A total of 2,021 valid questionnaires (95.78% response rate) were collected (1,034 mothers and 987 fathers) from seven public primary schools in urban and rural areas of Beijing, China. No statistically significant differences were found in age (t₁₈₆₀ = 0.562, p > 0.05) and education (t₂₀₆₆ = −0.988, p > 0.05) between respondents and non-respondents. These parents were biological parents of their children, and all lived with their children. The mean age of the mothers was 30.57 ± 3.84 years old (range, 31–52 years). The mean age of the fathers was 30.41 ± 4.73 years old (range, 30–64 years). The mean age of their children was 10.26 ± 0.32 years old (range, 9–12 years). In the sample of mothers, 3.20% completed middle school or below, 38.04% completed high school, 45.40% completed university, and 13.36% had a master’s education or above. In the sample of fathers, 5.10% completed middle school or below, 38.47% completed high school, 33.60% completed university, and 22.83% had a master’s education or above. In total, 85.11% of the mothers and 98.78% of the fathers were employed.

This study was approved by the Institutional Review Board at the study institution. First, standardized instructions about the purpose of the study were delivered to the sampled schools. The sampled schools were willing to support the implementation of this study. Then, written informed consent was obtained from the students’ parents. Finally, the questionnaires were provided to the students in sealed envelopes, and the students took them home for their parents to fill out. The questionnaires were returned by the students in sealed envelopes after being completed. All participants completed the PRFQ, questionnaires measuring family functioning, parenting stress, and parental warmth, as well as some demographic questions.

Data analysis was conducted in SPSS Version 20.0, Mplus Version 8.3. The absolute value range of skewness is 0.08 to 1.70, and the absolute value range of kurtosis is 0.05 to 2.70. According to Curran et al. (1996), skewness absolute values of 0–2 and kurtosis absolute values of 0–7 can be considered sufficient normality. First, to test the factor structure of the original PRFQ-C in the sample of Chinese parents, CFA was conducted in Mplus 8.3 (Muthén and Muthén, 1998–2019) by means of maximum likelihood (ML) estimation, with missing data handled with the full information maximum likelihood. The factor loading of each item was examined in order to ensure that each item was reliable to the latent factor (Brown, 2015). Model fit was evaluated using root mean square error of approximation (RMSEA), comparative fit index (CFI), and Tucker–Lewis index (TLI), and an RMSEA below 0.08 and other indexes (e.g., CFI and TLI) above 0.9 indicated good model fit to the data (Bentler, 1990; Hu and Bentler, 1999). Besides, CFA was also conducted on other shortened versions of the PRFQ in the literature (Pazzagli et al., 2018; De Roo et al., 2019) to further verify the rationality of the modified PRFQ-C.

Second, to examine the reliability of the modified PRFQ-C, the omega (w) coefficients (Brunner et al., 2012) were reported for all subscales. Third, the discriminant validity of the modified PRFQ-C was tested through heterotrait-monotrait (HTMT) analysis (Henseler et al., 2015). In addition, correlation analysis on the relationship between the three factors of the modified PRFQ-C, the original PRFQ-C, family functioning, parenting stress, and parental warmth was conducted to examine the criterion validity of the PRFQ-C.

Finally, the measurement invariance of the PRFQ-C measurement model was tested across mothers and fathers using the multigroup CFA technique. In the current research, three models of measurement invariance were examined in hierarchical order: configural, metric, and scalar invariance (Vandenberg and Lance, 2000; Milfont and Fischer, 2010). The first model tested whether the same factor structure existed across parent gender groups (configural invariance). The second model investigated whether the factor loadings were equal across groups (metric invariance). The third model examined whether the item intercepts were equal across groups (scalar invariance) based on the metric invariance achieved. As the chi-square test is sensitive to large samples and even small differences in covariance matrices could result in a significant chi-square value, chi-square was not used as a model fit index, and Δχ² was not used as a criterion to detect measurement invariance (Cheung and Rensvold, 2002; Chen, 2007; Kline, 2015). Instead, measurement invariance was assessed by comparing ΔCFI, ΔTLI, and ΔRMSEA with the cutoff criteria (i.e., ΔCFI ≤ 0.01, ΔTLI ≤ 0.01, and ΔRMSEA ≤ 0.015) suggested by Chen (2007) and Cheung and Rensvold (2002).

The factor structure of the original PRFQ-C was examined by CFA. The initial model had a poor fit to the data, χ²(132) = 2538.507; CFI = 0.762; TLI = 0.724; RMSEA = 0.095, 90%CI = [0.092, 0.098]; SRMR = 0.083. The items with low loadings (i.e., below 0.4, included item 11, item 10, and item 18) and the items with cross-loading (i.e., item 3, item 14, and item 12) were removed (Ford et al., 1986). The results of the factor analysis of the original PRFQ-C were shown in the Supplemental Material. One item at a time was deleted to determine if the model fit improved. The model fit significantly improved after these items were removed, χ²(51) = 604.489; CFI = 0.907; TLI = 0.879; RMSEA = 0.073, 90%CI = [0.068, 0.079]; SRMR = 0.049. The modification indexes (MIs) revealed the items with error covariance. For example, item 17 was related to item 15 and item 16. The model fit the data well after these two covariance’s were held in the model, χ²(49) = 472.381; CFI = 0.929; TLI = 0.904; RMSEA = 0.065, 90%CI = [0.060, 0.071]; SRMR = 0.044. Finally, with these modifications, the three-factor PRFQ-C with 12 items (PRFQ-12C) was formed. As shown in Table 1, all items had factor loadings higher than 0.40 in the final model.

