35.2 Vincoli sulle medie
Trasformiamo ora il modello restrittivo specificato in precedenza allentando via via i vincoli che abbiamo introdotto. In questo modello rendiamo possibile la differenza tra le medie nei due gruppi.
mg_math_lavaan_fitM2 <- sem(mg_math_lavaan_model,
data = nlsy_math_wide,
meanstructure = TRUE,
estimator = "ML",
missing = "fiml",
group = "lb_wght", # to separate groups
group.equal = c(
"loadings", # for constraints
# "means", commented out so can differ
"lv.variances",
"lv.covariances",
"residuals"
)
)Esaminiamo il risultato.
summary(mg_math_lavaan_fitM2, fit.measures = TRUE)
#> lavaan 0.6.15 ended normally after 31 iterations
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 24
#> Number of equality constraints 16
#>
#> Number of observations per group: Used Total
#> 0 857 858
#> 1 75 75
#> Number of missing patterns per group:
#> 0 60
#> 1 25
#>
#> Model Test User Model:
#>
#> Test statistic 243.910
#> Degrees of freedom 62
#> P-value (Chi-square) 0.000
#> Test statistic for each group:
#> 0 191.440
#> 1 52.470
#>
#> Model Test Baseline Model:
#>
#> Test statistic 887.887
#> Degrees of freedom 42
#> P-value 0.000
#>
#> User Model versus Baseline Model:
#>
#> Comparative Fit Index (CFI) 0.785
#> Tucker-Lewis Index (TLI) 0.854
#>
#> Robust Comparative Fit Index (CFI) 1.000
#> Robust Tucker-Lewis Index (TLI) 0.326
#>
#> Loglikelihood and Information Criteria:
#>
#> Loglikelihood user model (H0) -7966.093
#> Loglikelihood unrestricted model (H1) -7844.138
#>
#> Akaike (AIC) 15948.185
#> Bayesian (BIC) 15986.884
#> Sample-size adjusted Bayesian (SABIC) 15961.477
#>
#> Root Mean Square Error of Approximation:
#>
#> RMSEA 0.079
#> 90 Percent confidence interval - lower 0.069
#> 90 Percent confidence interval - upper 0.090
#> P-value H_0: RMSEA <= 0.050 0.000
#> P-value H_0: RMSEA >= 0.080 0.472
#>
#> Robust RMSEA 0.000
#> 90 Percent confidence interval - lower 0.000
#> 90 Percent confidence interval - upper 0.000
#> P-value H_0: Robust RMSEA <= 0.050 1.000
#> P-value H_0: Robust RMSEA >= 0.080 0.000
#>
#> Standardized Root Mean Square Residual:
#>
#> SRMR 0.127
#>
#> Parameter Estimates:
#>
#> Standard errors Standard
#> Information Observed
#> Observed information based on Hessian
#>
#>
#> Group 1 [0]:
#>
#> Latent Variables:
#> Estimate Std.Err z-value P(>|z|)
#> eta_1 =~
#> math2 1.000
#> math3 1.000
#> math4 1.000
#> math5 1.000
#> math6 1.000
#> math7 1.000
#> math8 1.000
#> eta_2 =~
#> math2 0.000
#> math3 1.000
#> math4 2.000
#> math5 3.000
#> math6 4.000
#> math7 5.000
#> math8 6.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> eta_1 ~~
#> eta_2 (.17.) -0.035 1.144 -0.031 0.975
#>
#> Intercepts:
#> Estimate Std.Err z-value P(>|z|)
#> eta_1 35.488 0.369 96.080 0.000
#> eta_2 4.292 0.092 46.898 0.000
#> .math2 0.000
#> .math3 0.000
#> .math4 0.000
#> .math5 0.000
#> .math6 0.000
#> .math7 0.000
#> .math8 0.000
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> eta_1 (.15.) 63.704 5.637 11.301 0.000
#> eta_2 (.16.) 0.699 0.325 2.147 0.032
#> .math2 (thet) 36.321 1.871 19.413 0.000
#> .math3 (thet) 36.321 1.871 19.413 0.000
#> .math4 (thet) 36.321 1.871 19.413 0.000
#> .math5 (thet) 36.321 1.871 19.413 0.000
#> .math6 (thet) 36.321 1.871 19.413 0.000
#> .math7 (thet) 36.321 1.871 19.413 0.000
#> .math8 (thet) 36.321 1.871 19.413 0.000
#>
#>
#> Group 2 [1]:
#>
#> Latent Variables:
#> Estimate Std.Err z-value P(>|z|)
#> eta_1 =~
#> math2 1.000
#> math3 1.000
#> math4 1.000
#> math5 1.000
#> math6 1.000
#> math7 1.000
#> math8 1.000
#> eta_2 =~
#> math2 0.000
#> math3 1.000
#> math4 2.000
#> math5 3.000
#> math6 4.000
#> math7 5.000
#> math8 6.000
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|)
#> eta_1 ~~
#> eta_2 (.17.) -0.035 1.144 -0.031 0.975
#>
#> Intercepts:
#> Estimate Std.Err z-value P(>|z|)
#> eta_1 32.733 1.244 26.314 0.000
#> eta_2 4.905 0.320 15.320 0.000
#> .math2 0.000
#> .math3 0.000
#> .math4 0.000
#> .math5 0.000
#> .math6 0.000
#> .math7 0.000
#> .math8 0.000
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> eta_1 (.15.) 63.704 5.637 11.301 0.000
#> eta_2 (.16.) 0.699 0.325 2.147 0.032
#> .math2 (thet) 36.321 1.871 19.413 0.000
#> .math3 (thet) 36.321 1.871 19.413 0.000
#> .math4 (thet) 36.321 1.871 19.413 0.000
#> .math5 (thet) 36.321 1.871 19.413 0.000
#> .math6 (thet) 36.321 1.871 19.413 0.000
#> .math7 (thet) 36.321 1.871 19.413 0.000
#> .math8 (thet) 36.321 1.871 19.413 0.000Eseguiamo il confronto tra i due modelli.
anova(mg_math_lavaan_fitM1, mg_math_lavaan_fitM2)
#>
#> Chi-Squared Difference Test
#>
#> Df AIC BIC Chisq Chisq diff RMSEA Df diff
#> mg_math_lavaan_fitM2 62 15948 15987 243.91
#> mg_math_lavaan_fitM1 64 15949 15978 249.11 5.2005 0.058601 2
#> Pr(>Chisq)
#> mg_math_lavaan_fitM2
#> mg_math_lavaan_fitM1 0.07425 .
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Non vi è evidenza che consentire una differenza tra medie tra gruppi migliori l’adattamento del modello.