Example of EQS with ‘raw data’ Sintaxis generator Method = robust Multiple group Mean and covariances (MACS)

Purpose … We have two samples with 162 clients, car retail sellers, from Spain and 109 from USA For each client the percibed quality of service and the loyalty to the car retailer 7 points likert scale for quality items. 5 points likert scale for Loyalty items. We want to research if perceived quality is related with Loyalty, and whether the relation is the same in the two countries.

Modelo Calidad Q2 Q3 Q4 Q5 Q7 Q1 Q6 Lealtad L2 L3 L4 L5 L1

Modelo muestra española /TITLE modelo_espana /SPECIFICATIONS DATA='c:\eqs61\espa_auto.ess'; VARIABLES=12; CASES=162; METHOD=ML,ROBUST; ANALYSIS=COVARIANCE; MATRIX=RAW; /LABELS V1=Q1; V2=Q2; V3=Q3; V4=Q4; V5=Q5; V6=Q6; V7=Q7; V8=L1; V9=L2; V10=L3; V11=L4; V12=L5; /EQUATIONS V1 = 1F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; V4 = *F1 + E4; V5 = *F1 + E5; V6 = *F1 + E6; V7 = *F1 + E7; V8 = 1F2 + E8; V9 = *F2 + E9; V10 = *F2 + E10; V11 = *F2 + E11; V12 = *F2 + E12; F2 = *F1 + D2; /VARIANCES F1 = *; E1 = *; E2 = *; E3 = *; E4 = *; E5 = *; E6 = *; E7 = *; E8 = *; E9 = *; E10 = *; E11 = *; E12 = *; D2 = *; /COVARIANCES /PRINT EIS; FIT=ALL; TABLE=EQUATION; /END

Modelo muestra española GOODNESS OF FIT SUMMARY FOR METHOD = ML INDEPENDENCE MODEL CHI-SQUARE = ON 66 DEGREES OF FREEDOM INDEPENDENCE AIC = INDEPENDENCE CAIC = MODEL AIC = MODEL CAIC = CHI-SQUARE = BASED ON 53 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS THE NORMAL THEORY RLS CHI-SQUARE FOR THIS ML SOLUTION IS FIT INDICES BENTLER-BONETT NORMED FIT INDEX =.932 BENTLER-BONETT NON-NORMED FIT INDEX =.951 COMPARATIVE FIT INDEX (CFI) =.961 BOLLEN (IFI) FIT INDEX =.961 MCDONALD (MFI) FIT INDEX =.815 LISREL GFI FIT INDEX =.899 LISREL AGFI FIT INDEX =.851 ROOT MEAN-SQUARE RESIDUAL (RMR) =.119 STANDARDIZED RMR =.041 ROOT MEAN-SQUARE ERROR OF APPROXIMATION (RMSEA) = % CONFIDENCE INTERVAL OF RMSEA (.067,.109)

Non-normality When normality is violated, what are the implications (robustness)? –We tend to reject correct models more often χ2 is larger –We tend to consider significative parameters that in reality are not significant Inflate the precision of parameter estimates

Spanish sample GOODNESS OF FIT SUMMARY FOR METHOD = ROBUST ROBUST INDEPENDENCE MODEL CHI-SQUARE = ON 66 DEGREES OF FREEDOM INDEPENDENCE AIC = INDEPENDENCE CAIC = MODEL AIC = MODEL CAIC = SATORRA-BENTLER SCALED CHI-SQUARE = ON 53 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS RESIDUAL-BASED TEST STATISTIC = PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS YUAN-BENTLER RESIDUAL-BASED TEST STATISTIC = PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS YUAN-BENTLER RESIDUAL-BASED F-STATISTIC = DEGREES OF FREEDOM = 53, 107 PROBABILITY VALUE FOR THE F-STATISTIC IS FIT INDICES BENTLER-BONETT NORMED FIT INDEX =.917 BENTLER-BONETT NON-NORMED FIT INDEX =.954 COMPARATIVE FIT INDEX (CFI) =.963 BOLLEN (IFI) FIT INDEX =.964 MCDONALD (MFI) FIT INDEX =.887 ROOT MEAN-SQUARE ERROR OF APPROXIMATION (RMSEA) = % CONFIDENCE INTERVAL OF RMSEA (.043,.090)

