable X (i.e. being African-American or Caucasian) and the extracted g-loadings are used to perform the MER (and MCV) on this dataset.
In terms of the real data set (Naglieri & Jensen, 1987), the MER will provide more substantiated sup-port for the source of the group difference. The MCV will show a correlation that is likely to be significant and substantial, but is unable to support this appropriately. The MER will be able to support the source of group difference with more power and by showing that one 6
av. av.
Data Model MER est MCV cor
cross-loadings ’g-only’ 0.97 0.117 0.78 -0.667 ’g and first-order’ 1 0.142 0.98 -0.826 simple structure ’g-only’ 1 0.121 0.94 -0.785 ’g and first-order’ 1 0.141 0.96 -0.839 After extraction of g-loadings through PCA cross-loadings ’g-only’ 0.92 -0.061 0.54 0.345 ’g and first-order’ 0.97 -0.084 0.77 0.575 simple structure ’g-only’ 0.92 -0.009 0.66 0.461 ’g and first-order’ 0.43 -0.013 0.84 0.542
Table 1: Power to detect g difference with the full MER model and MCV.
(alternative) model is favored over all the other models.
5 Results
The analyses show that the MER is better in determin-ing the role of g in group differences (see table 1 & 2). First of all, the power to detect a difference mediated through g is larger for the MER than for the MCV (ta-ble 1). That is, when fitting the full-model, the MER shows significant paths from g to X (race) with rates of .97 and 1 in data generated with the ’g-only’ model and the ’g and first-order’ model respectively. The MCV finds significant vector correlations with rates of respec-tively .78 and .98.
In addition, fitting the full-model with the MER re-sults in a much lower false positives rate than the vec-tor correlation of the MCV (table 2). In data generated without differences mediated through g – the ’first-order only’ model and the ’no difference’ model – rates of find-ing a g-difference are respectively .15 and .04 for the MER and .39 and .17 for the MCV. This indicates that the MCV will more often incorrectly ascribe the group difference to a difference in g. The lower power and high false positives rate in the MCV correspond to what was already established in previous research (Lubke et al., 2001; Dolan & Hamaker, 2001).
The results from the data generating models that are generated with a simple structure yield similar results
av. av.
Data Model MER est MCV cor cross-loadings ’first-order only’ 0.15 0.025 0.39 -0.434 ’no difference’ 0.04 -0.002 0.17 -0.015 simple structure ’first-order only’ 0.13 0.023 .5 -0.539 ’no difference’ 0.09 0.001 0.14 0.013 After extraction of g-loadings through PCA cross-loadings ’first-order only’ 0.16 -0.019 0.3 0.290 ’no difference’ 0.02 -0.001 0.12 0.014 simple structure ’first-order only’ 0.28 -0.010 0.5 0.415 ’no difference’ 0.15 -0.001 0.11 0.021
Table 2: False positive rates of full MER model and MCV.
to data models with cross-loadings (see table 1 & 2 ). The MCV shows a slight increase in power, but the false positive rates are still very high compared to those of the MER.
Similar analyses were done on data generating mod-els that extract the g-loadings with a PCA instead of us-ing the g-loadus-ings provided in literature (Dolan, 2000). These analyses yield similar results in respect to the MER’s adequacy of determining the role of g (see table 1). That is, it correctly finds the difference mediated through g with a rate of .92 and .97 respectively, where these are .54 and .77 for the MCV. In addition, the false positive rates are .16 and .02, where this is .3 and .12 for the MCV. As these results replicate the previous anal-yses with prior established factor loadings, it assures us that the factor loadings do not need to be extracted with a PCA. Thus the use of raw data to extract factor loadings is not necessary for these analyses.
5.1 Alternative Models
Above described analyses were all performed by fitting the full-model with the MER. In addition, the MER was used to fit several competing models like the g-model and subtest-model to the data (see table 5 in Appendix A). However, we find that none of the models captured the role of g better than the full-model when analyz-ing the regression paths. The g-model, estimatanalyz-ing only the regression path from g to X (race), showed
lar result in regard to the power and false positives of the MER. When fitting the full-model and the subtest-model that only estimates the paths from subtests to X (race), both show very low power in detecting first-order factor differences (see table 5). That is, paths between X (race) and subtests loading on the verbal (first-order) factor are not often significant when the underlying model includes a first-order difference. This suggests that – although MER has an acceptable power to detect g-differences – the power to detect first-order differences is poor. It does show that the path estima-tions between g and subtests help reduce the false pos-itives rates. For the ’first-order only’ data generating model, the false positive rates are higher when fitting a g-only model than with the full-model.
