Estimation of a hierarchical Exploratory Structural Equation Model (ESEM) using ESEMwithin-CFA

Transcription

1 Estimation of a hierarchical Exploratory Structural Equation Model (ESEM) using ESEMwithin-CFA Alexandre J.S. Morin, Substantive Methodological Synergy Research Laboratory, Department of Psychology, Concordia University, Canada & Tihomir Asparouhov, Muthén & Muthén, Los Angeles, CA Cite as: Morin, A.J.S., & Asparouhov, T. (2018). Estimation of a hierarchical Exploratory Structural Equation Model (ESEM) using ESEM-within-CFA. Montreal, QC: Substantive Methodological Synergy Research Laboratory. Morin, Arens and Marsh (2016, also see Marsh, Morin, Parker, & Kaur, 2014; Morin, Marsh & Nagengast, 2013), based on earlier work by Marsh, Nagengast & Morin (2013) and Joreskög (1969; also see Muthén & Muthén, 2009, slides ) introduced ESEM-Within-CFA (EWC) as a workaround some limitations of exploratory Structural Equation Models (ESEM). Essentially, EWC starts with an ESEM model, and re-express it in the CFA framework using the starts values generated from the initial ESEM model. More precisely, using the exact starts values generated form the initial ESEM model, the EWC model is estimated by adding m 2 constraints for identification purposes. To do so, selected parameter estimates are fixed to the values obtained from the ESEM solution. Typically: (1) The m factor variances are fixed to 1 (in a single-group ESEM, for the first group in a multiple group solution, or the first time point for a longitudinal solution). (2) A referent indicator is selected for each factor that has a large (target) loading for the factor it is designed to measure and small (non-target) cross-loadings. Then, for purposes of identification, these small cross-loadings are fixed (@) to their estimated values from the ESEM solution. (3) For all other parameter estimates, the pattern of fixed and free estimates should be the same as in the selected ESEM solution. (4) It should be noted that the mean structure from the EWC solution can be identified as in a standard CFA model (while using the ESEM start values when possible). For most applications proposed by Morin and colleagues, this EWC approach works fine. However, we recently realized that this approach was suboptimal for the estimation of higher-order ESEM models reported in Morin, Arens and Marsh (2016). More precisely, to fully re-express the ESEM model, the first-order factor variances have to be set to a value of 1 [i.e., the constraints described under (1) above]. However, in the higher-order EWC models reported in Morin, Arens and Marsh (2016), the constraints described in (1) resulted in into a EWC solution in which the residual variance of the first-order factors were fixed to 1, rather than their variance. In addition, having the factor loading of the first indicator on the higher-order factor fixed to 1 created further interference with the model estimation, taking it even further from the initial ESEM model. More precisely, if we consider one of the first-order factor (F1) and the Higher-order factor (HF): Var(F1) = residual variance (F1) + loading_hf 2 * Var(HF) So that if the constraints described above are imposed: Var(F1) = * Var(HF) To circumvent this issue, we proposed an alternative (and simpler) EWC solution in which the variance of the first-order factors can be freely estimated: (1) A referent indicator is selected for each factor that has a large (target) loading for the factor it is designed to measure and small (non-target) cross-loadings. Then, for purposes of identification, these large main loadings and small cross-loadings are fixed (@) to their estimated values from the ESEM solution. All factor variances are freely estimated. Note here that it is important to use the referent indicator approach. Fixing some other arbitrary m 2 loadings may compromise the efficiency of the estimation.