Besides, we also conducted a CFA of the PRFQ-C for a second-order three-factor model, a bi-factor model, and other shortened versions in the literature, i.e., the three-factor model with 16 items in De Roo et al. (2019) and the three-factor model with 15 items in Pazzagli et al. (2018). The CFA results were shown in Table 2, which indicated that the fist-order three-factor model of the PRFQ-12C with 12 items fit best.

The omega (w) was 0.68 for the PM subscale, 0.82 for the CMS subscale, and 0.76 for the IC subscale, respectively (see Table 1). The reliability coefficients were satisfactory for the subscales of the PRFQ-12C except for the PM subscale.

Table 3 demonstrates that all the HTMT values were <0.85, establishing the discriminant validity for the three-factor model. Table 4 shows the correlation matrix between the three subscales of the PRFQ-12C and the original three subscales of the PRFQ-C. The correlation coefficients of the modified version and the original version were r = 0.95 for the PM subscale, r = 0.95 for the CMS subscale, and r = 0.88 for the IC subscale, respectively. Table 5 shows the correlation matrix between the PRFQ-12C and family functioning, parenting stress, and parental warmth. All the correlation coefficients were significant (ps < 0.01). Specifically, PM had moderate negative correlations with family functioning (r = −0.36) and parental warmth (r = −0.35) and positive correlations with parenting stress (r = 0.48). CMS and IC were positively correlated with family functioning (r = 0.27 and r = 0.29, respectively) and parental warmth (r = 0.42 and r = 0.42, respectively) and negatively correlated with parenting stress (r = −0.11 and r = −0.12, respectively).

First, goodness-of-fit statistics for mothers and fathers sample were χ²(49) = 264.754; CFI = 0.923; TLI = 0.896; RMSEA = 0.065, 90%CI = [0.058, 0.073]; SRMR = 0.046, and χ²(49) = 271.189; CFI = 0.926; TLI = 0.901; RMSEA = 0.068, 90%CI = [0.060, 0.076]; SRMR = 0.048, respectively. Then, invariance was tested across mothers and fathers (see Table 6). Testing for configural invariance showed adequate fit (χ² = 535.943; df = 98; CFI = 0.924; TLI = 0.898; RMSEA = 0.067; 90%CI = [0.061, 0.072]; SRMR = 0.047). We then examined the factor loading invariance (χ² = 544.117; df = 107; CFI = 0.925; TLI = 0.907; RMSEA = 0.064; 90%CI = [0.058, 0.069]; SRMR = 0.048). The factor loadings were invariant across mothers and fathers (ΔCFI = 0.001; ΔTLI = 0.009; ΔRMESA = 0.003). Finally, we examined the intercept invariance (χ² = 565.681; df = 116; CFI = 0.922; TLI = 0.912; RMSEA = 0.062; 90%CI = [0.057, 0.067]; SRMR = 0.048). The intercept was invariant across mothers and fathers (ΔCFI = −0.003; ΔTLI = 0.005; ΔRMESA = 0.002).

In this study, the three-factor structure of the original PRFQ-C was evaluated through CFA. The results of the CFA showed that the original model fit was poor. Specifically, there were items with low loadings in each subscale, such as item 11, item 18, and item 10. The results regarding the low loadings of item 11 and item 18 were consistent with previous research (Pazzagli et al., 2018; De Roo et al., 2019). This may be because item 11 (i.e., “I can sometimes misunderstand the reactions of my child.”) and item 18 (i.e., “I believe there is no point in trying to guess what my child feels.”) are reverse-worded items, and their meanings are easily misunderstood (Weems and Onwuegbuzie, 2001). The meaning of item 10 (“My child sometimes gets sick to keep me from doing what I want to do”) in the Chinese version may just describe a realistic situation that Chinese parents encounter in real life when raising children. For example, when the child is sick, the parents need to stay with the child or take the child to see the doctor, so that the parents cannot go to work. The expression of item 10 in the Chinese version may not have reflected parents’ misunderstanding of their children’s mental states. Therefore, item 10 had a weak loading on the PM subscale. In addition, there were three items with high cross-loadings (i.e., item 3, item 14, and item 12) according to the modification indexes in the original model.