… spanish sample(cnt.) MEASUREMENT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS STATISTICS SIGNIFICANT AT THE 5% LEVEL ARE MARKED (ROBUST STATISTICS IN PARENTHESES) Q1 =V1 = F E1 Q2 =V2 = 1.220*F E2.125 (.127) ( Q3 =V3 = 1.197*F E3.120 (.135) ( Q4 =V4 = 1.295*F E4.125 (.139) ( Q5 =V5 = 1.016*F E5.112 (.135) ( Q6 =V6 = 1.103*F E6.121 (.155) ( Q7 =V7 = 1.103*F E7.121 (.130) ( L1 =V8 = F E8 L2 =V9 =.786*F E9.079 (.066) ( L3 =V10 = 1.051*F E (.046) ( L4 =V11 = 1.078*F E (.047) ( L5 =V12 = 1.056*F E (.052) ( CONSTRUCT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS STATISTICS SIGNIFICANT AT THE 5% LEVEL ARE MARKED (ROBUST STATISTICS IN PARENTHESES) F2 =F2 = 1.147*F D2.125 (.144) (

… spanish sample(cnt.) STANDARDIZED SOLUTION: R-SQUARED Q1 =V1 =.686 F E1.470 Q2 =V2 =.836*F E2.699 Q3 =V3 =.860*F E3.739 Q4 =V4 =.898*F E4.807 Q5 =V5 =.771*F E5.594 Q6 =V6 =.777*F E6.603 Q7 =V7 =.777*F E7.604 L1 =V8 =.928 F E8.862 L2 =V9 =.650*F E9.423 L3 =V10 =.907*F E L4 =V11 =.918*F E L5 =V12 =.852*F E F2 =F2 =.829*F D2.688

USA sample /TITLE modelo_USA /SPECIFICATIONS DATA='c:\eqs61\usa_auto.ess'; VARIABLES=12; CASES=109; METHOD=ML,ROBUST; ANALYSIS=COVARIANCE; MATRIX=RAW; /LABELS V1=Q1; V2=Q2; V3=Q3; V4=Q4; V5=Q5; V6=Q6; V7=Q7; V8=L1; V9=L2; V10=L3; V11=L4; V12=L5; /EQUATIONS V1 = 1F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; V4 = *F1 + E4; V5 = *F1 + E5; V6 = *F1 + E6; V7 = *F1 + E7; V8 = 1F2 + E8; V9 = *F2 + E9; V10 = *F2 + E10; V11 = *F2 + E11; V12 = *F2 + E12; F2 = *F1 + D2; /VARIANCES F1 = *; E1 = *; E2 = *; E3 = *; E4 = *; E5 = *; E6 = *; E7 = *; E8 = *; E9 = *; E10 = *; E11 = *; E12 = *; D2 = *; /COVARIANCES /PRINT EIS; FIT=ALL; TABLE=EQUATION; /END

USA sample (cnt.) GOODNESS OF FIT SUMMARY FOR METHOD = ML INDEPENDENCE MODEL CHI-SQUARE = ON 66 DEGREES OF FREEDOM INDEPENDENCE AIC = INDEPENDENCE CAIC = MODEL AIC = MODEL CAIC = CHI-SQUARE = BASED ON 53 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS THE NORMAL THEORY RLS CHI-SQUARE FOR THIS ML SOLUTION IS FIT INDICES BENTLER-BONETT NORMED FIT INDEX =.922 BENTLER-BONETT NON-NORMED FIT INDEX =.945 COMPARATIVE FIT INDEX (CFI) =.956 BOLLEN (IFI) FIT INDEX =.956 MCDONALD (MFI) FIT INDEX =.749 LISREL GFI FIT INDEX =.857 LISREL AGFI FIT INDEX =.789 ROOT MEAN-SQUARE RESIDUAL (RMR) =.098 STANDARDIZED RMR =.037 ROOT MEAN-SQUARE ERROR OF APPROXIMATION (RMSEA) = % CONFIDENCE INTERVAL OF RMSEA (.078,.130)

USA sample (cnt.) GOODNESS OF FIT SUMMARY FOR METHOD = ROBUST ROBUST INDEPENDENCE MODEL CHI-SQUARE = ON 66 DEGREES OF FREEDOM INDEPENDENCE AIC = INDEPENDENCE CAIC = MODEL AIC = MODEL CAIC = SATORRA-BENTLER SCALED CHI-SQUARE = ON 53 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS RESIDUAL-BASED TEST STATISTIC = PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS YUAN-BENTLER RESIDUAL-BASED TEST STATISTIC = PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS YUAN-BENTLER RESIDUAL-BASED F-STATISTIC = DEGREES OF FREEDOM = 53, 53 PROBABILITY VALUE FOR THE F-STATISTIC IS FIT INDICES BENTLER-BONETT NORMED FIT INDEX =.910 BENTLER-BONETT NON-NORMED FIT INDEX =.970 COMPARATIVE FIT INDEX (CFI) =.976 BOLLEN (IFI) FIT INDEX =.976 MCDONALD (MFI) FIT INDEX =.922 ROOT MEAN-SQUARE ERROR OF APPROXIMATION (RMSEA) = % CONFIDENCE INTERVAL OF RMSEA (.000,.088)

