# Bifactor Models for Predicting Criteria by General and Specific Factors: Problems of Nonidentifiability and Alternative Solutions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### Bifactor Model

## 2. Description of the Empirical Study

#### 2.1. Participants and Materials

#### 2.2. Data Analysis

#### 2.3. Application of the Bifactor Model

^{2}= 10.121, df = 11, p = 0.520). These estimation problems are due to the fact that a bifactor model with equal loadings and covariates is not identified (i.e., it is not possible to get a unique solution for the parameter estimates). Their nonidentifiability can be explained as follows: In a bifactor model with equal loadings, the covariance of an observed indicator of intelligence and a criterion variable is additively decomposed into (a) the covariance of the criterion variable with the g factor and (b) the variance of the criterion variable with a specific factor. Next, a formal proof is presented.

_{ik}is decomposed in the following way (the first index i refers to the indicator, the second indicator k to the facet):

_{ik}and a criterion variable C can be decomposed in the following way:

^{2}= 17.862, df = 21, p = 0.658). In this model, the g factor was significantly correlated with the mathematics grades (r = 0.574) and the English grades (r = 0.344). Consequently, one would conclude that only g is necessary for predicting grades. However, when we fixed $Cov\left(G,C\right)=0$, the respective model was also identified and fitted the data very well (χ

^{2}= 14.373, df = 17, p = 0.641). In this model, the g factor was not correlated with the grades; instead all the specific factors were significantly correlated with the mathematics and the English grades (mathematics—NS: r = 0.519, AN: r = 0.572, UN: r = 0.452; English—NS: r = 0.319, AN: r = 0.434, UN: r = 0.184). Hence, this analysis led to exactly the opposite conclusion: The g factor is irrelevant for predicting grades, only specific factors are relevant. It is important to note that both conclusions are arbitrary, and that the model with equal loadings is in no way suitable for analyzing this research question.

^{2}= 8.318, df = 10, p = 0.598). The estimated parameters of this model are presented in Table 21. All estimated g factor loadings were very high. The correlations of the mathematics grades with the g factor and with the specific factors were similar, but not significantly different from 0. For the English grades, the correlations differed more: The specific factor of verbal analogies showed the highest correlation with the English grades. However, the correlations were also not significantly different from 0. The results showed that neither the g factor nor the specific factors were correlated with the grades. According to these results, cognitive ability would not be a predictor of grades—which would be in contrast to ample research (e.g., [41]). However, it is important to note that the standard errors for the covariances between the factors and the grades were very high, meaning that they were imprecisely estimated. After fixing the correlations between the specific factors and the grades to 0, the model fitted the data very well (χ

^{2}= 16.998, df = 16, p = 0.386). In this model, the standard errors for the estimated covariances between the g factor and the grades were much smaller (mathematics: 0.128, English: 0.18). As a result, the g factor was significantly correlated with both grades (mathematics: r = 0.568, English: r = 0.341). So, in this analysis, g showed strong correlations with the grades whereas the specific factors were irrelevant. However, fixing the correlations of g with the grades to 0 and letting the specific factors correlate with the grades, resulted in the very opposite conclusion. Again, this model showed a very good fit (χ

^{2}= 8.185, df = 12, p = 0.771) and the standard errors of the covariances between the specific factors and the grades were lower (between 0.126 and 0.136). This time, however, all specific factors were significantly correlated with all grades (Mathematics—NS: r = 0.570, AN: r = 0.522, UN: r = 0.450; English—NS: r = 0.350, AN: r = 0.396, UN: r = 0.183). While all specific factors were relevant, in this case the g factor was irrelevant for predicting individual differences in school grades.

