The demographic characteristics of the 9606 participants are presented in Table 2.

The descriptive information of each item in polytomous scores version (range 0–7), such as the ratio of responses to each option, mean, standard deviation, and ratio of correct answers in polytomous scores version, is presented in Table 3. Items 5 and 6 were not included since they are two impairment and distress questions, which are different from the remaining 11 symptom items. Please note that the remaining tables present the statistics based on these FA 11 items.

Table 4 presents the correlations between each item and the total score when the specific item is removed in both the polytomous and dichotomous modes. The correlations reported in the table are all above 0.30, indicating a relatively strong relationship between each item and the total score [37].

Additionally, the correlations between the score of each item and the total score of other items, excluding the item itself, in both the polytomous and dichotomous versions demonstrate a positive and significant relationship. This suggests that each item is closely related to the overall score and reflects the collective nature of the items in assessing the construct of interest. These findings provide evidence for the good relationship and similarity between the individual items and the total score, indicating the coherence and consistency of the items in measuring the construct being assessed.

Descriptive indicators of the total scale score in the whole sample and by gender in polytomous and dichotomous version are presented in Table 5. The independent t-test indicates that there is a significant difference between women and men in polytomous and dichotomous version, and this difference is in favor of women.

Table 6 presents the fit indices for the single-factor structure of the scale for the total sample, as well as for women and men separately, in both the polytomous and dichotomous versions of the data. The results indicate that the single-factor model demonstrates a good fit with the data in the total sample for both the polytomous and dichotomous versions. However, it is worth noting that the fit of the model is slightly better in the dichotomous version compared to the polytomous version, as indicated by the RMSEA index. The fit indices suggest that the single-factor model adequately captures the underlying structure of the scale in both data versions for the total sample. Furthermore, when examining the fit of the model separately for women and men, similar findings are observed. The model demonstrates a good fit with the data for both polytomous and dichotomous versions in both gender groups. Once again, the fit of the model is slightly better in the dichotomous version compared to the polytomous version. These results indicate that the single-factor structure of the scale is well-supported by the data, suggesting that the items of the scale are measuring a common underlying construct. The findings also suggest that the dichotomous version of the data provides a better fit to the single-factor model compared to the polytomous version.

Ordinal alpha, combined reliability, and discriminant validity statistics for the entire sample, men and women, in Table 7 show that the level of internal consistency and composite reliability in the entire sample and male and female groups is favorable. Based on the AVE index, whose values greater than 0.5 are usually acceptable, convergent validity is also acceptable; its value is higher in dichotomous data than in polytomous version. Therefore, in the polytomous version, it can be observed that the structural model explains approximately 51% of the variance in the total sample, with slightly lower values of 49% for men and 52% for women. In the dichotomous version, the structural model accounts for approximately 59% of the variance in the total sample, with values of 56% for men and 59% for women. Overall, the explained variance is generally higher in women compared to men. To establish the construct-level discriminant validity, it is necessary to examine whether the square root of the Average Variance Extracted (AVE) index exceeds the correlation between the FA variable and other variables (Table 6) [38].

BES, BIS, and CD-RISC scales were significantly associated with the mYFAS 2.0 diagnosis (Table 8). There was a significant correlation between the BES, BIS, and CD-RISC scales and the mYFAS 2.0-diagnosed FA symptom count.

To examine the factor structure invariance based on gender, four models were assessed: Configural invariance, metric or weak invariance, scalar or strong invariance, and exact invariance. The analysis aimed to determine if the factor structure of FA was consistent across male and female groups. In the Configural invariance model, no restrictions were imposed on any parameters, and the same factorial structure was fitted to both groups. A significant fit indicated that the factor structure was equivalent in both groups. The metric model assumed equality of factor loadings between the groups. Lack of significance in the comparison between the metric and Configural models suggested invariance, indicating that the construct had the same meaning for both groups. This implied that the items captured the same underlying construct, allowing for comparison of variance and covariance of scores between the groups. The scalar model, considering the ordinal nature of the data, enforced equality of item thresholds in addition to factor loadings. Lack of significance in the comparison between the scalar and metric models allowed for comparing the means of the latent variable between the groups. In the exact model, residual variances of the items were assumed to be equal between the groups. Lack of significance in the comparison between the exact and scalar models allowed for comparing the total scores between the groups based on the sum of observed item scores. This indicated that the reliability of the items was consistent between the two groups. Finally, the variance and mean of the latent structure among the groups were examined. The analysis was conducted using Mplus software, employing the WLSMV estimation method suitable for ranked data (response range of 0 to 7 in the polytomous version). Thus, the models were fitted based on the polychoric correlation matrix. Model comparison was performed using the DIFTEST method in Mplus, considering chi-square difference test, as well as fit indices such as CFI, TLI, SRMR, and RMSEA to evaluate the differences between the models (Table 9).

According to the findings presented in Table 9, a comparison between the Configural model and metric model 1 revealed a significant difference (p < 0.05). Further examination of modification indices indicated that the factor loadings of items 12 and 3 differed significantly between the two groups. Consequently, in metric models 2 and 3, these factor loadings were freely estimated for the respective groups. As a result, the difference between metric model 3 and the Configural model became non-significant (p > 0.05). Similarly, when comparing scalar model 1 with metric model 3, it was observed that the first threshold of item 4 and the second threshold of item 13 differed between men and women (p < 0.05). In scalar models 2 and 3, these thresholds were freely estimated for the respective groups. Consequently, the difference between scalar model 3 and metric model 3 became non-significant (p > 0.05).