USA sample (cnt.) MEASUREMENT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS STATISTICS SIGNIFICANT AT THE 5% LEVEL ARE MARKED (ROBUST STATISTICS IN PARENTHESES) Q1 =V1 = F E1 Q2 =V2 =.895*F E2.129 (.118) ( Q3 =V3 = 1.072*F E3.125 (.126) ( Q4 =V4 = 1.132*F E4.128 (.117) ( Q5 =V5 =.999*F E5.114 (.110) ( Q6 =V6 = 1.165*F E6.132 (.124) ( Q7 =V7 = 1.218*F E7.146 (.130) ( L1 =V8 = F E8 L2 =V9 = 1.046*F E9.078 (.085) ( L3 =V10 = 1.075*F E (.056) ( L4 =V11 = 1.052*F E (.049) ( L5 =V12 = 1.059*F E (.066) ( CONSTRUCT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS STATISTICS SIGNIFICANT AT THE 5% LEVEL ARE MARKED (ROBUST STATISTICS IN PARENTHESES) F2 =F2 = 1.092*F D2.143 (.126) (

USA sample (cnt.) STANDARDIZED SOLUTION: R-SQUARED Q1 =V1 =.715 F E1.511 Q2 =V2 =.691*F E2.477 Q3 =V3 =.851*F E3.725 Q4 =V4 =.878*F E4.771 Q5 =V5 =.873*F E5.761 Q6 =V6 =.876*F E6.768 Q7 =V7 =.830*F E7.689 L1 =V8 =.963 F E8.927 L2 =V9 =.821*F E9.674 L3 =V10 =.951*F E L4 =V11 =.966*F E L5 =V12 =.895*F E F2 =F2 =.781*F D2.609

Multiple group

España - USA /TITLE Model built by EQS 6 for Windows in Group 1 /SPECIFICATIONS DATA='c:\eqs61\espa_auto.ess'; VARIABLES=12; CASES=162; GROUPS=2; METHOD=ML,ROBUST; ANALYSIS=COVARIANCE; MATRIX=RAW; /LABELS V1=Q1; V2=Q2; V3=Q3; V4=Q4; V5=Q5; V6=Q6; V7=Q7; V8=L1; V9=L2; V10=L3; V11=L4; V12=L5; /EQUATIONS V1 = 1F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; V4 = *F1 + E4; V5 = *F1 + E5; V6 = *F1 + E6; V7 = *F1 + E7; V8 = 1F2 + E8; V9 = *F2 + E9; V10 = *F2 + E10; V11 = *F2 + E11; V12 = *F2 + E12; F2 = *F1 + D2; /VARIANCES F1 = *; E1 = *; E2 = *; E3 = *; E4 = *; E5 = *; E6 = *; E7 = *; E8 = *; E9 = *; E10 = *; E11 = *; E12 = *; D2 = *; /COVARIANCES /END /TITLE Model built by EQS 6 for Windows in Group 2 /SPECIFICATIONS DATA='C:\EQS61\usa_auto.ESS'; VARIABLES=12; CASES=109; METHOD=ML,ROBUST; ANALYSIS=COVARIANCE; MATRIX=RAW; /LABELS V1=Q1; V2=Q2; V3=Q3; V4=Q4; V5=Q5; V6=Q6; V7=Q7; V8=L1; V9=L2; V10=L3; V11=L4; V12=L5; /EQUATIONS V1 = 1F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; V4 = *F1 + E4; V5 = *F1 + E5; V6 = *F1 + E6; V7 = *F1 + E7; V8 = 1F2 + E8; V9 = *F2 + E9; V10 = *F2 + E10; V11 = *F2 + E11; V12 = *F2 + E12; F2 = *F1 + D2; /VARIANCES F1 = *; E1 TO E12 = *; D2 = *; /COVARIANCES /PRINT FIT=ALL; TABLE=EQUATION; /LMTEST PROCESS=SIMULTANEOUS; SET=PVV,PFV,PFF,PDD,GVV,GVF,GFV,GFF, BVF,BFF; /END

