The development of the test instrument in this study is grounded in a hierarchical model of mathematical abstraction, which includes three progressive levels: Perceptual Abstraction, Internalization, and Interiorization (Dreyfus, 2014; Dreyfus & Kidron, 2014; Hazzan & Zazkis, 2005). These levels reflect the cognitive transitions students undergo from recognizing patterns in concrete representations to constructing and manipulating abstract mathematical ideas (Hong & Kim, 2016; Ocy et al., 2023). Each level encompasses specific indicators that guide the construction of test items and the evaluation of student responses.

At this foundational level, students begin to identify visible patterns and structures in mathematical representations (Spiller et al., 2023). This stage is crucial as it enables learners to make initial connections between concrete objects and symbolic representations (Kinanti et al., 2023). Key competencies include recognizing recurring patterns in shapes and algebraic forms, interpreting visual representations such as graphs, and understanding simple transformations.

The theoretical foundation of perceptual abstraction is grounded in the work of Gray and Tall (2007), who highlight the importance of perceptual cues as cognitive anchors that support students in the early stages of abstract mathematical thinking. Their research suggests that these cues provide the initial structure upon which deeper understanding is built. Complementing this view, Mitchelmore and White (2007) argue that the ability to recognize and generalize patterns plays a critical role in the development of abstract reasoning. In line with these perspectives, Kellman et al. (2010) emphasize the role of perceptual learning in enhancing students’ sensitivity to mathematical structures and relationships, thereby facilitating their progression toward more sophisticated forms of mathematical thought.

At this intermediate level, students internalize abstract ideas by mentally constructing and applying them in various contexts (Abreu-Mendoza et al., 2018). Internalization involves the ability to manipulate symbolic representations, connect abstract concepts to real-life applications, and articulate the reasoning behind mathematical operations (Uygun & Guner, 2022).

The theoretical foundation of internalization is rooted in the idea that abstract concepts must be mentally assimilated for deeper understanding and application. Ferrari (2003) and White and Mitchelmore (2010) describe internalization as the cognitive process through which students embed abstract mathematical ideas into their mental frameworks. This process allows learners to move beyond surface-level comprehension toward meaningful engagement with mathematical concepts. Supporting this view, Cooley (2002) and Reed (2016) emphasizes the importance of representational strategies—such as diagrams, symbolic notations, or visual models—which serve as bridges between abstract theory and practical application. These representations help students internalize complex ideas by making them more accessible and relatable within diverse problem-solving contexts.

This is the highest level of abstraction, where students demonstrate the ability to synthesize, justify, and evaluate mathematical ideas across diverse contexts (Ahmadah, 2020; Balkan, 2023; Žakelj et al., 2024). They create mathematical models, validate their solutions, and justify their reasoning within theoretical frameworks. This level reflects mathematical maturity and creative problem-solving (Abreu-Mendoza et al., 2018).

The theoretical foundation of interiorization lies in the advanced cognitive processes that enable students to operationalize abstract concepts and apply them flexibly across various contexts. Dreyfus (2014) and Dreyfus and Kidron (2014) conceptualize interiorization as the stage at which learners not only understand abstract mathematical ideas but also use them fluidly in generalized and adaptive ways. This stage marks a significant shift from procedural knowledge to a more integrated and strategic application of mathematical thought. In support of this perspective, Cetin and Dubinsky (2017), Gold (2018), and Simon (2020) emphasize the role of cognitive integration, where students draw upon multiple mathematical domains to construct coherent solutions to complex problems. Such sophisticated problem-solving reflects the characteristics of the formal operational stage as outlined in Piagetian theory Moessinger and Poulin-Dubois (1981) and Reed (2016), where learners demonstrate abstract reasoning, hypothetical thinking, and mental coordination of multiple variables.