## 3. Alternatives to Extended Bifactor Models

#### 3.1. Application of the Extended First-Order Factor Model

^{2}= 13.929, df = 15, p = 0.531) and did not fit significantly worse than a model with unrestricted loadings (χ

^{2}= 9.308, df = 12, p = 0.676; scaled χ

^{2}-difference = 2.933, df = 3, p = 0.402). The results of this analysis are presented in Table 4. The standardized factor loadings and therefore also the reliabilities of the observed indicators were sufficiently high for all observed variables. The correlations between the three facet factors were relatively similar and ranged from r = 0.408 to r = 0.464. Hence, the facets were sufficiently distinct to consider them as different facets of intelligence. The correlations of the factors with the mathematics grades were all significantly different from 0 and ranged from r = 0.349 (unfolding) to r = 0.400 (verbal analogies) showing that they differed only slightly between the intelligence facets. The correlations with the English grades were also significantly different from 0, but they differed more strongly between the facets. The strongest correlation of r = 0.304 was found for verbal analogies, the correlations with the facets number series and unfolding were r = 0.242 and r = 0.142, respectively.

#### 3.2. Application of the Bifactor(S-1) Model

^{2}= 13.929, df = 15, p = 0.531). This result reflects that both models are simply reformulations of each other. In addition, the correlations between the reference facet and the two grades did not differ from the correlations that were observed in the first-order model. This shows that the meaning of the reference facet does not change from one model to the other. There is, however, an important difference between both models. In the bifactor(S-1) model, the non-reference factors are residualized with respect to the reference facet. Consequently, the meaning of the non-reference facets and their correlations with the criterion variables change. Specifically, the correlations between the specific factors of the bifactor(S-1) model and the grades indicate whether the non-reference factors contain variance that is not shared with the reference facet, but that is shared with the grades. The correlations between the specific factors of the bifactor(S-1) model and the grades are part (semi-partial) correlations (i.e., correlations between the grades, on the one hand side, and the non-reference facets that are residualized with respect to the reference facet, on the other hand side).

## 4. Discussion

## 5. Conclusions and Recommendations

## Author Contributions

## Conflicts of Interest

## Appendix A

_{k}, the observed variables with Y

_{ik}, and measurement error variables with E

_{ik}. The first index i refers to the indicator, the second indicator k to the facet. Hence, Y

_{11}is the first indicator of the first facet considered. A criterion variable is denoted with C. We consider only one criterion variable. We only consider models in which the criterion variables are correlated with the factors. Because the regression coefficients in a multiple regression model are functions of the covariances, the identification issues also apply to the multiple regression model. Moreover, we will only consider the identification of the covariances between the criterion variables and the general as well as specific factors because the identification of the bifactor model itself has been shown elsewhere (e.g., [54]). In the models applied, it is assumed that the criterion variables are categorical variables with underlying continuous variables. The variables C are the underlying continuous variables. If the criterion variable is a continuous variable, C denotes the continuous variable itself. In the model with free loadings on the general factor, the observed variables can be decomposed in the following way:

_{11}= 1. The covariance of an observed variable Y

_{ik}with the criterion can be decomposed in the following way:

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1 | For reasons of parsimony, we present standard errors and significance tests only for unstandardized solutions (across all analyses included in this paper). The corresponding information for the standardized solutions leads to the same conclusions. |

2 | From a historical point of view this early paper is also interesting for the debate on the role of general and specific factors. It showed that achievements in school subjects that do not belong to the science or language spectrum such as shops and crafts as well as drawing were more strongly correlated with the specific spatial ability factor (r = 0.461 and r = 0.692) than with the general factor (r = 0.219 and r = 0.412), whereas the g factor was more strongly correlated with all other school domains (between r = 0.374 and r = 0.586) than the specific factor (between r = −0.057 and r = 0.257). |

**Figure 1.**Bifactor model and its extensions to criterion variables. (

**a**) Bifactor model without criterion variables, (

**b**) bifactor model with correlating criterion variables (grades), and (

**c**) multiple latent regression bifactor model. The factors of the extended models depicted refer to the empirical application. G: general factor, S

_{k}: specific factors; NS-S: specific factor number series, AN-S: specific factor verbal analogies, UN-S: specific factor unfolding. E

_{ik}: measurement error variables, E

_{G1}/E

_{G2}: residuals, λ: loading parameters, β: regression coefficients, i: indicator, k: facet.

