December 7, 2015

The data:

PISA Items from 2003 PISA problem solving assessment:

  • Items were selected to analyze problem solving as it relates to computational thinking in CS students
  • Items were scored in a much more fine grained manner than were the original pisa items
  • Items were scored dichotomously, but weighted differently in order to create sum scores

Map PISA PS framework onto CT framework

PISA:

  • Exploring and understanding the information associated with the problem.
  • Representing and formulating the problem with various representations and formulating hypotheses about the relevant factors and relationships.
  • Planning and executing: devising a plan, setting goals and subgoals and executing the steps of the plan.
  • Monitoring and reflecting: monitoring progress, react- ing to feedback, reflecting on the solution, the strategy

CT: Automation Algorithms Pattern Recognition Debugging Abstraction

Goal of this project:

  • Determine a more systematic/optimal scoring algorithm for creating sum scores

Item response characteristics

qscoreswolast <- qscoresadjust[,1:ncol(qscoresadjust) - 1]

# Classical Item statistics
item.exam(qscoreswolast, discrim = T)
##            Sample.SD Item.total Item.Tot.woi Difficulty Discrimination
## Q23        0.4126360  0.3388201    0.2545494 0.78279267      0.3389831
## Q24_1_TEXT 0.4568990  0.5770741    0.5036513 0.29619182      0.5593220
## Q24_2_TEXT 0.4237544  0.4889182    0.4127458 0.23413258      0.4322034
## Q24_3_TEXT 0.4818550  0.5498045    0.4688187 0.36530324      0.5508475
## Q24_4_TEXT 0.4347351  0.5966750    0.5292925 0.25246827      0.5338983
## Q24_5_TEXT 0.4355363  0.6087677    0.5426507 0.25387870      0.5296610
## Q24_6_TEXT 0.4237544  0.5588795    0.4893031 0.23413258      0.4618644
## Q24_7_TEXT 0.4523593  0.5650232    0.4910410 0.28631876      0.5211864
## Q24_8_TEXT 0.4056833  0.5624118    0.4963674 0.20733427      0.4322034
## Q24_9_TEXT 0.3595915  0.5607760    0.5025814 0.15232722      0.3516949
## Q34_1_TEXT 0.4623601  0.2939904    0.1967800 0.30888575      0.2796610
## Q21_1_TEXT 0.4927660  0.4261112    0.3309105 0.41325811      0.5042373
## Q21_2_TEXT 0.4306374  0.4581116    0.3780316 0.24541608      0.4364407
## Q21_3_TEXT 0.4096999  0.4933999    0.4202892 0.21297602      0.4152542
## Q21_4_TEXT 0.3662879  0.4176545    0.3475001 0.15937941      0.3135593
## Q21_5_TEXT 0.2281550  0.3678694    0.3230899 0.05500705      0.1440678
## Q9         0.4982458  0.4041683    0.3060829 0.54583921      0.4576271
## Q10_1      0.4798249  0.4039545    0.3096848 0.64174894      0.4703390
## Q10_2      0.4806504  0.3774084    0.2810644 0.63892807      0.4237288
## Q10_3      0.4900702  0.3497493    0.2495924 0.60084626      0.4152542
## Q11_1      0.4994319  0.4062780    0.3081094 0.46967560      0.4703390
## Q1         0.4987334  0.2838626    0.1782212 0.54019746      0.3432203
## Q3         0.4611823  0.2796274    0.1820240 0.69393512      0.2838983
##            Item.Criterion Item.Reliab Item.Rel.woi Item.Validity
## Q23                    NA  0.13971075   0.10496213            NA
## Q24_1_TEXT             NA  0.26347857   0.22995540            NA
## Q24_2_TEXT             NA  0.20703506   0.17477947            NA
## Q24_3_TEXT             NA  0.26473916   0.22574327            NA
## Q24_4_TEXT             NA  0.25921255   0.22993968            NA
## Q24_5_TEXT             NA  0.26495336   0.23617733            NA
## Q24_6_TEXT             NA  0.23666057   0.20719806            NA
## Q24_7_TEXT             NA  0.25541320   0.22197029            NA
## Q24_8_TEXT             NA  0.22800009   0.20122589            NA
## Q24_9_TEXT             NA  0.20150801   0.18059650            NA
## Q34_1_TEXT             NA  0.13583355   0.09091903            NA
## Q21_1_TEXT             NA  0.20982497   0.16294641            NA
## Q21_2_TEXT             NA  0.19714083   0.16267970            NA
## Q21_3_TEXT             NA  0.20200327   0.17207097            NA
## Q21_4_TEXT             NA  0.15287388   0.12719531            NA
## Q21_5_TEXT             NA  0.08387202   0.07366256            NA
## Q9                     NA  0.20123311   0.15239695            NA
## Q10_1                  NA  0.19369071   0.14848966            NA
## Q10_2                  NA  0.18127355   0.13499842            NA
## Q10_3                  NA  0.17128080   0.12223151            NA
## Q11_1                  NA  0.20276508   0.15377112            NA
## Q1                     NA  0.14147187   0.08882218            NA
## Q3                     NA  0.12886822   0.08388701            NA
rmthing <- RM(X = qscoreswolast)