In this study, the test instrument for assessing mathematical abstraction skills among senior high school students was developed based on 10 carefully selected indicators ( Table 1). These indicators were chosen to represent different cognitive levels of mathematical abstraction, which include Perceptual Abstraction, Internalization, and Interiorization. Each indicator is designed to assess specific aspects of students' mathematical reasoning and problem-solving capabilities, capturing their progression from basic understanding to more complex mathematical reasoning.

The Perceptual Abstraction level emphasizes foundational mathematical skills, including pattern recognition and the identification of relationships in simple mathematical contexts (Hong & Kim, 2016).

This level is assessed through Indicators Q1 (recognizing patterns in geometric shapes and algebraic expressions), Q2 (identifying relationships between variables in simple equations or visual representations), and Q4 (observing transformations in graphs and understanding their effects on equations).

The theoretical grounding for these indicators is rooted in studies by (Gray & Tall, 2007) and (Mitchelmore & White, 2007), which highlight the importance of perceptual cues in the early stages of abstraction. These researchers argue that recognizing and describing patterns is a precursor to deeper mathematical reasoning. Additionally, (Kellman et al., 2010) emphasize the role of visual processing and pattern recognition in enhancing students' ability to detect mathematical relationships, a skill critical for advancing to higher levels of abstraction.

At the Internalization level, the focus shifts to students' ability to internalize abstract concepts and apply them to real-world and mathematical problems (Hong & Kim, 2016). This level is assessed through Indicators Q3 (applying abstract mathematical concepts to solve real-world problems), Q5 (developing strategies to represent problems algebraically or geometrically), and Q7 (explaining reasoning behind algebraic manipulations).

Theoretical support for these indicators is drawn from (Ferrari, 2003) and (White & Mitchelmore, 2010), who describe internalization as the process of embedding abstract concepts into students’ cognitive frameworks, allowing them to navigate practical problems effectively. (Cooley, 2002) further supports this perspective by highlighting the role of strategic representation in fostering deeper understanding, enabling students to bridge abstract ideas and practical applications.

The Interiorization level represents the highest order of abstraction, where students synthesize multiple mathematical concepts into cohesive approaches(Hong & Kim, 2016). Indicators at this level include Q6 (integrating multiple concepts for problem-solving), Q8 (justifying steps in complex problem-solving), Q9 (evaluating accuracy and consistency of mathematical models), and Q10 (creating mathematical models from abstract concepts).

The conceptual foundation for this level draws on (Dreyfus, 2014),(Hershkowitz et al., 2007), (Hershkowitz et al., 2007), (Kidron & Dreyfus, 2008), and(Raychaudhuri, 2014), who emphasize the importance of integrating diverse mathematical constructs for advanced problem-solving. (Mitchelmore & White, 2007) also highlight that interiorization enables students to operate within a network of interconnected concepts, fostering flexibility and innovation. These skills are critical for advanced mathematical reasoning, particularly in real-world scenarios and STEM applications.

The selection of these indicators is based on a well-established theoretical framework of mathematical abstraction as a developmental process. Research by (Gray & Tall, 2007) underscores the progression from perceptual abstraction to interiorization, where each level builds upon the cognitive abilities developed in the preceding stage. This framework aligns with the theories of cognitive development proposed by (Moessinger & Poulin-Dubois, 1981) and (Kamina & Iyer, 2009), particularly the transition from concrete operational to formal operational stages.

A purposive sampling strategy was applied to select research sites. Four senior high schools in Tanjungpinang—SMAN 1 Tanjungpinang, SMAN 2 Tanjungpinang, SMAS Maitreyawira, and SMAS Santa Maria—were chosen based on their active implementation of Higher-Order Thinking Skills (HOTS) in instructional practices. The rationale behind this purposive selection was that students in such schools are more likely to be familiar with abstract reasoning, making them suitable for participation in a study focused on mathematical abstraction. A total of 507 students participated in the study.

