An examination of model fit and measurement invariance of general mental ability and personality measures used in the multilingual context of the Swiss Armed Forces: A Bayesian structural equation modeling approach

Figures & data

Table 1. Scales and their features and Cronbach’s alphas of the previously (i.e., 2012) and the presently studied (i.e., 2019) assessment cycles.

				Cronbach’s α – 2012 assessment cycle		Cronbach’s α – 2019 assessment cycle
Scale / construct	Items	Facets	Range	German(n = 1696)	French(n = 487)	German(n = 1284)	French(n = 474)	Reference
Leadership Motivation
Leadership motivation	4	1	-2-+2	.90	.89	.87	.89	Giauque, Vaso, & Boss (2006)
General mental ability
Matrices test	20	1	0-1	.74	.72	.71	.72	Hornke, Etzel, & Küppers (2000)
Concentration-stress test	200	4	0-1	.97M/.81A/.77O/.73C	.96M/.73A/.73O/.70C	.96M/.79A/.78O/.72C	.96M/.70A/.78O/.68C	Boss & Fischer (2006)
Concept (information processing)	4	1	1-4	-	-	-	-	Swiss Armed Forces (2012)
Self-competence
Achievement motivation	34	6	1-6	.97	.96	.96	.96	Gürber & Skupnjak (2006)
Stress tolerance	29	5	1-6	.83	.81	.82	.80	Stoll (2006)
Conscientiousness	10	1	1-6	.87	.83	.86	.79	Boss & Brenner (2006)
Independence	17	3	1-6	.89	.87	.87	.84	Melliger & Boss (2006)
Influencing behavior	23	4	1-6	.93	.93	.92	.92	Lécher & Boss (2006)
Concept (communication)	5	1	1-4	-	-	-	-	Swiss Armed Forces (2012)
Social-competence
Social behavior	22	4	1-6	.92	.89	.91	.88	Egger & Boss (2006a)
Conflict behavior	24	4	1-6	.88PR/.85PA	.86PR/.77PA	.87PR/.84PA	.83PR/.80PA	Egger & Boss (2006b)
Extraversion	10	1	1-6	.88	.87	.87	.85	Boss & Brenner (2006)
Agreeableness (courtesy)	10	1	1-6	.80	.78	.89	.89	Boss & Brenner (2006)
Agreeableness (cooperation)	10	1	1-6	.91	.90	.78	.78	Boss & Brenner (2006)
Concept (appearance)	2	1	1-4	-	-	-	-	Swiss Armed Forces (2012)
Concept (audience contact)	2	1	1-4	-	-	-	-	Swiss Armed Forces (2012)
Extra dimensions
Integrity	64	8	1-6	.94	.93	.93	.92	Egger & Boss (2006C)
Neuroticism	9	1	1-6	.80	.84	.79	.81	Boss & Brenner (2006)

Note. Scales selected for the purpose of the present study are displayed in gray. Items = number of scale items. Facets = number of facets / subtests that the scales are consisted of. Range = range of possible response values. For comparison purposes, Cronbach’s alphas are reported for two different assessment cycles: For the 2012 assessment cycle (see Goldammer, 2019) and for the 2019 assessment cycle (i.e., the cycle that is used in the present study). Cronbach’s alphas of concentration-stress test modules are abbreviated as follows: M = math test, A = analogy test, O = coordinate test, C = cutting pattern test. For the math, analogy, and coordinate tests, the reported Cronbach’s alphas values are based on both parts of each test. Cronbach’s alphas of the two conflict behavior facets are abbreviated as follows: PR = proactive facet, PA = passive facet.

Table 2. Model fit results of models with different levels of residual correlations.