**Figure 2.**Modell with correlated first-order factors. (

**a**) Model without criterion variables, (

**b**) model with correlating criterion variables, (

**c**) multiple latent regression model, and (

**d**) multiple latent regression model with composite factors. F

_{k}: facet factors, E

_{ik}: measurement error variables, NS: facet factor number series, AN: facet factor verbal analogies, UN: facet factor unfolding, CO

_{1}/CO

_{2}: composite factors, E

_{G1}/E

_{G2}: residuals λ: loading parameters, β: regression coefficients, i: indicator, k: facet.

**Figure 3.**Bifactor(S-1) model and its extensions to criterion variables. (

**a**) Bifactor(S-1) model without criterion variables, (

**b**) bifactor(S-1) model with correlating criterion variables (grades), and (

**c**) multiple latent regression bifactor(S-1) model. The factors of the extended models depicted refer to the empirical application. G: general factor, S

_{k}: specific factors; NS-S: specific factor number series, AN-S: specific factor verbal analogies, UN-S: specific factor unfolding. E

_{ik}: measurement error variables, E

_{G1}/E

_{G2}: residuals, λ: loading parameters, β: regression coefficients, i: indicator, k: facet.

NS_{1} | NS_{2} | AN_{1} | AN_{2} | UN_{1} | UN_{2} | Math | Eng | |
---|---|---|---|---|---|---|---|---|

NS_{1} | 4.456 | |||||||

NS_{2} | 0.787 | 4.487 | ||||||

AN_{1} | 0.348 | 0.297 | 4.496 | |||||

AN_{2} | 0.376 | 0.347 | 0.687 | 4.045 | ||||

UN_{1} | 0.383 | 0.378 | 0.295 | 0.366 | 5.168 | |||

UN_{2} | 0.282 | 0.319 | 0.224 | 0.239 | 0.688 | 5.539 | ||

Math | 0.349 | 0.350 | 0.289 | 0.378 | 0.302 | 0.275 | ||

Eng | 0.225 | 0.205 | 0.263 | 0.241 | 0.135 | 0.097 | 0.469 | |

Means | 4.438 | 3.817 | 4.196 | 4.018 | 4.900 | 4.411 | ||

Proportions of the grades | 1: 0.123 2: 0.311 3: 0.297 4: 0.174 5: 0.096 | 1: 0.059 2: 0.393 3: 0.338 4: 0.174 5: 0.037 |

_{i}= number series, AN

_{i}= verbal analogies, UN

_{i}= unfolding, i = test half, Math = mathematics grade, Eng = English grade.

G-Factor Loadings | S-Factor Loadings | Residual Variances | Rel | Covariances | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

G | NS-S | AN-S | UN-S | Math | Eng | ||||||

NS_{1} | 10.651 | 10.615 | 0.882 (0.176) 0.198 | 0.802 | G | 1.887 (0.481) | 0 | 0 | 0 | 0.286 | 0.150 |

NS_{2} | 0.971 (0.098) 0.630 | 10.613 | 1.022 (0.199) 0.228 | 0.772 | NS-S | 0 | 1.687 (0.331) | 0 | 0 | 0.272 | 0.194 |

AN_{1} | 0.759 (0.161) 0.492 | 10.620 | 1.681 (0.255) 0.374 | 0.626 | AN-S | 0 | 0 | 1.726 (0.316) | 0 | 0.283 | 0.270 |

AN_{2} | 0.838 (0.162) 0.573 | 10.653 | 0.993 (0.217) 0.245 | 0.755 | UN-S | 0 | 0 | 0 | 2.207 (0.441) | 0.212 | 0.058 |

UN_{1} | 1.000 (0.199) 0.604 | 10.653 | 1.074 (0.215) 0.208 | 0.792 | Math | 0.393 (0.456) | 0.353 (0.445) | 0.371 (0.353) | 0.315 (0.428) | ||

UN_{2} | 0.781 (0.198) 0.456 | 10.631 | 2.181 (0.334) 0.394 | 0.606 | Eng | 0.206 (0.470) | 0.252 (0.475) | 0.355 (0.384) | 0.086 (0.460) | 0.469 (0.055) |

_{i}= number series, AN

_{i}= verbal analogies, UN

_{i}= unfolding, i = test half, Math = mathematics grade, Eng = English grade. All parameter estimates are significantly different from 0 (p < 0.05) with the exceptions of parameters that are set in italics.