Instrument development and data collection

The data collection was carried out using a constructed-response test, as opposed to a multiple-choice format. This test required students to provide written explanations, justifications, or mathematical arguments based on the problems presented. Each item was designed to measure one or more aspects of mathematical abstraction according to a theoretically grounded analytic rubric developed by the researcher.

Prior to the large-scale administration, the test items underwent a rigorous content validation process. In the first phase, 17 mathematics education teachers and practitioners reviewed the clarity, relevance, and alignment of the items with the abstraction constructs. Their feedback was analyzed using the Aiken’s V index. In the second phase, five expert validators in mathematics education evaluated the items further. Content validity indices, including the Item-Level Content Validity Index (I-CVI), Scale-Level CVI using the average method (S-CVI/Ave), Universal Agreement (S-CVI/UA), and the Content Validity Ratio (CVR), were calculated to ensure rigorous content alignment.

After content validation, the finalized instrument was administered to the student sample. Students’ responses to the constructed-response items were scored using a predefined analytic rubric, which allowed for the identification of various cognitive processes involved in abstraction. The scored data were subjected to Exploratory Factor Analysis (EFA) to uncover the underlying factor structure of the instrument. Subsequently, Confirmatory Factor Analysis (CFA) was performed to test the hypothesized factor model, using fit indices such as RMSEA, CFI, TLI.

In addition to construct validity, reliability was assessed through Cronbach’s Alpha and Composite Reliability for each subscale, representing the three levels of abstraction. These analyses ensured that the test instrument was both valid and internally consistent. This methodologically rigorous approach resulted in a validated constructed-response test instrument that can be reliably used to assess students’ mathematical abstraction skills in secondary education settings.

In the expert validation process, the content validity of the instrument was assessed using Aiken’s V, I-CVI (Item Content Validity Index), CVR (Content Validity Ratio), and scale-level indices (S-CVI/Ave and S-CVI/UA), as shown in Table 2.

The panel evaluation was conducted and calculated using Aiken's V based on feedback from 17 mathematics education experts. Meanwhile, expert judgment involving 5 experts was assessed using I-CVI, CVR, S-CVI/Ave, and S-CVI/UA. The Aiken’s V values ranged from 0.868 to 0.970, indicating strong agreement among the experts regarding the clarity and relevance of the items. All items achieved an I-CVI of 1.00, reflecting unanimous expert consensus on their relevance.

Additionally, the CVR values showed that the majority of items (Q1, Q2, Q3, Q6, Q8, and Q10) were deemed essential by all experts, achieving a perfect CVR of 1.00. At the scale level, both S-CVI/Ave and S-CVI/UA indices were 1.00, demonstrating overall strong content validity for the instrument. The CVI, which summarizes the content validity of the instrument as a whole, was calculated to be 0.801, surpassing the minimum acceptable threshold.

These findings validate the appropriateness and necessity of the items for measuring mathematical abstraction skills. The rigorous content validation process provides strong evidence of the instrument's quality, serving as a foundation for subsequent analyses, including exploratory and confirmatory factor analysis, to further establish its construct validity and reliability.

Conducting assumption tests is a critical prerequisite before performing Exploratory Factor Analysis (EFA) and Confirmatory Factor Analysis (CFA). These tests ensure that the data meet essential statistical conditions, such as normality, sampling adequacy, and the absence of multicollinearity, which are necessary for the validity and reliability of the analyses (Prihono et al., 2022).

As shown in Table 3, the Shapiro-Wilk test for normality indicates that all items (Q1 to Q10) are normally distributed (p>0.05), with p-values exceeding 0.05. This supports the assumption of univariate normality, which is advantageous for both EFA and CFA. Collinearity statistics further confirm the absence of multicollinearity, as tolerance values exceed 0.10 and Variance Inflation Factor (VIF) values remain below 2.0, ensuring sufficient independence among items.

The Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy reports a robust overall value of 0.8502, with individual Measures of Sampling Adequacy (MSA) ranging from 0.812 to 0.901. These results indicate that the dataset is well-suited for factor analysis. Bartlett’s test of sphericity (χ² = 716.731, p < 0.001) confirms the presence of significant correlations among items, validating the suitability of the data for factor analysis. Additionally, Mardia’s test of multivariate normality demonstrates acceptable skewness (0.233) and kurtosis (2.778) values, with non-significant p-values (p>0.05), further supporting the appropriateness of the data.

These findings collectively confirm that the dataset meets the necessary assumptions for conducting EFA and CFA. The comprehensive assessment of normality, sampling adequacy, and inter-item correlations ensures the robustness of the subsequent analyses, laying a strong foundation for interpreting the results and validating the test instrument for mathematical abstraction skills.

The purpose of Exploratory Factor Analysis is to identify the underlying factor structure of a set of measured variables (Fu et al., 2022). EFA is a data-driven approach that examines the correlations among the variables and groups them into factors, without any prior assumptions about the structure. This helps to reveal the latent constructs that are responsible for the observed correlations in the data. Evaluating model fit is a critical step in Exploratory Factor Analysis (EFA) to ensure that the proposed model adequately represents the data.

As shown in Table 4, the chi-square test resulted in a value of 80.869 with 26 degrees of freedom and a p-value of 0.067. This p-value, while slightly above the conventional threshold of 0.05, suggests that the model does not significantly differ from the observed data, indicating an acceptable fit. Additional fit indices provide further evidence of the model's adequacy. The Comparative Fit Index (CFI) and Tucker-Lewis Index (TLI) values are 0.938 and 0.941, respectively, both exceeding the 0.90 benchmark for a good model fit. The Root Mean Square Error of Approximation (RMSEA) is 0.037, with a 90% confidence interval of 0.00 to 0.016, indicating excellent model fit according to commonly accepted thresholds.

Similarly, the Standardized Root Mean Square Residual (SRMR) value of 0.037 confirms the model's accuracy in reproducing the observed correlations, as values below 0.08 are considered acceptable.

The factor loadings and characteristics of the three factors extracted through Exploratory Factor Analysis (EFA) are presented in Table 5. The extraction was conducted using the eigenvalue > 1 criterion, which identifies factors that explain a meaningful amount of variance in the data. To improve interpretability, the Promax rotation method was employed, as it is particularly suitable for identifying correlated factors.

The factor loadings indicate the strength of the association between each item and its respective factor. Items such as Q1, with a loading of 0.997 on Factor 1, demonstrate a very strong relationship, while items like Q6, with a loading of 0.802 on Factor 3, indicate a moderate relationship. High factor loadings suggest that the items are strongly aligned with the latent constructs represented by their respective factors, supporting the validity of the instrument.

Uniqueness values provide additional insights, indicating the proportion of variance in each item not accounted for by the factors. For example, Q1 exhibits a low uniqueness value of 0.021, meaning that 97.9% of its variance is explained by Factor 1. In contrast, Q6 has a uniqueness value of 0.128, indicating that 12.8% of its variance remains unexplained. These results suggest that the majority of the variance in the items is well-captured by the identified factors.

The eigenvalues further underscore the significance of the factors. Factor 1, with an eigenvalue of 3.000, explains a substantial portion of the variance in the data, while Factor 2 and Factor 3, with eigenvalues of 2.000 and 1.214, respectively, contribute meaningfully to the overall variance. The Promax rotation refined these values, providing clearer differentiation between the factors. In the rotated solution, Factors 1, 2, and 3 explain 2.137, 2.129, and 2.005, respectively, offering a more interpretable structure.

In terms of variance explained, the three factors account for a substantial portion of the data's variance, with cumulative proportions of 83.8%, 84.1%, and 86.9% in the rotated solution. These results highlight the robust explanatory power of the identified factors in capturing the underlying constructs measured by the instrument.

