Introduction
Epistemic beliefs are beliefs about knowledge and its acquisition. They have an important role in various processes related to learning, self-regulation and academic achievement, as different authors have highlighted and empirical evidence supports [1–6]. In addition, there is evidence that epistemic beliefs can be modified with specific interventions [7–9], so change naïve epistemic beliefs could be a way to optimize learning processes.
Despite there are various models of epistemic beliefs emphasizing some different aspects, as the evolution of thinking process about knowledge and knowing [10], [11] the differences between women and men in thinking about knowledge [12], [13], the role of epistemic perspective in decision making [14], the attitude or disposition of teachers regarding knowledge and knowing process [15], or the resources character of epistemic beliefs [16], the model most commonly used in research has been proposed by Marlene Schommer [17], [18]. For her, epistemic beliefs are a set of more or less independent dimensions, whose development does not necessarily follow a homogeneous sequence but evolves from a naive dualist position on knowledge and learning to a relativistic and sophisticated position. Based on this model, she developed the Epistemological Questionnaire (EQ), from which she set five dimensions: knowledge structure, stability or certainty of knowledge, source of knowledge, learning control and speed of learning
Method
Participants Participants were 1,785 high school students (from 7th to 12th grade) from public schools and private schools with public funding from the cities of Iquique and Arica, in northern Chile. Of these, 49.8% were female, and ages ranged from 12 to 19 years. Although there was more availability of intermediate classes than elementary or advanced ones, all high school classes were represented, from freshmen to senior; however, the distribution by age and grade was not homogeneous. This sample was divided randomly into two subsamples of approximately 60% and 40% respectively; the first sample had 1,039 participants (subsample 1), and the other had 746 participants (subsample 2). The first subsample was used for exploratory analyses, and the second, for confirmatory analyses.
Data Preparation
Import dataset
We’ll only use data from entries where the value in the data frame’s submuestra (subsample) column is equal to 1. This is because we’re only doing a CFA and the other data (with submuestra=0) had been used for prior exploratory factor analysis (EFA) by the article authors. This is a common pattern: EFA first with a subset of data to find a model, then CFA with the remaining data to put the model to the test.
Summary
Use skim function to figure out how much missing data is in each variable
## ce1 ce2 ce3 ce4
## Min. :1.000 Min. :1.000 Min. :1.000 Min. :1.000
## 1st Qu.:3.000 1st Qu.:3.000 1st Qu.:2.000 1st Qu.:3.000
## Median :4.000 Median :4.000 Median :3.000 Median :4.000
## Mean :3.442 Mean :3.633 Mean :3.012 Mean :3.584
## 3rd Qu.:4.000 3rd Qu.:4.000 3rd Qu.:4.000 3rd Qu.:5.000
## Max. :5.000 Max. :5.000 Max. :5.000 Max. :5.000
## ce5 ce6 ce7 ce8
## Min. :1.000 Min. :1.000 Min. :1.000 Min. :1.000
## 1st Qu.:1.000 1st Qu.:2.000 1st Qu.:3.000 1st Qu.:2.000
## Median :2.000 Median :3.000 Median :4.000 Median :2.000
## Mean :2.382 Mean :3.101 Mean :3.914 Mean :2.681
## 3rd Qu.:3.000 3rd Qu.:4.000 3rd Qu.:5.000 3rd Qu.:4.000
## Max. :5.000 Max. :5.000 Max. :5.000 Max. :5.000
## ce9 ce10 ce11 ce12
## Min. :1.000 Min. :1.000 Min. :1.000 Min. :1.000
## 1st Qu.:2.000 1st Qu.:2.000 1st Qu.:3.000 1st Qu.:3.000
## Median :3.000 Median :3.000 Median :4.000 Median :3.000
## Mean :2.973 Mean :3.239 Mean :3.587 Mean :3.517
## 3rd Qu.:4.000 3rd Qu.:4.000 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :5.000 Max. :5.000 Max. :5.000 Max. :5.000
## ce13 ce14 ce15 ce16 ce17
## Min. :1.000 Min. :1.000 Min. :1.000 Min. :1.000 Min. :1.00
## 1st Qu.:2.000 1st Qu.:2.000 1st Qu.:1.000 1st Qu.:3.000 1st Qu.:3.00
## Median :3.000 Median :3.000 Median :2.000 Median :4.000 Median :3.00
## Mean :3.275 Mean :2.764 Mean :1.945 Mean :3.504 Mean :3.21
## 3rd Qu.:4.000 3rd Qu.:4.000 3rd Qu.:2.000 3rd Qu.:4.000 3rd Qu.:4.00
## Max. :5.000 Max. :5.000 Max. :5.000 Max. :5.000 Max. :5.00
## ce18 ce19 ce20 ce21 ce22
## Min. :1.00 Min. :1.000 Min. :1.000 Min. :1.000 Min. :1.000
## 1st Qu.:2.00 1st Qu.:2.000 1st Qu.:1.000 1st Qu.:2.000 1st Qu.:3.000
## Median :3.50 Median :3.000 Median :2.000 Median :3.000 Median :4.000
## Mean :3.33 Mean :2.873 Mean :2.072 Mean :3.038 Mean :3.932
## 3rd Qu.:4.00 3rd Qu.:4.000 3rd Qu.:3.000 3rd Qu.:4.000 3rd Qu.:5.000
## Max. :5.00 Max. :5.000 Max. :5.000 Max. :5.000 Max. :5.000
## ce23 ce24 ce25 ce26
## Min. :1.000 Min. :1.000 Min. :1.000 Min. :1.000
## 1st Qu.:2.000 1st Qu.:2.000 1st Qu.:3.000 1st Qu.:2.000
## Median :3.000 Median :2.000 Median :3.000 Median :3.000
## Mean :3.265 Mean :2.592 Mean :3.338 Mean :2.989
## 3rd Qu.:4.000 3rd Qu.:4.000 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :5.000 Max. :5.000 Max. :5.000 Max. :5.000
## ce27 ce28
## Min. :1.000 Min. :1.000
## 1st Qu.:1.000 1st Qu.:3.000
## Median :2.000 Median :4.000
## Mean :2.303 Mean :3.556
## 3rd Qu.:3.000 3rd Qu.:5.000
## Max. :5.000 Max. :5.000Each item in the instrument used in the study is represented in the data frame by a column with a name according to the pattern ce
Missingness
After review, it is determined that the dataset is complete with no missing values.