España - USA GOODNESS OF FIT SUMMARY FOR METHOD = ML INDEPENDENCE MODEL CHI-SQUARE = ON 132 DEGREES OF FREEDOM INDEPENDENCE AIC = INDEPENDENCE CAIC = MODEL AIC = MODEL CAIC = CHI-SQUARE = BASED ON 106 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS FIT INDICES BENTLER-BONETT NORMED FIT INDEX =.927 BENTLER-BONETT NON-NORMED FIT INDEX =.948 COMPARATIVE FIT INDEX (CFI) =.959 BOLLEN (IFI) FIT INDEX =.959 MCDONALD (MFI) FIT INDEX =.788 LISREL GFI FIT INDEX =.882 LISREL AGFI FIT INDEX =.826 ROOT MEAN-SQUARE RESIDUAL (RMR) =.109 STANDARDIZED RMR =.039 ROOT MEAN-SQUARE ERROR OF APPROXIMATION (RMSEA) = % CONFIDENCE INTERVAL OF RMSEA (.055,.079)

España – USA GOODNESS OF FIT SUMMARY FOR METHOD = ROBUST ROBUST INDEPENDENCE MODEL CHI-SQUARE = ON 132 DEGREES OF FREEDOM INDEPENDENCE AIC = INDEPENDENCE CAIC = MODEL AIC = MODEL CAIC = SATORRA-BENTLER SCALED CHI-SQUARE = ON 106 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS RESIDUAL-BASED TEST STATISTIC = PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS YUAN-BENTLER RESIDUAL-BASED TEST STATISTIC = PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS YUAN-BENTLER RESIDUAL-BASED F-STATISTIC = DEGREES OF FREEDOM = 106, 160 PROBABILITY VALUE FOR THE F-STATISTIC IS FIT INDICES BENTLER-BONETT NORMED FIT INDEX =.915 BENTLER-BONETT NON-NORMED FIT INDEX =.962 COMPARATIVE FIT INDEX (CFI) =.969 BOLLEN (IFI) FIT INDEX =.970 MCDONALD (MFI) FIT INDEX =.905 ROOT MEAN-SQUARE ERROR OF APPROXIMATION (RMSEA) = % CONFIDENCE INTERVAL OF RMSEA (.029,.057)

Multiple group Equality restrictions

España - USA /TITLE Model built by EQS 6 for Windows in Group 1 /SPECIFICATIONS DATA='c:\eqs61\espa_auto.ess'; VARIABLES=12; CASES=162; GROUPS=2; METHOD=ML,ROBUST; ANALYSIS=COVARIANCE; MATRIX=RAW; /LABELS V1=Q1; V2=Q2; V3=Q3; V4=Q4; V5=Q5; V6=Q6; V7=Q7; V8=L1; V9=L2; V10=L3; V11=L4; V12=L5; /EQUATIONS V1 = 1F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; V4 = *F1 + E4; V5 = *F1 + E5; V6 = *F1 + E6; V7 = *F1 + E7; V8 = 1F2 + E8; V9 = *F2 + E9; V10 = *F2 + E10; V11 = *F2 + E11; V12 = *F2 + E12; F2 = *F1 + D2; /VARIANCES F1 = *; E1 = *; E2 = *; E3 = *; E4 = *; E5 = *; E6 = *; E7 = *; E8 = *; E9 = *; E10 = *; E11 = *; E12 = *; D2 = *; /COVARIANCES /END /TITLE Model built by EQS 6 for Windows in Group 2 /SPECIFICATIONS DATA='C:\EQS61\usa_auto.ESS'; VARIABLES=12; CASES=109; METHOD=ML,ROBUST; ANALYSIS=COVARIANCE; MATRIX=RAW; /LABELS V1=Q1; V2=Q2; V3=Q3; V4=Q4; V5=Q5; V6=Q6; V7=Q7; V8=L1; V9=L2; V10=L3; V11=L4; V12=L5; /EQUATIONS V1 = 1F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; V4 = *F1 + E4; V5 = *F1 + E5; V6 = *F1 + E6; V7 = *F1 + E7; V8 = 1F2 + E8; V9 = *F2 + E9; V10 = *F2 + E10; V11 = *F2 + E11; V12 = *F2 + E12; F2 = *F1 + D2; /VARIANCES F1 = *; E1 TO E12 = *; D2 = *; /COVARIANCES /PRINT FIT=ALL; TABLE=EQUATION; /LMTEST PROCESS=SIMULTANEOUS; SET=PVV,PFV,PFF,PDD,GVV,GVF,GFV,GFF, BVF,BFF; /CONSTRAINTS (1,F2,F1)=(2,F2,F1); /END