Model	Pars.	PPP	95% CI	DIC	BCFI(90%CI)	BRMSEA(90%CI)
General mental ability (concentration-stress test) (1 first order factor, 4 subtest scores as indicators)
German
Exact zero residual correlations	12	.035	- 1.13 – 27.85	14000.60	.979(.963 – .989)	.071(.051 – .094)
Approx. zero residual correlations (v < .0001)	18	.442	- 13.76 – 17.29	13988.21	.999(.985 – 1)	.013(0 – .064)
Approx. zero residual correlations (v = .0001)	18	.479	- 14.21 – 15.92	13987.05	1(.987 – 1)	0(0 – .052)
Approx. zero residual correlations (v = .0003)	18	.495	- 14.53 – 15.38	13984.13	1(.987 – 1)	0(0 – .037)
Approx. zero residual correlations (v = .001)	18	.502	- 14.83 – 15.27	13981.02	1(.988 – 1)	0(0 – .028)
Approx. zero residual correlations (v = .01)	18	.506	- 14.76 – 15.19	13966.53	1(.988 – 1)	0(0 – .016)
French
Exact zero residual correlations	12	.359	- 12.11 – 17.11	5177.95	.991(.957 – 1)	.046(0 – .104)
Approx. zero residual correlations (v < .0001)	18	.345	- 13.05 – 20.53	5179.61	.988(.946 – 1)	.072(0 – .158)
Approx. zero residual correlations (v = .0001)	18	.423	- 14.14 – 17.85	5177.74	.995(.957 – 1)	.042(0 – .137)
Approx. zero residual correlations (v = .0003)	18	.466	- 14.72 – 16.04	5176.37	1(.964 – 1)	.005(0 – .109)
Approx. zero residual correlations (v = .001)	18	.484	- 14.83 – 15.60	5175.37	1(.966 – 1)	0(0 – .090)
Approx. zero residual correlations (v = .01)	18	.492	- 15.03 – 14.81	5171.28	1(.968 – 1)	0(0 – .053)
Conscientiousness (1 first order factor, 10 items as indicators)
German
Exact zero residual correlations	30	.000	338.13 – 392.16	32713.77	.912(.909 – .915)	.090(.088 – .092)
Approx. zero residual correlations (v < .0001)	75	.400	- 27.57 – 36.74	32386.99	.999(.995 – 1)	.053(0 – .136)
Approx. zero residual correlations (v = .0001)	75	.447	- 29.13 – 34.33	32384.65	1(.996 – 1)	.035(0 – .128)
Approx. zero residual correlations (v = .0003)	75	.480	- 30.75 – 32.31	32383.02	1(.996 – 1)	.016(0 – .122)
Approx. zero residual correlations (v = .001)	75	.491	- 30.56 – 31.57	32382.43	1(.996 – 1)	0(0 – .121)
Approx. zero residual correlations (v = .01)	75	.491	- 30.71 – 32.11	32382.49	1(.996 – 1)	0(0 – .122)
French
Exact zero residual correlations	30	.000	74.53 – 129.20	12534.55	.904(.891 – .915)	.079(.074 – .084)
Approx. zero residual correlations (v < .0001)	75	.254	- 23.10 – 46.13	12479.23	.989(.972 – 1)	.155(0 – .264)
Approx. zero residual correlations (v = .0001)	75	.368	- 28.96 – 39.04	12472.65	.995(.979 – 1)	.108(0 – .230)
Approx. zero residual correlations (v = .0003)	75	.442	- 30.80 – 35.62	12469.33	.998(.983 – 1)	.070(0 – .213)
Approx. zero residual correlations (v = .001)	75	.489	- 30.69 – 33.62	12466.85	.999(.985 – 1)	.039(0 – .202)
Approx. zero residual correlations (v = .01)	75	.510	- 31.85 – 30.45	12465.25	1(.986 – 1)	0(0 – .197)
Extraversion (1 first order factor, 10 items as indicators)
German
Exact zero residual correlations	30	.000	644.39 – 698.80	32191.84	.865(.862 – .867)	.122(.121 – .123)
Approx. zero residual correlations (v < .0001)	75	.402	- 27.88 – 37.15	31558.49	.999(.996 – 1)	.053(0 – .138)
Approx. zero residual correlations (v = .0001)	75	.450	- 29.37 – 34.49	31556.16	1(.996 – 1)	.036(0 – .128)
Approx. zero residual correlations (v = .0003)	75	.476	- 30.84 – 32.70	31554.56	1(.997 – 1)	.014(0 – .119)
Approx. zero residual correlations (v = .001)	75	.491	- 30.44 – 32.42	31.554.02	1(.997 – 1)	0(0 – .118)
Approx. zero residual correlations (v = .01)	75	.491	- 31.33 – 31.91	31553.90	1(.997 – 1)	0(0 – .120)
French
Exact zero residual correlations	30	.000	282.08 – 336.76	12090.93	.820(.812 – .827)	.137(.134 – .139)
Approx. zero residual correlations (v < .0001)	75	.265	- 23.08 – 45.33	11828.33	.994(.983 – 1)	.152(0 – .263)
Approx. zero residual correlations (v = .0001)	75	.373	- 28.21 – 38.61	11821.77	.997(.987 – 1)	.107(0 – .230)
Approx. zero residual correlations (v = .0003)	75	.445	- 30.27 – 35.55	11818.71	.999(.989 – 1)	.069(0 – .217)
Approx. zero residual correlations (v = .001)	75	.481	- 30.30 – 33.48	11816.33	1(.991 – 1)	.042(0 – .208)
Approx. zero residual correlations (v = .01)	75	.515	- 31.56 – 30.91	11814.34	1(.992 – 1)	0(0 – .193)