2 (2) For all other parameter estimates, the pattern of fixed and free estimates should be the same as in the selected ESEM solution. (3) It should be noted that the mean structure from the EWC solution can be identified as in a standard CFA model (while using the ESEM start values when possible). For most other contexts, the typical EWC approach would work. For instance, Morin, Arens and Marsh (2016) show how to re-express a partial mediation ESEM model with EWC in order to obtain bias-corrected bootstrap confidence intervals. In this example, the residual variance of the outcome ESEM factors has to be fixed to 1 in the EWC model to ensure its equivalence with the ESEM model in which they are also fixed to 1. Likewise, Morin, Marsh and Nagengast (2013) illustrated how EWC could be used to estimate latent change scores in the context of a longitudinal ESEM solution. In this context, the residual variance of the Time 2 factors should not be fixed to 1 in the EWC solution. However, this constraint is not required for this model to match the ESEM solution in which, due to measurement invariance, the variance of the Time 2 factors was allowed to be freely estimated. Still, we urge readers to carefully consider their specific EWC application in order to find the approach that best work with their data. The revised approach proposed here would lead to new sample inputs for the examples provided by Morin, Arens and Marsh (2016) on pages 8 (middle) and X of their online supplemental materials, as well as different fit indices associated with their Higher-Order EWC solutions: Table 1. Adjusted Goodness of Fit Statistics and Information Criteria for the Re-Estimated Higher-Order ESEM models Model χ 2 df CFI TLI RMSEA RMSEA AIC CAIC BIC SBIC 90% CI Simulated data * ; SDQ-I * ; SDQ-I without method control * ; Note. df = Degrees of freedom; CFI = comparative fit index; TLI = Tucker-Lewis index; RMSEA = root mean square error of approximation; CI = confidence interval; AIC = Akaike information criterion; CAIC = Constant AIC; BIC = Bayesian information criterion; ABIC = Sample size adjusted BIC. ESEM models were conducted with target oblique rotation. * All χ² values are significant (p <.01). Title: Hierarchical ESEM using ESEM-Within-CFA (Simulated Data)! The previous ESEM model is re-expressed using CFA. No rotation is necessary.! The model section uses the exact values of the non-standardized loadings and cross loadings! estimated from the previous model as starts values (using *). First-order factor variances are also freely! estimated, whereas the variance of the higher-order factor is fixed to 1 for identification purposes.! For the first-order factors, one item per factor has all loadings and cross loadings! constrained to be exactly equal to their ESEM values These 3 factors define a higher-order factor HF, with all higher-order loadings freely estimated.. Model: f1 BY x1* ; f1 BY x2* ; f1 BY x3@ ; f1 BY x4* ; f1 BY y1* ; f1 BY y2* ; f1 BY y3* ; f1 BY y4@ ; f1 BY z1* ; f1 BY z2* ; f1 BY z3* ; f1 BY z4@ ; f2 BY y1* ; f2 BY y2* ; f2 BY y3* ; f2 BY y4@ ; f2 BY x1* ; f2 BY x2* ; f2 BY x3@ ; f2 BY x4* ; f2 BY z1* ; f2 BY z2* ; f2 BY z3* ; f2 BY z4@ ; f3 BY z1* ; f3 BY z2* ; f3 BY z3* ; f3 BY z4@ ; f3 BY x1* ; f3 BY x2* ; f3 BY x3@ ; f3 BY x4* ; f3 BY y1* ; f3 BY y2* ; f3 BY y3* ; f3 BY y4@ ; f1-f3*1; HF BY F1*1 F2 F3; HF@1;