Variable Names
## [1] "ce1" "ce2" "ce3" "ce4" "ce5" "ce6" "ce7" "ce8" "ce9" "ce10"
## [11] "ce11" "ce12" "ce13" "ce14" "ce15" "ce16" "ce17" "ce18" "ce19" "ce20"
## [21] "ce21" "ce22" "ce23" "ce24" "ce25" "ce26" "ce27" "ce28"The variable names represent each of the questions responses given by the participants
Identify Multivariate Outliers
Mahalanobis
Run Mahalanobis to identify multivariate outliers
Summary
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 2.798 18.283 25.646 27.962 35.217 90.054Cutoff Score
Calculate cutoff score for p < 0.001
## [1] 28## [1] 56.89229This indicates that at Df = 28 and p-value < 0.001, the cut-off value for outliers is 56.8922.
Outliers
Identify number of participants who exceed the cut-off score
## Mode FALSE TRUE
## logical 23 723This indicates that 23 participants exceed the cut-off (FALSE) and need to be excluded from the dataset.
Participants to be removed:
## [1] 4 15 130 162 278 279 292 310 353 400 402 403 405 420 496 536 595 646 661
## [20] 679 687 693 727New Dataset
Remove multivariate outliers from dataset
## ce1 ce2 ce3 ce4 ce5 ce6 ce7 ce8 ce9 ce10 ce11 ce12 ce13 ce14 ce15 ce16
## 1043 5 5 5 5 2 3 2 3 2 5 1 5 1 5 5 5
## 1054 5 5 1 1 1 1 5 5 1 1 5 5 5 1 5 1
## 1169 1 5 1 5 5 2 1 2 3 1 1 5 2 1 1 5
## 1201 5 5 2 5 1 3 5 5 5 5 5 3 1 5 1 4
## 1317 1 1 1 5 1 1 1 5 1 5 1 1 1 1 1 5
## 1318 5 4 2 1 1 1 1 5 5 1 1 5 5 1 1 5
## 1331 1 5 1 1 1 1 1 1 5 5 1 5 1 5 1 1
## 1349 5 5 1 1 5 1 1 5 5 5 5 5 2 2 2 2
## 1392 5 5 1 1 1 1 5 1 2 5 5 5 1 1 1 5
## 1439 1 5 5 5 1 1 5 5 5 5 1 5 1 5 5 1
## 1441 5 5 1 5 1 5 5 1 5 1 5 5 1 1 1 1
## 1442 5 4 1 5 2 1 5 4 1 5 4 2 4 1 5 5
## 1444 1 5 4 5 2 1 5 5 5 5 1 4 1 3 1 4
## 1459 5 5 1 3 5 2 5 1 5 1 5 5 5 1 1 5
## 1535 1 1 3 5 5 1 5 5 5 5 5 5 1 1 1 5
## 1575 3 5 3 5 5 3 5 5 5 3 4 3 5 1 4 5
## 1634 4 5 1 5 5 5 1 1 5 5 5 5 1 1 1 1
## 1685 5 5 1 5 1 5 5 1 1 5 2 5 5 3 5 3
## 1700 2 1 5 4 1 5 5 1 2 1 5 1 1 5 1 5
## 1718 5 5 5 1 1 5 5 1 5 5 5 5 1 5 1 1
## 1726 1 1 1 5 1 1 5 1 5 1 1 1 1 1 1 5
## 1732 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
## 1766 4 2 5 5 1 5 1 1 1 5 4 1 5 1 5 4
## ce17 ce18 ce19 ce20 ce21 ce22 ce23 ce24 ce25 ce26 ce27 ce28
## 1043 5 5 5 5 1 2 4 2 4 1 1 3
## 1054 5 1 5 1 5 5 5 1 1 1 1 1
## 1169 5 1 5 5 2 3 5 5 1 1 3 2
## 1201 1 5 5 1 5 5 1 1 4 5 1 1
## 1317 1 5 5 5 1 5 1 1 1 5 1 1
## 1318 5 5 5 5 5 5 1 1 5 1 1 1
## 1331 5 5 5 1 1 5 1 1 1 1 5 1
## 1349 2 2 5 1 5 5 5 5 2 2 1 5
## 1392 5 5 5 1 5 5 5 5 1 1 1 1
## 1439 1 1 5 5 1 5 5 5 1 5 5 5
## 1441 1 1 1 1 1 5 5 1 1 5 5 5
## 1442 1 4 5 2 5 2 5 1 2 5 4 4
## 1444 4 2 3 4 5 1 2 3 5 5 4 5
## 1459 1 5 5 5 5 1 5 5 2 1 3 5
## 1535 5 5 5 1 5 5 5 5 2 1 1 5
## 1575 5 1 5 1 1 4 5 1 5 5 1 5
## 1634 5 5 1 5 3 5 1 1 3 5 5 5
## 1685 5 5 4 3 1 3 5 3 3 5 1 3
## 1700 5 1 1 1 1 5 1 1 5 2 1 2
## 1718 5 1 5 1 5 5 1 1 5 1 1 5
## 1726 1 5 1 1 5 5 5 1 5 1 1 5
## 1732 1 1 5 5 5 1 1 1 5 1 5 1
## 1766 5 4 1 1 3 1 5 1 5 1 1 4A list of those observations removed along with their scores for each variable.
Examine Multivariate Assumption
Normality Assumption
##
## Shapiro-Wilk normality test
##
## data: Z
## W = 0.96567, p-value = 0.000000000005517Given the p-value < 0.05 is statistically significant, we can reject the null hypothesis that the data are normally distributed.
QQ plot
From the plot, we can confirm results from the Shapir-Wilks test above from the visual inspection. If dataset was coming from a multivariate normal distribution, the points would follow an imaginary diagonal line.
Histogram
Multivariate Normality Histogram 
Another visual inspection of the distribution of the dataset. If running factor analysis, would consider tranformation of the data.
