exametrika
Overview
The exametrika package provides comprehensive Test Data
Engineering tools for analyzing educational test data. Based on the
methods described in Shojima (2022), this package enables researchers
and practitioners to:
- Analyze test response patterns and item characteristics
- Classify respondents using various psychometric models
- Investigate latent structures in test data
- Examine local dependencies between items
- Perform network analysis of item relationships
The package implements both traditional psychometric approaches and advanced statistical methods, making it suitable for various assessment and research purposes.
Features
The package implements various psychometric models and techniques:
Classical Methods
- Classical Test Theory (CTT)
- Item difficulty and discrimination
- Test reliability and validity
- Item Response Theory (IRT)
- 2PL, 3PL, and 4PL models
- Item characteristic curves
- Test information functions
Latent Structure Analysis
- Latent Class Analysis (LCA)
- Class membership estimation
- Item response profiles
- Latent Rank Analysis (LRA)
- Ordered latent classes
- Rank transition probabilities
- Biclustering and Ranklustering
- Simultaneous clustering of items and examinees
- Field-specific response patterns
- Infinite Relational Model (IRM)
- Optimal class/field determination
- Nonparametric clustering
Advanced Network Models
- Bayesian Network Analysis
- Structure Learning
- Genetic Algorithm approach
- Population-Based Incremental Learning (PBIL)
- Conditional probability estimation
- Structure Learning
- Local Dependence Models
- Local Dependence Latent Rank Analysis (LDLRA)
- Local Dependence Biclustering (LDB)
- Bicluster Network Model (BINET)
Model Overview
Local Dependence Models
The package implements three complementary approaches to modeling local dependencies in test data:
- LDLRA (Local Dependence Latent Rank Analysis)
- Analyzes how item dependencies change across different proficiency ranks
- Suitable when item relationships are expected to vary by student ability level
- Combines the strengths of LRA and Bayesian Networks
- LDB (Local Dependence Biclustering)
- Focuses on relationships between item fields within each rank
- Optimal when items naturally form groups (fields) with hierarchical relationships
- Integrates biclustering with field-level dependency structures
- BINET (Bicluster Network Model)
- Examines class transitions within each field
- Best for understanding complex patterns of class progression
- Combines biclustering with class-level network analysis
Background
Exametrika was originally developed and published as a Mathematica and Excel Add-in. For additional information about Exametrika, visit:
Installation
The development version of Exametrika can be installed from GitHub:
# Install devtools if not already installed
if (!require("devtools")) install.packages("devtools")
# Install Exametrika
devtools::install_github("kosugitti/exametrika")Dependencies
The package requires:
- R (>= 4.1.0)
- igraph (for network analysis)
- Other dependencies are automatically installed
Data Format and Usage
Basic Usage
library(exametrika)Data Requirements
Exametrika accepts both binary and polytomous response data:
- Binary data (0/1)
- 0: Incorrect answer
- 1: Correct answer
- Polytomous data
- Ordinal response categories
- Multiple score levels
- Missing values
- NA values supported
- Custom missing value codes can be specified
Input Data Specifications
The package accepts data in several formats with the following features:
- Data Structure
- Matrix or data.frame format
- Response data (binary or polytomous)
- Flexible handling of missing values
- Support for various data types and structures
- Optional Components
- Examinee ID column (default: first column)
- Item weights (default: all weights = 1)
- Item labels (default: sequential numbers)
- Missing value indicator matrix
Note: Some analysis methods may have specific data type requirements. Please refer to each function’s documentation for detailed requirements.
Data Formatting
The dataFormat function processes input data before
analysis:
- Functions
- Extracts and validates ID vectors if present
- Processes item labels or assigns sequential numbers
- Creates response data matrix U
- Generates missing value indicator matrix Z
- Handles item weights
- Converts data to appropriate format for analysis
Example:
# Format raw data for analysis
data <- dataFormat(J15S500) # Using sample dataset
str(data) # View structure of formatted data
#> List of 7
#> $ ID : chr [1:500] "Student001" "Student002" "Student003" "Student004" ...
#> $ ItemLabel : chr [1:15] "Item01" "Item02" "Item03" "Item04" ...
