If we are interested in the genetic correlations between factors, we might also be interested in whether the overall genetic correlation between the factors sufficiently represents the patterning of the cross-trait, cross-factor genetic correlation. With this goal in mind, QFactor was validated and described in Breunig et al. (2025) to help identify local model misfit, by quantifying the degree to which significant inter-factor correlations may be driven by a subset of indicators (traits) that load onto either factor. Additionally, it can identify non-significant factor correlations that consist of indicators that are correlated, but are variable in the direction of the correlation, and consequently average out to a cumulative effect near 0 and a non-significant factor correlation. Below we illustrate this by presenting different scenarios, displayed as hypothetical genetic correlation matrices of six correlated traits loading onto two factors in Panels B-C. Note that these genetic correlation matrices are subset to only include the three traits defining Factor 1 (T1-T3) in the columns and the three traits defining Factor 2 (T4-T6) in the rows. Despite extremely different patterns of correlations across the traits that define each factor, all four scenarios would produce the same overall inter-factor correlation of .2 displayed in the path diagram in Panel A. QFactor would be expected to be significant for Panels B-C, and is functioning to identify there is significant divergence within the traits of the factor in how they are correlated with traits of the other factor. We note that when the factor loadings are different across traits that a single factor correlation can still be consistent with differently sized correlations across traits so long as the divergences across correlations scale with the divergences across loadings.

In Genomic SEM we can test whether our factor – factor relationships are QFactor significant for each of the tested factor correlations in the model. In the case of a model with three correlated factors, we can obtain a QFactor result for each of the three assessed factor – factor correlations, therefor making QFactor a measure of local model misfit, indicating significant misfit induced by modeling the correlations between the traits across the two factors using a single inter-factor correlation. Below is a standardized path diagram from Genomic SEM for three genetically correlated factors: Internalizing, Neurodevelopmental, and Immune (defined by GWAS of mixed-pattern immune-mediated diseases).

As is standard practice, we can evaluate these factor correlations for statistical significance. By setting the Q_Factor argument to TRUE we can also assess the etiological utility of these inter-factor correlations by quantifying if they adequately capture the genetic correlations between the traits that define any two factors.

load("./LDSC_Qexample.RData")


Model_All<-"
Immune=~a*PSO + a*PSA
Neuro=~PTSD+ADHD+TS+ASD
Int=~PTSD+MDD+ANX
"

# running model with Q factor
CFAallQ <- usermodel(Factor_LDSC, model = Model_All, std.lv = TRUE, imp_cov = TRUE, Q_Factor = TRUE)
CFAallQ$modelfit

In addition, to the regular model output, we then get a separate section of output that gives us the QFactor χ^2statistic (QFactor_chisq) and degrees of freedom (QFactor_df), and corresponding p-value for this heterogeneity statistics (QFactor_p). The table below has been altered to only show the factor-factor results and to adjust the p-value for multiple testing.

##       lhs op   rhs STD_Genotype    STD_Genotype_SE      FDRpval QfactorFDRpval
## 8  Immune ~~ Neuro   0.11982032 0.0364367148251297 1.204646e-03    0.004807549
## 7  Immune ~~   Int   0.08784561 0.0254611804779287 8.918097e-04    0.086566960
## 18  Neuro ~~   Int   0.65885405 0.0422248371321114 2.066376e-54    0.156329521

From these results we can see that all three factors are significantly genetically correlated, but that the Neurodevelopmental-Immune factor correlation is also QFactor significant, indicating that the single inter-factor correlation does not sufficiently capture the genetic interrelationships of the individual disorders that define these factors.