Agreeableness (cooperation) (1 first order factor, 10 items as indicators)
German
Exact zero residual correlations	30	.000	1023.38 – 1077.71	31359.95	.830(.828 – .832)	.153(.152 – .154)
Approx. zero residual correlations (v < .0001)	75	.404	- 27.82 – 36.66	30347.35	.999(.997 – 1)	.052(0 – .140)
Approx. zero residual correlations (v = .0001)	75	.452	- 29.74 – 34.20	30344.96	1(.997 – 1)	.033(0 – .124)
Approx. zero residual correlations (v = .0003)	75	.485	- 31.01 – 32.55	30343.34	1(.997 – 1)	.001(0 – .118)
Approx. zero residual correlations (v = .001)	75	.495	- 30.70 – 32.40	30342.96	1(.998 – 1)	0(0 – .119)
Approx. zero residual correlations (v = .01)	75	.489	- 30.35 – 31.42	30342.71	1(.998 – 1)	0(0 – .124)
French
Exact zero residual correlations	30	.000	395.71 – 449.71	11610.16	.817(.812 – .822)	.160(.158 – .162)
Approx. zero residual correlations (v < .0001)	75	.268	- 23.70 – 45.11	11234.96	.995(.988 – 1)	.151(0 – .260)
Approx. zero residual correlations (v = .0001)	75	.378	- 27.92 – 38.18	11227.81	.998(.991 – 1)	.105(0 – .232)
Approx. zero residual correlations (v = .0003)	75	.450	- 30.85 – 35.96	11224.66	.999(.992 – 1)	.066(0 – .212)
Approx. zero residual correlations (v = .001)	75	.494	- 30.82 – 33.72	11222.45	1(.993 – 1)	.037(0 – .206)
Approx. zero residual correlations (v = .01)	75	.515	- 32.50 – 30.46	11220.65	1(.994 – 1)	0(0 – .187)
Integrity (1 first order factor, 8 subtest scores as indicators)
German
Exact zero residual correlations	24	.000	299.90 – 345.11	25406.08	.922(.919 – .924)	.112(.110 – .114)
Approx. zero residual correlations (v < .0001)	52	.406	- 23.93 – 30.92	25106.30	.999(.996 – 1)	.046(0 – .122)
Approx. zero residual correlations (v = .0001)	52	.453	- 24.83 – 28.83	25104.44	1(.996 – 1)	.028(0 – .117)
Approx. zero residual correlations (v = .0003)	52	.482	- 25.60 – 26.90	25103.07	1(.997 – 1)	.008(0 – .109)
Approx. zero residual correlations (v = .001)	52	.497	- 26.39 – 26.61	25102.31	1(.997 – 1)	0(0 – .109)
Approx. zero residual correlations (v = .01)	52	.501	- 26.39 – 26.43	25100.21	1(.997 – 1)	0(0 – .064)
French
Exact zero residual correlations	24	.000	61.79 – 108.68	9588.77	.934(.925 – .943)	.095(.089 – .101)
Approx. zero residual correlations (v < .0001)	52	.284	- 19.38 – 38.39	9532.22	.994(.982 – 1)	.131(0 – .237)
Approx. zero residual correlations (v = .0001)	52	.377	- 23.49 – 32.12	9527.72	.997(.986 – 1)	.091(0 – .212)
Approx. zero residual correlations (v = .0003)	52	.457	- 25.47 – 29.53	9524.32	.999(.989 – 1)	.042(0 – .193)
Approx. zero residual correlations (v = .001)	52	.483	- 25.91 – 27.68	9523.09	1(.990 – 1)	.014(0 – .185)
Approx. zero residual correlations (v = .01)	52	.494	- 26.85 – 26.92	9522.47	1(.990 – 1)	0(0 – .181)
Neuroticism (1 first order factor, 9 items as indicators)
German
Exact zero residual correlations	27	.000	146.49 – 196.89	30630.67	.928(.923 – .933)	.071(.068 – .073)
Approx. zero residual correlations (v < .0001)	63	.404	- 26.61 – 33.97	30488.00	.999(.992 – 1)	.050(0 – .132)
Approx. zero residual correlations (v = .0001)	63	.455	- 27.74 – 31.46	30485.76	.999(.993 – 1)	.032(0 – .122)
Approx. zero residual correlations (v = .0003)	63	.490	- 29.07 – 29.39	30484.07	1(.994 – 1)	.004(0 – .098)
Approx. zero residual correlations (v = .001)	63	.501	- 29.40 – 28.92	30483.25	1(.994 – 1)	0(0 – .087)
Approx. zero residual correlations (v = .01)	63	.508	- 29.76 – 28.23	30482.10	1(.994 – 1)	0(0 – .070)
French
Exact zero residual correlations	27	.000	56.38 – 106.00	11105.62	.928(.916 – .938)	.079(.074 – .085)
Approx. zero residual correlations (v < .0001)	63	.262	- 21.39 – 42.33	11062.45	.991(.976 – 1)	.143(0 – .256)
Approx. zero residual correlations (v = .0001)	63	.363	- 25.08 – 36.20	11057.03	.996(.982 – 1)	.103(0 – .225)
Approx. zero residual correlations (v = .0003)	63	.455	- 27.61 – 31.73	11053.19	.998(.985 – 1)	.060(0 – .201)
Approx. zero residual correlations (v = .001)	63	.481	- 27.88 – 30.05	11051.79	1(.987 – 1)	.030(0 – .196)
Approx. zero residual correlations (v = .01)	63	.500	- 28.89 – 29.29	11050.61	1(.988 – 1)	0(0 – .193)