In strict model 1, the variance of the remaining items was freely estimated between the two groups, while in strict model 2, this parameter was constrained to be the same across the two groups. The results indicated a statistically significant chi-square difference between the two models (p < 0.05). Subsequently, in strict models 2–9, the error variances of items 13, 1, 4, 7, 12, 2, and 11 were freely estimated between the two groups. The comparison of strict model 9 with strict model 1 yielded a non-significant chi-square difference (p > 0.05). Moreover, the last two rows of Table 8 demonstrate that while the mean and variance of the women's group (reference group) were fixed at zero and one, respectively, constraining these parameters in the men's group resulted in a significant chi-square difference (p < 0.05). This indicates that the mean and variance differ between women and men in the structure of FA. Specifically, the average score for men (target group) in the FA structure was -0.328, with a variance of 0.96. In comparison, the women's group (reference group) had scores ranging from zero to one, indicating higher mean and variance. Thus, it can be concluded that the mean and variance of the men's group are lower than those of the women's group.

The reported parameters in Table 10 are based on the results of strict model 2. Considering the sensitivity of the chi-square test to sample size, based on the ΔCFI index ≤ 0.01, it can be inferred that the most influential models in terms of gender are metric 1, scalar 1, and average models. However, based on the ΔRMSEA ≥ 0.015 and ΔSRMR ≥ 0.03 criteria, none of the identified effects are considered significant [36].

Table 9 shows the results of the strict model 9. In this model, the mean and variance of women are fixed at zero and one, respectively, while the men's group is freely estimated with previously mentioned values. The factor loading of items 3 and 12 in the male group is freely estimated, while the factor loading of other items remains the same in both groups. The first threshold of item 4 and the second threshold of item 13 have been freely estimated for both women and men. Additionally, the error variances of items 13, 1, 4, 7, 12, 2, and 11 between the two groups were freely estimated. The factor loadings of the items, along with the seven threshold values (corresponding to the spectrum of eight items, resulting in seven threshold values for each item), as well as the variance of the remaining items in raw form, are presented in Table 8. It should be noted that in the women's group, the error variances were fixed at one, hence their specific values are not reported.

The results presented in Table 9 indicate that the comparison between the Configural model and metric model 1 yielded a statistically significant difference (p < 0.05). Further analysis of the modification indices revealed that the factor loadings of items 12 and 3 significantly differed between the two groups, with item 12 having a higher modification index than item 3. Consequently, the factor loading of item 12 was freely estimated between the two groups (metric model 2). This adjustment resulted in a non-significant difference in the chi-square test between metric model 2 and the Configural model (p > 0.05), indicating metric invariance. Next, the invariance of threshold values for the items was examined. Comparing scalar model 1 with metric model 2 revealed significant modification indices for the threshold values of items 13, 4, and 9 between women and men (p < 0.05). Subsequently, the threshold parameter for item 13 (scalar model 2) and item 4 (scalar model 3) were freely estimated for both groups. The comparison between scalar model 2 and metric model 2, as well as scalar model 3 and metric model 2, did not result in a significant difference (p > 0.05). It is worth noting that none of the threshold values for the remaining items exhibited significant differences between the two groups, and thus, were not considered for free estimation (likely due to the large sample size). Moving forward, the remaining item values were examined for invariance.

In the strict model 1, the variance of the residual errors for the items was freely estimated between the two groups, while in the strict model 2, this parameter was constrained to be the same across the groups. The chi-square difference test between the two models did not yield a statistically significant result (p ≥ 0.05). Further examination of the strict model 2 indicated that the highest modification indices were associated with the error covariance between items 1 and 2 (169.180) and items 4 and 10 (101.827) in women. However, these results were not observed in men. Despite these findings, the lack of significance in the chi-square test limited the free estimation of these error covariances between the two groups. Next, the invariance of the variance and mean of the latent structure was assessed between the two groups. The results from the variance and mean model (Table 7) showed that while the mean and variance of the women's group were fixed at zero and one, respectively, adjusting the variance of the men's group (the target group) with the women's group (the reference group) resulted in statistical significance (p < 0.05), indicating a difference in variance between the groups. The variance of the male group in the FA structure was estimated to be 1.314, which is higher than the variance of the female group. Based on the mean model results, it can be observed that setting the mean of the male group in the FA structure equal to the mean of the female group led to chi-square significance (p < 0.05), indicating that the mean of the male group (− 0.575) is lower than that of the female group in the FA structure.

The estimates presented in Table 11 are derived from detailed model 2. Considering the sensitivity of the chi-square test to sample size, based on a ΔCFI index ≤ 0.01, it can be concluded that gender has a limited influence in the two-value mode [39]. However, based on ΔRMSEA ≥ 0.015 and ΔSRMR ≥ 0.03, the effects related to the invariance of factor variances (that is variability in a latent variable and the relationships among multiple latent variables is equivalent across groups) and mean between the two groups are statistically significant. Table 12 provides the factor loadings of the questionnaire items in the round value mode (with a threshold for each item) and the variance estimates of the remaining items in raw mode. In the women's group, the error variances have been fixed at one, and thus their specific values are not reported.