Approximate Measurement Invariance Research Articles

Cross-National Measurement of Mathematics Intrinsic Motivation: An Investigate of Measurement Invariance with MG-CFA and Aligment Method Across Fourteen Countries

One of the main objectives of international large-scale assessments is to make comparisons between different countries, education policies, education systems, or subgroups. One of the main criteria for making comparisons between different groups is to ensure measurement invariance. The purpose of this study was to test the measurement invariance of the mathematics intrinsic motivation scale across 14 countries. For this purpose, the "students like learning mathematics" scale, which measures intrinsic motivation for mathematics, was included in the TIMSS 2019 cycle. The study sample consisted of a total of 152992 students, 70192 4th grade and 82800 8th grade students from 14 different countries participating in the TIMSS 2019 cycle. Measurement invariance was tested with Multi-Group Confirmatory Factor Analysis (MG-CFA) and Alignment Method. The mathematics intrinsic motivation scale provides only configural invariance according to MG-CFA at the 4th grade level, whereas the scale provides approximate invariance according to the alignment method. At the 8th grade level, the scale provides configural and metric invariance according to MG-CFA, whereas the scale provides approximate invariance according to the alignment method. The results indicate that the mathematics intrinsic motivation scale provides approximate measurement invariance at both grade levels and that comparisons can be made between the scores of the identified countries.

An examination of factorial invariance of the Asthma Control Questionnaire among adults with severe asthma

Background The Asthma Control Questionnaire (ACQ) is used to assess asthma symptom control. The relationship between the questionnaire items and symptom control has not been fully studied in severe asthmatic patients, and its validity for making comparisons between subgroups of patients is unknown. Methods Data was obtained from patients in the United Kingdom Severe Asthma Registry whose symptom control was assessed using the five-item ACQ (ACQ5) (n = 2,951). Confirmatory factor analysis determined whether a latent factor for asthma symptom control, as measured by the ACQ5, was consistent with the data. Measurement invariance was examined in relation to ethnicity, sex and age; this included testing for approximate measurement invariance using Bayesian Structural Equation Modelling (BSEM). The fitted models were used to estimate the internal consistency reliability of the ACQ5. Invariance of factor means across subgroups was assessed. Results A one-factor construct with residual correlations for the ACQ5 was an excellent fit to the data in all subgroups (Root Mean Square Error Approximation 0.03 [90%CI 0.02,0.05], p-close fit 0.93, Comparative Fit Index 1.00, Tucker Lewis Index 1.00}. Expected item responses were consistent for Caucasian and non-Caucasian patients with the same absolute level of symptom control. There was some evidence that females and younger adults reported wakening more frequently during the night than males and older adults respectively with the same absolute level of symptom control (p<0.001). However approximate measurement invariance was tenable and any failure to observe strong measurement invariance had minimal impact when comparing mean levels of asthma symptom control between patients of different sexes or ages. Average levels of asthma symptom control were lower for non-Caucasians (p = 0.001), females (p<0.01)and increased with age (p<0.01). Reliability of the instrument was high (over 88%) in all subgroups studied. Conclusion The ACQ5 is informative in comparing levels of symptom control between severe asthmatic patients of different ethnicities, sexes and ages. It is important that analyses are replicated in other severe asthma registries to determine whether measurement invariance is observed.

Measurement invariance of the driving inattention scale (ARDES) across 7 countries

The Attention-Related Driving Errors Scale (ARDES) is a self-report measure of individual differences in driving inattention. ARDES was originally developed in Spanish (Argentina), and later adapted to other countries and languages. Evidence supporting the reliability and validity of ARDES scores has been obtained in various different countries. However, no study has been conducted to specifically examine the measurement invariance of ARDES measures across countries, thus limiting their comparability. Can different language versions of ARDES provide comparable measures across countries with different traffic regulations and cultural norms? To what extent might cultural differences prevent researchers from making valid inferences based on ARDES measures? Using Alignment Analysis, the present study assessed the approximate invariance of ARDES measures in seven countries: Argentina (n = 603), Australia (n = 378), Brazil (n = 220), China (n = 308). Spain (n = 310), UK (n = 298), and USA (n = 278). The three-factor structure of ARDES scores (differentiating driving errors occurring at Navigation, Manoeuvring and Control levels) was used as the target theoretical model. A fixed alignment analysis was conducted to examine approximate measurement invariance. 12.3 % of the intercepts and 0.8 % of the item-factor loadings were identified as non-invariant, averaging 8.6 % of non-invariance. Despite substantial differences among the countries, sample recruitment or representativeness, study results support resorting to ARDES measures to make comparisons across the country samples. Thus, the range of cultures, laws and collision risk across these 7 countries provides a demanding assessment for a cultural-free inattention while-driving. The alignment analysis results suggest that ARDES measures reach near equivalence among the countries in the study. We hope this study will serve as a basis for future cross-cultural research on driving inattention using ARDES.