España - USA GOODNESS OF FIT SUMMARY FOR METHOD = ML INDEPENDENCE MODEL CHI-SQUARE = ON 132 DEGREES OF FREEDOM INDEPENDENCE AIC = INDEPENDENCE CAIC = MODEL AIC = MODEL CAIC = CHI-SQUARE = BASED ON 107 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS FIT INDICES BENTLER-BONETT NORMED FIT INDEX =.927 BENTLER-BONETT NON-NORMED FIT INDEX =.949 COMPARATIVE FIT INDEX (CFI) =.959 BOLLEN (IFI) FIT INDEX =.959 MCDONALD (MFI) FIT INDEX =.790 LISREL GFI FIT INDEX =.882 LISREL AGFI FIT INDEX =.828 ROOT MEAN-SQUARE RESIDUAL (RMR) =.110 STANDARDIZED RMR =.039 ROOT MEAN-SQUARE ERROR OF APPROXIMATION (RMSEA) = % CONFIDENCE INTERVAL OF RMSEA (.055,.078)

España - USA GOODNESS OF FIT SUMMARY FOR METHOD = ROBUST ROBUST INDEPENDENCE MODEL CHI-SQUARE = ON 132 DEGREES OF FREEDOM INDEPENDENCE AIC = INDEPENDENCE CAIC = MODEL AIC = MODEL CAIC = SATORRA-BENTLER SCALED CHI-SQUARE = ON 107 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS RESIDUAL-BASED TEST STATISTIC = PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS YUAN-BENTLER RESIDUAL-BASED TEST STATISTIC = PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS YUAN-BENTLER RESIDUAL-BASED F-STATISTIC = DEGREES OF FREEDOM = 107, 159 PROBABILITY VALUE FOR THE F-STATISTIC IS FIT INDICES BENTLER-BONETT NORMED FIT INDEX =.914 BENTLER-BONETT NON-NORMED FIT INDEX =.962 COMPARATIVE FIT INDEX (CFI) =.970 BOLLEN (IFI) FIT INDEX =.970 MCDONALD (MFI) FIT INDEX =.906 ROOT MEAN-SQUARE ERROR OF APPROXIMATION (RMSEA) = % CONFIDENCE INTERVAL OF RMSEA (.028,.056)

Spain - USA Unrestricted model: - chi-square = df= 106 Restricted model (nested): - chi-square = df= 107 Difference testing (Δ χ 2 ): (H o = Equality among groups) -chi-cuadrado (dif.) = 0.08 ( – ) -Grados de libertad (dif.) = 1 (107 – 106) -P-value ≈ 1 The Δ χ 2 is not statistically significant, so we accept the model of equality of of parameters across groups Equality among groups

Means and covariances (MACS)

Model with the means Calidad Q2 Q3 Q4 Q5 Q7 Q1 Q6 Lealtad L2 L3 L4 L5 L1 1

Spanish sample /TITLE modelo_espana /SPECIFICATIONS DATA='E:\raw_data\espa_auto.ess'; VARIABLES=12; CASES=162; METHOD=ML,ROBUST; ANALYSIS=MOMENTS; MATRIX=RAW; /LABELS V1=Q1; V2=Q2; V3=Q3; V4=Q4; V5=Q5; V6=Q6; V7=Q7; V8=L1; V9=L2; V10=L3; V11=L4; V12=L5; /EQUATIONS V1 = *V F1 + E1; V2 = *V999 + *F1 + E2; V3 = *V999 + *F1 + E3; V4 = *V999 + *F1 + E4; V5 = *V999 + *F1 + E5; V6 = *V999 + *F1 + E6; V7 = *V999 + *F1 + E7; V8 = *V F2 + E8; V9 = *V999 + *F2 + E9; V10 = *V999 + *F2 + E10; V11 = *V999 + *F2 + E11; V12 = *V999 + *F2 + E12; F2 = *F1 + D2; /VARIANCES F1 = *; E1 TO E12 = *; D2 = *; /COVARIANCES /lmtest /PRINT EIS; FIT=ALL; TABLE=EQUATION; /END

Spanish sample (with means) GOODNESS OF FIT SUMMARY FOR METHOD = ROBUST ROBUST INDEPENDENCE MODEL CHI-SQUARE = ON 66 DEGREES OF FREEDOM INDEPENDENCE AIC = INDEPENDENCE CAIC = MODEL AIC = MODEL CAIC = SATORRA-BENTLER SCALED CHI-SQUARE = ON 53 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS RESIDUAL-BASED TEST STATISTIC = PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS YUAN-BENTLER RESIDUAL-BASED TEST STATISTIC = PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS YUAN-BENTLER RESIDUAL-BASED F-STATISTIC = DEGREES OF FREEDOM = 53, 107 PROBABILITY VALUE FOR THE F-STATISTIC IS FIT INDICES (BASED ON COVARIANCE MATRIX ONLY, NOT THE MEANS) BENTLER-BONETT NORMED FIT INDEX =.917 BENTLER-BONETT NON-NORMED FIT INDEX =.954 COMPARATIVE FIT INDEX (CFI) =.963 BOLLEN (IFI) FIT INDEX =.964 MCDONALD (MFI) FIT INDEX =.887 ROOT MEAN-SQUARE ERROR OF APPROXIMATION (RMSEA) = % CONFIDENCE INTERVAL OF RMSEA (.043,.090) The same fit than without means!!!