#> $ Z : num [1:500, 1:15] 1 1 1 1 1 1 1 1 1 1 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : NULL
#> .. ..$ : chr [1:15] "Item01" "Item02" "Item03" "Item04" ...
#> $ w : num [1:15] 1 1 1 1 1 1 1 1 1 1 ...
#> $ response.type: chr "binary"
#> $ categories : Named int [1:15] 2 2 2 2 2 2 2 2 2 2 ...
#> ..- attr(*, "names")= chr [1:15] "Item01" "Item02" "Item03" "Item04" ...
#> $ U : num [1:500, 1:15] 0 1 1 1 1 1 0 0 1 1 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : NULL
#> .. ..$ : chr [1:15] "Item01" "Item02" "Item03" "Item04" ...
#> - attr(*, "class")= chr [1:2] "exametrika" "exametrikaData"Sample Datasets
The package includes various sample datasets from Shojima (2022) for testing and learning:
- Naming Convention: JxxSxxx format
- J: Number of items (e.g., J15 = 15 items)
- S: Sample size (e.g., S500 = 500 examinees)
Available datasets:
- J5S10: Very small dataset (5 items, 10 examinees)
- Useful for quick testing and understanding basic concepts
- J12S5000: Large sample dataset (12 items, 5000 examinees)
- Suitable for LDLRA and other advanced analyses
- J14S500: Medium dataset (14 items, 500 examinees)
- J15S500: Medium dataset (15 items, 500 examinees)
- Often used in IRT and LCA examples
- J20S400: Medium dataset (20 items, 400 examinees)
- J35S515: Large item dataset (35 items, 515 examinees)
- Used in Biclustering and network model examples
- J15S3810: Ordinal scale dataset (15 items with 4-point scale, 3810
examinees)
- Used in ordinal latent rank model examples
- J35S5000: Multiple-choice dataset (35 items, 5000 examinees)
- Includes both response categories and correct answer data
- Used in nominal scale latent rank model examples
Examples
Test Statistics
TestStatistics(J15S500)
#> Test Statistics
#> value
#> TestLength 15.0000000
#> SampleSize 500.0000000
#> Mean 9.6640000
#> SEofMean 0.1190738
#> Variance 7.0892826
#> SD 2.6625707
#> Skewness -0.4116220
#> Kurtosis -0.4471624
#> Min 2.0000000
#> Max 15.0000000
#> Range 13.0000000
#> Q1.25% 8.0000000
#> Median.50% 10.0000000
#> Q3.75% 12.0000000
#> IQR.75% 4.0000000
#> Stanine.4% 5.0000000
#> Stanine.11% 6.0000000
#> Stanine.23% 7.0000000
#> Stanine.40% 9.0000000
#> Stanine.60% 11.0000000
#> Stanine.77% 12.0000000
#> Stanine.89% 13.0000000
#> Stanine.96% 14.0000000ItemStatistics
ItemStatistics(J15S500)
#> Item Statistics
#> ItemLabel NR CRR ODDs Threshold Entropy ITCrr
#> 1 Item01 500 0.746 2.937 -0.662 0.818 0.375
#> 2 Item02 500 0.754 3.065 -0.687 0.805 0.393
#> 3 Item03 500 0.726 2.650 -0.601 0.847 0.321
#> 4 Item04 500 0.776 3.464 -0.759 0.767 0.503
#> 5 Item05 500 0.804 4.102 -0.856 0.714 0.329
#> 6 Item06 500 0.864 6.353 -1.098 0.574 0.377
#> 7 Item07 500 0.716 2.521 -0.571 0.861 0.483
#> 8 Item08 500 0.588 1.427 -0.222 0.978 0.405
#> 9 Item09 500 0.364 0.572 0.348 0.946 0.225
#> 10 Item10 500 0.662 1.959 -0.418 0.923 0.314
#> 11 Item11 500 0.286 0.401 0.565 0.863 0.455
#> 12 Item12 500 0.274 0.377 0.601 0.847 0.468