Cross-cultural invariance of the Spanish version of the COVID-19 Assessment Scorecard to measure the perception of government actions against COVID-19 in Latin America

ObjectivesThe present study aimed to evaluate the measurement invariance of a general measure of the perception of governmental responses to COVID-|19 (COVID-SCORE-10) in the general population of 13 Latin American countries.MethodsA total of 5780 individuals from 13 Latin American and Caribbean countries selected by non-probabilistic snowball sampling participated. A confirmatory factor analysis was performed and the alignment method was used to evaluate invariance. Additionally, a graded response model was used for the assessment of item characteristics.ResultsThe results indicate that there is approximate measurement invariance of the COVID-SCORE-10 among the participating countries. Furthermore, IRT results suggest that the COVID-SCORE-10 measures with good psychometric ability a broad spectrum of the construct assessed, especially around average levels. Comparison of COVID-SCORE-10 scores indicated that participants from Cuba, Uruguay and El Salvador had the most positive perceptions of government actions to address the pandemic. Thus, the underlying construct of perception of government actions was equivalent in all countries.ConclusionThe results show the importance of initially establishing the fundamental measurement properties and MI before inferring the cross-cultural universality of the construct to be measured.

TEST OF INVARIANCE OF THE ANOMIE BRIEF SCALE WITH THE ALIGNMENT METHOD IN 12 LATIN AMERICAN COUNTRIES

The Social Anomie Brief Scale (SAS10) is a self-report measure of social anomie against new social standards implemented during the current COVID-19 pandemic. The main objective of this study was to evaluate the invariance of the cross-cultural measurement of the SAS10 in a sample of 12 Latin American countries. Additionally, the difficulty and discrimination characteristics of the items were evaluated using the IRT and social anomie was compared between the participating countries. A total of 4,911 people from 12 Latin American countries (Argentina, Bolivia, Chile, Colombia, Cuba, Ecuador, Guatemala, Mexico, Paraguay, Peru, Uruguay, and Venezuela) selected by non-probabilistic snowball sampling participated. The results indicated that the original two-factor model of the SAS10 shows estimation problems and low fit indices in several countries. However, a model of two related dimensions (behavioral and affective) of nine items (SAS-9) presented adequate fit indices in all countries. This model presents adequate estimation of reliability and approximate cross-cultural measurement invariance. Peru was the country with the highest score in the behavioral dimension of social anomie; while Venezuela and Bolivia presented the highest scores in the affective dimension. Mexico was one of the countries with the lowest anomie scores. The results of the IRT would indicate that the characteristics of the SAS-9 items were appropriate. It is concluded that the SAS-9 has shown good psychometric properties when evaluated in 12 Latin American countries and can be used in future studies that compare social anomie cross-culturally.

Towards a conceptualisation of intensive parenting norms: Testing exact and approximate measurement invariance across social and country contexts

Objective: We examine the measurement invariance (MI) of intensive parenting norms across three European countries as well as across gender, education, and parental status. Background: The social norms of intensive parenting have become increasingly dominant in recent decades. However, there is no measurement model of intensive parenting norms in large-scale social surveys and it is unknown whether these norms convey the same meanings across different social and cultural contexts. Method: This study draws on data from the European Social Survey Cross-national Online Survey panel (2017) administered in Estonia, Slovenia, and Great Britain. We conducted a multi-group confirmatory factor analysis and alignment optimization to assess the exact and approximate MI of intensive parenting norms across three countries and social contexts. Results: Due to a poor fit of the previously suggested four-factor model of intensive parenting norms, a revised two-factor model with stimulation and child-centred dimensions - excluding items that measure expert guidance and parental responsibility - was proposed. The two-factor model attained scalar invariance between educational groups and between parents of children aged under 12 years and others; however, only metric invariance was achieved among countries, gender, and parental status. The alignment optimization results suggest that the reduced scale is approximately invariant across all examined groups. Conclusion: This study highlights that the MI of parenting norms should not just be assumed but tested. This study provides insights into the conceptualisation of intensive parenting norms and recommendations for future research and development on measurement.