ICC Plots

plotICC(rmthing)

Analysis Part 1: EFA with items

Principal components solution

Scree Plot

EFA rotation solutions: 4 & 5 factor models, oblimin rotation (oblique)

## Factor Analysis using method =  minres
## Call: psych::fa(r = corMatrix$correlations, nfactors = 4, n.obs = 710)
## Standardized loadings (pattern matrix) based upon correlation matrix
##              MR2   MR1   MR4   MR3    h2    u2 com
## Q23        -0.05  0.48 -0.07  0.28 0.304 0.696 1.7
## Q24_1_TEXT -0.01  0.86  0.09 -0.03 0.822 0.178 1.0
## Q24_2_TEXT -0.01  0.88 -0.04 -0.07 0.731 0.269 1.0
## Q24_3_TEXT  0.01  0.81  0.10 -0.05 0.745 0.255 1.0
## Q24_4_TEXT -0.01  0.80  0.18 -0.01 0.819 0.181 1.1
## Q24_5_TEXT  0.10  0.55  0.36  0.05 0.708 0.292 1.8
## Q24_6_TEXT  0.03  0.41  0.51  0.04 0.666 0.334 1.9
## Q24_7_TEXT  0.01  0.06  0.92  0.03 0.925 0.075 1.0
## Q24_8_TEXT  0.03  0.05  0.92  0.03 0.921 0.079 1.0
## Q24_9_TEXT  0.06  0.09  0.89  0.05 0.935 0.065 1.0
## Q34_1_TEXT  0.21  0.44 -0.44  0.28 0.335 0.665 3.1
## Q21_1_TEXT  0.90 -0.03  0.02 -0.06 0.775 0.225 1.0
## Q21_2_TEXT  0.99 -0.02 -0.06 -0.02 0.940 0.060 1.0
## Q21_3_TEXT  0.95  0.04  0.03 -0.04 0.922 0.078 1.0
## Q21_4_TEXT  0.92 -0.06  0.03  0.00 0.845 0.155 1.0
## Q21_5_TEXT  0.89  0.01  0.04  0.12 0.888 0.112 1.0
## Q9          0.05  0.32 -0.18  0.59 0.471 0.529 1.8
## Q10_1       0.06  0.01  0.00  0.76 0.605 0.395 1.0
## Q10_2      -0.01 -0.10  0.04  0.85 0.715 0.285 1.0
## Q10_3      -0.02 -0.19  0.19  0.75 0.590 0.410 1.3
## Q11_1       0.06  0.19  0.05  0.45 0.302 0.698 1.4
## Q1          0.02  0.18  0.03  0.20 0.097 0.903 2.0
## Q3         -0.06  0.30 -0.12  0.36 0.206 0.794 2.2
## Q4         -0.05  0.01  0.25  0.35 0.207 0.793 1.9
## 
##                        MR2  MR1  MR4  MR3
## SS loadings           4.48 4.39 3.60 3.01
## Proportion Var        0.19 0.18 0.15 0.13
## Cumulative Var        0.19 0.37 0.52 0.64
## Proportion Explained  0.29 0.28 0.23 0.19
## Cumulative Proportion 0.29 0.57 0.81 1.00
## 
##  With factor correlations of 
##      MR2  MR1  MR4  MR3
## MR2 1.00 0.22 0.22 0.24
## MR1 0.22 1.00 0.50 0.15
## MR4 0.22 0.50 1.00 0.16
## MR3 0.24 0.15 0.16 1.00
## 
## Mean item complexity =  1.4
## Test of the hypothesis that 4 factors are sufficient.
## 
## The degrees of freedom for the null model are  276  and the objective function was  80.47 with Chi Square of  56341.46
## The degrees of freedom for the model are 186  and the objective function was  61.27 
## 
## The root mean square of the residuals (RMSR) is  0.07 
## The df corrected root mean square of the residuals is  0.08 
## 
## The harmonic number of observations is  710 with the empirical chi square  1735.3  with prob <  2.9e-249 
## The total number of observations was  710  with MLE Chi Square =  42734.43  with prob <  0 
## 
## Tucker Lewis Index of factoring reliability =  -0.13
## RMSEA index =  0.573  and the 90 % confidence intervals are  0.563 NA
## BIC =  41513.29
## Fit based upon off diagonal values = 0.96
## Measures of factor score adequacy             
##                                                MR2 MR1 MR4 MR3