Examine Additive Assumption
“In multivariate regression, on of the assumptions is the additive assumption. This assumption states that the influence of a predictor variable on the dependent variable is independent of any other influence. Violations of the additive assumption can be addressed through the use of interactions in a regression model.” source
Bivariate Correlations
Run bivariate correlations on all relevant variables
Correlaton Table
Tables with correlations followed by table identifying correlations that are higher than 0.90
## ce1 ce2 ce3 ce4 ce5
## ce1 1.000000000 0.217218890 0.11818961 0.1371432442 0.029549808
## ce2 0.217218890 1.000000000 0.02288536 0.0498222281 -0.003515857
## ce3 0.118189607 0.022885361 1.00000000 0.1870967737 0.238161614
## ce4 0.137143244 0.049822228 0.18709677 1.0000000000 0.038031198
## ce5 0.029549808 -0.003515857 0.23816161 0.0380311983 1.000000000
## ce6 -0.028746234 0.061214668 -0.03910482 -0.0321662348 0.018852307
## ce7 0.163369073 0.038027104 0.08883263 0.1626445873 0.097340113
## ce8 0.037552640 0.086286958 0.30725706 0.0117665387 0.229964644
## ce9 0.049377267 0.067572960 0.13497887 0.0220769830 0.118848167
## ce10 -0.004541209 0.003347013 0.15511637 -0.0497397829 0.083721100
## ce11 0.108871933 0.187966050 0.09237803 0.0758156363 -0.030582202
## ce12 0.123600986 0.055031393 0.10514195 0.0661899924 0.094944447
## ce13 0.046695769 0.030106549 0.10040545 -0.0371318405 0.194013018
## ce14 0.156146385 0.036034263 0.31725248 0.0474127408 0.345804167
## ce15 0.067723030 0.010770865 0.26385173 0.0462978421 0.296741247
## ce16 0.050909731 -0.012858800 0.14089272 0.0699756593 0.130843181
## ce17 0.063703378 0.102289705 -0.04236900 -0.0288249008 -0.011701542
## ce18 0.048753678 0.006102704 0.18552190 0.1341833657 0.184138120
## ce19 0.071424459 0.050701383 0.14780768 -0.1388822917 0.091249416
## ce20 0.066706084 -0.044802431 0.13429953 0.0313669501 0.223734105
## ce21 0.148676253 0.143722926 0.11829450 0.0997761071 0.120678502
## ce22 0.065857610 0.157661577 -0.02061325 0.1155329095 0.018004359
## ce23 0.116431484 0.021420682 0.21486638 0.1908626539 0.117891754
## ce24 0.063323547 0.024759178 0.24775991 0.0146419478 0.382779402
## ce25 0.172716683 0.059360941 0.12426361 0.3953034174 0.041797221
## ce26 0.128627580 0.004666603 0.15008535 0.3439057631 0.124977211
## ce27 0.039651185 -0.068313710 0.20825108 -0.0107532448 0.225123436
## ce28 0.063444854 0.118932615 0.09493581 -0.0007775686 0.054630170
## ce6 ce7 ce8 ce9 ce10
## ce1 -0.028746234 0.16336907 0.03755264 0.049377267 -0.004541209
## ce2 0.061214668 0.03802710 0.08628696 0.067572960 0.003347013
## ce3 -0.039104822 0.08883263 0.30725706 0.134978874 0.155116373
## ce4 -0.032166235 0.16264459 0.01176654 0.022076983 -0.049739783
## ce5 0.018852307 0.09734011 0.22996464 0.118848167 0.083721100
## ce6 1.000000000 -0.03710203 0.06620102 0.087242843 -0.045554369
## ce7 -0.037102031 1.00000000 -0.01024475 0.028178012 0.036139273
## ce8 0.066201023 -0.01024475 1.00000000 0.150300130 0.111624427
## ce9 0.087242843 0.02817801 0.15030013 1.000000000 0.226294102
## ce10 -0.045554369 0.03613927 0.11162443 0.226294102 1.000000000
## ce11 0.022477420 0.04364369 0.01537100 0.143103906 0.116264804
## ce12 -0.002114716 0.06363623 0.03611680 0.071040155 0.275061804
## ce13 0.099110483 0.06805288 0.17610727 0.014575249 0.049456999
## ce14 -0.072711853 0.11547885 0.26104658 0.143619821 0.100976377
## ce15 0.032449349 0.03456229 0.19741951 0.196064560 0.118257713
## ce16 0.007281880 0.04234016 0.17087456 0.128085117 0.182230367
## ce17 0.096519355 -0.04483648 0.03422201 0.181586588 0.044611038
## ce18 -0.064639666 0.16372762 0.06846537 0.114409206 0.138525582
## ce19 0.067828251 -0.07970461 0.13465765 0.097691185 0.024785653
## ce20 0.002308970 -0.01743245 0.09554205 0.136000502 0.068904723
## ce21 0.099082702 0.06584538 0.06731378 0.080424895 -0.071911220
## ce22 0.068461278 0.14446381 -0.03814030 0.002977984 0.019092362
## ce23 -0.137285666 0.16969356 0.05540570 0.128629614 0.095242817
## ce24 0.035440887 0.01597511 0.23027116 0.114836916 0.053221552
## ce25 -0.010654894 0.16961200 0.06432695 -0.075015313 -0.085354428
## ce26 -0.087023381 0.19439363 0.03676539 0.040440603 -0.018314869
## ce27 -0.028744785 0.02589988 0.16171011 0.194311173 0.102791790
## ce28 0.135887585 0.04004276 0.07541833 0.150674004 0.040165742
## ce11 ce12 ce13 ce14 ce15 ce16
## ce1 0.108871933 0.123600986 0.046695769 0.15614638 0.06772303 0.050909731
## ce2 0.187966050 0.055031393 0.030106549 0.03603426 0.01077086 -0.012858800
## ce3 0.092378030 0.105141952 0.100405447 0.31725248 0.26385173 0.140892725
## ce4 0.075815636 0.066189992 -0.037131840 0.04741274 0.04629784 0.069975659
## ce5 -0.030582202 0.094944447 0.194013018 0.34580417 0.29674125 0.130843181
## ce6 0.022477420 -0.002114716 0.099110483 -0.07271185 0.03244935 0.007281880
## ce7 0.043643693 0.063636227 0.068052884 0.11547885 0.03456229 0.042340163
## ce8 0.015371000 0.036116804 0.176107275 0.26104658 0.19741951 0.170874561
## ce9 0.143103906 0.071040155 0.014575249 0.14361982 0.19606456 0.128085117
## ce10 0.116264804 0.275061804 0.049456999 0.10097638 