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. 2014 Sep 19;5:983. doi: 10.3389/fpsyg.2014.00983

Table 2.

Goodness of fit, estimated latent mean differences and their estimated bias for Bayesian full and partial invariance models.

Model 95% CI χ2 PPP Median absolute intercept difference (range) Mean difference estimate Simulation Bias
α SE p d AVG SD SE Avg 95% Cover % Sig. M Bias (%) SE Bias (%)
FULL INVARIANCE
FMI1 configural −16.183 58.471 0.106 – – – – – – – –
FMI2 weak −12.794 57.036 0.111 0.131 (0.010–0.409) – – – – – – – –
FMI3 strong 20.836 93.107 0.001 0.000 (0.000–0.000) −0.340 0.066 <0.001 0.27 −0.348 3.161 0.593 0.314 0.682 2.24 −81.23
AFMI1 σ2 = 0.5 −14.605 59.226 0.119 0.088 (0.014–0.224) −0.279 0.533 0.284 0.03 −0.340 0.360 0.425 0.952 0.123 22.01 18.01
AFMI2 σ2 = 0.25 −14.219 58.635 0.119 0.071 (0.007–0.196) −0.315 0.377 0.194 0.04 −0.344 0.194 0.346 0.995 0.080 9.24 78.25
AFMI3 σ2 = 0.125 −14.811 59.127 0.122 0.062 (0.001–0.184) −0.333 0.280 0.123 0.06 −0.340 0.117 0.266 1.000 0.119 1.98 127.76
AFMI4 σ2 = 0.05 −15.240 61.016 0.107 0.051 (0.003–0.183) −0.354 0.185 0.027 0.10 −0.342 0.077 0.186 1.000 0.410 −3.50 142.00
AFMI5 σ2 = 0.025 −15.163 61.056 0.108 0.048 (0.004–0.178) −0.355 0.138 0.005 0.13 −0.342 0.069 0.142 1.000 0.827 −3.80 104.18
AFMI6 σ2 = 0.01 −14.855 61.681 0.106 0.044 (0.003–0.158) −0.350 0.101 <0.001 0.18 −0.340 0.066 0.104 0.997 0.976 −2.94 56.97
AFMI7 σ2 = 0.005 −13.530 63.776 0.086 0.036 (0.001–0.134) −0.346 0.084 <0.001 0.21 −0.338 0.065 0.087 0.987 0.995 −2.43 34.41
AFMI8 σ2 = 0.001 −0.366 77.232 0.028 0.016 (0.000–0.062) −0.341 0.069 <0.001 0.26 −0.333 0.064 0.071 0.967 0.999 −2.38 9.97
AFMI9 σ2 = 0.0005 6.400 83.511 0.012 0.009 (0.000–0.038) −0.341 0.067 <0.001 0.26 −0.333 0.064 0.068 0.961 1.000 −2.29 6.88
PARTIAL INVARIANCE*
PMI strong −13.704 62.416 0.095 −0.214 −0.264 −0.499 0.071 <0.001 0.36 −0.494 0.070 0.075 0.967 1.000 −1.10 6.28
APMI1 σ2 = 0.5 −11.802 61.498 0.090 −0.213 −0.263 −0.499 0.071 <0.001 0.36 −0.492 0.071 0.074 0.967 1.000 −1.34 5.09
APMI2 σ2 = 0.25 −11.718 61.594 0.090 −0.212 −0.262 −0.497 0.074 <0.001 0.35 −0.489 0.071 0.074 0.965 1.000 −1.59 5.25
APMI3 σ2 = 0.125 −11.799 61.592 0.090 −0.209 −0.259 −0.495 0.074 <0.001 0.35 −0.485 0.070 0.074 0.964 1.000 −2.12 5.12
APMI4 σ2 = 0.05 −12.175 61.725 0.087 −0.199 −0.250 −0.489 0.074 <0.001 0.34 −0.472 0.069 0.073 0.959 1.000 −3.46 6.24
APMI5 σ2 = 0.025 −11.671 62.672 0.087 −0.186 −0.236 −0.479 0.074 <0.001 0.33 −0.454 0.068 0.072 0.954 1.000 −5.14 6.33
APMI6 σ2 = 0.01 −9.953 64.009 0.081 −0.155 −0.203 −0.457 0.072 <0.001 0.33 −0.419 0.066 0.071 0.934 1.000 −8.29 6.49
APMI7 σ2 = 0.005 −7.646 67.449 0.062 −0.122 −0.165 −0.433 0.071 <0.001 0.32 −0.388 0.065 0.069 0.921 1.000 −10.39 6.45
APMI8 σ2 = 0.001 6.166 81.574 0.017 −0.045 −0.067 −0.374 0.069 <0.001 0.28 −0.340 0.064 0.067 0.933 0.999 −9.14 4.35
APMI9 σ2 = 0.0005 12.325 87.639 0.008 −0.025 −0.039 −0.358 0.068 <0.001 0.27 −0.334 0.064 0.067 0.945 1.000 −6.70 4.07

95% CI χ2, 95% confidence interval for the difference between the observed and the replicated χ2; PPP, posterior predictive p-value; α, mean latent score difference parameter; SE, standard error of α; p, significance of α; d, effect size; AVG, average of estimated mean latent score differences over the replications; SD, standard deviation of mean difference parameter estimate over the replications; SE avg, average of the estimated standard errors for the mean difference parameter estimate over the replications; 95% cover, proportion of replications for which the 95% CI included the hypothesized population value α; % sig., proportion of datasets for which the 95% CI did not include zero, i.e., the percentage of datasets for which it can be concluded that AVG is larger than zero in the population; M bias, (AVG-α)/α*100; SE bias, (SE avg-SD)/SD; FMI, Full measurement invariance; AFMI, Approximate full measurement invariance; PMI, partial measurement invariance; APMI, approximate partial measurement invariance; σ2, prior variance of intercepts for all pairs of items;

*

Intercepts of items 3 and 4 were not invariant, values in the intercept difference column are the differences between women's–men's intercepts for items 3 and 4, respectively.