Note. Factor variance was fixed to 1 for identification purposes. Factor indicators were standardized so that the prior variance has similar interpretation across items. Pars. = estimated model parameters; PPP = posterior predictive p-value; 95% CI = 95% confidence interval for the difference between the observed and the replicated model χ² values; DIC = deviance information criterion; BCFI = Bayesian comparative fit index; BRMSEA = Bayesian root mean square error of approximation; For BCFI and BRMSEA, the median is reported as point estimate with a highest density 90% credibility interval. v = prior variance for the approximate zero residual correlations. Models considered for further investigation are shown in gray.

Table 3. Model fit results of models with different levels of scalar invariance across language groups.

Model	Pars.	PPP	95% CI	DIC	BCFI(90%CI)	BRMSEA(90%CI)
General mental ability (concentration-stress test)(Model with approximate zero residual covariances with a prior variance of 6.25*10^−6)^a
Exact scalar invariance	30	.103	-8.65 – 42.23	-7671.53	.981(.963 – 1)	.010(0 – .015)
Approx. scalar invariance (v = .001)	38	.247	-14.79 – 31.74	-7517.72	.991(.976 – 1)	.030(0 – .053)
Approx. scalar invariance (v = .005)	38	.249	-14.68 – 31.43	-7831.70	.991(.976 – 1)	.005(0 – .009)
Approx. scalar invariance (v = .01)	38	.249	-14.70 – 31.60	-8371.04	.991(.976 – 1)	.003(0 – .005)
Approx. scalar invariance (v = .05)	38	.246	-14.51 – 31.96	-8893.27	.991(.976 – 1)	.002(0 – .004)
Approx. scalar invariance (v = .1)	38	.245	-14.47 – 32.07	-8861.40	.991(.976 – 1)	.003(0 – .004)
Conscientiousness(Model with approximate zero residual covariances with a prior variance of .001)
Exact scalar invariance	132	.427	-41.54 – 50.73	45711.80	.999(.995 – 1)	.063(0 – .165)
Approx. scalar invariance (v = .001)	152	.456	-42.03 – 46.59	45707.72	1(.995 – 1)	.033(0 – .116)
Approx. scalar invariance (v = .005)	152	.132	-21.74 – 73.74	45730.50	.995(.990 – 1)	.083(0 – .119)
Approx. scalar invariance (v = .01)	152	.321	-36.74 – 55.52	45715.12	.998(.994 – 1)	.062(0 – .116)
Approx. scalar invariance (v = .05)	152	.483	-44.07 – 43.44	45706.06	1(.996 – 1)	.023(0 – .105)
Approx. scalar invariance (v = .1)	152	.486	-44.16 – 45.11	45705.16	1(.996 – 1)	.016(0 – .097)
Extraversion(Model with approximate zero residual covariances with a prior variance of .001)
Exact scalar invariance	132	.439	-42.58 – 50.77	48422.83	.999(.996 – 1)	.060(0 – .170)
Approx. scalar invariance (v = .001)	152	.458	-4288 – 46.71	48420.88	1(.996 – 1)	.041(0 – .134)
Approx. scalar invariance (v = .005)	152	.461	-43.37 – 47.12	48419.31	1(.996 – 1)	.029(0 – .098)
Approx. scalar invariance (v = .01)	152	.294	-33.63 – 58.50	48429.43	.998(.995 – 1)	.063(0 – .110)
Approx. scalar invariance (v = .05)	152	.481	-43.94 – 43.57	48419.20	1(.997 – 1)	.026(0 – .127)
Approx. scalar invariance (v = .1)	152	.484	-45.18 – 44.86	48418.25	1(.997 – 1)	.020(0 – .108)
Agreeableness (cooperation)(Model with approximate zero residual covariances with a prior variance of .01)