0.11825771 0.182230367
## ce11 1.000000000 0.248034289 0.059293042 0.02941340 0.07912510 0.089929850
## ce12 0.248034289 1.000000000 0.158830889 0.09438288 0.05890369 0.115038964
## ce13 0.059293042 0.158830889 1.000000000 0.22773084 0.07359778 0.245080901
## ce14 0.029413400 0.094382881 0.227730842 1.00000000 0.35304494 0.185511924
## ce15 0.079125095 0.058903688 0.073597784 0.35304494 1.00000000 0.196901131
## ce16 0.089929850 0.115038964 0.245080901 0.18551192 0.19690113 1.000000000
## ce17 0.227758841 0.139163360 0.092660002 0.07188699 0.03194868 0.098831129
## ce18 0.079941512 0.122590036 -0.002888260 0.21724072 0.20377305 0.131256344
## ce19 0.007083212 0.090965235 0.043028577 0.11489497 0.20563990 0.061876822
## ce20 0.004869405 0.035518415 -0.023335455 0.19149159 0.33774811 0.008985802
## ce21 0.107315492 0.064393267 0.064152345 0.12772663 0.09532219 -0.012290977
## ce22 0.196720507 0.119209554 0.143530000 0.01507166 0.03051956 0.067850348
## ce23 0.082715332 0.119593740 -0.017674134 0.19102492 0.22064119 0.067270210
## ce24 0.031206041 0.113077903 0.357915366 0.30103915 0.24604873 0.204391986
## ce25 0.088131615 -0.008221376 0.042834086 0.08058905 0.01039184 0.045515543
## ce26 0.068900375 0.045335758 0.003687336 0.12997950 0.08144706 0.054570512
## ce27 -0.003443380 0.071062266 0.020191270 0.23857968 0.33673978 0.077481040
## ce28 0.095509621 0.063215566 0.085587060 0.09243219 0.10931346 0.134471481
## ce17 ce18 ce19 ce20 ce21
## ce1 0.06370338 0.048753678 0.071424459 0.066706084 0.14867625
## ce2 0.10228971 0.006102704 0.050701383 -0.044802431 0.14372293
## ce3 -0.04236900 0.185521902 0.147807683 0.134299534 0.11829450
## ce4 -0.02882490 0.134183366 -0.138882292 0.031366950 0.09977611
## ce5 -0.01170154 0.184138120 0.091249416 0.223734105 0.12067850
## ce6 0.09651936 -0.064639666 0.067828251 0.002308970 0.09908270
## ce7 -0.04483648 0.163727616 -0.079704610 -0.017432446 0.06584538
## ce8 0.03422201 0.068465366 0.134657648 0.095542046 0.06731378
## ce9 0.18158659 0.114409206 0.097691185 0.136000502 0.08042490
## ce10 0.04461104 0.138525582 0.024785653 0.068904723 -0.07191122
## ce11 0.22775884 0.079941512 0.007083212 0.004869405 0.10731549
## ce12 0.13916336 0.122590036 0.090965235 0.035518415 0.06439327
## ce13 0.09266000 -0.002888260 0.043028577 -0.023335455 0.06415234
## ce14 0.07188699 0.217240719 0.114894970 0.191491593 0.12772663
## ce15 0.03194868 0.203773048 0.205639895 0.337748109 0.09532219
## ce16 0.09883113 0.131256344 0.061876822 0.008985802 -0.01229098
## ce17 1.00000000 0.036747442 0.073975881 0.072725306 0.12375172
## ce18 0.03674744 1.000000000 0.074297656 0.149828539 0.10779971
## ce19 0.07397588 0.074297656 1.000000000 0.185547155 0.02375307
## ce20 0.07272531 0.149828539 0.185547155 1.000000000 0.09295264
## ce21 0.12375172 0.107799708 0.023753068 0.092952644 1.00000000
## ce22 0.15379957 0.047516588 -0.022012900 -0.098568600 0.09598961
## ce23 -0.07934250 0.294241641 0.036935375 0.120982645 0.06097312
## ce24 0.04677297 0.091667802 0.114392298 0.180383809 0.04213247
## ce25 -0.01065640 0.063756907 -0.137805425 -0.099559435 0.13294565
## ce26 0.02818866 0.122014517 -0.176295870 0.078137903 0.10733234
## ce27 0.04380017 0.212413235 0.200344972 0.321037514 0.05784386
## ce28 0.07023174 0.110466460 0.126760811 -0.004050114 0.09535591
## ce22 ce23 ce24 ce25 ce26 ce27
## ce1 0.065857610 0.11643148 0.06332355 0.172716683 0.128627580 0.03965118
## ce2 0.157661577 0.02142068 0.02475918 0.059360941 0.004666603 -0.06831371
## ce3 -0.020613254 0.21486638 0.24775991 0.124263610 0.150085354 0.20825108
## ce4 0.115532909 0.19086265 0.01464195 0.395303417 0.343905763 -0.01075324
## ce5 0.018004359 0.11789175 0.38277940 0.041797221 0.124977211 0.22512344
## ce6 0.068461278 -0.13728567 0.03544089 -0.010654894 -0.087023381 -0.02874478
## ce7 0.144463813 0.16969356 0.01597511 0.169612001 0.194393625 0.02589988
## ce8 -0.038140302 0.05540570 0.23027116 0.064326945 0.036765387 0.16171011
## ce9 0.002977984 0.12862961 0.11483692 -0.075015313 0.040440603 0.19431117
## ce10 0.019092362 0.09524282 0.05322155 -0.085354428 -0.018314869 0.10279179
## ce11 0.196720507 0.08271533 0.03120604 0.088131615 0.068900375 -0.00344338
## ce12 0.119209554 0.11959374 0.11307790 -0.008221376 0.045335758 0.07106227
## ce13 0.143530000 -0.01767413 0.35791537 0.042834086 0.003687336 0.02019127
## ce14 0.015071660 0.19102492 0.30103915 0.080589053 0.129979496 0.23857968
## ce15 0.030519561 0.22064119 0.24604873 0.010391839 0.081447058 0.33673978
## ce16 0.067850348 0.06727021 0.20439199 0.045515543 0.054570512 0.07748104
## ce17 0.153799566 -0.07934250 0.04677297 -0.010656399 0.028188664 0.04380017
## ce18 0.047516588 0.29424164 0.09166780 0.063756907 0.122014517 0.21241323
## ce19 -0.022012900 0.03693537 0.11439230 -0.137805425 -0.176295870 0.20034497
## ce20 -0.098568600 0.12098264 0.18038381 -0.099559435 0.078137903 0.32103751
## ce21 0.095989609 0.06097312 0.04213247 0.132945653 0.107332337 0.05784386
## ce22 1.000000000 0.13274682 0.01335708 0.082599023 -0.028605821 -0.07285574
## ce23 0.132746819 1.00000000 0.13791988 0.125576267 0.163987797 0.15474056
## ce24 0.013357075 