Comparing the fitted models in terms of their fit indices did not add much valuable information. Fit in-dices like the AIC and BIC have penalties for the num-ber of parameters in the model. Accordingly, the model with the least parameters (g-only) was almost always selected over the other models, even when this was not in accordance with the true underlying model.
A likelihood ratio test determined the functionality of the MER in detecting not only a g-difference but also the effects of first-order factor differences (see table 3). The fitted models that estimate a path from g to X (race) do not perform well in estimating first-order fac-tor differences. Fitted models that do not estimate the path from g to X (race) do detect first-order differences. However, the power of correctly detecting these differ-ences is very low (.24). In addition, we see that when a difference is mediated through g in the data, fitted models that are less restrictive (a subtest-model that estimate all subtest to X paths) are preferred over more restrictive models. This can be explained by the factor-loadings of the subtests. The factor-factor-loadings essentially capture the differences caused by g. Thus, a more elab-orate model will capture more of the g-difference. 5.2 Strength of Relationship
One of the main advantages of the MER over the MCV is that the MER is able to estimate the strength of the relation between g and X. Table 1 and 2 also show the mean estimation of the paths between g and X (av. est). The data that is generated to have a difference mediated though g is generated with a mean g-difference of 0.357. Thus it seems that the MER slightly underestimates the strength of the relationship between g and X as we find estimates of .117 and .142. However, it does correctly indicate when paths are significant. For the MCV the results are limited to the strength of the vector correlation. This correlation does not directly indicate the strength of the relationship between g and X.
Previous research has claimed vector correlations around .4 as g-differences (Armstrong et al., 2014; Rushton & Jensen, 2003), where we now see a similar correlation in data with only a first-order difference (see table 2). A g-difference was not present in this data, again empha-sizing the limitations of the MCV.
5.3 Real data example
After the performance and adequacy of the MER was es-tablished, the real dataset (Naglieri & Jensen, 1987) was analyzed. The dataset provides data from the WISC-R and the K-ABC for two groups of children; 86 African-American and 86 Caucasian. The subtest correlations, g-loadings, and subtest differences between groups are provided in Naglieri and Jensen (1987). This informa-tion is enough to calculate the MCV. For the MER we need the correlation between X (race) and subtests. By simulating testscores for the two groups based on the subtest correlations, we can extract the correlation be-tween group membership and the subtests. This vector V can then be added to the correlation matrix.
The augmented correlation matrix that follows is then used to fit the full MER model that estimates all regression paths to X. That is, from all subtests to X as well as from g to X. The MER finds a significant path between g and X for the WISC-R, K-ABC, and the WISC-R and K-ABC together (see table 4). The MCV also finds significant vector correlations for these datasets. This would indicate that there is indeed a relationship between g and the external variable X (in this case race). The basis on which this relationship is established is more convincing with the MER than the MCV, as we have seen that the MER reflects a better power.
Contrary to the MCV, the MER also shows the strength of the relationship. The strength of the relationship is most likely stronger than the MER indicates here, as the simulation found it underestimates the strength of the regression path.
Dataset
MER
MCV
WISC-R
0.321**
0.807*
K-ABC
0.292**
0.749*
WISC-R & K-ABC
0.364**
0.774**
Table 4: MER and MCV performed on data of Naglieri and Jensen (1987). MER path estimate between g and X and MCV vector correlation. *p-value < 0.01, **p-*p-value < 0.001
Cross loadings Simple Structure p o w er to detect path comparing ’g-only’ ’first-order only’ ’g and first-order’ ’no difference’ ’g-only’ ’first-order only’ ’g and first-order’ ’no difference’ g to X full-mo del & subtest-mo del 0 0 0 0 0 0 0 0 f1 to X full-mo del & full-f1 0 0 0 0 0 0.01 0.03 0 f2 to X full-mo del & full-f2 0 0 0 0 0 0 0 0 f3 to X full-mo del & full-f3 4 0 0 0 0 .01 0 0 0 f1 to X subtest-mo del& subtest-f1 0.96 0.24 1 0.02 0.97 0.15 1 0 f2 to X subtest-mo del& subtest-f2 0.56 0.07 0.48 0.03 0.68 0.02 0.54 0.02 f3 to X subtest-mo del& subtest-f3 0.48 0.03 0.48 1 0.48 0.07 0.48 0.04 T able 3: P o w er to detect paths to X with log lik eliho o d ratio test 9
6 Discussion
This paper has shown that the Method of Explicit Rela-tions (MER) is a more adequate method to describe the relation between g and an external variable X than the Method of Correlated Vectors (MCV). The MER can-not be subjected to the same criticism as the MCV as it is more specific, has a higher power and lower number of false positives.