#> 13 Item13 500 0.634 1.732 -0.342 0.948 0.471
#> 14 Item14 500 0.764 3.237 -0.719 0.788 0.485
#> 15 Item15 500 0.706 2.401 -0.542 0.874 0.413CTT
CTT(J15S500)
#> Realiability
#> name value
#> 1 Alpha(Covariance) 0.625
#> 2 Alpha(Phi) 0.630
#> 3 Alpha(Tetrachoric) 0.771
#> 4 Omega(Covariance) 0.632
#> 5 Omega(Phi) 0.637
#> 6 Omega(Tetrachoric) 0.779
#>
#> Reliability Excluding Item
#> IfDeleted Alpha.Covariance Alpha.Phi Alpha.Tetrachoric
#> 1 Item01 0.613 0.618 0.762
#> 2 Item02 0.609 0.615 0.759
#> 3 Item03 0.622 0.628 0.770
#> 4 Item04 0.590 0.595 0.742
#> 5 Item05 0.617 0.624 0.766
#> 6 Item06 0.608 0.613 0.754
#> 7 Item07 0.594 0.600 0.748
#> 8 Item08 0.611 0.616 0.762
#> 9 Item09 0.642 0.645 0.785
#> 10 Item10 0.626 0.630 0.773
#> 11 Item11 0.599 0.606 0.751
#> 12 Item12 0.597 0.603 0.748
#> 13 Item13 0.597 0.604 0.753
#> 14 Item14 0.593 0.598 0.745
#> 15 Item15 0.607 0.612 0.759IRT
The IRT function estimates the number of parameters using a logistic
model, which can be specified using the model option. It
supports 2PL, 3PL, and 4PL models.
result.IRT <- IRT(J15S500, model = 3)
result.IRT
#> Item Parameters
#> slope location lowerAsym PSD(slope) PSD(location) PSD(lowerAsym)
#> Item01 0.818 -0.834 0.2804 0.182 0.628 0.1702
#> Item02 0.860 -1.119 0.1852 0.157 0.471 0.1488
#> Item03 0.657 -0.699 0.3048 0.162 0.798 0.1728
#> Item04 1.550 -0.949 0.1442 0.227 0.216 0.1044
#> Item05 0.721 -1.558 0.2584 0.148 0.700 0.1860
#> Item06 1.022 -1.876 0.1827 0.171 0.423 0.1577
#> Item07 1.255 -0.655 0.1793 0.214 0.289 0.1165
#> Item08 0.748 -0.155 0.1308 0.148 0.394 0.1077
#> Item09 1.178 2.287 0.2930 0.493 0.423 0.0440
#> Item10 0.546 -0.505 0.2221 0.131 0.779 0.1562
#> Item11 1.477 1.090 0.0628 0.263 0.120 0.0321
#> Item12 1.479 1.085 0.0462 0.245 0.115 0.0276
#> Item13 0.898 -0.502 0.0960 0.142 0.272 0.0858
#> Item14 1.418 -0.788 0.2260 0.248 0.291 0.1252
#> Item15 0.908 -0.812 0.1531 0.159 0.383 0.1254
#>
#> Item Fit Indices
#> model_log_like bench_log_like null_log_like model_Chi_sq null_Chi_sq
#> Item01 -262.979 -240.190 -283.343 45.578 86.307
#> Item02 -253.405 -235.436 -278.949 35.937 87.025
#> Item03 -280.640 -260.906 -293.598 39.468 65.383
#> Item04 -204.884 -192.072 -265.962 25.623 147.780
#> Item05 -232.135 -206.537 -247.403 51.196 81.732
#> Item06 -173.669 -153.940 -198.817 39.459 89.755
#> Item07 -250.905 -228.379 -298.345 45.053 139.933
#> Item08 -314.781 -293.225 -338.789 43.111 91.127
#> Item09 -321.920 -300.492 -327.842 42.856 54.700
#> Item10 -309.318 -288.198 -319.850 42.240 63.303
#> Item11 -248.409 -224.085 -299.265 48.647 150.360
#> Item12 -238.877 -214.797 -293.598 48.160 157.603
#> Item13 -293.472 -262.031 -328.396 62.882 132.730
#> Item14 -223.473 -204.953 -273.212 37.040 136.519
#> Item15 -271.903 -254.764 -302.847 34.279 96.166
#> model_df null_df NFI RFI IFI TLI CFI RMSEA AIC CAIC
#> Item01 11 13 0.472 0.376 0.541 0.443 0.528 0.079 23.578 -22.805