## Correlation of scores with factors               1   1   1   1
## Multiple R square of scores with factors         1   1   1   1
## Minimum correlation of possible factor scores    1   1   1   1
## Factor Analysis using method =  minres
## Call: psych::fa(r = corMatrix$correlations, nfactors = 5, n.obs = 710)
## Standardized loadings (pattern matrix) based upon correlation matrix
##              MR2   MR1   MR4   MR3   MR5   h2    u2 com
## Q23        -0.05  0.43 -0.06  0.17  0.23 0.30 0.696 2.0
## Q24_1_TEXT -0.01  0.87  0.06 -0.05  0.08 0.83 0.173 1.0
## Q24_2_TEXT -0.01  0.89 -0.07 -0.09  0.06 0.74 0.262 1.0
## Q24_3_TEXT  0.01  0.89  0.03  0.05 -0.10 0.81 0.195 1.0
## Q24_4_TEXT -0.01  0.84  0.13  0.03 -0.01 0.84 0.158 1.1
## Q24_5_TEXT  0.10  0.59  0.32  0.07 -0.01 0.72 0.280 1.6
## Q24_6_TEXT  0.03  0.47  0.46  0.11 -0.08 0.69 0.313 2.2
## Q24_7_TEXT  0.01  0.08  0.92  0.02  0.00 0.93 0.071 1.0
## Q24_8_TEXT  0.03  0.05  0.92 -0.01  0.05 0.93 0.065 1.0
## Q24_9_TEXT  0.06  0.09  0.89  0.01  0.06 0.95 0.054 1.0
## Q34_1_TEXT  0.21  0.28 -0.37  0.02  0.46 0.40 0.604 3.1
## Q21_1_TEXT  0.89 -0.04  0.03 -0.07  0.01 0.78 0.223 1.0
## Q21_2_TEXT  0.99 -0.02 -0.06 -0.02 -0.01 0.94 0.060 1.0
## Q21_3_TEXT  0.95  0.05  0.03 -0.02 -0.02 0.92 0.078 1.0
## Q21_4_TEXT  0.92 -0.03  0.01  0.05 -0.07 0.85 0.150 1.0
## Q21_5_TEXT  0.89 -0.01  0.05  0.08  0.09 0.89 0.111 1.0
## Q9          0.05  0.19 -0.14  0.36  0.46 0.49 0.506 2.5
## Q10_1       0.05  0.09 -0.07  0.85 -0.03 0.75 0.254 1.0
## Q10_2      -0.02 -0.03 -0.02  0.91  0.03 0.82 0.177 1.0
## Q10_3      -0.02 -0.12  0.13  0.82 -0.02 0.68 0.318 1.1
## Q11_1       0.06 -0.02  0.15  0.09  0.62 0.49 0.508 1.2
## Q1          0.02 -0.02  0.14 -0.12  0.53 0.30 0.703 1.2
## Q3         -0.06  0.10 -0.02  0.03  0.58 0.36 0.635 1.1
## Q4         -0.05 -0.20  0.36  0.00  0.58 0.45 0.548 1.9
## 
##                        MR2  MR1  MR4  MR3  MR5
## SS loadings           4.47 4.33 3.51 2.54 2.01
## Proportion Var        0.19 0.18 0.15 0.11 0.08
## Cumulative Var        0.19 0.37 0.51 0.62 0.70
## Proportion Explained  0.27 0.26 0.21 0.15 0.12
## Cumulative Proportion 0.27 0.52 0.73 0.88 1.00
## 
##  With factor correlations of 
##      MR2  MR1  MR4  MR3  MR5
## MR2 1.00 0.22 0.21 0.22 0.17
## MR1 0.22 1.00 0.50 0.14 0.20
## MR4 0.21 0.50 1.00 0.15 0.16
## MR3 0.22 0.14 0.15 1.00 0.25
## MR5 0.17 0.20 0.16 0.25 1.00
## 
## Mean item complexity =  1.3
## Test of the hypothesis that 5 factors are sufficient.
## 
## The degrees of freedom for the null model are  276  and the objective function was  80.47 with Chi Square of  56341.46
## The degrees of freedom for the model are 166  and the objective function was  60.57 
## 
## The root mean square of the residuals (RMSR) is  0.06 
## The df corrected root mean square of the residuals is  0.08 
## 
## The harmonic number of observations is  710 with the empirical chi square  1420.27  with prob <  6e-198 
## The total number of observations was  710  with MLE Chi Square =  42207.09  with prob <  0 
## 
## Tucker Lewis Index of factoring reliability =  -0.253
## RMSEA index =  0.603  and