Submitted:
06 July 2025
Posted:
07 July 2025
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Abstract
Keywords:
1. Introduction
2. Materials and Methods
3. Results
3.1. Factorial Analysis

4. Discussion
5. Conclusions
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
| AGFI | Adjusted Goodness-of-Fit Index |
| ASD | Autism Spectrum Disorder |
| CFI | Comparative Fit Index |
| EFA | Exploratory Factor Analysis |
| GFI | Goodness-of-Fit Index |
| ICIPES | International COVID-19 Impact on Parental Engagement Study |
| KMO | Kaiser-Meyer-Olkin |
| MIF | Minimum Items Per Factor |
| MSA | Measure Of Sampling Adequacy |
| NNFI | Non-Normalized Fit Index |
| PCIT | Parent-Child Interaction Therapy |
| RMSEA | Root Mean Square Error of Approximation |
| RMSR | Root Mean Square Root of Residuals |
| ULS | Unweighted Least Squares |
| χ2/df | Chi-Square/Degree of Freedom |
Appendix A
| Item (Variable) |
N | Minimum | Maximum | Mean | Std. Deviation | Skewness | Kurtosis | ||
| Statistic | Statistic | Statistic | Statistic | Statistic | Statistic | Std. Error | Statistic | Std. Error | |
| PUTR_1 | 1432 | .00 | 4.00 | 1.2346 | 1.34460 | .837 | .065 | -.527 | .129 |
| PE_1 | 1432 | .00 | 4.00 | 1.7430 | 1.25321 | .268 | .065 | -.927 | .129 |
| PE_2 * | 1432 | .00 | 4.00 | 1.7374 | 1.32156 | .320 | .065 | -1.029 | .129 |
| PUTS_1 | 1432 | .00 | 4.00 | 2.7982 | 1.29199 | -.760 | .065 | -.611 | .129 |
| PUTS_2 | 1432 | .00 | 4.00 | 2.6662 | 1.30324 | -.607 | .065 | -.781 | .129 |
| PUTS_3 * | 1432 | .00 | 4.00 | 2.3624 | 1.29475 | -.239 | .065 | -1.048 | .129 |
| PUTBC_1 | 1432 | .00 | 4.00 | 1.7605 | 1.29905 | .324 | .065 | -.938 | .129 |
| PEIS | 1406 | .00 | 4.00 | 2.0149 | 1.21048 | .032 | .065 | -.889 | .130 |
| PE_3 * | 1432 | .00 | 4.00 | 2.0321 | 1.37323 | .018 | .065 | -1.221 | .129 |
| PE_4 | 1432 | .00 | 4.00 | 1.5126 | 1.33380 | .545 | .065 | -.858 | .129 |
| WHSB_1 * | 1432 | .00 | 4.00 | 1.7123 | 1.41111 | .323 | .065 | -1.189 | .129 |
| WHSB_2 * | 1432 | .00 | 4.00 | 2.5279 | 1.38750 | -.408 | .065 | -1.151 | .129 |
| PE_5 | 1432 | .00 | 4.00 | 1.3645 | 1.17917 | .635 | .065 | -.394 | .129 |
| SHS | 1432 | .00 | 4.00 | 2.9183 | 1.37651 | -.949 | .065 | -.482 | .129 |
| PUTR_2 * | 1432 | .00 | 4.00 | 2.0077 | 1.37282 | -.149 | .065 | -1.240 | .129 |
| PUTBC_2 | 1432 | .00 | 4.00 | 1.3568 | 1.26299 | .529 | .065 | -.823 | .129 |
| PUTBC_3 * | 1432 | .00 | 4.00 | 1.5992 | 1.33436 | .298 | .065 | -1.108 | .129 |
| PUTBC_4 * | 1432 | .00 | 4.00 | 2.0985 | 1.39959 | -.199 | .065 | -1.242 | .129 |
| PUTR_3 * | 1432 | .00 | 4.00 | 1.8443 | 1.33968 | .068 | .065 | -1.164 | .129 |
| PUTBC_5 * | 1432 | .00 | 4.00 | 1.8596 | 1.36462 | .080 | .065 | -1.200 | .129 |
| PUTBC_6 * | 1432 | .00 | 4.00 | 1.9406 | 1.33017 | -.043 | .065 | -1.163 | .129 |
| PUTBC_7 * | 1432 | .00 | 4.00 | 1.8059 | 1.32967 | .075 | .065 | -1.161 | .129 |
| PUTBC_8 * | 1432 | .00 | 4.00 | 1.9309 | 1.34537 | -.022 | .065 | -1.170 | .129 |
| PUTBC_9 * | 1432 | .00 | 4.00 | 1.8855 | 1.32155 | .044 | .065 | -1.099 | .129 |