#> Item02 11 13 0.587 0.512 0.672 0.602 0.663 0.067 13.937 -32.446
#> Item03 11 13 0.396 0.287 0.477 0.358 0.457 0.072 17.468 -28.915
#> Item04 11 13 0.827 0.795 0.893 0.872 0.892 0.052 3.623 -42.759
#> Item05 11 13 0.374 0.260 0.432 0.309 0.415 0.086 29.196 -17.186
#> Item06 11 13 0.560 0.480 0.639 0.562 0.629 0.072 17.459 -28.924
#> Item07 11 13 0.678 0.620 0.736 0.683 0.732 0.079 23.053 -23.330
#> Item08 11 13 0.527 0.441 0.599 0.514 0.589 0.076 21.111 -25.272
#> Item09 11 13 0.217 0.074 0.271 0.097 0.236 0.076 20.856 -25.527
#> Item10 11 13 0.333 0.211 0.403 0.266 0.379 0.075 20.240 -26.143
#> Item11 11 13 0.676 0.618 0.730 0.676 0.726 0.083 26.647 -19.735
#> Item12 11 13 0.694 0.639 0.747 0.696 0.743 0.082 26.160 -20.222
#> Item13 11 13 0.526 0.440 0.574 0.488 0.567 0.097 40.882 -5.501
#> Item14 11 13 0.729 0.679 0.793 0.751 0.789 0.069 15.040 -31.343
#> Item15 11 13 0.644 0.579 0.727 0.669 0.720 0.065 12.279 -34.104
#> BIC
#> Item01 -22.783
#> Item02 -32.424
#> Item03 -28.893
#> Item04 -42.737
#> Item05 -17.164
#> Item06 -28.902
#> Item07 -23.308
#> Item08 -25.250
#> Item09 -25.505
#> Item10 -26.121
#> Item11 -19.713
#> Item12 -20.200
#> Item13 -5.479
#> Item14 -31.321
#> Item15 -34.082
#>
#> Model Fit Indices
#> value
#> model_log_like -3880.769
#> bench_log_like -3560.005
#> null_log_like -4350.217
#> model_Chi_sq 641.528
#> null_Chi_sq 1580.424
#> model_df 165.000
#> null_df 195.000
#> NFI 0.594
#> RFI 0.520
#> IFI 0.663
#> TLI 0.594
#> CFI 0.656
#> RMSEA 0.076
#> AIC 311.528
#> CAIC -384.212
#> BIC -383.883The estimated population of subjects is included in the returned object.
head(result.IRT$ability)
#> ID EAP PSD
#> 1 Student001 -0.75526692 0.5805700
#> 2 Student002 -0.17398753 0.5473605
#> 3 Student003 0.01382307 0.5530502
#> 4 Student004 0.57628160 0.5749107
#> 5 Student005 -0.97449499 0.5915605
#> 6 Student006 0.85233023 0.5820543The plots offer options for Item Response Function(also known as Item
Characteristic Curves (ICC)),Test Response Function, Item Information
Curves (IIC), and Test Information Curves (TIC), which can be specified
through options. Items can be specified using the items
argument, and if not specified, plots will be drawn for all items. The
number of rows and columns for dividing the plotting area can be
specified using nr and nc, respectively.
plot(result.IRT, type = "IRF", items = 1:6, nc = 2, nr = 3)
plot(result.IRT, type = "IRF", overlay = TRUE)
plot(result.IRT, type = "IIC", items = 1:6, nc = 2, nr = 3)
plot(result.IRT, type = "TRF")
plot(result.IRT, type = "TIC")
GRM: IRT for Polytomous Cases
The Graded Response Model (Samejima, 1969) can be considered an extension of IRT to polytomous response models. In this package, it can be implemented using the GRM function. However, the estimation accuracy is somewhat inferior to packages such as ltm, so it might be better to use different packages for more sophisticated analyses.