the 90 % confidence intervals are  0.592 NA
## BIC =  41117.26
## Fit based upon off diagonal values = 0.97
## Measures of factor score adequacy             
##                                                MR2 MR1 MR4 MR3  MR5
## Correlation of scores with factors               1   1   1   1 1.00
## Multiple R square of scores with factors         1   1   1   1 1.00
## Minimum correlation of possible factor scores    1   1   1   1 0.99
## 
## Loadings:
##            MR2    MR1    MR4    MR3   
## Q23                0.479         0.282
## Q24_1_TEXT         0.864              
## Q24_2_TEXT         0.882              
## Q24_3_TEXT         0.811  0.103       
## Q24_4_TEXT         0.801  0.183       
## Q24_5_TEXT         0.554  0.361       
## Q24_6_TEXT         0.408  0.511       
## Q24_7_TEXT                0.924       
## Q24_8_TEXT                0.918       
## Q24_9_TEXT                0.892       
## Q34_1_TEXT  0.207  0.436 -0.439  0.277
## Q21_1_TEXT  0.895                     
## Q21_2_TEXT  0.989                     
## Q21_3_TEXT  0.951                     
## Q21_4_TEXT  0.924                     
## Q21_5_TEXT  0.894                0.118
## Q9                 0.320 -0.182  0.592
## Q10_1                            0.761
## Q10_2                            0.853
## Q10_3             -0.191  0.186  0.748
## Q11_1              0.188         0.448
## Q1                 0.176         0.202
## Q3                 0.300 -0.119  0.361
## Q4                        0.249  0.352
## 
##                  MR2   MR1   MR4   MR3
## SS loadings    4.415 4.043 3.291 2.904
## Proportion Var 0.184 0.168 0.137 0.121
## Cumulative Var 0.184 0.352 0.490 0.611
## 
## Loadings:
##            MR2    MR1    MR4    MR3    MR5   
## Q23                0.426         0.173  0.234
## Q24_1_TEXT         0.866                     
## Q24_2_TEXT         0.886                     
## Q24_3_TEXT         0.887               -0.105
## Q24_4_TEXT         0.842  0.135              
## Q24_5_TEXT         0.593  0.322              
## Q24_6_TEXT         0.472  0.462  0.108       
## Q24_7_TEXT                0.917              
## Q24_8_TEXT                0.922              
## Q24_9_TEXT                0.894              
## Q34_1_TEXT  0.208  0.279 -0.372         0.463
## Q21_1_TEXT  0.895                            
## Q21_2_TEXT  0.988                            
## Q21_3_TEXT  0.950                            
## Q21_4_TEXT  0.923                            
## Q21_5_TEXT  0.893                            
## Q9                 0.191 -0.136  0.359  0.465
## Q10_1                            0.850       
## Q10_2                            0.909       
## Q10_3             -0.118  0.127  0.822       
## Q11_1                     0.153         0.620
## Q1                        0.137 -0.117  0.526
## Q3                                      0.583
## Q4                -0.195  0.362         0.584
## 
##                  MR2   MR1   MR4   MR3   MR5
## SS loadings    4.407 3.997 3.200 2.451 1.877
## Proportion Var 0.184 0.167 0.133 0.102 0.078
## Cumulative Var 0.184 0.350 0.483 0.586 0.664