Exact scalar invariance	132	.488	-44.81 – 45.61	44488.78	1(.997 – 1)	.015(0 – .108)
Approx. scalar invariance (v = .001)	152	.479	-44.84 – 53.53	44495.19	1(.997 – 1)	.040(0 – .180)
Approx. scalar invariance (v = .005)	152	.467	-44.59 – 56.41	44473.28	1(.997 – 1)	.009(0 – .042)
Approx. scalar invariance (v = .01)	152	.304	-33.93 – 58.71	44499.92	.999(.996 – 1)	.065(0 – .120)
Approx. scalar invariance (v = .05)	152	.503	-44.46 – 44.89	44489.16	1(.997 – 1)	.009(0 – .130)
Approx. scalar invariance (v = .1)	152	.513	-44.87 – 43.93	44488.40	1(.997 – 1)	0(0 – .118)
Integrity(Model with approximate zero residual covariances with a prior variance of .01)
Exact scalar invariance	90	.458	-36.40 – 41.08	26000.22	1(.996 – 1)	.041(0 – .149)
Approx. scalar invariance (v = .001)	106	.162	-19.38 – 59.34	26011.23	.996(.993 – 1)	.053(0 – .080)
Approx. scalar invariance (v = .005)	106	.482	-36.22 – 38.51	25997.36	1(.996 – 1)	.015(0 – .096)
Approx. scalar invariance (v = .01)	106	.509	-37.17 – 37.12	25996.36	1(.997 – 1)	0(0 – .098)
Approx. scalar invariance (v = .05)	106	.514	-37.45 – 36.80	25994.15	1(.997 – 1)	0(0 – .073)
Approx. scalar invariance (v = .1)	106	.511	-37.49 – 36.97	25991.88	1(.997 – 1)	0(0 – .060)
Neuroticism(Model with approximate zero residual covariances with a prior variance of .001)
Exact scalar invariance	110	.460	-39.04 – 43.79	44650.40	.999(.994 – 1)	.049(0 – .172)
Approx. scalar invariance (v = .001)	128	.461	-39.40 – 44.92	44650.32	.999(.994 – 1)	.049(0 – .177)
Approx. scalar invariance (v = .005)	128	.458	-39.13 – 43.89	44648.82	.999(.994 – 1)	.033(0 – .122)
Approx. scalar invariance (v = .01)	128	.453	-39.56 – 44.72	44646.85	.999(.993 – 1)	.024(0 – .084)
Approx. scalar invariance (v = .05)	128	.484	-39.98 – 44.26	44647.25	1(.994 – 1)	.021(0 – .103)
Approx. scalar invariance (v = .1)	128	.498	-40.40 – 43.05	44645.65	1(.994 – 1)	.013(0 – .084)

Note. For identification purposes, the factor variance and factor mean were fixed in the French-speaking group to 1 and 0, respectively. In all scalar invariance models, the residual (co)variances were freely estimated across the language groups. Unstandardized model estimates were used for comparisons across groups. Subtest scores were used as indicators for the models of general mental ability (concentration-stress test) and integrity. Pars. = estimated model parameters; PPP = posterior predictive p-value; 95% CI = 95% confidence interval for the difference between the observed and the replicated model χ² values; DIC = deviance information criterion; BCFI = Bayesian comparative fit index; BRMSEA = Bayesian root mean square error of approximation; For BCFI and BRMSEA, the median is reported as point estimate with a highest density 90% credibility interval. v = prior variance for the difference between the measurement parameters (i.e., loadings and intercepts) of the German language group and those of the French language group. Models considered for further investigation are shown in gray.