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4 Title: Hierarchical ESEM Model of the SDQ-I Using ESEM-Within-CFA (Real Data)! [ ] Analysis and Model sections only Model: esteem BY sdq_1* ; esteem BY sdq_2* ; esteem BY sdq_3* ; esteem BY sdq_4@ ; esteem BY sdq_5* ; esteem BY sdq_6* ; esteem BY sdq_7* ; esteem BY sdq_8* ; esteem BY sdq_9* ; esteem BY sdq_10* ; esteem BY sdq_11* ; esteem BY sdq_12* ; esteem BY sdq_13* ; esteem BY sdq_14* ; esteem BY sdq_15* ; esteem BY sdq_16@ ; esteem BY sdq_17* ; esteem BY sdq_18* ; esteem BY sdq_19* ; esteem BY sdq_20* ; esteem BY sdq_21@ ; esteem BY sdq_22@ ; esteem BY sdq_23* ; esteem BY sdq_24* ; esteem BY sdq_25* ; esteem BY sdq_26* ; esteem BY sdq_27@ ; esteem BY sdq_28* ; esteem BY sdq_29* ; esteem BY sdq_30* ; esteem BY sdq_31* ; esteem BY sdq_32* ; esteem BY sdq_33* ; esteem BY sdq_34* ; esteem BY sdq_35* ; esteem BY sdq_36* ; esteem BY sdq_37* ; esteem BY sdq_38* ; esteem BY sdq_39@ ; esteem BY sdq_40* ; esteem BY sdq_41* ; esteem BY sdq_42* ; esteem BY sdq_43* ; esteem BY sdq_44* ; esteem BY sdq_45* ; esteem BY sdq_46* ; esteem BY sdq_47* ; esteem BY sdq_48* ; esteem BY sdq_49* ; esteem BY sdq_50@ ; esteem BY sdq_51@ ; esteem BY sdq_52* ; esteem BY sdq_53* ; esteem BY sdq_54* ; esteem BY sdq_55* ; esteem BY sdq_56@ ; esteem BY sdq_57@ ; esteem BY sdq_58* ; esteem BY sdq_59* ; esteem BY sdq_60* ; esteem BY sdq_61* ; esteem BY sdq_62* ; esteem BY sdq_63* ; esteem BY sdq_64* ; esteem BY sdq_65* ; esteem BY sdq_66* ; esteem BY sdq_67* ; esteem BY sdq_68* ; esteem BY sdq_69* ; esteem BY sdq_70* ; esteem BY sdq_71* ; esteem BY sdq_72@ ; esteem BY sdq_73* ; esteem BY sdq_74* ; esteem BY sdq_75* ; esteem BY sdq_76* ; peer BY sdq_1* ; peer BY sdq_2* ; peer BY sdq_3* ; peer BY sdq_4@ ; peer BY sdq_5* ; peer BY sdq_6* ; peer BY sdq_7* ; peer BY sdq_8* ; peer BY sdq_9* ; peer BY sdq_10* ; peer BY sdq_11* ; peer BY sdq_12* ; peer BY sdq_13* ; peer BY sdq_14* ; peer BY sdq_15* ; peer BY sdq_16@ ; peer BY sdq_17* ; peer BY sdq_18* ; peer BY sdq_19* ; peer BY sdq_20* ; peer BY sdq_21@ ; peer BY sdq_22@ ; peer BY sdq_23* ; peer BY sdq_24* ; peer BY sdq_25* ; peer BY sdq_26* ; peer BY sdq_27@ ; peer BY sdq_28* ; peer BY sdq_29* ; peer BY sdq_30* ; peer BY sdq_31* ; peer BY sdq_32* ; peer BY sdq_33* ; peer BY sdq_34* ; peer BY sdq_35* ; peer BY sdq_36* ; peer BY sdq_37* ; peer BY sdq_38* ; peer BY sdq_39@ ; peer BY sdq_40* ; peer BY sdq_41* ; peer BY sdq_42* ; peer BY sdq_43* ; peer BY sdq_44* ; peer BY sdq_45* ; peer BY sdq_46* ; peer BY sdq_47* ; peer BY sdq_48* ; peer BY sdq_49* ; peer BY sdq_50@ ; peer BY sdq_51@ ; peer BY sdq_52* ; peer BY sdq_53* ; peer BY sdq_54* ; peer BY sdq_55* ; peer BY sdq_56@ ; peer BY sdq_57@ ; peer BY sdq_58* ; peer BY sdq_59* ; peer BY sdq_60* ; peer BY sdq_61* ; peer BY sdq_62* ; peer BY sdq_63* ; peer BY sdq_64* ; peer BY sdq_65* ; peer BY sdq_66* ; peer BY sdq_67* ; peer BY sdq_68* ; peer BY sdq_69* ; peer BY sdq_70* ; peer BY sdq_71* ; peer BY sdq_72@ ; peer BY sdq_73* ; peer BY sdq_74* ; peer BY sdq_75* ; peer BY sdq_76* ;