| PUTBC_10 * | 1432 | .00 | 4.00 | 1.8366 | 1.31427 | .070 | .065 | -1.108 | .129 |
| PUTS_4 | 1432 | .00 | 4.00 | 1.1697 | 1.08279 | .731 | .065 | .044 | .129 |
| Valid N (listwise) |
1406 | ||||||||
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| Article | Sample | Method | MIF | χ2/df | RMSEA | AGFI | GFI | CFI | NNFI | RMSR |
| Schermelleh-Engel et al. [4] | ≥ 200 | Good fit | NR | ≥ 0 ≤ 2 | ≤ .05 | ≥ .90 ≤ 1.00 | ≥ .95 ≤ 1.00 | ≥ .97 ≤ 1.00 | ≥ .97 ≤ 1.00 | < .05+ |
| Acceptable fit | ≥ 3 | > 2 ≤ 3 | > .05 ≤ .08 | ≥ .85 < .90 | ≥ .90 < .95 | ≥ .95 < .97 | ≥ .95 < .97 | ≥ .05 ≤ .08++ |
| Name (ID) | Categories | Value | N |
| Parent/carer gender (PGEN) N = 1432 |
Mother, Stepmother, Grandmother, adoptive/foster mother or female guardian | 0 | 1171 |
| Father, Stepfather, Grandfather, adoptive/foster father or male guardian | 1 | 261 | |
| Parent/carer years of schooling (PYS) N = 1432 |
8 | 12 | |
| 9 | 0 | ||
| 10 | 55 | ||
| 11 | 0 | ||
| 12 | 29 | ||
| 13 | 104 | ||
| 14 | 33 | ||
| 15 | 689 | ||
| 16 | 0 | ||
| 17 | 428 | ||
| 18 | 0 | ||
| 19 | 81 | ||
| 20 | 0 | ||
| 21 | 1 | ||
| Parent/carer Age (PAG) N = 1432 |
25-34 years old | 2 | 191 |
| 35-44 years old | 3 | 674 | |
| 45-54 years old | 4 | 469 | |
| 55-64 years old | 5 | 98 | |
| Family structure/composition (FAMC) N = 1283 |
Living with the father/mother of the child | 0 | 913 |
| Living with a partner who is not the father/mother of the child | 1 | 113 | |
| Raising a child without a partner | 2 | 257 | |
| Child’s age (CHAG) N = 1432 |
6-year-old | 0 | 204 |
| 7-year-old | 1 | 129 | |
| 8-year-old | 2 | 149 | |
| 9-year-old | 3 | 114 | |
| 10-year-old | 4 | 128 | |
| 11-year-old | 5 | 116 | |
| 12-year-old | 6 | 137 | |
| 13-year-old | 7 | 96 | |
| 14-year-old | 8 | 96 | |
| 15-year-old | 9 | 84 | |
| 16-year-old | 10 | 179 | |
| Children in the household (CHH) (Number of siblings) N = 1432 |
0 | 453 | |
| 1 | 609 | ||
| 2 | 275 | ||
| 3 | 67 | ||
| 4 | 20 | ||
| 5 | 5 | ||
| 6 | 2 | ||
| 7 | 1 | ||
| School Home communication: Learning Plan (SHC_LP) N = 1432 |
Yes | 0 | 1335 |
| No | 1 | 97 | |
| School Home communication (SHC_FLP) N = 1335 |
Everyday | 0 | 640 |
| Between two to three times per week | 1 | 207 | |
| Once per week | 2 | 350 | |
| Fortnight / every two weeks | 3 | 103 | |
| Once per month | 4 | 35 | |
| Parental engagement: homeschooling (PEHS) N = 1432 |
Yes | 0 | 1115 |
| No | 1 | 317 | |
| Parental engagement: time (PETT) N = 1115 |
Less than 10 hours per week | 0 | 793 |
| Between 11 and 20 hours per week | 1 | 279 | |
| Between 21 and 30 hours per week | 2 | 35 | |
| More than 31 hours per week | 3 | 8 |
| Item ID | Questions | Factor |
Factor Name |
|
| 1 | 2 | |||
| PUTS_1 | Follow on social media what other parents do and try to do exactly the same. | .885 | F1: Social Learning for Parenting |
|
| PUTS_2 | Follow on social media what other parents do and use it as an inspiration. | .845 | ||
| PUTBC_1 | Look for ideas on the internet using different websites. | .487 | ||