^a= The unstandardized variances of the four indicators of the concentration-stress were remarkably different from 1 (i.e., on average only 0.0265). Thus, a smaller prior variance had to be chosen than for the single-group analyses (see also footnote 5). In order to ensure the same ratio between the indicator variance (i.e., 1) and the prior variance for the residual covariances (i.e., .0003) as we had observed it in the single-group analysis, we aimed for a prior variance of 7.95*10⁻⁶ in the unstandardized case. The inverse Wishart (IW) setting of IW(0,60) with a prior variance of 6.25*10⁻⁶ approached this value best. In addition, runs with larger IW settings (i.e., [0,65], [0,75], [0,85]) resulted in only marginal improvement of model fit. Thus IW(0,60) was selected as prior variance for the residual covariances in the multiple-group analyses of the concentration-stress test.

Table 4. Parameter difference tests for the extraversion construct across the full sample and the subsamples.

	Full sample comparisons
	German (n = 1,284)		French (n = 474)			Subsample comparisons (k = 100[n = 474 ; n = 474])
Parameters	Estimate	SD	Estimate	SD	Diff(95% CI)	MDiff	SDDiff	% sig Diff
Loadings
λ1	0.909	0.141	0.937	0.108	-0.023(-0.346 – 0.284)	-0.027	0.025	0
λ2	0.645	0.123	0.570	0.070	0.079(-0.181 – 0.337)	0.077	0.032	0
λ3	0.943	0.150	0.805	0.105	0.140(-0.192 – 0.479)	0.157	0.026	0
λ4	0.806	0.117	0.771	0.073	0.037(-0.214 – 0.287)	0.030	0.026	0
λ5	0.897	0.137	0.835	0.098	0.070(-0.232 – 0.369)	0.076	0.034	0
λ6	0.597	0.095	0.503	0.059	0.096(-0.113 – 0.304)	0.095	0.026	0
λ7	0.618	0.136	0.487	0.142	0.140(-0.226 – 0.490)	0.140	0.036	0
λ8	0.992	0.151	0.785	0.080	0.210(-0.099 – 0.516)	0.217	0.024	0
λ9	0.637	0.129	0.687	0.130	-0.036(-0.356 – 0.284)	-0.031	0.035	0
λ10	0.905	0.152	1.066	0.102	-0.156(-0.470 – 0.186)	-0.133	0.030	0
Intercepts
τ1	4.776	0.130	4.686	0.061	0.091(-0.156 – 0.349)	0.068	0.030	0
τ2	4.598	0.094	5.077	0.042	-0.478*(-0.653 – -0.284)	-0.485	0.029	100
τ3	3.873	0.133	4.090	0.057	-0.217(-0.471 – 0.047)	-0.230	0.032	9
τ4	4.967	0.114	4.917	0.049	0.049(-0.166 – 0.277)	0.034	0.019	0
τ5	4.191	0.128	3.995	0.056	0.196(-0.051 – 0.450)	0.174	0.028	0
τ6	5.099	0.085	5.276	0.037	-0.177*(-0.341 – -0.009)	-0.190	0.024	79
τ7	4.257	0.091	3.800	0.056	0.458*(0.264 – 0.656)	0.438	0.034	100
τ8	4.401	0.140	4.661	0.051	-0.261(-0.521 – 0.013)	-0.276	0.029	52
τ9	4.988	0.095	4.574	0.058	0.415*(0.221 – 0.620)	0.398	0.030	100
τ10	4.590	0.126	4.524	0.064	0.063(-0.184 – 0.304)	0.053	0.029	0

Note. The full sample and the subsample comparisons were based on the approximate scalar invariance model (with a prior variance of .1) with approximate zero residual covariances (with a prior variance of .001). The residual (co)variances were freely estimated across the language groups. For identification purposes, the factor variance and factor mean were fixed in the French-speaking group to 1 and 0, respectively. Estimate = unstandardized posterior point estimate (i.e., median); SD = standard deviation of the estimate’s posterior distribution; Diff_{(95% CI)} = posterior difference estimate with 95% highest density credibility interval. k = number of subsamples (i.e., 100); M_Diff = average parameter difference across the subsample comparisons; SD_Diff = standard deviation of the parameter difference across the subsample comparisons; % sig Diff = percentage of subsample comparisons in which the parameter difference was significantly larger than zero (i.e., 95% highest density credibility interval did not include zero).

* = 95% highest density credibility interval of the parameter difference does not include zero.

The row with the additional potential non-invariant parameter across the language groups (i.e., τ₈) that was detected through the subsampling comparisons is highlighted in gray.

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Data availability statement

The data that support the findings of this study are available from the first author, upon reasonable request.