Global mental health: an improved measure of well-being in multiple languages.

BACKGROUND: An increasing number of international organisations and national governments have committed to well-being promotion. Unfortunately, important questions regarding how to assess well-being are still unresolved, making policy implementation and evaluation difficult.
METHODS: This research expanded on Huppert and So's (Soc Indic Res. 110, 837-861 2013) multidimensional subjective well-being framework by investigating the replicability of the model in two non-European regions (South America, represented by Brazil and Colombia, and Eastern Africa, represented by Uganda), and the United Kingdom. Additionally, previous limitations of the framework were also addressed. ESS Round Six items were crucially improved in terms of temporal and response scale consistency. Bayesian approximate measurement invariance was applied on a sample of 381 young adult participants to test for consistency across countries.
RESULTS: The Huppert & So (Soc Indic Res. 110, 837-861 2013) model was found to fairly replicate across non-European regions, where meaningful differences in well-being patterns across regions were observed. Additionally, estimated well-being was related with other well-being measures (Five Ways): Learn and Connect were the strongest predictors of general well-being, with Take Notice and Give being associated with specific aspects of it.
CONCLUSIONS: Based on this narrow sample of young adults, it appears the ten-item measure proposed by Huppert & So (Soc Indic Res. 110, 837-861 2013) is useful for assessing population mental health outside of Europe. This is only an initial attempt to assess qualities, so further testing should be done before applying at scale for identifying policy opportunities to address well-being of populations.

Background

There is an increasing prevalence in arguments highlighting the limitations of traditional economic measurements as indicators of population well-being [1]. Major supranational organisations, including the OECD, WHO and UN, have acknowledged the need for the direct measurement of well-being due to such evidence-based assertions. Significantly, for the first time the promotion of well-being has been recognised as part of the global development agenda in the United Nation’s 2030 Sustainable Development Goals outlined in 2016. Progress towards the direct measurement of well-being has concurrently been made on a national-level, with more than 40 countries reportedly measuring citizens’ well-being [1]. For example, there have been initiatives in the United Kingdom assessing the impact of specific policies on well-being since 2010. Nevertheless, although there have been a number of attempts to develop a cross-culturally validated well-being measurement tool for international use and comparisons, such attempts have had limited success (e.g. [2, 3]).

Determining the standards by which to measure or define “well-being” has proven to be a persistent challenge. However, one increasingly common framework distinguishes between hedonic well-being, which corresponds to “positive feeling”, and eudaimonic well-being, which corresponds to “positive functioning” [4]. A number of multidimensional scales integrating these two components have been developed, such as the “Satisfaction with Life Scale” [5], Lyubomirsky and Lepper’s “Subjective Happiness Scale” [6], the “Flourishing Scale” [7], the PANAS scale [8], and the Oxford Happiness Questionnaire [9]. Such multidimensional approaches to the measurement of well-being are increasingly favoured because they offer a more holistic assessments of an individual’s experience, as well as a robust framework upon which improvements can be made. Cross-cultural validation is a particularly important consideration since the integrity of international comparisons will rely on the premise that the same construct is being adequately captured across diverse populations [5, 10].

This paper seeks to build on previous work conducted on multidimensional well-being assessments. Specifically, the well-being module developed for the European Social Survey [11] represents a unique undertaking because of the scope of its sample (more than 43,000 Europeans), the cross-cultural validity of the model derived (including representative samples of 23 countries), and the anticipated applications of their findings to policy. Within this framework, 19 items were later identified by Huppert and So [4] as markers of ten crucial well-being dimensions. These dimensions were derived as opposites of the diagnostic criteria for Major Depressive Episode, Depressive Episode, and Generalized Anxiety Disorder as defined in the Diagnostic and Statistical Manual of Mental Disorders (DSM-IV) of the International Classification of the American Psychiatric Association (1994) and the International Classification of Diseases (ICD-10) of the World Health Organization (1993). From their analyses of the ESS data, Huppert and So [4] developed a two-factor model to account for their ten proposed dimensions of well-being. The first factor, which they termed “Positive Characteristics” (PC) included: emotional stability, vitality, resilience, optimism, positive emotion and self-esteem. The second, which they termed “Positive Functioning” (PF) included: engagement, meaning, positive relationships and competence.