Why Full, Partial, or Approximate Measurement Invariance Are Not a Prerequisite for Meaningful and Valid Group Comparisons

It is frequently stated in the literature that measurement invariance is a prerequisite for the comparison of group means or standard deviations of the latent variable in factor models. This article argues that measurement invariance is not necessary for meaningful and valid comparisons across groups. There is unavoidable ambiguity in how researchers can define comparisons if measurement invariance is violated. Moreover, there is no support for preferring the partial invariance approach over competing approaches, such as invariance alignment, robust linking, or Bayesian approximate invariance. Furthermore, we also argue why an intentionally misspecified multiple-group factor model with invariant item parameters can be justified if measurement invariance is violated.

Measurement Invariance of the Meaning in Life Questionnaire Across 17 Countries

The Meaning in Life Questionnaire assesses presence of and search for meaning in life. Although the questionnaire has shown promising psychometric properties in samples from different countries, the scale’s measurement invariance across a large number of nations has yet to be assessed. This study is aimed at addressing this gap, providing insight into how meaning in life is constructed and experienced across countries and into the extent to which cross-country comparisons can be made. A total of 3867 adult participants from 17 countries, aged 30–60, balanced by gender, and with at least secondary education, completed the questionnaire as part of the Eudaimonic and Hedonic Happiness Investigation. Single sample confirmatory factor analysis, multigroup confirmatory factor analysis, and alignment optimization were applied to investigate the scale’s performance across the samples. Good psychometric properties and high levels of approximate measurement invariance emerged for the Presence subscale after removal of item 9, the only reverse-phrased item. Performance of the Search subscale varied more across samples, suggesting caution in interpreting related results supporting approximate measurement invariance. The conceptualization of presence of meaning operationalized in the corresponding subscale (without item 9) appears consistent across countries, whereas search for meaning seems to be less universally homogenous and requires further exploration. Moreover, the Meaning in Life Questionnaire does not reflect the conceptual distinction between “purpose” and “meaning” currently acknowledged by researchers. This issue should be further explored in studies addressing the scale’s performance across cultures.

Cross-cultural measurement of social withdrawal motivations across 10 countries using multiple-group factor analysis alignment

The goal of this study was to evaluate the measurement invariance of an adapted assessment of motivations for social withdrawal (Social Preference Scale–Revised; SPS-R) across cultural contexts and explore associations with loneliness. Participants were a large sample of university students (N = 4,397; Mage = 20.08 years, SD = 2.96; 66% females) from 10 countries (Argentina, Australia, Canada, China, India, Italy, South Korea, Norway, Turkey, and the United States). With this cross-cultural focus, we illustrate the multiple-group factor analysis alignment method, an approach developed to assess measurement invariance when there are several groups. Results indicated approximate measurement invariance across the 10 country groups. Additional analyses indicated that overall, shyness, avoidance, and unsociability are three related, but distinct factors, with some notable country differences evident (e.g., in China, India, and Turkey). Shyness and avoidance were related positively to loneliness in all countries, but the strength of the association between shyness and loneliness differed in Italy and India relative to the other countries. Results also indicated that unsociability was related positively to loneliness in the United States only. Theoretical and assessment implications are discussed.

Measurement invariance in the social sciences: Historical development, methodological challenges, state of the art, and future perspectives

This review summarizes the current state of the art of statistical and (survey) methodological research on measurement (non)invariance, which is considered a core challenge for the comparative social sciences. After outlining the historical roots, conceptual details, and standard procedures for measurement invariance testing, the paper focuses in particular on the statistical developments that have been achieved in the last 10 years. These include Bayesian approximate measurement invariance, the alignment method, measurement invariance testing within the multilevel modeling framework, mixture multigroup factor analysis, the measurement invariance explorer, and the response shift-true change decomposition approach. Furthermore, the contribution of survey methodological research to the construction of invariant measurement instruments is explicitly addressed and highlighted, including the issues of design decisions, pretesting, scale adoption, and translation. The paper ends with an outlook on future research perspectives.

Measuring Loneliness in Different Age Groups: The Measurement Invariance of the UCLA Loneliness Scale.