The Scree Plot, displayed in Figure 1, depicts the distribution of eigenvalues for the extracted components, with eigenvalues plotted on the y-axis and the number of factors on the x-axis. Each point on the figure indicates a given factor's eigenvalue, or the proportion of variation explained by that factor. The plot shows a sharp decrease in eigenvalues for the beginning factors, indicating that they account for a significant percentage of the variance in the data. Beyond the third component, the eigenvalues plateau, creating a prominent “elbow” in the plot. This moment of inflection is critical for choosing the best amount of elements to keep. In this study, the top three factors have eigenvalues greater than one, meeting the Kaiser criterion for factor retention, whereas the other factors contribute minimally to overall variance. A horizontal dashed line at an eigenvalue of 1 is included as a reference point, emphasizing the factors that meet the threshold for significance. Based on the ScreePlot, retaining the first three factors is appropriate, as they explain the most variance and are likely the most meaningful in capturing the underlying structure of the data.

The results from the Scree Plot, along with the findings in Tables 4 and 5, confirm the validity of retaining three factors for the Exploratory Factor Analysis (EFA), aligning with theoretical expectations and demonstrating a well-structured model. The test instrument exhibits strong factor loadings, low uniqueness values, and meaningful contributions of each factor to the variance, supported by the eigenvalue > 1 criterion and Promax rotation. These fit indices collectively indicate that the EFA model adequately represents the underlying constructs and provides strong evidence for the validity of the instrument. This robust foundation supports the subsequent Confirmatory Factor Analysis (CFA) to further validate the instrument’s structure and effectiveness in assessing mathematical abstraction skills among senior high school students.

Confirmatory Factor Analysis (CFA) is a robust statistical method used to evaluate the validity of a hypothesized factor structure within a dataset. It serves to confirm whether the observed variables align with an underlying theoretical model, often derived from prior research or an Exploratory Factor Analysis (EFA) (Xiao et al., 2019). In the context of educational research, CFA is indispensable for verifying the structure and construct validity of assessment instruments, ensuring that they accurately measure the intended latent constructs (Fu et al., 2022) (Xiao et al., 2019). This method also provides critical fit indices that determine the alignment between the proposed model and the observed data, establishing a solid foundation for assessing the instrument's reliability and effectiveness.

The Confirmatory Factor Analysis (CFA) model, as shown in Figure 2, visually encapsulates the relationships between the latent constructs (Fc1, Fc2, and Fc3) and their corresponding observed indicators (Q1 to Q10), providing a robust framework for validating the structure of the test instrument designed to measure mathematical abstraction skills in senior high school students.

In this model, the latent factors—depicted as circles labeled Fc1, Fc2, and Fc3—represent hierarchical cognitive domains of mathematical abstraction. Specifically:

The observed indicators—illustrated as rectangles labeled Q1 to Q10—serve as the measurable items linked to each latent factor. These indicators evaluate distinct dimensions of mathematical reasoning and problem-solving, aligning with the theoretical framework underlying the test instrument.

Table 6shows the fit indices derived from the Confirmatory Factor Analysis (CFA), which show that the proposed factor model accurately represents the underlying structure of the data. The factor model's chi-square value (91.973, df = 27, p = 0.106) indicates no significant difference between observed and predicted data, implying a good model fit. This finding is confirmed by multiple robust fit indices.

The Comparative Fit Index (CFI) of 0.904 and the Tucker-Lewis Index (TLI) of 0.940 both above the frequently recognized threshold of 0.90, indicating that the factor model fits the data well. The RMSEA value of 0.021, with a 90% confidence interval (0.038-0.025), shows excellent model fit and falls inside the conventional threshold of RMSEA < 0.05. The Standardized Root Mean Square Residual (SRMR) of 0.039 supports this conclusion, as values less than 0.08 are considered acceptable. The Goodness of Fit Index (GFI) of 0.988 indicates that the model explains a significant fraction of the data's variation, and the McDonald Fit Index (MFI) of 0.939 verifies an adequate fit. Additionally, the Expected Cross Validation Index (ECVI) of 0.126 suggests reasonable predictive validity, emphasizing the model's potential reliability in multiple samples.