Unfortunately, the Huppert and So [4] approach presented relevant limitations, as some items referred to different time windows and made use of different response scales (p.843, [4]). Additionally, this framework was only assessed using European samples, hindering its generalisability to alternative populations. In our application, we aim to amend both issues: whereas questions from the ESS survey use a range of words to indicate time-period, from specific phrases such as ‘in the past week’ to more general ones such as ‘often’, this questionnaire prompted participants to answer with reference to ‘in general’ for every question. Additionally, all items were presented using a common response scale. We tested this improved questionnaire in countries across different geographical regions (following United Nation’s regions) outside the ESS application region: Brazil, Colombia (South America) and Uganda (Easter Africa). Accordingly, the primary purpose of this study was to assess the feasibility of the measures in new settings, not to conclude the overall fit of items or a final recommendation for application at scale. Lastly, original scale characteristics (i.e., the presence of two well-being factors) were retained in this study so the results obtained with the improved scale were directly comparable with those of the original publication.

We further assessed the criterion validity of the new scale version by investigating how the dimensions proposed by Huppert and So’s model [4] were linked to alternative well-being behavioural markers. Among those, the Five Ways to Wellbeing (Five Ways, namely Connect, Be Active, Take Notice, Give and Learn; Government Office for Science, 2008), reflect some specific behaviours associated with improved well-being [12, 13]. Accordingly, this article will evaluate how each of the Five Ways impacted the different well-being components across the explored regions.

Methods

Participants

Ethical approval for this study was obtained from the Department of Psychology Ethics Committee (PEC), University of Cambridge. Consent was obtained from all participants, and a debrief was presented upon completion of the study. Recruitment was conducted over a period of 3 weeks between March and April 2017 in Brazil, Uganda, Colombia, and the United Kingdom with the support from local non-governmental organizations and academics. Qualtrics was used to recruit participants from Colombia due to difficulties in collecting complete questionnaires through other methods. While a larger sample had been targeted for a full-scale validation, the aim of this study is to provide initial evidence of the psychometric properties of an improved scale in alternative contexts outside European regions.

Importantly, the diverse geographical regions under investigation were selected to represent areas not previously tested within the context of the ESS. The geographic area selection intentionally aimed to avoid introducing systematic bias of presenting highly similar countries belonging to similar cultural, geographical and economic backgrounds. We additionally control for age differences by only including participants who were aged 18 to 24 years. Lastly, even though respondents were requested to have some proficiency in English (to respond to opening demographic questions), well-being and the Five Ways items were translated to local languages.

Questionnaire

The questionnaire was administered using the Qualtrics survey platform, and participants were granted access to the questionnaire through an emailed link. The self-report questionnaire (Appendix A) consisted of sociodemographic questions, ten items assessing different ‘dimensions’ of well-being [4, 11] and the Five Ways items as follows:

Firstly, sociodemographic measures for age, gender, primary nationality, years of education, and employment status were included at the start of the questionnaire and were written in English. Following the initial sociodemographic questions, participants were asked to select their native language before accessing the main questionnaire. The main questionnaire contained the ten well-being dimension’s items plus the Five Ways questions. The well-being dimension items were those designed for ESS Round Three [11] and later selected by Huppert and So [4] to develop their well-being model. It is noteworthy that these items were additionally found in ESS Round Six (2012), where they presented minor changes in the former due to floor effects observed in Round Three. Additionally, the Five Ways to Wellbeing [14] were measured using the items also included in the ESS Round Six well-being module. Accordingly, the original questions that we aimed to improve were those of the ESS Round Six well-being model.

Two major changes were conducted: firstly, all items were placed on the same seven-point Likert scale to ensure better internal consistency and to ameliorate the negatively skewed responses that were found in the ESS Round Three and Six. Secondly, all items were adjusted to achieve temporal consistency across items such that all questions were answered with reference to the prompt “in general” instead of referring to specific time periods (e.g. in the last week, in the last year). We further modified the wording of the resilience item to ensure the same directionality of all questions. The specific questions associated with each of these items are displayed in Table 1.