0.13791988 1.00000000 0.060419320 0.095337561 0.18542283
## ce25 0.082599023 0.12557627 0.06041932 1.000000000 0.299030157 -0.07239409
## ce26 -0.028605821 0.16398780 0.09533756 0.299030157 1.000000000 0.09998435
## ce27 -0.072855740 0.15474056 0.18542283 -0.072394093 0.099984351 1.00000000
## ce28 0.195772943 0.07729420 0.03763356 0.010599064 0.062291432 0.06452121
## ce28
## ce1 0.0634448542
## ce2 0.1189326145
## ce3 0.0949358123
## ce4 -0.0007775686
## ce5 0.0546301700
## ce6 0.1358875851
## ce7 0.0400427624
## ce8 0.0754183342
## ce9 0.1506740040
## ce10 0.0401657420
## ce11 0.0955096214
## ce12 0.0632155661
## ce13 0.0855870604
## ce14 0.0924321861
## ce15 0.1093134590
## ce16 0.1344714811
## ce17 0.0702317421
## ce18 0.1104664600
## ce19 0.1267608114
## ce20 -0.0040501141
## ce21 0.0953559066
## ce22 0.1957729435
## ce23 0.0772942036
## ce24 0.0376335586
## ce25 0.0105990644
## ce26 0.0622914324
## ce27 0.0645212080
## ce28 1.0000000000## ce1 ce2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 c15 c16 c17 c18 c19 c20
## ce1 1
## ce2 1
## ce3 1
## ce4 1
## ce5 1
## ce6 1
## ce7 1
## ce8 . 1
## ce9 1
## ce10 1
## ce11 1
## ce12 1
## ce13 1
## ce14 . . 1
## ce15 . 1
## ce16 1
## ce17 1
## ce18 1
## ce19 1
## ce20 . 1
## ce21
## ce22
## ce23
## ce24 . . .
## ce25 .
## ce26 .
## ce27 . .
## ce28
## c21 c22 c23 c24 c25 c26 c27 c28
## ce1
## ce2
## ce3
## ce4
## ce5
## ce6
## ce7
## ce8
## ce9
## ce10
## ce11
## ce12
## ce13
## ce14
## ce15
## ce16
## ce17
## ce18
## ce19
## ce20
## ce21 1
## ce22 1
## ce23 1
## ce24 1
## ce25 1
## ce26 1
## ce27 1
## ce28 1
## attr(,"legend")
## [1] 0 ' ' 0.3 '.' 0.6 ',' 0.8 '+' 0.9 '*' 0.95 'B' 1Interpretation
Provide interpretation of whether additivity assumption is met
From the symbolnum rendering of the correlation table, pairs with * symbol indicate correlations that are higher than 0.90 and ‘B’ indicates those with correlations above 0.95. none of the pairs are correlated at a level above 0.95, however there is a smattering of those above 0.90
Examine Linearity Assumption
Chi-square Test
Create random chi-square values for the same number of participants in data
Run Fake Regression
Run fake regression with random chi-square values as DV, predicted by all of the relevant variables in dataset
Studentized Residuals
Produce studentized residuals based on fake regression
QQ Plot
Make qqplot, integrated with a straight line through plot 
Interpretation
Provide interpretation of whether linearity assumption is met
There might be some slight issues with linearity, but we seem to be okay for the most part. Given the sample size, there do not appear to be any gross deviations from the line.
Examine Multivariate Normality
Histogram
Make histogram of standardized residuals 
Interpretation
Provide interpretation of whether normality assumption is met
While slightly positively skewed, it appears fairly normal
Examine Homogeneity and Homoscedasticity
Scatterplot
Make scatterplot of standardized fitted values (i.e., predicted score for each person) predicting standardized residuals 
Interpretation
The assumption of homogeneity appers to be met.
For homogeneity, we basically want the points evenly distributed among the four quadrants. So this is mostly okay. There are a few participants that are located on the far ends, given the sample size. Since there are no shapes, there does not appear to be an issue with heteroscedasticity.
Analysis
Preamble
A sample of high school students (N = 1785) from public and private schools participated to complete a battery of measures that included the instrument. The instrument is composed of 28 five-point Likert scale items,, each one consisting of a statement regarding epistemic beliefs and following the model of five dimensions of Schommer-Aikins [18] described in the introduction. The dimensions, as described in [20], are:
- Omniscient Authority (e.g., "People shouldn’t question Epistemic Beliefs Inventory cross-validation
- Certain Knowledge (e.g., “What is true today will be true tomorrow”);
- Quick Learning (e.g., “Working on a problem with no quick solution is a waste of time”);
- Simple Knowledge (e.g., Too many theories just complicate things");
- Innate Ability (e.g., “Smart people are born that way”).
The principal author conducted a translation from English intoSpanish, which was back-translated into English by an independent English-speaker to ensure equivalence
Citation: PLOS ONE | DOI:10.1371/journal.pone.0173295 March 9, 2017 4 / 16 authority
Mahalanobis values were calculated to identify multivariate outliers. A total of 23 participants were identified as multivariate outliers based on cut-off value of 56.89, = = 0.001. These participants were discarded from further analysis, leaving a total sample size of N = 723. Bivariate correlations between all items demonstrated only a few correlations above 0.90, suggesting that the additivity assumption is met. Visual inspection of qq plots, histogram of standardized residuals, and scatterplot of standardized fitted values predicting standardized residuals suggest that multivariate linearity, normality, and homoskedasticity assumptions were met.