| PE_1 | Follow my ideas about what my children need to learn. | .474 | ||
| SHS | I ask my older child(ren) to be in charge of homeschooling the little one(s). | .450 | ||
| PEIS | I try to replicate the way I was taught when I was at school. | .441 | ||
| PE_4 | My children and I have a set homeschooling timetable. | .578 | F2: Parenting in Homeschooling |
|
| PUTR_1 | Check the school’s emails, blog, and website to follow the activities they suggest for the children. | .549 | ||
| PE_5 | I develop with my children spontaneous learning activities not necessarily school-related such as cooking, woodwork, online games, physical activities, etc. | .445 | ||
| Rotation Sums of Squared Loadings a |
2.539 | 1.295 | ||
| Factor Correlation |
Factor 1 | 1.000 | .129 | |
| Factor 2 | .129 | 1.000 | ||
| Item ID | Factor | Factor | Factor | ||
| 1 | 2 | ||||
| PUTS_1 | .989 (0.956; 1.015) a | F1: Social Learning for Parenting |
|||
| PUTS_2 | .909 (0.876; 0.939) | ||||
| SHS | .473 (0.409; 0.529) | ||||
| PUTBC_1 | .444 (0.398; 0.485) | ||||
| PE_1 | .439 (0.396; 0.490) | ||||
| PE_4 | .720 (0.660; 0.777) | F2: Parenting in Homeschooling |
|||
| PUTR_1 | .649 (0.582; 0.725) | ||||
| PE_5 | .538 (0.481; 0.595) | ||||
| Eigenvalues of the Reduced Correlation Matrix |
2.972 | 1.137 | Sum: 4.109 | ||
| Weighted Eigenvalues | 0.723 | 0.277 | |||
| Factor Correlation | 1.000 | .344 (0.290; 0.397) | Factor 1 | ||
| .344 (0.290; 0.397) | 1.000 | Factor 2 | |||
| Article | Sample | Fit | MIF | χ2/df | RMSEA | AGFI | GFI | CFI | NNFI | RMSR |
| Schermelleh-Engel et al. [38] | ≥ 200 | Good | NR | ≥ 0 ≤ 2 | ≤ .05 | ≥ .90 ≤ 1.00 | ≥ .95 ≤ 1.00 | ≥ .97 ≤ 1.00 | ≥ .97 ≤ 1.00 | < .05+ |
| Acceptable | ≥ 3 | > 2 ≤ 3 | > .05 ≤ .08 | ≥ .85 < .90 | ≥ .90 < .95 | ≥ .95 < .97 | ≥ .95 < .97 | ≥ .05 ≤ .08++ | ||
| Osorio-Saez et al [29] by Chile | 1597 | NR | NR | .153 | NR | NR | .869 | NR | NR | |
| Proposed Model | 1406 | 3 | 3.611 | .054* (.030; .061) |
.989** (.986; .995) | .994** (.993; .998) | .986** (.982; .996) | .973** (.966; .992) | .037** (.024; .041) |
| Name (ID) | F1 | F2 | FT | ||||||||||
| Val. |
Asym. Std. Error |
Approx. T |
Approx. Sig. | Val. |
Asym. Std. Error |
Approx. T |
Approx. Sig. |
Val. |
Asym. Std. Error |
Approx. T |
Approx. Sig. | ||
| Parent/carer gender (PGEN) N = 1432 |
Kendall’s tau-c | .031 | .023 | 1.382 | .167 | .022 | .023 | .962 | .336 | .033 | .022 | 1.492 | .136 |
| Gamma | .075 | .054 | 1.382 | .167 | .052 | .054 | .962 | .336 | .085 | .057 | 1.492 | .136 | |
| Parent/carer years of schooling (PYS) N = 1432 |
Kendall’s tau-c | -.006 | .019 | -.338 | .736 | .011 | .019 | .591 | .554 | -.010 | .019 | -.544 | .587 |
| Gamma | -.011 | .033 | -.338 | .736 | .019 | .032 | .591 | .554 | -.019 | .034 | -.544 | .587 | |
| Parent/carer Age (PAG) N = 1432 |
Kendall’s tau-c | .038 | .021 | 1.863 | .062 | .014 | .020 | .676 | .499 | .025 | .020 | 1.246 | .213 |
| Gamma | .063 | .034 | 1.863 | .062 | .022 | .033 | .676 | .499 | .045 | .036 | 1.246 | .213 | |