Table 1

Items applied to measure Hupper and So [4] scale and Five Ways to Well-being

Items	Formulation
Huppert & So [4] scale
Competence	I feel a sense of accomplishment from what I do.
Emotional stability	I feel calm and peaceful.
Engagement	I feel absorbed in what I am doing.
Meaning	I feel what I do in my life is valuable and worthwhile.
Optimism	I am optimistic about my future.
Positive emotion	I feel happy.
Positive relationships	I receive help and support from people I am close to when I need it.
Resilience	I recover quickly from things that go wrong in my life
Self-esteem	I feel positive about myself.
Vitality	I feel full of energy.
Five Ways
Learn	I pursue opportunities to try new things.
Take notice	I take time during my daily activities to appreciate my surroundings.
Give	I give help and support to those close to me.
Connect	I am spending time socialising with friends, peers and other people close to me.
Be active	In a typical week, how many days are you active for at least 30 min? Active means are doing enough to raise your breathing rate.

Huppert and So’s [4] and the Five Ways questionnaire items were translated from English to Spanish, and Portuguese, following World Health Organization (WHO) translation guidelines (Appendix B).

Statistical analyses

Bayesian Structural Equation Modelling (BSEM) was used to study whether: a) evidence supported previous findings regarding multidimensional well-being (whether the dimensions and well-being factors were found); b) to assess meaningful cross-country differences. To this end, we employed approximate measurement invariance. BSEM represents a critical improvement over traditional SEM and CFA models, where cross-loadings and residual correlations are not fixed to zero, but given “informative, small-variance priors” ([15], p.316). This flexibility represents a substantive improvement in terms of model fit and parameter estimation, and its use has been widely adopted in cross-cultural survey analysis (see references in Appendix C). Following the guidelines described in [16-19], a nested-model approach was followed for estimating the BSEM models. Additionally, a comparison of BSEM with traditional estimation frameworks (confirmatory and exploratory factor analysis) was performed. A detailed report of the analyses can be found in Appendix C.

All models were estimated in Mplus 7 [20, 21]. Bayesian confirmatory factor analysis (BCFA) models were sampled in four different chains, with a maximum of 500,000. Each parameter convergence was confirmed through visual inspections of the trace plots and autocorrelation plots. Additionally, the potential scale reduction (PSR) criterion [22] was lower than 1.05 for each parameter (where values lower than 1.10 assure chain convergence). BSEM model fit was assessed by means of the Posterior Predictive Checking (PPC). A PPC lower than .05 indicates poor fit, while values close to .50 and 95% PPC CIs that include zero values indicate a good model fit. DIC and BIC statistics are also reported, where lower values represent a better model fit.

Results

Sample characteristics by country are described in Table 1. Of the 700 survey respondents, 381 (54.4%) fulfilled the participation criteria. These included 161 Brazilian respondents, 78 Ugandan respondents, 86 Colombian respondents, and 56 British respondents.

Bayesian structural equation modeling

A partial approximate measurement invariance (PAMI) model was found to fit the data best and was preferred over several alternative models (exploratory factor analysis, classical confirmatory factor analysis, BCFA with informative priors over cross-loadings and BCFA with informative priors for cross-loadings and residual correlations; Appendix C). The PAMI model held factor loadings and items intercept equally across countries. There were two exceptions for Meaning and Competence, which were shown to be higher and lower for the United Kingdom, respectively. Therefore, the PAMI model resembled traditional partial scalar invariance models. The PAMI model successfully reproduced the factor pattern hypothesized by Huppert and So [4], including two additional cross-loadings: meaning for positive characteristics (PC) and positive emotion in positive functioning (PF; Table 2). Additionally, several minor residual correlations were found, as reported in Appendix C. Sensitivity analyses revealed that under more informative priors over item intercepts (e.g., using σ2 = .001 instead of σ2 = .01), these could be considered as equal across countries. Nevertheless, only the PAMI model applying σ2 = .01 prior is presented depicting the most conservative results found (Table 3).