Spanish sample (with means) MEASUREMENT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS STATISTICS SIGNIFICANT AT THE 5% LEVEL ARE MARKED (ROBUST STATISTICS IN PARENTHESES) Q1 =V1 = 4.881*V F E1.130 (.130) ( Q2 =V2 = 4.837*V *F E (.130) (.127) ( ( Q3 =V3 = 5.106*V *F E (.124) (.135) ( ( Q4 =V4 = 5.244*V *F E (.129) (.139) ( ( Means of the Vs

Spanish sample (with means) STANDARDIZED SOLUTION: R-SQUARED Q1 =V1 =.000*V F E1.470 Q2 =V2 =.000*V *F E2.699 Q3 =V3 =.000*V *F E3.739 Q4 =V4 =.000*V *F E4.807 Q5 =V5 =.000*V *F E5.594 Q6 =V6 =.000*V *F E6.604 Q7 =V7 =.000*V *F E7.604 L1 =V8 =.928 F *V E8.862 L2 =V9 =.650*F *V E9.423 L3 =V10 =.907*F *V E L4 =V11 =.918*F *V E L5 =V12 =.852*F *V E F2 =F2 =.829*F D2.687 En la solución estandarizada, los intercepts son = 0 !!

Spanish sample: equality of means /TITLE modelo_espana /SPECIFICATIONS DATA='E:\raw_data\espa_auto.ess'; VARIABLES=12; CASES=162; METHOD=ML,ROBUST; ANALYSIS=MOMENTS; MATRIX=RAW; /LABELS V1=Q1; V2=Q2; V3=Q3; V4=Q4; V5=Q5; V6=Q6; V7=Q7; V8=L1; V9=L2; V10=L3; V11=L4; V12=L5; /EQUATIONS V1 = *V F1 + E1; V2 = *V999 + *F1 + E2; V3 = *V999 + *F1 + E3; V4 = *V999 + *F1 + E4; V5 = *V999 + *F1 + E5; V6 = *V999 + *F1 + E6; V7 = *V999 + *F1 + E7; V8 = *V F2 + E8; V9 = *V999 + *F2 + E9; V10 = *V999 + *F2 + E10; V11 = *V999 + *F2 + E11; V12 = *V999 + *F2 + E12; F2 = *F1 + D2; /VARIANCES F1 = *; E1 TO E12 = *; D2 = *; /COVARIANCES /CONSTRAINTS (V1,V999) = (V2,V999); (V1,V999) = (V3,V999); (V1,V999) = (V4,V999); (V1,V999) = (V5,V999); (V1,V999) = (V6,V999); (V1,V999) = (V7,V999); /lmtest /PRINT EIS; FIT=ALL; TABLE=EQUATION; /END GOODNESS OF FIT SUMMARY FOR METHOD = ROBUST ROBUST INDEPENDENCE MODEL CHI-SQUARE = ON 72 DEGREES OF FREEDOM INDEPENDENCE MODEL HAS BEEN MODIFIED TO INCLUDE 6 CONSTRAINTS FROM THE SPECIFIED MODEL. INDEPENDENCE AIC = INDEPENDENCE CAIC = MODEL AIC = MODEL CAIC = SATORRA-BENTLER SCALED CHI-SQUARE = ON 59 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS RESIDUAL-BASED TEST STATISTIC = PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS YUAN-BENTLER RESIDUAL-BASED TEST STATISTIC = PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS YUAN-BENTLER RESIDUAL-BASED F-STATISTIC = DEGREES OF FREEDOM = 59, 101 PROBABILITY VALUE FOR THE F-STATISTIC IS FIT INDICES (BASED ON MODIFIED INDEPENDENCE MODEL, AND BASED ON COVARIANCE MATRIX ONLY, NOT THE MEANS) BENTLER-BONETT NORMED FIT INDEX =.913 BENTLER-BONETT NON-NORMED FIT INDEX =.939 COMPARATIVE FIT INDEX (CFI) =.955 BOLLEN (IFI) FIT INDEX =.956 MCDONALD (MFI) FIT INDEX =.856 ROOT MEAN-SQUARE ERROR OF APPROXIMATION (RMSEA) = % CONFIDENCE INTERVAL OF RMSEA (.081,.119)