Model
Model Definition
The three factors and respective questions for the model are;
- Factor 1: IA ‘Innate Ability’: CE3, CE5, CE8, CE9, CE14, CE15, CE20, CE24, CE27
- Factor 2: OA ‘Omniscient Ability’: CE4, CE25, CE26
- Factor 3: SC ’Simple and Certain knowledge: CE1. CE2. CE11. CE17. CE22
Factor loadings were determined in Exploratory Factor Analysis Project. Naming conventions were takaen from author of published report.
Model Fit
Run factor model using Lavaan’s cfa function
SEM Path Diagram
Diagram Interpretation
Check for Heywoood cases
## lavaan 0.6-8 ended normally after 53 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 38
##
## Number of observations 723
##
## Model Test User Model:
##
## Test statistic 318.844
## Degrees of freedom 115
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## IA =~
## ce3 1.000 0.621 0.465
## ce5 1.134 0.124 9.154 0.000 0.705 0.557
## ce8 0.818 0.101 8.076 0.000 0.508 0.386
## ce9 0.575 0.096 6.003 0.000 0.357 0.291
## ce14 1.212 0.129 9.386 0.000 0.753 0.588
## ce15 1.086 0.115 9.450 0.000 0.675 0.597
## ce20 0.878 0.112 7.871 0.000 0.546 0.425
## ce24 1.002 0.115 8.691 0.000 0.623 0.503
## ce27 0.936 0.111 8.403 0.000 0.582 0.473
## OA =~
## ce4 1.000 0.787 0.664
## ce25 0.828 0.100 8.266 0.000 0.651 0.587
## ce26 0.775 0.094 8.228 0.000 0.610 0.520
## SC =~
## ce1 1.000 0.382 0.325
## ce2 1.078 0.231 4.675 0.000 0.412 0.407
## ce11 1.390 0.287 4.847 0.000 0.531 0.499
## ce17 0.862 0.197 4.388 0.000 0.329 0.342
## ce22 0.995 0.220 4.520 0.000 0.380 0.368
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .ce3 ~~
## .ce8 0.225 0.060 3.762 0.000 0.225 0.157
## IA ~~
## OA 0.082 0.028 2.926 0.003 0.168 0.168
## SC 0.033 0.016 2.095 0.036 0.141 0.141
## OA ~~
## SC 0.080 0.024 3.280 0.001 0.267 0.267
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .ce3 1.403 0.082 17.106 0.000 1.403 0.784
## .ce5 1.106 0.069 16.017 0.000 1.106 0.690
## .ce8 1.475 0.083 17.757 0.000 1.475 0.851
## .ce9 1.376 0.075 18.401 0.000 1.376 0.915
## .ce14 1.074 0.069 15.483 0.000 1.074 0.655
## .ce15 0.823 0.054 15.308 0.000 0.823 0.644
## .ce20 1.352 0.077 17.551 0.000 1.352 0.819
## .ce24 1.146 0.068 16.761 0.000 1.146 0.747
## .ce27 1.170 0.068 17.092 0.000 1.170 0.776
## .ce4 0.784 0.083 9.503 0.000 0.784 0.559
## .ce25 0.807 0.065 12.373 0.000 0.807 0.655
## .ce26 1.000 0.069 14.562 0.000 1.000 0.729
## .ce1 1.235 0.074 16.709 0.000 1.235 0.894
## .ce2 0.855 0.057 15.087 0.000 0.855 0.835
## .ce11 0.847 0.068 12.453 0.000 0.847 0.751
## .ce17 0.820 0.050 16.425 0.000 0.820 0.883
## .ce22 0.921 0.058 15.932 0.000 0.921 0.865
## IA 0.386 0.068 5.638 0.000 1.000 1.000
## OA 0.619 0.094 6.580 0.000 1.000 1.000
## SC 0.146 0.048 3.040 0.002 1.000 1.000
##
## R-Square:
## Estimate
## ce3 0.216
## ce5 0.310
## ce8 0.149
## ce9 0.085
## ce14 0.345
## ce15 0.356
## ce20 0.181
## ce24 0.253
## ce27 0.224
## ce4 0.441
## ce25 0.345
## ce26 0.271
## ce1 0.106
## ce2 0.165
## ce11 0.249
## ce17 0.117
## ce22 0.135For R2, there are no values > 1 so that is good. Also, there are no negative coefficients. Therefore, there does not appear to be the presense of Heywood Cases.
Get Global Fit indices
| x | |
|---|---|
| npar | 38.00000000 |
| fmin | 0.22050100 |
| chisq | 318.84445016 |
| df | 115.00000000 |
| pvalue | 0.00000000 |
| baseline.chisq | 1552.32049671 |
| baseline.df | 136.00000000 |
| baseline.pvalue | 0.00000000 |
| cfi | 0.85607463 |
| tli | 0.82979261 |
| nnfi | 0.82979261 |
| rfi | 0.75709383 |
| nfi | 0.79460140 |
| pnfi | 0.67190560 |
| ifi | 0.85817746 |
| rni | 0.85607463 |
| logl | -18779.90670115 |
| unrestricted.logl | -18620.48447607 |
| aic | 37635.81340231 |
| bic | 37809.98295275 |
| ntotal | 723.00000000 |
| bic2 | 37689.32187964 |
| rmsea | 0.04951438 |
| rmsea.ci.lower | 0.04309952 |
| rmsea.ci.upper | 0.05601692 |
| rmsea.pvalue | 0.53791954 |
| rmr | 0.07286396 |
| rmr_nomean | 0.07286396 |
| srmr | 0.05219344 |
| srmr_bentler | 0.05219344 |
| srmr_bentler_nomean | 0.05219344 |
| crmr | 0.05535950 |
| crmr_nomean | 0.05535950 |
| srmr_mplus | 0.05219344 |
| srmr_mplus_nomean | 0.05219344 |
| cn_05 | 320.79379335 |
| cn_01 | 348.36940500 |
| gfi | 0.94782623 |
| agfi | 0.93058621 |
| pgfi | 0.71241841 |
| mfi | 0.86851427 |
| ecvi | 0.54611957 |
The fit indices tell us how well the theoretical factor model is able to be produced the correlation matrix that is coming from the actual data itself. The pertinent information needed for model comparison later are listed below.