5 appear BY sdq_1* ; appear BY sdq_2* ; appear BY sdq_3* ; appear BY appear BY sdq_5* ; appear BY sdq_6* ; appear BY sdq_7* ; appear BY sdq_8* ; appear BY sdq_9* ; appear BY sdq_10* ; appear BY sdq_11* ; appear BY sdq_12* ; appear BY sdq_13* ; appear BY sdq_14* ; appear BY sdq_15* ; appear BY appear BY sdq_17* ; appear BY sdq_18* ; appear BY sdq_19* ; appear BY sdq_20* ; appear BY appear BY appear BY sdq_23* ; appear BY sdq_24* ; appear BY sdq_25* ; appear BY sdq_26* ; appear BY appear BY sdq_28* ; appear BY sdq_29* ; appear BY sdq_30* ; appear BY sdq_31* ; appear BY sdq_32* ; appear BY sdq_33* ; appear BY sdq_34* ; appear BY sdq_35* ; appear BY sdq_36* ; appear BY sdq_37* ; appear BY sdq_38* ; appear BY appear BY sdq_40* ; appear BY sdq_41* ; appear BY sdq_42* ; appear BY sdq_43* ; appear BY sdq_44* ; appear BY sdq_45* ; appear BY sdq_46* ; appear BY sdq_47* ; appear BY sdq_48* ; appear BY sdq_49* ; appear BY appear BY appear BY sdq_52* ; appear BY sdq_53* ; appear BY sdq_54* ; appear BY sdq_55* ; appear BY appear BY appear BY sdq_58* ; appear BY sdq_59* ; appear BY sdq_60* ; appear BY sdq_61* ; appear BY sdq_62* ; appear BY sdq_63* ; appear BY sdq_64* ; appear BY sdq_65* ; appear BY sdq_66* ; appear BY sdq_67* ; appear BY sdq_68* ; appear BY sdq_69* ; appear BY sdq_70* ; appear BY sdq_71* ; appear BY appear BY sdq_73* ; appear BY sdq_74* ; appear BY sdq_75* ; appear BY sdq_76* ; phy BY sdq_1* ; phy BY sdq_2* ; phy BY sdq_3* ; phy BY phy BY sdq_5* ; phy BY sdq_6* ; phy BY sdq_7* ; phy BY sdq_8* ; phy BY sdq_9* ; phy BY sdq_10* ; phy BY sdq_11* ; phy BY sdq_12* ; phy BY sdq_13* ; phy BY sdq_14* ; phy BY sdq_15* ; phy BY phy BY sdq_17* ; phy BY sdq_18* ; phy BY sdq_19* ; phy BY sdq_20* ; phy BY phy BY phy BY sdq_23* ; phy BY sdq_24* ; phy BY sdq_25* ; phy BY sdq_26* ; phy BY phy BY sdq_28* ; phy BY sdq_29* ; phy BY sdq_30* ; phy BY sdq_31* ; phy BY sdq_32* ; phy BY sdq_33* ; phy BY sdq_34* ; phy BY sdq_35* ; phy BY sdq_36* ; phy BY sdq_37* ; phy BY sdq_38* ; phy BY phy BY sdq_40* ; phy BY sdq_41* ; phy BY sdq_42* ; phy BY sdq_43* ; phy BY sdq_44* ; phy BY sdq_45* ; phy BY sdq_46* ; phy BY sdq_47* ; phy BY sdq_48* ; phy BY sdq_49* ; phy BY phy BY phy BY sdq_52* ; phy BY sdq_53* ; phy BY sdq_54* ; phy BY sdq_55* ; phy BY phy BY phy BY sdq_58* ; phy BY sdq_59* ; phy BY sdq_60* ; phy BY sdq_61* ; phy BY sdq_62* ; phy BY sdq_63* ; phy BY sdq_64* ; phy BY sdq_65* ; phy BY sdq_66* ; phy BY sdq_67* ; phy BY sdq_68* ; phy BY sdq_69* ; phy BY sdq_70* ; phy BY sdq_71* ; phy BY phy BY sdq_73* ; phy BY sdq_74* ; phy BY sdq_75* ; phy BY sdq_76* ; parent BY sdq_1* ; parent BY sdq_2* ; parent BY sdq_3* ; parent BY parent BY sdq_5* ; parent BY sdq_6* ; parent BY sdq_7* ; parent BY sdq_8* ; parent BY sdq_9* ; parent BY sdq_10* ; parent BY sdq_11* ; parent BY sdq_12* ;