Factor loadings are crucial in determining the strength and direction of the interactions between indicators (items) and the variables they represent. They show how much of each item's variance is explained by the underlying component, with larger values indicating a stronger link. As seen in Table 7, the loadings for Factor 1 components Q1 (0.883), Q2 (0.879), and Q4 (0.771) are strikingly high, with Q1 and Q2 exhibiting particularly strong relationships. This implies that these elements are closely related to the construct denoted by Factor 1. Similarly, for Factor 2, item Q5 has a loading of 0.890, indicating a strong link, since items Q3 (0.862) and Q7 (0.735) make significant contributions to this factor. Factor 3 has substantial loadings from items Q8 (0.796), Q10 (0.814), Q9 (0.835), and Q6 (0.861), indicating their consistency with Factor 3's underlying construct.

The standard error values for the factor loadings are small, indicating precise estimates. For instance, the standard error for Q1 is 0.031, suggesting a reliable estimate. The z-values, which assess the statistical significance of each loading, are also high. For example, Q5 has a z-value of 11.206, indicating a highly significant relationship with Factor 2. The p-values for all items are less than 0.001, further confirming the statistical significance of the factor loadings and the meaningful relationships between the items and their respective factors. Additionally, the narrow 95% confidence intervals for each loading further support the reliability of the estimates, with the true factor loadings likely to fall within the provided ranges.

The intercepts obtained from the Confirmatory Factor Analysis (CFA) provide important insights into the baseline levels of the indicators when the latent factors are at zero. These intercept estimates reflect the expected value of each indicator when the corresponding latent factor is absent, thereby offering an understanding of the initial or baseline responses of the items.

As shown in Table 8, for Factor 1, the intercepts for items Q1 (0.781), Q2 (0.672), and Q4 (0.614) suggest positive baseline values for these items, even in the absence of the latent factor. This indicates that these items have inherent baseline levels that are not explained by the underlying factor.

The standard error values for these intercepts are small, with Q1 having a standard error of 0.035, which indicates precise estimates of the intercepts.

The z-values, which assess the statistical significance of each intercept, are also high, indicating that the intercepts are statistically significant. For example, the intercept for Q5 has a z-value of 11.345, highlighting its substantial baseline level. Furthermore, the p-values for all intercepts are less than 0.001, confirming the statistical significance of these baseline levels and ensuring that they are meaningful and not due to random variation.

The 95% confidence intervals for the intercepts are narrow, providing a range within which the true intercepts are likely to fall. For instance, the confidence interval for Q1 ranges from 0.701 to 0.834, ensuring that the true intercept is reliably estimated within this range.

The residual variances from the Confirmatory Factor Analysis (CFA) provide vital insights into the fraction of variance in each indicator that cannot be explained by the latent factors. These residual variances show the unexplained variance that remains after accounting for the underlying structures, providing an indication of how well the model fits the data.

As indicated in Table 9, the residual variance for Q1 is 0.157, suggesting that the model fails to account for 15.7% of the variance in this item. Similarly, residual variances for other items indicate how much of their volatility is unexplained. The standard error values for these residual variances are tiny, with Q1 having a standard error of 0.012, indicating precise estimations of unexplained variance.

The z-values, which assess the significance of each residual variance, are notably high for most items. For example, Q2 has a z-value of 12.456, indicating that the residual variance for this item is highly significant. However, some items, such as Q2 (p = 0.072) and Q7 (p = 0.059), have p-values close to the 0.05 threshold, suggesting that there may be slight unexplained variance worth further investigation.

The 95% confidence intervals for the residual variances are generally narrow, with the confidence interval for Q1 ranging from 0.067 to 0.257. This indicates that the true residual variance is likely to fall within this range with 95% confidence, further supporting the reliability of the estimates. The results presented in Table 9 suggest that the CFA model accounts for a substantial portion of the variance in the indicators, with some residual variances remaining that warrant further consideration.