Original 3-factor Model
- chi-square: 318.844
- Df: 115
- RMSEA: 0.0495 (95% CI = 0.043 - 0.056)
- SRMR: 0.05219
- CFI: 0.856
- TLI: 0.830
- AIC: 37635.813
- BIC: 37809.983
Note: RMSEA p-value = 0.5379. Additionally, other values look pretty bad, i.e. CFI and TLI, so we might need to revise the model.
Modification Indices
| lhs | op | rhs | mi | epc | sepc.lv | sepc.all | sepc.nox | |
|---|---|---|---|---|---|---|---|---|
| 50 | OA | =~ | ce3 | 22.26422 | 0.3495111 | 0.2750147 | 0.2056225 | 0.2056225 |
| 96 | ce5 | ~~ | ce24 | 21.72430 | 0.2360195 | 0.2360195 | 0.2095831 | 0.2095831 |
| 157 | ce20 | ~~ | ce27 | 20.28312 | 0.2348175 | 0.2348175 | 0.1866787 | 0.1866787 |
| 131 | ce9 | ~~ | ce17 | 17.21803 | 0.1717775 | 0.1717775 | 0.1617101 | 0.1617101 |
| 159 | ce20 | ~~ | ce25 | 17.02386 | -0.1839309 | -0.1839309 | -0.1761106 | -0.1761106 |
| 59 | OA | =~ | ce1 | 16.97517 | 0.3180891 | 0.2502901 | 0.2130067 | 0.2130067 |
| 145 | ce15 | ~~ | ce20 | 13.88471 | 0.1750666 | 0.1750666 | 0.1659715 | 0.1659715 |
| 83 | ce3 | ~~ | ce4 | 12.94532 | 0.1695900 | 0.1695900 | 0.1617159 | 0.1617159 |
| 67 | SC | =~ | ce9 | 12.36886 | 0.5913158 | 0.2257427 | 0.1840801 | 0.1840801 |
| 44 | IA | =~ | ce26 | 11.83703 | 0.2798399 | 0.1738911 | 0.1484535 | 0.1484535 |
| 201 | ce1 | ~~ | ce2 | 11.45857 | 0.1653540 | 0.1653540 | 0.1609194 | 0.1609194 |
The largest modification index (MI = 22.264; EPC = 0.35) does not meet the threshhold for consideration of a model revision. From the standardized EPC values column, anything above .5 would be worth considering for model modification. Based on the output, model modification is not justified.
Inspect residual correlations
|
|
| ce3 | ce5 | ce8 | ce9 | ce14 | ce15 | ce20 | ce24 | ce27 | ce4 | ce25 | ce26 | ce1 | ce2 | ce11 | ce17 | ce22 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ce3 | 0.0000 | -0.8966 | 0.0000 | -0.0104 | 1.9429 | -0.6394 | -2.4192 | 0.5659 | -0.4618 | 4.0590 | 2.3071 | 3.2008 | 2.6978 | -0.1049 | 1.7532 | -1.8201 | -1.2608 |
| ce5 | -0.8966 | 0.0000 | 0.6098 | -1.6538 | 0.9477 | -1.9450 | -0.5346 | 4.2766 | -1.7172 | -0.7783 | -0.4044 | 2.2898 | 0.1162 | -1.0335 | -2.1256 | -1.0947 | -0.3102 |
| ce8 | 0.0000 | 0.6098 | 0.0000 | 1.2076 | 1.4108 | -1.4625 | -2.4553 | 1.3443 | -0.7797 | -0.9121 | 0.7527 | 0.0870 | 0.5503 | 1.7927 | -0.3351 | 0.4351 | -1.6116 |
| ce9 | -0.0104 | -1.6538 | 1.2076 | 0.0000 | -1.0997 | 0.8853 | 0.4034 | -1.1269 | 1.9131 | -0.2934 | -2.8773 | 0.4192 | 0.9936 | 1.4018 | 3.3926 | 4.5449 | -0.3314 |
| ce14 | 1.9429 | 0.9477 | 1.4108 | -1.0997 | 0.0000 | 0.1185 | -2.6278 | 0.2614 | -1.8772 | -0.6020 | 0.7187 | 2.3944 | 3.6555 | 0.0725 | -0.3701 | 1.2626 | -0.4477 |
| ce15 | -0.6394 | -1.9450 | -1.4625 | 0.8853 | 0.1185 | 0.0000 | 3.5098 | -2.7451 | 2.4229 | -0.6806 | -1.5391 | 0.8936 | 1.1528 | -0.6957 | 1.1653 | 0.0948 | -0.0106 |
| ce20 | -2.4192 | -0.5346 | -2.4553 | 0.4034 | -2.6278 | 3.5098 | 0.0000 | -1.3082 | 4.2507 | -0.4782 | -4.0716 | 1.1751 | 1.3044 | -1.9432 | -0.7216 | 1.4516 | -3.3410 |
| ce24 | 0.5659 | 4.2766 | 1.3443 | -1.1269 | 0.2614 | -2.7451 | -1.3082 | 0.0000 | -2.1946 | -1.2908 | 0.3287 | 1.5161 | 1.1315 | -0.1149 | -0.1222 | 0.6405 | -0.3601 |
| ce27 | -0.4618 | -1.7172 | -0.7797 | 1.9131 | -1.8772 | 2.4229 | 4.2507 | -2.1946 | 0.0000 | -1.9478 | -3.5141 | 1.7003 | 0.4994 | -2.7067 | -1.0788 | 0.5889 | -2.7285 |
| ce4 | 4.0590 | -0.7783 | -0.9121 | -0.2934 | -0.6020 | -0.6806 | -0.4782 | -1.2908 | -1.9478 | 0.0000 | 1.2454 | -0.3094 | 2.3651 | -0.7283 | -0.4579 | -2.7769 | 1.5626 |
| ce25 | 2.3071 | -0.4044 | 0.7527 | -2.8773 | 0.7187 | -1.5391 | -4.0716 | 0.3287 | -3.5141 | 1.2454 | 0.0000 | -0.8075 | 3.5503 | -0.1380 | 0.3260 | -1.9358 | 0.7521 |
| ce26 | 3.2008 | 2.2898 | 0.0870 | 0.4192 | 2.3944 | 0.8936 | 1.1751 | 1.5161 | 1.7003 | -0.3094 | -0.8075 | 0.0000 | 2.3887 | -1.5576 | -0.0171 | -0.5630 | -2.3457 |
| ce1 | 2.6978 | 0.1162 | 0.5503 | 0.9936 | 3.6555 | 1.1528 | 1.3044 | 1.1315 | 0.4994 | 2.3651 | 3.5503 | 2.3887 | 0.0000 | 3.1825 | -2.7753 | -1.7512 | -2.0882 |
| ce2 | -0.1049 | -1.0335 | 1.7927 | 1.4018 | 0.0725 | -0.6957 | -1.9432 | -0.1149 | -2.7067 | -0.7283 | -0.1380 | -1.5576 | 3.1825 | 0.0000 | -0.9086 | -1.5301 | 0.3382 |
| ce11 | 1.7532 | -2.1256 | -0.3351 | 3.3926 | -0.3701 | 1.1653 | -0.7216 | -0.1222 | -1.0788 | -0.4579 | 0.3260 | -0.0171 | -2.7753 | -0.9086 | 0.0000 | 2.6611 | 0.6754 |
| ce17 | -1.8201 | -1.0947 | 0.4351 | 4.5449 | 1.2626 | 0.0948 | 1.4516 | 0.6405 | 0.5889 | -2.7769 | -1.9358 | -0.5630 | -1.7512 | -1.5301 | 2.6611 | 0.0000 | 1.0409 |