6 parent BY sdq_13* ; parent BY sdq_14* ; parent BY sdq_15* ; parent BY parent BY sdq_17* ; parent BY sdq_18* ; parent BY sdq_19* ; parent BY sdq_20* ; parent BY parent BY parent BY sdq_23* ; parent BY sdq_24* ; parent BY sdq_25* ; parent BY sdq_26* ; parent BY parent BY sdq_28* ; parent BY sdq_29* ; parent BY sdq_30* ; parent BY sdq_31* ; parent BY sdq_32* ; parent BY sdq_33* ; parent BY sdq_34* ; parent BY sdq_35* ; parent BY sdq_36* ; parent BY sdq_37* ; parent BY sdq_38* ; parent BY parent BY sdq_40* ; parent BY sdq_41* ; parent BY sdq_42* ; parent BY sdq_43* ; parent BY sdq_44* ; parent BY sdq_45* ; parent BY sdq_46* ; parent BY sdq_47* ; parent BY sdq_48* ; parent BY sdq_49* ; parent BY parent BY parent BY sdq_52* ; parent BY sdq_53* ; parent BY sdq_54* ; parent BY sdq_55* ; parent BY parent BY parent BY sdq_58* ; parent BY sdq_59* ; parent BY sdq_60* ; parent BY sdq_61* ; parent BY sdq_62* ; parent BY sdq_63* ; parent BY sdq_64* ; parent BY sdq_65* ; parent BY sdq_66* ; parent BY sdq_67* ; parent BY sdq_68* ; parent BY sdq_69* ; parent BY sdq_70* ; parent BY sdq_71* ; parent BY parent BY sdq_73* ; parent BY sdq_74* ; parent BY sdq_75* ; parent BY sdq_76* ; schocom BY sdq_1* ; schocom BY sdq_2* ; schocom BY sdq_3* ; schocom BY schocom BY sdq_5* ; schocom BY sdq_6* ; schocom BY sdq_7* ; schocom BY sdq_8* ; schocom BY sdq_9* ; schocom BY sdq_10* ; schocom BY sdq_11* ; schocom BY sdq_12* ; schocom BY sdq_13* ; schocom BY sdq_14* ; schocom BY sdq_15* ; schocom BY schocom BY sdq_17* ; schocom BY sdq_18* ; schocom BY sdq_19* ; schocom BY sdq_20* ; schocom BY schocom BY schocom BY sdq_23* ; schocom BY sdq_24* ; schocom BY sdq_25* ; schocom BY sdq_26* ; schocom BY schocom BY sdq_28* ; schocom BY sdq_29* ; schocom BY sdq_30* ; schocom BY sdq_31* ; schocom BY sdq_32* ; schocom BY sdq_33* ; schocom BY sdq_34* ; schocom BY sdq_35* ; schocom BY sdq_36* ; schocom BY sdq_37* ; schocom BY sdq_38* ; schocom BY schocom BY sdq_40* ; schocom BY sdq_41* ; schocom BY sdq_42* ; schocom BY sdq_43* ; schocom BY sdq_44* ; schocom BY sdq_45* ; schocom BY sdq_46* ; schocom BY sdq_47* ; schocom BY sdq_48* ; schocom BY sdq_49* ; schocom BY schocom BY schocom BY sdq_52* ; schocom BY sdq_53* ; schocom BY sdq_54* ; schocom BY sdq_55* ; schocom BY schocom BY schocom BY sdq_58* ; schocom BY sdq_59* ; schocom BY sdq_60* ; schocom BY sdq_61* ; schocom BY sdq_62* ; schocom BY sdq_63* ; schocom BY sdq_64* ; schocom BY sdq_65* ; schocom BY sdq_66* ; schocom BY sdq_67* ; schocom BY sdq_68* ; schocom BY sdq_69* ; schocom BY sdq_70* ; schocom BY sdq_71* ; schocom BY schocom BY sdq_73* ; schocom BY sdq_74* ; schocom BY sdq_75* ; schocom BY sdq_76* ; schoaff BY sdq_1* ; schoaff BY sdq_2* ; schoaff BY sdq_3* ; schoaff BY schoaff BY sdq_5* ; schoaff BY sdq_6* ; schoaff BY sdq_7* ; schoaff BY sdq_8* ; schoaff BY sdq_9* ; schoaff BY sdq_10* ; schoaff BY sdq_11* ; schoaff BY sdq_12* ; schoaff BY sdq_13* ; schoaff BY sdq_14* ; schoaff BY sdq_15* ; schoaff BY schoaff BY sdq_17* ; schoaff BY sdq_18* ; schoaff BY sdq_19* ; schoaff BY sdq_20* ; schoaff BY