Meanwhile, discussion with psychological experts and interview with parents also showed that these items were redundant in expression. Therefore, they were removed. The model fit changed from poor to acceptable after those items with low loadings and cross-loadings were removed. Finally, the Chinese version of the PRFQ with 12 items (PRFQ-12C) was formed, in which the PM subscale contains five items, the CMS subscale has four items, and the IC subscale includes three items. The model fit improved when the error covariance’s between item 17 and items 16 and 15 were added to the final CFA model. Considering the meanings of those items, it is reasonable that item 17 (“I always know why my child acts the way he or she does”) was positively related to item 15 (“I try to understand the reasons why my child misbehaves”), and negatively related to item 16 (“Often, my child’s behavior is too confusing to bother figuring out”), because parents’ efforts to understand the reasons of their children’s behavior in general should lead to greater knowledge about why their children behave and thus less confusions. Besides, the dimension of CMS is negatively correlated with PM and positively correlated with IC (Cooke et al., 2017; Luyten et al., 2017), and items 17, 16, and 15 belong to the dimensions of CMS, PM, and IC, respectively. Therefore, it can be considered reasonable to add the corresponding error covariance.

In summary, the three-factor structure of the Chinese version of the PRFQ was verified in Chinese parents after removing some items, which was consistent with previous studies in Western countries (Luyten et al., 2017; Pazzagli et al., 2018; De Roo et al., 2019; Gordo et al., 2020) but different from the results in a Korean sample (Lee et al., 2021). The three-factor structure of the original PRFQ was not appropriate in the Korean sample even after removing some items as the modification indices suggested (Lee et al., 2021). There are some possible reasons for this result. First, although it has a collectivistic culture, in recent years, China has imported and integrated individualistic values from the Western world, which influences parents’ theories about raising children. Secondly, the sample in this study comprised parents with school-aged children, while Lee et al. (2021)’s research focused on Korean parents with children 0 to 5 years of age. The structure of PRF may be different in parents with children of different ages.

The omega (w) coefficients of the revised PRFQ-12C factors ranged from 0.68 to 0.82, which indicated good reliability except for the PM subscale (w = 0.68). Generally, values greater or equal to 0.60, 0.70, and 0.80 are considered as marginal, acceptable, and good, respectively, and a value less than 0.60 indicates insufficient (Barker et al., 1994). Therefore, the reliability of the PM subscale is marginally acceptable and cannot be improved by deleting items in the PM subscale. This is consistent with Pazzagli et al. (2018) that reported a Cronbach’s α coefficient less than 0.70 for the PM subscale (α = 0.67). Besides, the evaluation of the reliability of the scale cannot be completely dependent on this numerical standard and also depends on the operational definition of the variables themselves (Crutzen and Peters, 2015). We suspect that the construct measured by the PM subscale may have broad connotations and is not easy to be operationalized, which may result in a low reliability value. However, this is beyond the scope of the current paper. Future studies may need to pay attention to this issue.

All the HTMT values of the PRFQ-12C were low (ranging from 0.18 to 0.59). HTMT values < 0.85 establish discriminant validity for the constructs (Henseler et al., 2015). The correlation coefficients between the three subscales of the PRFQ-12C and the original PRFQ-C were very high (ranging from 0.88 to 0.95, see Table 4), which indicated that they are conceptually equivalent. The significant correlation coefficients between all three subscales of the PRFQ-12C and family functioning, parenting stress, and parental warmth indicated that the PRFQ-12C has good empirical validity. Specifically, PM had negative correlations with family functioning, but CMS and IC had positive correlations with family functioning, which was consistent with previous studies (e.g., Cooke et al., 2017). The results suggest that both mothers and fathers who have difficulties understanding their children’s mental states might have more family problems and have poor family relationships; in contrast, parents who are interested in and curious about their children’s mental states and are certain about them have better family relationships.

PM was positively associated with parenting stress, but CMS and IC were negatively associated with parenting stress. Similarly, Rutherford et al. (2013, 2015) found that PM and IC were related to mothers’ stress tolerance in a study following the BSIM paradigm in which mothers were asked to soothe an inconsolable baby simulator. High PRF might be an important buffer for stress associated with parenting (Luyten et al., 2017). Parents who have high PRF could regulate their own mental states toward their children’s distress and thus have less parenting stress.

Finally, there was a significant correlation between PRF and parental warmth, with PM being negatively associated with parental warmth but CMS and IC being positively associated with parental warmth. In other words, parents with low PM, high CMS and IC showed warmth and love to their children. This result provided empirical evidence for the theory that PRF is the basis of parental sensitivity (Slade, 2005, 2007), such that parents who have a high level of PRF are able to identify children’s needs and respond to them appropriately and in a timely manner. Taken together, the significant correlations between the three factors of the PRFQ-12C and family functioning, parenting stress, and parental warmth described above supported the empirical validity of the PRFQ-12C.

To test the measurement invariance at three restrictive levels across fathers and mothers, configural, metric, and scalar models were compared. The findings indicated that there were no significant differences among the fit indexes of configural, metric, and scalar invariance across genders. Therefore, the Chinese parent sample data supported the full configural, full metric, and full scalar invariance models. These results revealed that the PRFQ-12C had the same factor structure, factor loadings, and item intercepts for the mother and the father groups. Thus, it was concluded that the PRFQ-12C can adequately measure PRF in both mothers and fathers. The three-factor PRFQ-12C can be used to compare the mean PRF scores of mothers and fathers.