It’s unfortunate that we encounter difficulty to de-termine the relation between g and first-order factors with the MER. The power to detect these differences is somewhat disappointing and is not substantial enough to responsibly base major conclusion on. However, it also emphasizes that finding first-order differences or detecting Spearman’s weak hypothesis (over the strong hypothesis) is proving difficult in all respects. The MCV has no appropriate measures to account for these differ-ences, and the MER is still limited. This shortcoming can be resolved by analyzing a full data set with, for ex-ample, methods suggested by Dolan (2000) and Dolan and Hamaker (2001). For cases in which raw data is not available, the MER will need to be further devel-oped to make more substantiated inferences on first-order effects. The estimation of these first-first-order factor differences should nonetheless be part of the model as it helps reduce false positives.
This naturally leads us to encourage the use of the MER instead of the MCV when full datasets are not accessible. When a general understanding of the SEM framework is present, the MER has proven no more diffi-cult than trying to understand the MCV. The MER is a more explicit and accurate way to describe the relation-ship between g and an external variable. In addition, the MER captures the strength of the relationship. Ev-idently, the MER produces elaborate results that allow more detailed inferences and assures that more accurate conclusions – and solutions – can be established. References
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Appendix A
Table 5 shows the results of all fitted models. In the main analyses only the results of fitting a full-model are taken into account, as the other models did not substan-tially add to these results. The power to detect paths from subtests to X is also shown in table 5.
Data Mo del Fitted mo del P aths from X (race) to: g I S A V C DS TS PC P A BD O A CO MA cr oss-lo adings ’g-only’ full-mo del 0.97 0.13 0.17 0.06 0.25 0.11 0.06 0.05 0.04 0.06 0.1 0.03 0.1 0.06 subtest-mo del 0.94 0.88 0.74 0.93 0.89 0.51 0.31 0.54 0.53 0.75 0.45 0.4 0.33 g-mo del 0.98 ’first-order only’ full-mo del 0.15 0.13 0.06 0.06 0.19 0.08 0.07 0.08 0.04 0.04 0.05 0.07 0.07 0.08 subtest-mo del 0.19 0.14 0.07 0.26 0.18 0.07 0.1 0.04 0.05 0.05 0.05 0.04 0.06 g-mo del 0.19 ’g and first-order’ full-mo del 1 0.31 0.31 0.09 0.69 0.34 0.06 0.09 0.08 0.08 0.05 0.04 0.05 0.05 subtest-mo del 1 0.98 0.88 1 1 0.47 0.33 0.69 0.7 0.71 0.54 0.34 0.32 g-mo del 1 ’no difference’ full-mo del 0.04 0.02 0.05 0.02 0.09 0.06 0.06 0.02 0.03 0.03 0.05 0.03 0.03 0.06 subtest-mo del 0.01 0.04 0.04 0.06 0.04 0.06 0.04 0.05 0.05 0.08 0.04 0.04 0.05 g-mo del 0.03 simple structur e ’g-only’ full-mo del 1 0.21 0.17 013 0.26 0.24 0.08 0.04 0.08 0.12 0.13 0.09 0.07 0.07 subtest-mo del 0.94 0.92 0.47 0.98 0.94 0.46 0.27 0.42 0.32 0.57 0.53 0.28 0.27 g-mo del 1 ’first-order only’ full-mo del 0.13 0.11 0.1 0.03 0.13 0.08 0.05 0.05 0.06 0.06 0.03 0.06 0.06 0.07 subtest-mo del 0.19 0.22 0.02 0.21 0.13 002 0.04 0.04 0.05 0.02 0.05 0.07 0.06 g-mo del 0.19 ’g and first-order’ full-mo del 1 0.44 0.43 0.07 0.61 0.38 0.09 0.08 0.09 0.06 0.02 0.05 0.04 0.08 subtest-mo del 0.98 1 0.5 1 1 0.53 0.35 0.4 0.32 0.58 0.43 0.34 0.24 g-mo del 1 ’no difference’ full-mo del 0.09 0.07 0.03 0.03 0.03 0.04 0.03 0.08 0.05 0.07 0.05 0.01 0.04 0.06 subtest-mo del 0.08 0.06 0.05 0.06 0.02 0.05 0.07 0.05 0.07 0.06 0.02 0.03 0.06 g-mo del 0.07 T able 5: P o w er to detect significan t paths to X . In de data mo dels that gener ate a first-or der differ enc e, the subtests I,S,A,V,C, DS, and TS ar e affe cte d when ther e ar e cr oss-lo adings. F or the simple structur e only subtest I, S, V, and C ar e affe cte d. Note that the subtest-mo del and g-mo del that ar e fitte d do not estimate al l p aths 12