Table 2

Descriptive values for all the items included in the questionnaire, divided by country of origin of participants

Variable / Level	Overall	Country-specific
		Brazil	Uganda	Colombia	United Kingdom
Participants	381	161 (42.3%)	78 (20.5%)	86 (22.6%)	56 (14.7%)
Gender
Female	230 (60.4%)	106 (65.8%)	30 (38.5%)	66 (76.7%)	26 (48.1%)
Age	21.65 (1.79)	21.13 (1.88)	22.46 (1.38)	21.87 (1.96)	21.70 (1.18)
Employment
Employed	160 (42.0%)	50 (31.1%)	18 (23.1%)	79 (91.9%)	13(23.2%)
Education	159 (41.7%)	73 (45.3%)	38 (48.7%)	7 (8.1%)	41(73.2%)
Seeking employment	37 (9.7%)	16 (9.9%)	8 (10.3%)	0 (0%)	1 (1.8%)
Huppert & So[4]scale
Competence	4.94 (1.23)	4.89 (1.17)	4.69 (1.19)	5.02 (1.51)	5.32 (.77)
Emotional stability	4.50 (1.25)	4.31 (1.27)	4.74 (1.10)	4.97 (1.26)	4.09 (1.13)
Engagement	4.71 (1.20)	4.67 (1.11)	4.40 (1.28)	5.58(1.28)	4.18 (1.07)
Meaning	5.33 (1.13)	5.26 (1.20)	5.30 (1.18)	4.77 (.95)	4.93 (.89)
Optimism	5.34 (1.15)	5.15 (1.27)	5.39 (1.01)	5.71 (1.09)	5.27 (.96)
Positive emotion	5.13 (1.1)	5.07 (1.14)	5.09 (.91)	5.47 (1.09)	4.88 (1.13)
Positive relationships	5.31 (1.25)	5.42 (1.31)	4.99 (1.15)	5.40 (1.21)	5.30 (1.22)
Resilience	4.66 (1.18)	4.55 (1.22)	4.50 (1.16)	4.93 (1.13)	4.67 (1.13)
Self esteem	5.02 (1.22)	4.76 (1.20)	5.37 (1.06)	5.31 (1.20)	4.82 (1.34)
Vitality	4.62 (1.27)	4.19 (1.36)	4.95 (1.01)	5.06 (1.18)	4.75 (1.13)
Five Ways
Learn	5.42 (1.05)	5.11 (1.04)	5.49 (.95)	6.04 (.95)	5.29 (.96)
Take notice	4.68 (1.26)	4.35 (1.25)	4.90 (1.24)	4.98 (1.19)	4.84 (1.23)
Give	5.56 (.92)	5.71 (.86)	5.31 (.96)	5.58 (.99)	5.46 (.89)
Connect	5.00 (1.29)	4.72 (1.41)	5.05 (1.26)	5.20 (1.20)	5.41 (.91)
Be active	5.20 (1.96)	5.11 (2.08)	5.20 (1.82)	5.40 (1.88)	5.20 (1.95)

Note. Values presented are M (SD) for continuous variables and number of participants and percentage over total sample for categorical variables. The highest average for each item is bolded

Table 3

Partial approximate invariance model (PAMI) estimated parameters and model fit

Dimension	Positive Functioning		Positive Characteristics
Emotional stability	.74 (.54,.94)*		.01 (−.19, .19)
Vitality	1.00 (.83, 1.17)*		−.17 (−.34, .03)
Resilience	.76 (.53, .94)*		−.04 (−.21, .14)
Optimism	.68 (.51, .83)*		.12 (−.04, .26)
Positive emotion	.59 (.46, .73)*		.19 (.04, .33)*
Self esteem	.98 (.83, 1.12)*		−.04 (−.21, .11)
Engagement	−.05 (−.24, 12)		.92 (.70, 1.10)*
Meaning	.22(.03, .38)*		.54 (.39, .74)*
Positive relationships	−.09 (−.27, .12)		.76 (.49, 1.01)*
Competence	−.04 (−.21, .13)		.74 (.56, .92)*
Factor correlation	.60 (.39, .76)
Model	PPp	95%CI	DIC	BIC
PAMI	.34	−50.34 – 79.85	10,628.37	11,769.57