Spanish sample: mean on factors Calidad Q2 Q3 Q4 Q5 Q7 Q1 Q6 Lealtad L2 L3 L4 L5 L1 1 mx a

Spanish sample: mean on factors /TITLE modelo_espana /SPECIFICATIONS DATA='E:\raw_data\espa_auto.ess'; VARIABLES=12; CASES=162; METHOD=ML,ROBUST; ANALYSIS=MOMENTS; MATRIX=RAW; /LABELS V1=Q1; V2=Q2; V3=Q3; V4=Q4; V5=Q5; V6=Q6; V7=Q7; V8=L1; V9=L2; V10=L3; V11=L4; V12=L5; /EQUATIONS V1 = 1F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; V4 = *F1 + E4; V5 = *F1 + E5; V6 = *F1 + E6; V7 = *F1 + E7; V8 = 1F2 + E8; V9 = *F2 + E9; V10 = *F2 + E10; V11 = *F2 + E11; V12 = *F2 + E12; F1 = *V999 + D1; F2 = *V999 + *F1 + D2; /VARIANCES ! F1 = *; !COMANDO ELIMINADO E1 TO E12 = *; D1 TO D2 = *; !F1 “VARIABLE DEPENDIENTE” /COVARIANCES /CONSTRAINTS /lmtest /PRINT EFFECT = YES; EIS; FIT=ALL; TABLE=EQUATION; /END

Spanish sample: mean on factors GOODNESS OF FIT SUMMARY FOR METHOD = ROBUST ROBUST INDEPENDENCE MODEL CHI-SQUARE = ON 78 DEGREES OF FREEDOM INDEPENDENCE AIC = INDEPENDENCE CAIC = MODEL AIC = MODEL CAIC = SATORRA-BENTLER SCALED CHI-SQUARE = ON 63 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS RESIDUAL-BASED TEST STATISTIC = PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS YUAN-BENTLER RESIDUAL-BASED TEST STATISTIC = PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS YUAN-BENTLER RESIDUAL-BASED F-STATISTIC = DEGREES OF FREEDOM = 63, 97 PROBABILITY VALUE FOR THE F-STATISTIC IS FIT INDICES (BASED ON COVARIANCE MATRIX ONLY, NOT THE MEANS) BENTLER-BONETT NORMED FIT INDEX =.902 BENTLER-BONETT NON-NORMED FIT INDEX =.917 COMPARATIVE FIT INDEX (CFI) =.944 BOLLEN (IFI) FIT INDEX =.945 MCDONALD (MFI) FIT INDEX =.825 ROOT MEAN-SQUARE ERROR OF APPROXIMATION (RMSEA) = % CONFIDENCE INTERVAL OF RMSEA (.051,.093) The difference of d.f. …

Spanish sample: mean on factors CONSTRUCT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS STATISTICS SIGNIFICANT AT THE 5% LEVEL ARE MARKED (ROBUST STATISTICS IN PARENTHESES) F1 =F1 = 4.852*V D1.135 (.137) ( F2 =F2 = 1.085*F *V D (.062) (.322) ( ( -.190) Mean of F1 Intercept

Spanish sample: mean on factors DECOMPOSITION OF EFFECTS WITH NONSTANDARDIZED VALUES STATISTICS SIGNIFICANT AT THE 5% LEVEL ARE MARKED PARAMETER TOTAL EFFECTS F1 =F1 = 4.852*V D1 F2 =F2 = 1.085*F *V D D (.062) (.652) (.062) ( ( ( Mean of F2

Spanish sample: mean on factors STANDARDIZED SOLUTION: R-SQUARED Q1 =V1 =.722 F E1.521 Q2 =V2 =.804*F E2.646 Q3 =V3 =.853*F E3.728 Q4 =V4 =.877*F E4.769 Q5 =V5 =.817*F E5.668 Q6 =V6 =.801*F E6.642 Q7 =V7 =.766*F E7.587 L1 =V8 =.937 F E8.878 L2 =V9 =.657*F E9.431 L3 =V10 =.904*F E L4 =V11 =.915*F E L5 =V12 =.821*F E F1 =F1 =.000*V D1.000 F2 =F2 =.827*F *V D2.684 ¡ The intercepts are 0 in the standardized solution !