result.GRM <- GRM(J5S1000)
#> initial value 6497.015125
#> iter 10 value 6046.073326
#> iter 20 value 6013.787923
#> iter 30 value 6008.297277
#> final value 6008.297277
#> converged
result.GRM
#> Item Parameter
#> Descriminate Threshold1 Threshold2 Threshold3
#> V1 0.927 -1.54 0.0512 1.53
#> V2 1.234 -1.21 1.3936 NA
#> V3 0.917 -1.60 -0.0757 1.27
#> V4 1.479 -1.44 1.3166 NA
#> V5 0.947 -1.37 0.0286 1.54
#>
#> Item Fit Indices
#> NFI RFI IFI TLI CFI RMSEA AIC CAIC BIC
#> V1 0.248 0.265 0.271 0.288 0.272 0.090 324.819 99.017 99.063
#> V2 0.219 0.244 0.236 0.263 0.238 0.098 263.864 111.692 111.723
#> V3 0.253 0.269 0.276 0.293 0.277 0.089 321.824 96.021 96.067
#> V4 0.000 0.000 0.000 0.000 0.000 0.132 512.023 359.852 359.883
#> V5 0.255 0.272 0.278 0.295 0.279 0.090 327.598 101.795 101.841
#>
#> Model Fit Indices
#> value
#> NFI 0.165
#> RFI 0.186
#> IFI 0.179
#> TLI 0.201
#> CFI 0.180
#> RMSEA 0.099
#> AIC 1750.128
#> CAIC 768.377
#> BIC 768.577Similar output to IRT is also possible.
plot(result.GRM, type = "IRF")
plot(result.GRM, type = "IIF")
plot(result.GRM, type = "TIF")
LCA
Latent Class Analysis requires specifying the dataset and the number of classes.
LCA(J15S500, ncls = 5)
#>
#> Item Reference Profile
#> IRP1 IRP2 IRP3 IRP4 IRP5
#> Item01 0.5185 0.6996 0.76358 0.856 0.860
#> Item02 0.5529 0.6276 0.81161 0.888 0.855
#> Item03 0.7959 0.3205 0.93735 0.706 0.849
#> Item04 0.5069 0.5814 0.86940 0.873 1.000
#> Item05 0.6154 0.7523 0.94673 0.789 0.886
#> Item06 0.6840 0.7501 0.94822 1.000 0.907
#> Item07 0.4832 0.4395 0.83377 0.874 0.900
#> Item08 0.3767 0.3982 0.62563 0.912 0.590
#> Item09 0.3107 0.3980 0.26616 0.165 0.673
#> Item10 0.5290 0.5341 0.76134 0.677 0.781
#> Item11 0.1007 0.0497 0.00132 0.621 0.623
#> Item12 0.0355 0.1673 0.15911 0.296 0.673
#> Item13 0.2048 0.5490 0.89445 0.672 0.784
#> Item14 0.3508 0.7384 0.77159 0.904 1.000
#> Item15 0.3883 0.6077 0.82517 0.838 0.823
#>
#> Test Profile
#> Class 1 Class 2 Class 3 Class 4 Class 5
#> Test Reference Profile 6.453 7.613 10.415 11.072 12.205
#> Latent Class Ditribution 87.000 97.000 125.000 91.000 100.000
#> Class Membership Distribution 90.372 97.105 105.238 102.800 104.484
#>
#> Item Fit Indices
#> model_log_like bench_log_like null_log_like model_Chi_sq null_Chi_sq
#> Item01 -264.179 -240.190 -283.343 47.978 86.307
#> Item02 -256.363 -235.436 -278.949 41.853 87.025
#> Item03 -237.888 -260.906 -293.598 -46.037 65.383
#> Item04 -208.536 -192.072 -265.962 32.928 147.780
#> Item05 -226.447 -206.537 -247.403 39.819 81.732
#> Item06 -164.762 -153.940 -198.817 21.644 89.755
#> Item07 -249.377 -228.379 -298.345 41.997 139.933
#> Item08 -295.967 -293.225 -338.789 5.483 91.127
#> Item09 -294.250 -300.492 -327.842 -12.484 54.700
#> Item10 -306.985 -288.198 -319.850 37.574 63.303
#> Item11 -187.202 -224.085 -299.265 -73.767 150.360
#> Item12 -232.307 -214.797 -293.598 35.020 157.603
#> Item13 -267.647 -262.031 -328.396 11.232 132.730
#> Item14 -203.468 -204.953 -273.212 -2.969 136.519