The residual covariances from the Confirmatory Factor Analysis (CFA) provide useful insights into the relationships between pairs of indicators that are not explained by the latent factors. These residual covariances show the extent to which two indicators share variance that the model does not account for, providing a more in-depth view of item interdependence after adjusting for latent components.

As seen in Table 10, the residual covariance between Q1 and Q2 is -0.026, demonstrating a modest negative link between these two items that the model does not account for. The standard error values for these covariances are often minimal, such as 0.004 for the covariance between Q1 and Q2, implying that the residual covariance estimates are credible. The z-values measure the significance of each residual covariance, with higher absolute values suggesting stronger correlations.

For example, the covariance between Q4 and Q5 has a z-value of 3.289, indicating a strong link between the two variables. The 95% confidence intervals for the residual covariances are relatively narrow, such as the interval for the covariance between Q1 and Q2, which ranges from -0.053 to -0.009. This indicates a high level of confidence in the estimates of these residual covariances.

However, the p-values for all pairings of indicators are greater than 0.05, indicating that the residual covariances are not statistically significant and that the model effectively accounts for the relationships between the majority of indicators. The correlation between Q10 and Q6, with a p-value of 0.057, is an exception since it is close to the threshold, indicating a possible relationship that deserves further examination.

The factor covariances derived from the Confirmatory Factor Analysis (CFA) shed light on the relationships between the model's latent factors. Factor covariances measure the extent to which two latent factors share variance, reflecting the strength and direction of their interactions. As indicated in Table 11, the calculated covariance between Factor 1 and Factor 2 is 0.705, indicating a strong positive association between these two components. Similarly, the covariance between Factor 1 and Factor 3 is highly significant (z-value = 11.507), indicating a strong relationship between these components. The standard errors for these covariances, such as 0.069 for the covariance between Factor 1 and Factor 2, suggest exact estimations, which adds to the results' reliability. All factor covariances have p-values less than 0.001, indicating that the factors' correlations are statistically significant. This implies that the detected factors have significant relationships with one another and contribute to the underlying constructs assessed by the instrument. Furthermore, the 95% confidence intervals for the covariances, such as the range of 0.602 to 0.798 for the covariance between Factor 1 and Factor 2, are quite small, indicating that the estimates are accurate.

The reliability coefficients for the identified factors and the overall measurement instrument are presented, which are crucial for assessing the consistency and stability of the instrument. Reliability is an essential aspect of psychometric evaluation, ensuring that the instrument consistently measures the intended constructs.

As shown in Table 12, the coefficients ω (McDonald’s Omega) are reported for each factor and the total instrument, providing an estimate of internal consistency. Coefficient ω is a robust measure of reliability, particularly in cases where the items may not be unidimensional. For Factor 1, the coefficient ω is 0.959, indicating excellent reliability and suggesting that the items within this factor are highly consistent in measuring the underlying construct. Factor 2 has a coefficient ω of 0.880, indicating good reliability, while Factor 3 has a coefficient ω of 0.820, which is considered acceptable reliability, though slightly lower than the other two factors. The total coefficient ω for the instrument is 0.886, reflecting good overall internal consistency.

Cronbach's alpha (α), a commonly used metric of internal consistency, is also included. Cronbach's alpha levels more than 0.70 are generally considered satisfactory, while those greater than 0.90 are considered exceptional. Factor 1 has a Cronbach's alpha of 0.879, indicating strong dependability. Factor 2 has a Cronbach's alpha of 0.816, which is likewise acceptable, showing that the items measure the same construct consistently. Factor 3 has a Cronbach's alpha of 0.941, indicating that it is quite reliable. The total Cronbach's alpha for the instrument is 0.879, supporting the conclusion that the instrument has good internal consistency.