Four Factor CFA Model

## Warning in lav_model_vcov(lavmodel = lavmodel, lavsamplestats = lavsamplestats, : lavaan WARNING: could not compute standard errors!
##   lavaan NOTE: this may be a symptom that the model is not identified.
## lavaan (0.5-18) converged normally after  86 iterations
## 
##   Number of observations                           710
## 
##   Estimator                                         ML
##   Minimum Function Test Statistic            42837.387
##   Degrees of freedom                               242
##   P-value (Chi-square)                           0.000
## 
## Model test baseline model:
## 
##   Minimum Function Test Statistic            57132.739
##   Degrees of freedom                               276
##   P-value                                        0.000
## 
## User model versus baseline model:
## 
##   Comparative Fit Index (CFI)                    0.251
##   Tucker-Lewis Index (TLI)                       0.146
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -17019.028
##   Loglikelihood unrestricted model (H1)       4399.665
## 
##   Number of free parameters                         58
##   Akaike (AIC)                               34154.057
##   Bayesian (BIC)                             34418.842
##   Sample-size adjusted Bayesian (BIC)        34234.678
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.498
##   90 Percent Confidence Interval          0.494  0.502
##   P-value RMSEA <= 0.05                          0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.111
## 
## Parameter estimates:
## 
##   Information                                 Expected
##   Standard Errors                             Standard
## 
##                    Estimate  Std.err  Z-value  P(>|z|)
## Latent variables:
##   F1 =~
##     Q34_1_TEXT        0.296
##     Q21_1_TEXT        1.083
##     Q21_2_TEXT        1.229
##     Q21_3_TEXT        1.245
##     Q21_4_TEXT        1.185
##     Q21_5_TEXT        1.193
##   F2 =~
##     Q23               0.468
##     Q24_1_TEXT        1.152
##     Q24_2_TEXT        1.031
##     Q24_3_TEXT        1.350
##     Q24_4_TEXT        1.390
##     Q24_5_TEXT        1.066
##   F3 =~
##     Q24_6_TEXT        0.824
##     Q24_7_TEXT        1.286
##     Q24_8_TEXT        1.299
##     Q24_9_TEXT        1.277
##   F4 =~
##     Q9                0.922
##     Q10_1             1.558
##     Q10_2             1.726
##     Q10_3             1.403
##     Q11_1             0.625
##     Q1                0.280
##     Q3                0.485
##     Q4                0.452
## 
## Covariances:
##   F1 ~~
##     F2                0.129
##     F3                0.151
##     F4                0.104
##   F2 ~~
##     F3                0.307
##     F4                0.073
##   F3 ~~
##     F4                0.079
## 
## Variances:
##     Q34_1_TEXT        0.946
##     Q21_1_TEXT        0.299
##     Q21_2_TEXT        0.099
##     Q21_3_TEXT        0.075
##     Q21_4_TEXT        0.162
##     Q21_5_TEXT        0.151
##     Q23               0.891
##     Q24_1_TEXT        0.346
##     Q24_2_TEXT        0.477
##     Q24_3_TEXT        0.103
##     Q24_4_TEXT        0.050
##     Q24_5_TEXT        0.440
##     Q24_6_TEXT        0.599
##     Q24_7_TEXT        0.025
##     Q24_8_TEXT        0.006
##     Q24_9_TEXT        0.038
##     Q9                0.778
##     Q10_1             0.367
##     Q10_2             0.224
##     Q10_3             0.487
##     Q11_1             0.897
##     Q1                0.978
##     Q3                0.937
##     Q4                0.946
##     F1                0.596
##     F2                0.491
##     F3                0.589
##     F4                0.260