#> Item15 -268.616 -254.764 -302.847 27.705 96.166
#> model_df null_df NFI RFI IFI TLI CFI RMSEA AIC CAIC
#> Item01 9 13 0.444 0.197 0.496 0.232 0.468 0.093 29.978 -7.972
#> Item02 9 13 0.519 0.305 0.579 0.359 0.556 0.086 23.853 -14.097
#> Item03 9 13 1.000 1.000 1.000 1.000 1.000 0.000 -64.037 -101.987
#> Item04 9 13 0.777 0.678 0.828 0.744 0.822 0.073 14.928 -23.022
#> Item05 9 13 0.513 0.296 0.576 0.352 0.552 0.083 21.819 -16.130
#> Item06 9 13 0.759 0.652 0.843 0.762 0.835 0.053 3.644 -34.305
#> Item07 9 13 0.700 0.566 0.748 0.625 0.740 0.086 23.997 -13.952
#> Item08 9 13 0.940 0.913 1.000 1.000 1.000 0.000 -12.517 -50.466
#> Item09 9 13 1.000 1.000 1.000 1.000 1.000 0.000 -30.484 -68.433
#> Item10 9 13 0.406 0.143 0.474 0.179 0.432 0.080 19.574 -18.375
#> Item11 9 13 1.000 1.000 1.000 1.000 1.000 0.000 -91.767 -129.716
#> Item12 9 13 0.778 0.679 0.825 0.740 0.820 0.076 17.020 -20.930
#> Item13 9 13 0.915 0.878 0.982 0.973 0.981 0.022 -6.768 -44.717
#> Item14 9 13 1.000 1.000 1.000 1.000 1.000 0.000 -20.969 -58.919
#> Item15 9 13 0.712 0.584 0.785 0.675 0.775 0.065 9.705 -28.244
#> BIC
#> Item01 -7.954
#> Item02 -14.079
#> Item03 -101.969
#> Item04 -23.004
#> Item05 -16.112
#> Item06 -34.287
#> Item07 -13.934
#> Item08 -50.448
#> Item09 -68.415
#> Item10 -18.357
#> Item11 -129.698
#> Item12 -20.912
#> Item13 -44.699
#> Item14 -58.901
#> Item15 -28.226
#>
#> Model Fit Indices
#> Number of Latent class: 5
#> Number of EM cycle: 73
#> value
#> model_log_like -3663.994
#> bench_log_like -3560.005
#> null_log_like -4350.217
#> model_Chi_sq 207.977
#> null_Chi_sq 1580.424
#> model_df 135.000
#> null_df 195.000
#> NFI 0.868
#> RFI 0.810
#> IFI 0.950
#> TLI 0.924
#> CFI 0.947
#> RMSEA 0.033
#> AIC -62.023
#> CAIC -631.265
#> BIC -630.995The returned object contains the Class Membership Matrix, which indicates which latent class each subject belongs to. The Estimate includes the one with the highest membership probability.
result.LCA <- LCA(J15S500, ncls = 5)
head(result.LCA$Students)
#> Membership 1 Membership 2 Membership 3 Membership 4 Membership 5
#> Student001 0.7839477684 0.171152798 0.004141844 4.075759e-02 3.744590e-12
#> Student002 0.0347378747 0.051502214 0.836022799 7.773694e-02 1.698776e-07
#> Student003 0.0146307878 0.105488644 0.801853496 3.343026e-02 4.459682e-02
#> Student004 0.0017251650 0.023436459 0.329648386 3.656488e-01 2.795412e-01
#> Student005 0.2133830569 0.784162066 0.001484616 2.492073e-08 9.702355e-04
#> Student006 0.0003846482 0.001141448 0.001288901 8.733869e-01 1.237981e-01
#> Estimate
#> Student001 1
#> Student002 3
#> Student003 3
#> Student004 4
#> Student005 2
#> Student006 4The plots offer options for IRP, CMP, TRP, and LCD. For more details on each, please refer to Shojima (2022).
plot(result.LCA, type = "IRP", items = 1:6, nc = 2, nr = 3)
plot(result.LCA, type = "CMP", students = 1:9, nc = 3, nr = 3)
plot(result.LCA, type = "TRP")
plot(result.LCA, type = "LCD")
LRA
Latent Rank Analysis requires specifying the dataset and the number of classes.