7 schoaff BY schoaff BY sdq_23* ; schoaff BY sdq_24* ; schoaff BY sdq_25* ; schoaff BY sdq_26* ; schoaff BY schoaff BY sdq_28* ; schoaff BY sdq_29* ; schoaff BY sdq_30* ; schoaff BY sdq_31* ; schoaff BY sdq_32* ; schoaff BY sdq_33* ; schoaff BY sdq_34* ; schoaff BY sdq_35* ; schoaff BY sdq_36* ; schoaff BY sdq_37* ; schoaff BY sdq_38* ; schoaff BY schoaff BY sdq_40* ; schoaff BY sdq_41* ; schoaff BY sdq_42* ; schoaff BY sdq_43* ; schoaff BY sdq_44* ; schoaff BY sdq_45* ; schoaff BY sdq_46* ; schoaff BY sdq_47* ; schoaff BY sdq_48* ; schoaff BY sdq_49* ; schoaff BY schoaff BY schoaff BY sdq_52* ; schoaff BY sdq_53* ; schoaff BY sdq_54* ; schoaff BY sdq_55* ; schoaff BY schoaff BY schoaff BY sdq_58* ; schoaff BY sdq_59* ; schoaff BY sdq_60* ; schoaff BY sdq_61* ; schoaff BY sdq_62* ; schoaff BY sdq_63* ; schoaff BY sdq_64* ; schoaff BY sdq_65* ; schoaff BY sdq_66* ; schoaff BY sdq_67* ; schoaff BY sdq_68* ; schoaff BY sdq_69* ; schoaff BY sdq_70* ; schoaff BY sdq_71* ; schoaff BY schoaff BY sdq_73* ; schoaff BY sdq_74* ; schoaff BY sdq_75* ; schoaff BY sdq_76* ; germcom BY sdq_1* ; germcom BY sdq_2* ; germcom BY sdq_3* ; germcom BY germcom BY sdq_5* ; germcom BY sdq_6* ; germcom BY sdq_7* ; germcom BY sdq_8* ; germcom BY sdq_9* ; germcom BY sdq_10* ; germcom BY sdq_11* ; germcom BY sdq_12* ; germcom BY sdq_13* ; germcom BY sdq_14* ; germcom BY sdq_15* ; germcom BY germcom BY sdq_17* ; germcom BY sdq_18* ; germcom BY sdq_19* ; germcom BY sdq_20* ; germcom BY germcom BY germcom BY sdq_23* ; germcom BY sdq_24* ; germcom BY sdq_25* ; germcom BY sdq_26* ; germcom BY germcom BY sdq_28* ; germcom BY sdq_29* ; germcom BY sdq_30* ; germcom BY sdq_31* ; germcom BY sdq_32* ; germcom BY sdq_33* ; germcom BY sdq_34* ; germcom BY sdq_35* ; germcom BY sdq_36* ; germcom BY sdq_37* ; germcom BY sdq_38* ; germcom BY germcom BY sdq_40* ; germcom BY sdq_41* ; germcom BY sdq_42* ; germcom BY sdq_43* ; germcom BY sdq_44* ; germcom BY sdq_45* ; germcom BY sdq_46* ; germcom BY sdq_47* ; germcom BY sdq_48* ; germcom BY sdq_49* ; germcom BY germcom BY germcom BY sdq_52* ; germcom BY sdq_53* ; germcom BY sdq_54* ; germcom BY sdq_55* ; germcom BY germcom BY germcom BY sdq_58* ; germcom BY sdq_59* ; germcom BY sdq_60* ; germcom BY sdq_61* ; germcom BY sdq_62* ; germcom BY sdq_63* ; germcom BY sdq_64* ; germcom BY sdq_65* ; germcom BY sdq_66* ; germcom BY sdq_67* ; germcom BY sdq_68* ; germcom BY sdq_69* ; germcom BY sdq_70* ; germcom BY sdq_71* ; germcom BY germcom BY sdq_73* ; germcom BY sdq_74* ; germcom BY sdq_75* ; germcom BY sdq_76* ; germaff BY sdq_1* ; germaff BY sdq_2* ; germaff BY sdq_3* ; germaff BY germaff BY sdq_5* ; germaff BY sdq_6* ; germaff BY sdq_7* ; germaff BY sdq_8* ; germaff BY sdq_9* ; germaff BY sdq_10* ; germaff BY sdq_11* ; germaff BY sdq_12* ; germaff BY sdq_13* ; germaff BY sdq_14* ; germaff BY sdq_15* ; germaff BY germaff BY sdq_17* ; germaff BY sdq_18* ; germaff BY sdq_19* ; germaff BY sdq_20* ; germaff BY germaff BY germaff BY sdq_23* ; germaff BY sdq_24* ; germaff BY sdq_25* ; germaff BY sdq_26* ; germaff BY germaff BY sdq_28* ; germaff BY sdq_29* ; germaff BY sdq_30* ;