Five Factor CFA Model

## Warning in lav_model_vcov(lavmodel = lavmodel, lavsamplestats = lavsamplestats, : lavaan WARNING: could not compute standard errors!
##   lavaan NOTE: this may be a symptom that the model is not identified.
## lavaan (0.5-18) converged normally after  74 iterations
## 
##   Number of observations                           710
## 
##   Estimator                                         ML
##   Minimum Function Test Statistic            42377.288
##   Degrees of freedom                               237
##   P-value (Chi-square)                           0.000
## 
## Model test baseline model:
## 
##   Minimum Function Test Statistic            57132.739
##   Degrees of freedom                               276
##   P-value                                        0.000
## 
## User model versus baseline model:
## 
##   Comparative Fit Index (CFI)                    0.259
##   Tucker-Lewis Index (TLI)                       0.137
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -16788.979
##   Loglikelihood unrestricted model (H1)       4399.665
## 
##   Number of free parameters                         63
##   Akaike (AIC)                               33703.957
##   Bayesian (BIC)                             33991.569
##   Sample-size adjusted Bayesian (BIC)        33791.529
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.500
##   90 Percent Confidence Interval          0.496  0.504
##   P-value RMSEA <= 0.05                          0.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.078
## 
## Parameter estimates:
## 
##   Information                                 Expected
##   Standard Errors                             Standard
## 
##                    Estimate  Std.err  Z-value  P(>|z|)
## Latent variables:
##   F1 =~
##     Q21_1_TEXT        1.003
##     Q21_2_TEXT        1.139
##     Q21_3_TEXT        1.155
##     Q21_4_TEXT        1.099
##     Q21_5_TEXT        1.106
##   F2 =~
##     Q23               0.447
##     Q24_1_TEXT        1.098
##     Q24_2_TEXT        0.978
##     Q24_3_TEXT        1.285
##     Q24_4_TEXT        1.340
##     Q24_5_TEXT        1.049
##     Q24_6_TEXT        0.995
##   F3 =~
##     Q24_7_TEXT        1.216
##     Q24_8_TEXT        1.231
##     Q24_9_TEXT        1.207
##   F4 =~
##     Q10_1             1.108
##     Q10_2             1.282
##     Q10_3             1.004
##   F5 =~
##     Q34_1_TEXT        0.830
##     Q9                1.337
##     Q11_1             1.338
##     Q1                0.661
##     Q3                0.836
##     Q4                0.939
## 
## Covariances:
##   F1 ~~
##     F2                0.146
##     F3                0.172
##     F4                0.137
##     F5                0.133
##   F2 ~~
##     F3                0.341
##     F4                0.085
##     F5                0.135
##   F3 ~~
##     F4                0.089
##     F5                0.142
##   F4 ~~
##     F5                0.169
## 
## Variances:
##     Q21_1_TEXT        0.302
##     Q21_2_TEXT        0.100
##     Q21_3_TEXT        0.074
##     Q21_4_TEXT        0.161
##     Q21_5_TEXT        0.151
##     Q23               0.892
##     Q24_1_TEXT        0.356
##     Q24_2_TEXT        0.489
##     Q24_3_TEXT        0.119
##     Q24_4_TEXT        0.042
##     Q24_5_TEXT        0.412
##     Q24_6_TEXT        0.471
##     Q24_7_TEXT        0.027
##     Q24_8_TEXT        0.003
##     Q24_9_TEXT        0.041
##     Q10_1             0.378
##     Q10_2             0.168
##     Q10_3             0.489
##     Q34_1_TEXT        0.843
##     Q9                0.594
##     Q11_1             0.594
##     Q1                0.900
##     Q3                0.841
##     Q4                0.799
##     F1                0.693
##     F2                0.533
##     F3                0.657
##     F4                0.505
##     F5                0.226

Factor Score Methods

Refined (Distefano et al. 2009):

  • Sum scores by factor
  • Sum scores above cut off
  • Sum scores - standardized variables
  • Sum scores - weighted sum scores

Non-refined (Distefano et al. 2009):

  • Thurstone: Regression based factor scores; biased estimate; maximizes validity
  • Bartlett: Uses shared factors only; produces unbiased estimates of true factor scores
  • Anderson: Uncorrelated factor scores that are orthogonal
##            thurstSum   bartSum  anderSum egr100PISA
## thurstSum  1.0000000 0.9825627 0.9796678  0.8848186
## bartSum    0.9825627 1.0000000 0.9950304  0.8944919
## anderSum   0.9796678 0.9950304 1.0000000  0.9101660
## egr100PISA 0.8848186 0.8944919 0.9101660  1.0000000
## $thurstSum
## [1] 9.467218
## 
## $bartSum
## [1] 6.252382
## 
## $anderSum
## [1] 5
## 
## $egr100PISA
## [1] 21.4867

Thurstone scores

summary(thurstSum)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -6.0270 -2.1190 -0.5309  0.0000  1.5630 11.3800
hist(thurstSum)

Bartlett scores

summary(bartSum)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -4.6760 -1.7920 -0.2929  0.0000  1.3230  8.6810
hist(bartSum)

Anderson scores

summary(anderSum)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  -4.412  -1.621  -0.265   0.000   1.276   7.702
hist(anderSum)

Previous scores

summary(egr100PISA, na.rm = T)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.000   6.167   9.333   9.420  12.500  22.000
hist(egr100PISA)