Age differences in the prevalence of loneliness have been a key focus among researchers, practitioners, and policy makers. However, the degree to which those reflect genuine differences in the experience of loneliness or the way individuals understand and respond to loneliness measures is yet to be examined. The current study explored the age measurement invariance of the 20-item Revised University of California Los Angeles, Loneliness Scale (UCLA-LSR) and its shorter forms in a U.K. sample of adults aged 18 to 99 years (M = 50.6, SD = 19.7). The fit of different structures/versions was explored through multigroup confirmatory factor analysis (CFA; N = 4,375) and local structural equation modeling (N = 19,521). Results indicated a poor and/or inconsistent structure for the 20-item UCLA-LSR and many of its shorter forms. Of the structures considered, 12 showed acceptable model fit and received age measurement invariance testing through multigroup CFA and alignment; 10 of these achieved full, partial, or approximate measurement invariance. Our findings suggest that the age measurement invariance of loneliness measures should not be assumed, and crucially, this must be explored before accurate and meaningful age comparisons can be made. Implications for measurement research, and clinical and community practice, are discussed.

Cross-cultural measurement invariance of the Quality of Life Enjoyment and Satisfaction Questionnaire-Short form across ten countries: the application of Bayesian approximate measurement invariance

BackgroundThe Quality of Life Enjoyment and Satisfaction Questionnaire-Short Form (Q-LES-Q-SF) is the most frequently used generic quality of life (QOL) measure in many countries and cultures worldwide. However, no single study has been carried out to investigate whether this questionnaire performs similarly across diverse cultures/countries. Accordingly, this study aimed to assess the cross-cultural measurement invariance of the Q-LES-Q-SF across ten different countries.MethodsThe Q-LES-Q-SF was administrated to a sample of 2822 university students from ten countries: Bangladesh, Brazil, Croatia, India, Nepal, Poland, Serbia, Turkey, the United Arab Emirates, and Vietnam. The Bayesian approximate measurement invariance approach was used to assess the measurement invariance of the Q-LES-Q-SF.ResultsApproximate measurement invariance did not hold across the countries for the Q-LES-Q-SF, with only two out of 14 items being non-invariant; namely items related to doing household and leisure time activities.ConclusionsOur findings did not support the cross-cultural measurement invariance of the Q-LES-Q-SF; thus, considerable caution is warranted when comparing QOL scores across different countries with this measure. Item rewording and adaptation along with calibrating non-invariant items may narrow these differences and help researchers to create an invariant questionnaire for reliable and valid QOL comparisons across different countries.

Measurement Invariance and Relationships Among School Connectedness, Cyberbullying, and Cybervictimization: A Comparison Among Canadian, Chinese, and Tanzanian Adolescents

The aim of the present study was to evaluate the measurement invariance of the School Connectedness Scale for Chinese, Canadian, and Tanzanian adolescents, and to explore the inter association between school connectedness and cyberbullying/cybervictimization. Participants included 3872 adolescents from urban settings in China ( N= 2053, M age=16.36 years, SD = 1.14 years; 44.6% boys), Canada ( N = 642, M age = 12.13 years, SD = 0.77 years; 50.1% boys), and Tanzania ( N = 1056 , M age=15.87 years, SD = 2.03 years; 52.8% boys). Adolescents self-reported their cybervictimization and cyberbullying experiences, as well as their perceived school connectedness. Multigroup Confirmatory Factor Analysis revealed an approximate measurement invariance of the scale across the three countries. Chinese students showed the lowest levels of school connectedness while Tanzanian students showed the highest. The findings of the multivariate multigroup regression analyses across the three countries revealed similar relationships between school connectedness and cyberbullying/cybervictimization, thus broadening our understanding of school connectedness and its relationship to cyberbullying/cybervictimization across these three different countries.

Measurement invariance of non-cognitive measures in TIMSS across countries and across time. An application and comparison of Multigroup Confirmatory Factor Analysis, Bayesian approximate measurement invariance and alignment optimization approach

Measurement invariance is a key topic in international large-scale assessments (ILSA). The study investigates measurement invariance across countries and over time of noncognitive measures in TIMSS, as well as comparing different approaches of measurement invariance analyses. Multigroup Confirmatory Factor Analysis (MGCFA) is the standard procedure to test exact measurement invariance. Due to criticism of this approach in the context of ILSAs, the novel approaches alignment optimization and Bayesian approximate measurement invariance method were applied. Principal, teacher, and student data of the TIMSS cycles 2007, 2011 and 2015 of 26 countries was used. MGCFA and Bayesian approximate measurement invariance analyses indicates configural invariance in all analyses across countries and a variety of invariance levels (mostly metric) over time. The results of the alignment optimization method show for most scales that latent means are comparable across countries - under the assumption of approximate measurement invariance

Global measurement of intimate partner violence to monitor Sustainable Development Goal 5