| ce22 | -1.2608 | -0.3102 | -1.6116 | -0.3314 | -0.4477 | -0.0106 | -3.3410 | -0.3601 | -2.7285 | 1.5626 | 0.7521 | -2.3457 | -2.0882 | 0.3382 | 0.6754 | 1.0409 | 0.0000 |
With z-scores,
- anything larger than +/-1.96 is significant at a p <.05 level
- anything larger than +/-2.58 is significant at a p < .01 level
- anything larger than +/-
We set fit.measures=TRUE to obtain various global fit indices. We set standardized=TRUE which is used to obtain standardized factor loadings in addition to unstandardized factor loadings. summary(fit, fit.measures=TRUE, standardized=TRUE)
Interpretation
Fit statistics interpretation Chi-Square test (Model Test User Model) Historically, a significant p-value for the Chi-Square test was taken as an indication of a lack of fit. However, Chi-Square tests are impacted by sample size, and given that CFA/SEM are generally large-sample procedures, very often the p-value for the Chi-Square test will be significant. For this reason, nowadays much less weight is given to this test.
Typically values for the CFI and TLI are interpreted as indicative of the following:
- Above .95: Very good fit.
- Above .90: Acceptable fit.
Since the values we get (.856 and .83) are low relative to .90, they indicate a lack of fit.
Root Mean Square Error of Approximation Root Mean Square Error of Approximation (RMSEA) values are generally interpreted as indicative of the following:
Below 0.05: Close fit. Below 0.08: Acceptable fit. The RMSEA value we got is 0.0495. The ‘PCLOSE test’ of the RMSEA (P-value RMSEA <= 0.05) is non-significant, which is also taken as an indicator of close fit.
Standardized Root Mean Square Residual Standardized Root Mean Square Residual (SRMR) values at 0.05 or below are generally considered indicative of a well-fitting model. Here, we have SRMR of 0.052.
Conclusion The statistics point in different directions with regard to fit. SRMR, CFI and TLI suggest less than optimal levels of fit. The RMSEA indicates close fit. One could also note here how it becomes obvious that binary ‘significant’/‘non-significant’ evaluations of statistics are inherently problematic.
Factor loading-related statistics interpretation Factor loading (Estimate) Note that for each set of variables that were specified for a latent variable, the first variable’s factor loading is fixed to 1.This is used to scale the latent factors’ variances, and the other factor loadings. This is also why there are no standard error/z-value/p-value estimates for the first variable in each set.
p-values (P(>|z|)) All p-values indicate significant factor loadings.
Standardized factor loadings (Std.all) The standardized factor loadings are basically analogous to the loadings that you might see within e. g. a standard exploratory factor analysis (EFA) rotated loading matrix. These would essentially be the correlations between each of the indicator variables and their corresponding latent variables.
Covariances interpretation The syntax LV1 ~~ LV2 specifies the correlation between latent variable LV1 and latent variable LV2. Note that this is the same syntax as for all other correlations, such as the correlation we specified for two individual variables, ce3 ~~ ce8.
‘Innate Ability’ ~~ ‘Omniscient Authority’ (IA ~~ OA) The covariance (Estimate - 0.082) is statistically significant (P(>|z|) - 0.001). The Std.all value (0.168) is essentially the correlation between the two latent variables.
‘Innate Ability’ ~~ ‘Simple and Certain knowledge’ (IA ~~ SC) The covariance (0.033) is statistically significant (p=0.036). The correlation is 0.141.
‘Omniscient Authority’ ~~ ‘Simple and Certain knowledge’ (OA ~~ SC) The covariance (0.080) is statistically significant (p=0.001). The correlation is 0.267.
Instrument item 3 ~~ Instrument item 8 (ce3 ~~ ce8) The covariance (0.225) is statistically significant (p<0.001). The correlation is 0.157.
Variances interpretation Variance estimates (Estimate) for indicator (‘non-latent’) variables such as ce3 or ce5 represent estimates of error variance. The variance values for latent variables represent estimates of factor variance, for the unstandardized solution. In the standardized solution (Std.all) the variances are 1 for each of these latent factors.
Revised Model
There is not a theoretical basis for a model modification
References
Leal-Soto F, Ferrer-Urbina R. (2017). Three-factor structure for Epistemic Belief Inventory: A cross-validation study. PLoS ONE 12 (3): e0173295. doi:10.1371/journal.pone.0173295
Appendix
Exhibit 1
Source: Leal-Soto & Ferrer-Urbina, 2017
Exhibit 2
Source: Leal-Soto & Ferrer-Urbina, 2017
Exhibit 3
Source: Leal-Soto & Ferrer-Urbina, 2017