**Table 3.**Multivariate Regression Analyses with the Mathematics and English Grades as Dependent Variables and the g Factor and the Three Specific Factors as Independent Variables.

Mathematics (R ^{2} = 0.284) | English (R ^{2} = 0.113) | |||
---|---|---|---|---|

b | b_{s} | B | b_{s} | |

G | 0.205 (0.234) | 0.282 | 0.115 (0.246) | 0.158 |

NS-S | 0.213 (0.264) | 0.276 | 0.143 (0.283) | 0.186 |

AN-S | 0.218 (0.207) | 0.286 | 0.200 (0.223) | 0.264 |

UN-S | 0.145 (0.198) | 0.216 | 0.035 (0.208) | 0.051 |

_{s}), and coefficient of determination (R

^{2}). G = general factor, NS-S = number series specific factor, AN-S = verbal analogies specific factor, UN-S = unfolding specific factor, Math = Mathematics grade, Eng = English grade. None of the estimated parameters are significantly different from 0 (all p > 0.05).

Factor Loadings | Residual Variances | Rel | Covariances | ||||||
---|---|---|---|---|---|---|---|---|---|

NS | AN | UN | Math | Eng | |||||

NS_{1} | 10.889 | 0.938 (0.200) 0.211 | 0.789 | NS | 3.519 (0.425) | 0.464 | 0.461 | 0.394 | 0.242 |

NS_{2} | 10.886 | 0.967 (0.197) 0.215 | 0.785 | AN | 1.490 (0.274) | 2.927 (0.394) | 0.408 | 0.400 | 0.304 |

AN_{1} | 10.807 | 1.569 (0.290) 0.349 | 0.651 | UN | 1.661 (0.302) | 1.338 (0.277) | 3.680 (0.493) | 0.349 | 0.142 |

AN_{2} | 10.851 | 1.118 (0.257) 0.276 | 0.724 | Math | 0.740 (0.127) | 0.685 (0.126) | 0.669 (0.134) | 0.469 | |

UN_{1} | 10.844 | 1.487 (0.365) 0.288 | 0.712 | Eng | 0.455 (0.136) | 0.520 (0.128) | 0.272 (0.133) | 0.469 | |

UN_{2} | 10.815 | 1.859 (0.390) 0.336 | 0.664 |

_{i}= number series, AN

_{i}= verbal analogies, UN

_{i}= unfolding, i = test half, Math = mathematics grade, Eng = English grade. All parameter estimates are significantly different from 0 (p < 0.05).

**Table 5.**Multivariate Regression Analyses with Mathematics and English Grades as Dependent Variables and the Three Intelligence Factors as Independent Variables.

Mathematics (R ^{2} = 0.233) | English (R ^{2} = 0.106) | |||
---|---|---|---|---|

b | b_{s} | b | b_{s} | |

NS | 0.113 ** (0.039) | 0.213 | 0.073 (0.046) | 0.137 |

AN | 0.140 ** (0.046) | 0.239 | 0.146 ** (0.050) | 0.250 |

UN | 0.080 * (0.037) | 0.153 | −0.012 (0.041) | −0.023 |

_{s}), and coefficient of determination (R

^{2}). NS = number series, AN = verbal analogies, UN = unfolding, Math = Mathematics grade, Eng = English grade. ** p < 0.01, * p < 0.05.