8 germaff BY sdq_31* ; germaff BY sdq_32* ; germaff BY sdq_33* ; germaff BY sdq_34* ; germaff BY sdq_35* ; germaff BY sdq_36* ; germaff BY sdq_37* ; germaff BY sdq_38* ; germaff BY germaff BY sdq_40* ; germaff BY sdq_41* ; germaff BY sdq_42* ; germaff BY sdq_43* ; germaff BY sdq_44* ; germaff BY sdq_45* ; germaff BY sdq_46* ; germaff BY sdq_47* ; germaff BY sdq_48* ; germaff BY sdq_49* ; germaff BY germaff BY germaff BY sdq_52* ; germaff BY sdq_53* ; germaff BY sdq_54* ; germaff BY sdq_55* ; germaff BY germaff BY germaff BY sdq_58* ; germaff BY sdq_59* ; germaff BY sdq_60* ; germaff BY sdq_61* ; germaff BY sdq_62* ; germaff BY sdq_63* ; germaff BY sdq_64* ; germaff BY sdq_65* ; germaff BY sdq_66* ; germaff BY sdq_67* ; germaff BY sdq_68* ; germaff BY sdq_69* ; germaff BY sdq_70* ; germaff BY sdq_71* ; germaff BY germaff BY sdq_73* ; germaff BY sdq_74* ; germaff BY sdq_75* ; germaff BY sdq_76* ; mathcom BY sdq_1* ; mathcom BY sdq_2* ; mathcom BY sdq_3* ; mathcom BY mathcom BY sdq_5* ; mathcom BY sdq_6* ; mathcom BY sdq_7* ; mathcom BY sdq_8* ; mathcom BY sdq_9* ; mathcom BY sdq_10* ; mathcom BY sdq_11* ; mathcom BY sdq_12* ; mathcom BY sdq_13* ; mathcom BY sdq_14* ; mathcom BY sdq_15* ; mathcom BY mathcom BY sdq_17* ; mathcom BY sdq_18* ; mathcom BY sdq_19* ; mathcom BY sdq_20* ; mathcom BY mathcom BY mathcom BY sdq_23* ; mathcom BY sdq_24* ; mathcom BY sdq_25* ; mathcom BY sdq_26* ; mathcom BY mathcom BY sdq_28* ; mathcom BY sdq_29* ; mathcom BY sdq_30* ; mathcom BY sdq_31* ; mathcom BY sdq_32* ; mathcom BY sdq_33* ; mathcom BY sdq_34* ; mathcom BY sdq_35* ; mathcom BY sdq_36* ; mathcom BY sdq_37* ; mathcom BY sdq_38* ; mathcom BY mathcom BY sdq_40* ; mathcom BY sdq_41* ; mathcom BY sdq_42* ; mathcom BY sdq_43* ; mathcom BY sdq_44* ; mathcom BY sdq_45* ; mathcom BY sdq_46* ; mathcom BY sdq_47* ; mathcom BY sdq_48* ; mathcom BY sdq_49* ; mathcom BY mathcom BY mathcom BY sdq_52* ; mathcom BY sdq_53* ; mathcom BY sdq_54* ; mathcom BY sdq_55* ; mathcom BY mathcom BY mathcom BY sdq_58* ; mathcom BY sdq_59* ; mathcom BY sdq_60* ; mathcom BY sdq_61* ; mathcom BY sdq_62* ; mathcom BY sdq_63* ; mathcom BY sdq_64* ; mathcom BY sdq_65* ; mathcom BY sdq_66* ; mathcom BY sdq_67* ; mathcom BY sdq_68* ; mathcom BY sdq_69* ; mathcom BY sdq_70* ; mathcom BY sdq_71* ; mathcom BY mathcom BY sdq_73* ; mathcom BY sdq_74* ; mathcom BY sdq_75* ; mathcom BY sdq_76* ; mathaff BY sdq_1* ; mathaff BY sdq_2* ; mathaff BY sdq_3* ; mathaff BY mathaff BY sdq_5* ; mathaff BY sdq_6* ; mathaff BY sdq_7* ; mathaff BY sdq_8* ; mathaff BY sdq_9* ; mathaff BY sdq_10* ; mathaff BY sdq_11* ; mathaff BY sdq_12* ; mathaff BY sdq_13* ; mathaff BY sdq_14* ; mathaff BY sdq_15* ; mathaff BY mathaff BY sdq_17* ; mathaff BY sdq_18* ; mathaff BY sdq_19* ; mathaff BY sdq_20* ; mathaff BY mathaff BY mathaff BY sdq_23* ; mathaff BY sdq_24* ; mathaff BY sdq_25* ; mathaff BY sdq_26* ; mathaff BY mathaff BY sdq_28* ; mathaff BY sdq_29* ; mathaff BY sdq_30* ; mathaff BY sdq_31* ; mathaff BY sdq_32* ; mathaff BY sdq_33* ; mathaff BY sdq_34* ; mathaff BY sdq_35* ; mathaff BY sdq_36* ; mathaff BY sdq_37* ; mathaff BY sdq_38* ; mathaff BY