| Family structure/composition (FAMC) N = 1283 |
Kendall’s tau-c | -.045 | .021 | -2.117 | .034* | -.016 | .021 | -.755 | .450 | -.029 | .020 | -1.450 | .147 |
| Gamma | -.094 | .044 | -2.117 | .034* | -.033 | .044 | -.755 | .450 | -.068 | .046 | -1.450 | .147 | |
| Child’s age (CHAG) N = 1432 |
Kendall’s tau-c | .034 | .021 | 1.622 | .105 | .008 | .022 | .367 | .714 | .025 | .020 | 1.233 | .218 |
| Gamma | .042 | .026 | 1.622 | .105 | .010 | .027 | .367 | .714 | .034 | .028 | 1.233 | .218 | |
| Children in the household (CHH) (Number of siblings) N = 1432 |
Kendall’s tau-c | .015 | .020 | .764 | .445 | -.006 | .020 | -,314 | .753 | -.005 | .019 | -.255 | .799 |
| Gamma | .025 | .033 | .764 | .445 | -.010 | .032 | -.314 | .753 | -.009 | .034 | -.255 | .799 | |
| School Home communication: Learning Plan (SHC_LP) N = 1432 |
Kendall’s tau-c | -.032 | .015 | -2.084 | .037* | .005 | .016 | .327 | .744 | -.016 | .016 | -1.036 | .300 |
| Gamma | -.177 | .082 | -2.084 | .037* | .028 | .084 | .327 | .744 | -.095 | .090 | 1.036 | .300 | |
| School Home communication (SHC_FLP) N = 1335 |
Kendall’s tau-c | .012 | .020 | .614 | .539 | -.018 | .020 | -.882 | .378 | .009 | .020 | .447 | .655 |
| Gamma | .021 | .034 | .614 | .539 | -.030 | .034 | -.882 | .378 | .017 | .037 | .447 | .655 | |
| Parental engagement: homeschooling (PEHS) N = 1432 |
Kendall’s tau-c | -.005 | .025 | -.191 | .849 | -.014 | .025 | -.544 | .586 | .001 | .024 | .039 | .969 |
| Gamma | -.010 | .050 | -.191 | .849 | -.027 | .050 | -.544 | .586 | .002 | .053 | .039 | .969 | |
| Parental engagement: time (PETT) N = 1115 |
Kendall’s tau-c | -.018 | .020 | -.887 | .375 | -.033 | .020 | -1.624 | .104 | -.031 | .019 | -1.581 | .114 |
| Gamma | -.044 | .050 | -.887 | .375 | -.081 | .050 | -1.624 | .104 | -.083 | .052 | -1.581 | .114 | |
| Name (ID) | F1 | F2 | FT | |||
| Value |
Asymptotic Significance (2 - sided) |
Value |
Asymptotic Significance (2 - sided) |
Value |
Asymptotic Significance (2 - sided) |
|
| Parent/carer gender (PGEN) N = 1432 | 2.672 | .614 | 8.304 | .081 | 5.545 | .236 |
| Parent/carer years of schooling (PYS) N = 1432 | 23.346 | .867 | 21.851 | .911 | 27.375 | .700 |
| Parent/carer Age (PAG) N = 1432 | 14.795 | .253 | 6.050 | .914 | 7.907 | .792 |
| Family structure/composition (FAMC) N = 1283 | 9.230 | .323 | 6.818 | .556 | 12.660 | .124 |
| Child’s age (CHAG) N = 1432 | 70.192 | .002** | 85.920 | .000** | 95.144 | .000** |
| Children in the household (CHH) (Number of siblings) N = 1432 | 27.141 | .511 | 21.278 | .814 | 30.729 | .329 |
| School Home communication: Learning Plan (SHC_LP) N = 1432 | 10.208 | .037* | 7.045 | .134 | 12.447 | .014* |
| School Home communication (SHC_FLP) N = 1335 | 20.963 | .180 | 13.756 | .617 | 21.616 | .156 |
| Parental engagement: homeschooling (PEHS) N = 1432 | 5.032 | .284 | 4.560 | .336 | 2.631 | .621 |
| Parental engagement: time (PETT) N = 1115 | 19.649 | .074 | 21.912 | .039* | 21.972 | .038* |
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