Note. PAMI Approximate Partial Measurement Invariance Model. PPp Prior-posterior checking p-value (values close to .50 and 95% Credible Interval containing zero indicate good fit). * 95% Credible Interval of posterior density does not include zero. DIC Deviance Information Criterion. BIC Bayesian Information Criterion

Differences in latent means are further explored using the PAMI model (Table 4). This reflects how participants in each country scored on average on each well-being dimension. Firstly, participants from Colombia scored the highest in both PC and PF. Participants from Uganda scored the second highest in PC, but the lowest of all countries in PF. Participants from Brazil and the United Kingdom showed a similar response pattern, scoring lower than those from Uganda and Colombia in PC, but higher than those from Uganda and as high as those from Colombia in PF.

Table 4

Factor latent means for each country

	Uganda*	Brazil	Colombia	United Kingdom
PC	.00	−.59 (−.90, −.26)	.14 (−.21,.50)	−.42 (−.82, −.02)
PF	.00	.24 (−.07,.59)	.59 (.23, .98)	.34 (−.06, .75)

Note: Uganda means are fixed to zero due to identification constraints. Thus, Uganda serves as a baseline to compare other countries against. PC Positive Characteristics. PF Positive Functioning

Five ways to wellbeing

In order to understand the relationship between the Five Ways and the two well-being factors, we ran two regression models that predicted the well-being score on each factor from the Five Ways. As before, approximate invariance was considered for the regression slopes (PPp = .06 (− 17.75, 162.91), DIC = 16,215.76, BIC = 18,058.79). Regression parameters for each country are presented in Table 5 and Fig. 1. Analysis showed that no regression coefficient significantly varied across countries. Previous findings regarding invariance of factor loadings and intercepts, and factor latent intercept interpretation remained unchanged.

Table 5

Parameter and model fit for the PAMI SEM model including Five Ways to Wellbeing as a predictor of well-being factors

	Brazil		Uganda		Colombia		United Kingdom
	PF	PC	PF	PC	PF	PC	PF	PC
Learn	.29*	.45*	.31*	.55*	.30*	.53*	.29*	.56*
Take notice	.00	.30*	−.07	.25*	.01	.31*	−.03	.26*
Give	.31*	.17	.28*	.18	.29*	.19	.29*	.11
Connect	.26*	.32*	.24*	.20*	.28*	.30*	.25*	.25*
Be active	−.03	.10*	−.01	−.02	−.02	.07	.01	.06

Note: * means a difference whose credible interval does not cover zero. PC Positive Characteristics. PF Positive Functioning

Fig. 1

Parameter and model fit for the PAMI SEM model including Five Ways to Wellbeing as a predictor of well-being factors

Overall, the patterns found were similar for all countries. Learn and Connect were strong predictors of both PF and PC well-being factors for all countries. Give predicted PF in all countries, while Take Notice predicted PC in all countries. Lastly, Be Active was not a significant predictor of any of the well-being factors, except for PC in Brazil.

Discussion

This study investigated the properties of an improved version of Huppert and So [4] multidimensional well-being framework, with a particular emphasis in its cross-cultural and criterion validity. We expanded previous findings in the field in three main areas: a) the psychometric properties of the scale items, as we ensured item time and response scale consistency across items; b) the generalizability of the model to non-European areas, by testing Huppert and So [4] model in geographically diverse regions beyond those that participated in the original study; c) we investigated the extent that each well-being factor was connected with five different behavioural markers of well-being. This work suggested that improving the assessments of well-being allowed for more nuanced evaluations and better identification of areas for improvement, and its usefulness to inform future policies and well-being interventions.

One of the key strengths of this research is that we have adapted the items used in the ESS Round Six to improve internal consistency. Temporal consistency was achieved by setting all items within the same timeframe, addressing concerns that temporal inconsistencies between items may lead to a distorted measure of life satisfaction that unduly combines information from different time periods [23]. As a result, the survey items presented here, while comparable to the ESS Round Six items, represents an improvement in terms of consistency, which makes them preferable for future use and data collection. We further replicated the original model, including the distinction between “Positive Functioning” (PF) and “Positive Characteristics” (PC), as well as their respective item loadings. Only two domains (“Positive Emotion” and “Meaning”) were found to load on both factors. However, the presence of such cross-loadings was already suggested in the exploratory solution presented by Huppert and So’s (Table 3 [4];). Thus, the proposed scale was able to capture the original theoretical model while improving its psychometric properties.