BackgroundOne third of women experience intimate partner violence (IPV) and potential sequelae. Sustainable Development Goal (SDG) 5.2—to eliminate violence against women, including IPV—compels states to monitor such violence. We conducted the first global measurement-invariance assessment of standardised item sets for IPV.MethodsDemographic and Health Surveys (DHS) from 36 Lower−/Middle-Income Countries (LMICs) administering 18 IPV items during 2012–2018 were included. Analyses were performed separately for two items sets: lifetime physical IPV (seven items) and controlling behaviours (five items). We performed country-specific exploratory and confirmatory factor analyses (EFA/CFA). Datasets meeting benchmarks for acceptable item loadings and model-fit statistics were included in multiple-group CFA (MGCFA) to test for exact measurement invariance. Based on findings, alignment optimization (AO) was performed to assess approximate measurement invariance (< 25% of model parameters non-invariant). For each item set, national rankings based on AO-derived scores and on prevalence estimates were compared. AO-derived scores were correlated with type-specific IPV prevalences to assess correspondence.ResultsNational rates of physical IPV (5.6–50.5%) and controlling behavior (25.9–84.7%) varied. For each item set, item loadings and model-fit statistics were adequate in country-specific, unidimensional EFAs and CFAs. Both unidimensional constructs lacked exact invariance in MGCFA but achieved approximate invariance in AO analysis (12.3% of model parameters for physical IPV and 6.7% for controlling behaviour non-invariant). For both item sets, national rankings based on AO-derived scores were distributed similarly to rankings based on prevalence. However, estimates often were not significantly different cross-nationally, precluding national-level comparisons regardless of estimation strategy. Three physical-IPV items (slap, twist, choke) and two controlling-behaviour items (meet female friends; contact with family) warrant cognitive testing to improve their psychometric properties. Correlations of AO-derived scores for physical IPV (0.48–0.66) and controlling behaviours (0.49–0.87) with prevalences of lifetime physical, sexual, psychological IPV as well as controlling behaviour varied.ConclusionsSeven DHS lifetime physical-IPV items and five DHS controlling-behaviour items were approximately invariant across 36 LMICs spanning five world regions, such that cross-national comparisons of factor means are reasonable. Measurement-invariance testing over time will inform their utility to monitor SDG5.2.1; cross-national, cross-time measurement-invariance testing of improved sexual and psychological IPV item-sets is needed.

Longitudinal measurement invariance in urbanization index of Chinese communities across 2000 and 2015: a Bayesian approximate measurement invariance approach

BackgroundThe Urbanicity Scale was developed based on the China Health and Nutrition Survey (CHNS) to measure the urbanization index of communities according to 12 components. The present study was designed to systematically investigate the factorial validity, reliability, and longitudinal measurement invariance (LMI) of the Urbanicity Scale.MethodsSix waves of CHNS data from 2000 to 2015 were adopted. The factor structure and reliability of the Urbanicity Scale for 301 communities were examined using Bayesian exploratory factor analysis. Metric and scalar LMIs were evaluated using both the conventional exact and a novel approximate LMI approach via Bayesian structural equation modeling across various timeframes.ResultsThe findings verified the one-factor structure for the Urbanicity Scale, with adequate reliability. LMI was established for the Urbanicity Scale only over a shorter timeframe from 2006 to 2009 but not over a longer timeframe from 2000 to 2015. Partial LMI was found in the factor loadings and item intercepts for the Urbanicity Scale over the 2004 to 2011 period.ConclusionInterpretation of the temporal change in urbanicity was supported only for a shorter (2006 to 2009) but not a longer timeframe (2000 to 2015). Adjustments addressing the partial non-invariance of the measurement parameters are needed for the analysis of temporal changes in urbanicity between 2004 and 2011.

Measurement Invariance of the SQWLi Instrument Over Time

The SQWLi questionnaire was developed for the long-term measurement of subjectively perceived quality of working life. The aim of this study is to test the instrument's measurement invariance between 2009 and 2019 and determine whether - despite the modifications made to the instrument over the years - the results remain comparable. Data from eight representative surveys of the economically active population in the Czech Republic were analysed (total N = 6909) using the MG CFA method (configural, metric, and scalar invariance) and the alignment method (approximate measurement invariance). The findings from the MG CFA tests for measurement invariance indicate that the SQWLi instrument achieves configural and metric invariance over time but not full scalar invariance. Achieving a partial scalar invariance would be challenging because of the many high modification indices; therefore, an approximate measurement invariance approach, namely the Alignment Method, was applied. The results suggest that comparisons of latent means across all years can be made. Consequently, it is possible to make meaningful comparisons of overall indices of dimensions (batteries) and of more general domains. However, not all the individual items can be compared. The results confirm that the biggest risk of invariance is caused by conceptual changes to items and by substantial or frequent modifications to item wording. Conversely, the results also show that a conceptual change to an entire dimension may not necessarily cause any problem on the general level, and that a disruption of invariance caused by changes to the range of scales used can be rectified by means of their harmonisation ex post.