Multi-group mean and covariance structures MG-MACS

Multi-group MACS Calidad Q2 Q3 Q4 Q5 Q7 Q1 Q6 Lealtad L2 L3 L4 L5 L1 1 Mx = a=a= = Iguales entre grupos

Multi-group MACS /TITLE Model built by EQS 6 for Windows in Group 1 /SPECIFICATIONS DATA='E:\Raw_data\espa_auto.ess'; VARIABLES=12; CASES=162; GROUPS=2; METHOD=ML,ROBUST; ANALYSIS=MOMENTS; MATRIX=RAW; /LABELS V1=Q1; V2=Q2; V3=Q3; V4=Q4; V5=Q5; V6=Q6; V7=Q7; V8=L1; V9=L2; V10=L3; V11=L4; V12=L5; /EQUATIONS V1 = 1F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; V4 = *F1 + E4; V5 = *F1 + E5; V6 = *F1 + E6; V7 = *F1 + E7; V8 = 1F2 + E8; V9 = *F2 + E9; V10 = *F2 + E10; V11 = *F2 + E11; V12 = *F2 + E12; F1 = *V999 + D1; F2 = *V999 + *F1 + D2; /VARIANCES ! F1 = *; !COMANDO ELIMINADO E1 TO E12 = *; D1 TO D2 = *; !F1 VARIABLE DEPENDIENTE /COVARIANCES /END /TITLE Model built by EQS 6 for Windows in Group 2 /SPECIFICATIONS DATA='E:\Raw_data\usa_auto.ESS'; VARIABLES=12; CASES=109; METHOD=ML,ROBUST; ANALYSIS=MOMENTS; MATRIX=RAW; /LABELS V1=Q1; V2=Q2; V3=Q3; V4=Q4; V5=Q5; V6=Q6; V7=Q7; V8=L1; V9=L2; V10=L3; V11=L4; V12=L5; /EQUATIONS V1 = 1F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; V4 = *F1 + E4; V5 = *F1 + E5; V6 = *F1 + E6; V7 = *F1 + E7; V8 = 1F2 + E8; V9 = *F2 + E9; V10 = *F2 + E10; V11 = *F2 + E11; V12 = *F2 + E12; F1 = *V999 + D1; F2 = *V999 + *F1 + D2; /VARIANCES ! F1 = *; !COMANDO ELIMINADO E1 TO E12 = *; D1 TO D2 = *; !F1 VARIABLE DEPENDIENTE /COVARIANCES /PRINT FIT=ALL; TABLE=EQUATION; /LMTEST PROCESS=SIMULTANEOUS; SET=PVV,PFV,PFF,PDD,GVV,GVF,GFV,GFF, BVF,BFF; /CONSTRAINTS (1,F2,F1)=(2,F2,F1); !same slope across groups (1,F1,V999) = (2,F1,V999); !same mean across groups (1,F2,V999) = (2,F2,V999); !same intercept across groups /END Across-group constraints

Multisample MACS GOODNESS OF FIT SUMMARY FOR METHOD = ROBUST ROBUST INDEPENDENCE MODEL CHI-SQUARE = ON 156 DEGREES OF FREEDOM INDEPENDENCE AIC = INDEPENDENCE CAIC = MODEL AIC = MODEL CAIC = SATORRA-BENTLER SCALED CHI-SQUARE = ON 129 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS RESIDUAL-BASED TEST STATISTIC = PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS YUAN-BENTLER RESIDUAL-BASED TEST STATISTIC = PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS YUAN-BENTLER RESIDUAL-BASED F-STATISTIC = DEGREES OF FREEDOM = 129, 137 PROBABILITY VALUE FOR THE F-STATISTIC IS FIT INDICES (BASED ON COVARIANCE MATRIX ONLY, NOT THE MEANS) BENTLER-BONETT NORMED FIT INDEX =.896 BENTLER-BONETT NON-NORMED FIT INDEX =.920 COMPARATIVE FIT INDEX (CFI) =.945 BOLLEN (IFI) FIT INDEX =.947 MCDONALD (MFI) FIT INDEX =.828 ROOT MEAN-SQUARE ERROR OF APPROXIMATION (RMSEA) = % CONFIDENCE INTERVAL OF RMSEA (.037,.061)

Multisample MACS CONSTRUCT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS STATISTICS SIGNIFICANT AT THE 5% LEVEL ARE MARKED (ROBUST STATISTICS IN PARENTHESES) F1 =F1 = 5.062*V D1.103 (.105) ( F2 =F2 = 1.086*F *V D (.052) (.271) ( ( -.801) CONSTRUCT EQUATIONS WITH STANDARD ERRORS AND TEST STATISTICS STATISTICS SIGNIFICANT AT THE 5% LEVEL ARE MARKED (ROBUST STATISTICS IN PARENTHESES) F1 =F1 = 5.062*V D1.103 (.105) ( F2 =F2 = 1.086*F *V D (.052) (.271) ( ( -.801) ¡¡¡Same means, intercepts and slopes across groups!!!