By employing a novel, methodologically sound framework (i.e., Bayesian approximate measurement invariance) for testing cross-cultural invariance, this research advanced that the multidimensional well-being framework proposed by Huppert and So [4] could potentially replicate in non-European regions. Our results suggest that the multidimensional well-being framework originally suggested by Huppert and So [4] could be explored in future research including larger, representative and diverse samples with a higher degree of confidence, given the psychometrics improvements here presented. Moreover, this research highlights the necessity of continuing to improve well-being assessments tools under different contexts.

Our results also suggest the existence of regional differences for both well-being dimensions, which could be further explored for local policy precision, but are still useful for macro level monitoring. What would further add to local policy is that results indicate that not all Five Ways were similarly related with both well-being factors. Moreover, the specificity with which each “way” affects the two well-being domains highlights this well-being measurement’s potential for evaluating policy actions. Naturally, this should only be applied for fully powered and focused samples. Such research endeavours should aim to confirm or discard the differences in patterns observed here. Nevertheless, the proposed scale and the Huppert and So’s [4] model (with various revisions and iterations) remains useful in broad policy research investigating the relationship between different well-being predictors and specific components of this construct.

Limitations

This research is subject to a number of limitations, many of which have been previously outlined in Huppert and So’s [4] original research. For example, the scales have not been extended to include constructs fundamental to certain conceptualisations of well-being, including Autonomy, which is considered central to certain theoretical models [24-27], the psychodynamic domains of “personal-growth”, and “self-acceptance” [26]. As formerly noted by Huppert and So, Autonomy might be a dimension that is particularly sensitive to different cultural and societal norms, specifically when considering the balance between individualism and collectivism, and as such might not be considered as necessary to well-being in all societies. Although the questionnaire was also translated into Arabic and French with the aim of collecting data in North Africa, this intention proved impossible to realise within the timeframe of this research.

The study was designed such that the opening demographic questions were in English, which necessarily excluded individuals who do not speak the languages in certain countries and thus could have been a source of bias. Furthermore, the age requirement for the participants means that whilst an equivalency has been found for a sub-population, the scales might not be equivalent within the whole population. Lastly, it would be important to study in further detail cross-country differences observed in the PAMI model, such as United Kingdom individuals scoring higher in Meaning and Competence items.

Conclusions

In summary, this research aimed to improve multidimensional well-being assessments by enhancing Huppert and So’s [4] proposed items. The proposed assessment tool has a number of benefits: (a) the items in the scale have a theoretically sound rationale for inclusion, and have been (b) critically evaluated and refined across different studies, (c) the scale itself is short, reliable and valid, (d) and has been found to be cross-culturally invariant across limited European, South American and Eastern African populations. As a result, it provides a time-efficient measure that can capture how different policies may influence specific aspects of well-being, offering insights beyond single-item well-being measures (e.g., life satisfaction or happiness).

One of the reasons why valid, reliable, and robust measures of well-being are critical is that such instruments are necessary to identify potentially unmet needs in a population. Naturally, truly comprehensive measures would cover wider and culturally or contextually specific items, but for high level national surveying, the instrument presented in this study does offer insight for policy and other interventions to tackle unmet needs on a population level. These findings contribute to the improvement of well-being measures for informing policy decisions on local, national, and international levels. We argue that using this consistent, fully aligned, and temporally coherent approach to measurement offers an improvement on existing measures. Moreover, evidence suggests that the scale represents a valid tool for assessing well-being, with further testing needed in additional countries and cultures.