LRA(J15S500, nrank = 6)
#> estimating method is GTM
#> Item Reference Profile
#> IRP1 IRP2 IRP3 IRP4 IRP5 IRP6
#> Item01 0.5851 0.6319 0.708 0.787 0.853 0.898
#> Item02 0.5247 0.6290 0.755 0.845 0.883 0.875
#> Item03 0.6134 0.6095 0.708 0.773 0.801 0.839
#> Item04 0.4406 0.6073 0.794 0.882 0.939 0.976
#> Item05 0.6465 0.7452 0.821 0.837 0.862 0.905
#> Item06 0.6471 0.7748 0.911 0.967 0.963 0.915
#> Item07 0.4090 0.5177 0.720 0.840 0.890 0.900
#> Item08 0.3375 0.4292 0.602 0.713 0.735 0.698
#> Item09 0.3523 0.3199 0.298 0.282 0.377 0.542
#> Item10 0.4996 0.5793 0.686 0.729 0.717 0.753
#> Item11 0.0958 0.0793 0.136 0.286 0.472 0.617
#> Item12 0.0648 0.0982 0.156 0.239 0.421 0.636
#> Item13 0.2908 0.4842 0.715 0.773 0.750 0.778
#> Item14 0.4835 0.5949 0.729 0.849 0.933 0.977
#> Item15 0.3981 0.5745 0.756 0.827 0.835 0.834
#>
#> Item Reference Profile Indices
#> Alpha A Beta B Gamma C
#> Item01 3 0.0786 1 0.585 0.0 0.00000
#> Item02 2 0.1264 1 0.525 0.2 -0.00787
#> Item03 2 0.0987 2 0.610 0.2 -0.00391
#> Item04 2 0.1864 1 0.441 0.0 0.00000
#> Item05 1 0.0987 1 0.647 0.0 0.00000
#> Item06 2 0.1362 1 0.647 0.4 -0.05198
#> Item07 2 0.2028 2 0.518 0.0 0.00000
#> Item08 2 0.1731 2 0.429 0.2 -0.03676
#> Item09 5 0.1646 6 0.542 0.6 -0.07002
#> Item10 2 0.1069 1 0.500 0.2 -0.01244
#> Item11 4 0.1867 5 0.472 0.2 -0.01650
#> Item12 5 0.2146 5 0.421 0.0 0.00000
#> Item13 2 0.2310 2 0.484 0.2 -0.02341
#> Item14 2 0.1336 1 0.484 0.0 0.00000
#> Item15 2 0.1817 2 0.574 0.2 -0.00123
#>
#> Test Profile
#> Rank 1 Rank 2 Rank 3 Rank 4 Rank 5 Rank 6
#> Test Reference Profile 6.389 7.675 9.496 10.631 11.432 12.144
#> Latent Rank Ditribution 96.000 60.000 91.000 77.000 73.000 103.000
#> Rank Membership Distribution 83.755 78.691 81.853 84.918 84.238 86.545
#>
#> Item Fit Indices
#> model_log_like bench_log_like null_log_like model_Chi_sq null_Chi_sq
#> Item01 -264.495 -240.190 -283.343 48.611 86.307
#> Item02 -253.141 -235.436 -278.949 35.409 87.025
#> Item03 -282.785 -260.906 -293.598 43.758 65.383
#> Item04 -207.082 -192.072 -265.962 30.021 147.780
#> Item05 -234.902 -206.537 -247.403 56.730 81.732
#> Item06 -168.218 -153.940 -198.817 28.556 89.755
#> Item07 -250.864 -228.379 -298.345 44.970 139.933
#> Item08 -312.621 -293.225 -338.789 38.791 91.127
#> Item09 -317.600 -300.492 -327.842 34.216 54.700
#> Item10 -309.654 -288.198 -319.850 42.910 63.303
#> Item11 -242.821 -224.085 -299.265 37.472 150.360
#> Item12 -236.522 -214.797 -293.598 43.451 157.603
#> Item13 -287.782 -262.031 -328.396 51.502 132.730
#> Item14 -221.702 -204.953 -273.212 33.499 136.519
#> Item15 -267.793 -254.764 -302.847 26.059 96.166
#> model_df null_df NFI RFI IFI TLI CFI RMSEA AIC CAIC
#> Item01 9.233 13 0.437 0.207 0.489 0.244 0.463 0.092 30.146 -8.785