Longitudinal Measurement Invariance of the PHQ-9 and GAD-7

Intraindividual change over time is commonly used to estimate treatment effectiveness. However, patients may not respond similarly to a scale after treatment, rendering pre-post change an unreliable metric. The current objective was to investigate longitudinal measurement invariance of the Patient Health Questionnaire-9 and Generalized Anxiety Disorder Scale-7 among 4,323 patients completing a partial hospital program. We used confirmatory factor analysis to determine (1) factor structure at pretreatment and posttreatment and (2) longitudinal invariance, accounting for dependent observations, using both classical and approximate measurement invariance approaches. Results indicated a two-factor solution for both scales. Longitudinal invariance was not established for either scale, thus, using raw score differences from the Patient Health Questionnaire-9 and Generalized Anxiety Disorder Scale-7 for measuring symptom change over time may be problematic. The most longitudinally consistent items captured somatic as opposed to affective/cognitive symptoms. We discuss the potential use of these measures for diagnostic screening and between-group comparisons and suggest alternative ways to monitor client progress over time. Limitations included a majority White sample and uniqueness of a partial hospital setting.

Measurement invariance of the Childhood Autism Rating Scale (CARS) across six countries.

The Childhood Autism Rating Scale (CARS) is a simple and inexpensive tool for Autism spectrum disorder (ASD) assessments, with evidenced psychometric data from different countries. However, it is still unclear whether ASD symptoms are measured the same way across different societies and world regions with this tool, since data on its cross-cultural validity are lacking. This study evaluated the cross-cultural measurement invariance of the CARS among children with ASD from six countries, for whom data were aggregated from previous studies in India (n=101), Jamaica (n=139), Mexico (n=72), Spain (n=99), Turkey (n=150), and the United States of America (n=186). We analyzed the approximate measurement invariance based on Bayesian structural equation modeling. The model did not fit the data and its measurement invariance did not hold, with all items found non-invariant across the countries. Items related to social communication and interaction (i.e., relating to people, imitation, emotional response, and verbal and nonverbal communication) displayed lower levels of cross-country non-invariance compared to items about stereotyped behaviors/sensory sensitivity (i.e., body and object use, adaptation to change, or taste, smell, and touch response). This study found that the CARS may not provide cross-culturally valid ASD assessments. Thus, cross-cultural comparisons with the CARS should consider first which items operate differently across samples of interest, since its cross-cultural measurement non-invariance could be a source of cross-cultural variability in ASD presentations. Additional studies are needed before drawing valid recommendations in relation to the cultural sensitivity of particular items.

Measurement Invariance of Liberal and Authoritarian Notions of Democracy: Evidence From the World Values Survey and Additional Methodological Considerations

This article offers a gentle introduction to the measurement invariance (MI) literature with a focus on its relevance to comparative political research. It reviews 1) the conceptual foundations of MI; 2) standard procedures of testing for MI in practical applications within the multiple-group confirmatory factor analysis (MGCFA) paradigm; and 3) two novel approaches to MI, Bayesian approximate measurement invariance, and MGCFA alignment optimization, which are especially suitable for dealing with extremely heterogeneous data from large-scale comparative surveys typical for modern political science. It then provides an empirical illustration of the key concepts and methods from the MGCFA-MI literature by applying them to testing for MI of two recently introduced measures of democracy attitudes, so-called liberal and authoritarian notions of democracy, across 60 countries in the sixth round of the World Values Survey. These analyses show that both measures can be considered reliable comparative measures of democratic attitudes, although for different reasons. Finally, this study emphasizes that some survey-based constructs, e.g., authoritarian notions of democracy, do not follow the reflective (correlation-based) logic of construct development. These alternative measures, known as formative measures, do not assume strong correlations between their indicators, for which reason it is inappropriate to test their comparability using the reflective MGCFA approach. Instead, their comparability can be tied to their correlations with theoretically relevant external variables.