Table 6

Model fit for traditional CFA and EFA model

Model	χ2	CFI	TLI	RMSEA
CFA	123.60	.89	.86	.09 (.07–.10)
EFA	58.06	.96	.93	.07 (.05–.09)

Note. CFA Confirmatory Factor Analysis model; EFA Exploratory Factor Analysis; CFI Comparative Fit Index; TLI Tucker-Lewis index; RMSEA Root mean square error of approximation

Table 7

Model fit for Bayesian Confirmatory Factor Analyses

	PPp	95%CI	DIC	BIC
BCFA 1	.001	94.26–152.42	10,786.76	10,906.43
BCFA 2	.001	28.34–94.94	10,726.86	10,898.73
BCFA 3	.48	−34.46 –30.31	10,690.81	11,079.87

Note. BCFA1 Bayesian Confirmatory Factor model with cross-loadings and residual covariances fixed to zero; BCFA 2 Bayesian Confirmatory Factor model with informative priors for cross-loadings and residual covariances fixed to zero; BCFA 3 Bayesian Confirmatory Factor model with informative priors for cross-loadings and residual covariances fixed; PPp Prior-posterior checking p-value (values close to .50 and 95% Credible Interval containing zero indicate good fit). * 95% Credible Interval of posterior density does not include zero. DIC Deviance Information Criterion. BIC Bayesian Information Criterion

Table 8

Model fit for Bayesian Confirmatory Factor Analyses

	PPp	95%CI	DIC	BIC
AMI	.50	− 65.91 – 65.45	10,621.16	12,064.79
PAMI	.34	−50.34 – 79.85	10,628.37	11,769.57

Note. AMI Approximate Measurement Invariance Model; APMI Approximate Partial Measurement Invariance Model; PPp Prior-posterior checking p-value (values close to .50 and 95% Credible Interval containing zero indicate good fit). * 95% Credible Interval of posterior density does not include zero; DIC Deviance Information Criterion. BIC Bayesian Information Criterion

Table 9

Model fit for traditional CFA and ESEM models

Model	χ2	CFI	TLI	RMSEA
SEM	220.58	.86	.83	.07 (.06–.09)
ESEM	176.09	.90	.85	.07 (.06–.08)
ESEM-MI	131.90	.94	.90	.06 (.04–.07)

Note. SEM Traditional Structural Equation Modelling; ESEM Exploratory Structural Equation Modelling; ESEM-MI Exploratory Structural Equation Modelling after freeing parameters with higher modification index statistics; CFI Comparative Fit Index. TLI Tucker-Lewis index; RMSEA Root mean square error of approximation

Table 10

Sensitivity analysis for BCFA model

	PPp	95%CI	DIC	BIC
BCFA.16	.48	−33.97 – 30.41	10,691.81	11,097.87
BCFA.20	.44	−31.92 – 31.85	10,692.58	11,080.97
BCFA.30	.35	−27.21 – 37.16	10,698.13	11,084.92
BCFA.50	.11	−15.98 – 53.89	10,714.10	11,095.94
BCFA.70	.02	1.79–78.29	10,734.80	11,113.45
BCFA.100	.00	35.49–121.73	10,773.80	11,161.96

BCFA Bayesian Confirmatory Factor Analysis. The number following the BCFA. Indicates the df of the Inverse Wishart distribution applied as a prior. PPp: Prior-posterior checking p-value (values close to .50 and 95% Credible Interval containing zero indicate good fit). * 95% Credible Interval of posterior density does not include zero. DIC Deviance Information Criterion; BIC Bayesian Information Criterion.

Table 11

Sensitivity analysis for the PAMI model

	PPp	95%CI	DIC	BIC
PAMI.01	.50	−65.91 – 65.45	10,621.16	12,064.79
PAMI.001	.16	−32.62 – 98.94	10,646.07	12,108.01
PAMI.0001	.11	−35.34 – 106.43	10,651.94	12,115.97
PAMI.00001	.11	−24.09 – 105.94	10,652.31	12,116.97

PAMI Partial Approximate Measurement Invariance Model. The number following the AMI. Indicates the variance parameter of the normal distribution applied as a prior. PPp: Prior-posterior checking p-value (values close to .50 and 95% Credible Interval containing zero indicate good fit). * 95% Credible Interval of posterior density does not include zero. DIC Deviance Information Criterion; BIC Bayesian Information Criterion.