G-Factor Loadings | S-Factor Loadings | Residual Variances | Rel | Covariances | ||||||
---|---|---|---|---|---|---|---|---|---|---|

NS-S | AN | UN-S | Math | Eng | ||||||

NS_{1} | 0.509 (0.083) 0.412 | 10.787 | 0.938 (0.200) 0.211 | 0.789 | NS-S | 2.760 (0.333) | 0 | 0.337 | 0.235 | 0.114 |

NS_{2} | 0.509 (0.083) 0.411 | 10.784 | 0.968 (0.197) 0.216 | 0.784 | AN | 0 | 2.928 (0.394) | 0 | 0.400 | 0.304 |

AN_{1} | 10.807 | 1.568 (0.290) 0.349 | 0.651 | UN-S | 0.980 (0.244) | 0 | 3.069 (0.442) | 0.203 | 0.020 | |

AN_{2} | 10.851 | 1.117 (0.257) 0.276 | 0.724 | Math | 0.391 (0.110) | 0.685 (0.126) | 0.356 (0.124) | |||

UN_{1} | 0.457 (0.084) 0.344 | 10.771 | 1.487 (0.365) 0.288 | 0.712 | Eng | 0.190 (0.121) | 0.520 (0.128) | 0.035 (0.123) | 0.469 (0.055) | |

UN_{2} | 0.781 (0.084) 0.332 | 10.744 | 1.858 (0.390) 0.336 | 0.664 |

_{i}= number series, AN

_{i}= verbal analogies, UN

_{i}= unfolding, i = test half, AN = verbal analogies reference facet factor, NS-S = number series specific factor, UN-S = unfolding specific factor, Math = Mathematics grade, Eng = English grade. All parameter estimates are significantly different from 0 (p < 0.05) with the exceptions of parameters that are set in italics.

**Table 7.**Multivariate Regression analyses with the Mathematics and English Grades as Dependent Variables and the Three Factors of the Bifactor(S-1) Model as Independent Variables (Reference Facet = Verbal Analogies).

Mathematics (R ^{2} = 0.233) | English (R ^{2} = 0.106) | |||
---|---|---|---|---|

b | b_{s} | b | b_{s} | |

AN | 0.234 ** (0.038) | 0.400 | 0.178 ** (0.040) | 0.304 |

NS-S | 0.113 ** (0.046) | 0.188 | 0.073 (0.046) | 0.122 |

UN-S | 0.080 * (0.037) | 0.140 | −0.012 (0.041) | −0.021 |

_{s}), and coefficient of determination (R

^{2}). AN = verbal analogies reference facet factor, NS-S = number series specific factor, UN-S = unfolding specific factor, Math = Mathematics grade, Eng = English grade. ** p < 0.01, * p < 0.05.

**Table 8.**Multivariate Regression analyses with the Mathematics and English Grades as Dependent Variables and the Three Factors of the Bifactor(S-1) Model as Independent Variables (Reference Facet = Number Series).

Mathematics (R ^{2} = 0.233) | English (R ^{2} = 0.106) | |||
---|---|---|---|---|

b | b_{s} | b | b_{s} | |

NS | 0.210 ** (0.031) | 0.394 | 0.129 ** (0.037) | 0.242 |

AN-S | 0.140 ** (0.046) | 0.212 | 0.146 ** (0.050) | 0.221 |

UN-S | 0.080 * (0.037) | 0.136 | −0.012 (0.041) | −0.021 |

_{s}), and coefficient of determination (R

^{2}). NS = number series reference facet factor, AS-S = verbal analogies specific factor, UN-S = unfolding specific factor, Math = Mathematics grade, Eng = English grade. ** p < 0.01, * p < 0.05.

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**MDPI and ACS Style**

Eid, M.; Krumm, S.; Koch, T.; Schulze, J.
Bifactor Models for Predicting Criteria by General and Specific Factors: Problems of Nonidentifiability and Alternative Solutions. *J. Intell.* **2018**, *6*, 42.
https://doi.org/10.3390/jintelligence6030042

**AMA Style**

Eid M, Krumm S, Koch T, Schulze J.
Bifactor Models for Predicting Criteria by General and Specific Factors: Problems of Nonidentifiability and Alternative Solutions. *Journal of Intelligence*. 2018; 6(3):42.
https://doi.org/10.3390/jintelligence6030042

**Chicago/Turabian Style**

Eid, Michael, Stefan Krumm, Tobias Koch, and Julian Schulze.
2018. "Bifactor Models for Predicting Criteria by General and Specific Factors: Problems of Nonidentifiability and Alternative Solutions" *Journal of Intelligence* 6, no. 3: 42.
https://doi.org/10.3390/jintelligence6030042