9 mathaff BY sdq_40* ; mathaff BY sdq_41* ; mathaff BY sdq_42* ; mathaff BY sdq_43* ; mathaff BY sdq_44* ; mathaff BY sdq_45* ; mathaff BY sdq_46* ; mathaff BY sdq_47* ; mathaff BY sdq_48* ; mathaff BY sdq_49* ; mathaff BY mathaff BY mathaff BY sdq_52* ; mathaff BY sdq_53* ; mathaff BY sdq_54* ; mathaff BY sdq_55* ; mathaff BY mathaff BY mathaff BY sdq_58* ; mathaff BY sdq_59* ; mathaff BY sdq_60* ; mathaff BY sdq_61* ; mathaff BY sdq_62* ; mathaff BY sdq_63* ; mathaff BY sdq_64* ; mathaff BY sdq_65* ; mathaff BY sdq_66* ; mathaff BY sdq_67* ; mathaff BY sdq_68* ; mathaff BY sdq_69* ; mathaff BY sdq_70* ; mathaff BY sdq_71* ; mathaff BY mathaff BY sdq_73* ; mathaff BY sdq_74* ; mathaff BY sdq_75* ; mathaff BY sdq_76* ; esteem*1 peer*1 appear*1 phy*1 parent*1; schocom*1 schoaff*1 Germcom*1 Germaff*1 MathAff*1 MathCom*1 ;! Higher-order factor general by esteem*1 peer appear phy parent schocom schoaff Germcom Germaff MathAff MathCom ; general@1;! Method Factor (negative items) MF BY SDQ_30* SDQ_17 SDQ_12 SDQ_21 SDQ_47 SDQ_23 SDQ_33 SDQ_65 SDQ_75 SDQ_6 SDQ_37 SDQ_61; MF@1; [MF@0]; MF WITH general@0 esteem@0 peer@0 appear@0 phy@0 parent@0 schocom@0 schoaff@0 Germcom@0 Germaff@0 MathAff@0 MathCom@0;! correlated uniquenesses between parallel worded items SDQ_11 with SDQ_51 SDQ_71; SDQ_51 with SDQ_71; SDQ_25 with SDQ_35 SDQ_39; SDQ_35 with SDQ_39; SDQ_41 with SDQ_68 SDQ_9; SDQ_68 with SDQ_9; SDQ_57 with SDQ_20 SDQ_55; SDQ_20 with SDQ_55; SDQ_23 with SDQ_6 SDQ_65; SDQ_6 with SDQ_65; SDQ_4 with SDQ_27 SDQ_16; SDQ_27 with SDQ_16; SDQ_18 with SDQ_59 SDQ_2; SDQ_59 with SDQ_2; SDQ_49 with SDQ_13 SDQ_63; SDQ_13 with SDQ_63; SDQ_73 with SDQ_43 SDQ_31; SDQ_43 with SDQ_31; SDQ_47 with SDQ_75 SDQ_33; SDQ_75 with SDQ_33; References: Jöreskog, K. G. (1969). A general approach to confirmatory maximum likelihood factor analysis. Psychometrika, 34, Morin, A. J. S., Arens, A., & Marsh, H. (2016). A bifactor exploratory structural equation modeling framework for the identification of distinct sources of construct-relevant psychometric multidimensionality. Structural Equation Modeling, 23, Marsh, H.W., Morin, A.J.S., Parker, P.D., & Kaur, G. (2014). Exploratory structural equation modelling: An integration of the best features of exploratory and confirmatory factor analyses. Annual Review of Clinical Psychology, 10, Marsh, H.W., Nagengast, B., & Morin, A.J.S. (2013). Measurement invariance of big-five factors Construct-Relevant Multidimensionality over the life span: ESEM tests of gender, age, plasticity, maturity, and La Dolce Vita effects. Developmental Psychology, 49, Morin, A. J. S., Marsh, H. W., & Nagengast, B. (2013). Exploratory structural equation modeling. In Hancock, G. R., & Mueller, R. O. (Eds.). Structural equation modeling: A second course (2nd ed., pp ). Charlotte, NC: Information Age Publishing, Inc. Muthén, L. K., & Muthén, B. O. (2009). Mplus short courses: Topic 1: Exploratory factor analysis, confirmatory factor analysis, and structural equation modeling for continuous outcomes. Los Angeles CA: Muthén & Muthén. Retrieved from