#> Item02 9.233 13 0.593 0.427 0.664 0.502 0.646 0.075 16.944 -21.987
#> Item03 9.233 13 0.331 0.058 0.385 0.072 0.341 0.087 25.293 -13.638
#> Item04 9.233 13 0.797 0.714 0.850 0.783 0.846 0.067 11.555 -27.375
#> Item05 9.233 13 0.306 0.023 0.345 0.027 0.309 0.102 38.264 -0.667
#> Item06 9.233 13 0.682 0.552 0.760 0.646 0.748 0.065 10.091 -28.840
#> Item07 9.233 13 0.679 0.548 0.727 0.604 0.718 0.088 26.504 -12.427
#> Item08 9.233 13 0.574 0.401 0.639 0.467 0.622 0.080 20.326 -18.605
#> Item09 9.233 13 0.374 0.119 0.451 0.156 0.401 0.074 15.751 -23.180
#> Item10 9.233 13 0.322 0.046 0.377 0.057 0.330 0.085 24.445 -14.486
#> Item11 9.233 13 0.751 0.649 0.800 0.711 0.794 0.078 19.006 -19.925
#> Item12 9.233 13 0.724 0.612 0.769 0.667 0.763 0.086 24.985 -13.946
#> Item13 9.233 13 0.612 0.454 0.658 0.503 0.647 0.096 33.037 -5.894
#> Item14 9.233 13 0.755 0.654 0.809 0.723 0.804 0.073 15.034 -23.897
#> Item15 9.233 13 0.729 0.618 0.806 0.715 0.798 0.060 7.593 -31.338
#> BIC
#> Item01 -8.767
#> Item02 -21.969
#> Item03 -13.620
#> Item04 -27.357
#> Item05 -0.648
#> Item06 -28.822
#> Item07 -12.408
#> Item08 -18.587
#> Item09 -23.162
#> Item10 -14.467
#> Item11 -19.906
#> Item12 -13.927
#> Item13 -5.875
#> Item14 -23.879
#> Item15 -31.319
#>
#> Model Fit Indices
#> Number of Latent rank: 6
#> Number of EM cycle: 17
#> value
#> model_log_like -3857.982
#> bench_log_like -3560.005
#> null_log_like -4350.217
#> model_Chi_sq 595.954
#> null_Chi_sq 1580.424
#> model_df 138.491
#> null_df 195.000
#> NFI 0.623
#> RFI 0.469
#> IFI 0.683
#> TLI 0.535
#> CFI 0.670
#> RMSEA 0.081
#> AIC 318.973
#> CAIC -264.989
#> BIC -264.712The estimated subject rank membership probabilities and plots are almost the same as those in LCA (Latent Class Analysis). Since a ranking is assumed for the latent classes, rank-up odds and rank-down odds are calculated.
result.LRA <- LRA(J15S500, nrank = 6)
head(result.LRA$Students)
#> Membership 1 Membership 2 Membership 3 Membership 4 Membership 5
#> Student001 0.2704649921 0.357479353 0.27632327 0.084988078 0.010069050
#> Student002 0.0276546965 0.157616072 0.47438958 0.279914853 0.053715813
#> Student003 0.0228189795 0.138860955 0.37884545 0.284817610 0.120794858
#> Student004 0.0020140858 0.015608542 0.09629429 0.216973334 0.362406292
#> Student005 0.5582996437 0.397431414 0.03841668 0.003365601 0.001443909
#> Student006 0.0003866603 0.003168853 0.04801344 0.248329964 0.428747502
#> Membership 6 Estimate Rank-Up Odds Rank-Down Odds
#> Student001 0.0006752546 2 0.7729769 0.7565891
#> Student002 0.0067089816 3 0.5900527 0.3322503
#> Student003 0.0538621490 3 0.7518042 0.3665372
#> Student004 0.3067034562 5 0.8462973 0.5987019
#> Student005 0.0010427491 1 0.7118604 NA
#> Student006 0.2713535842 5 0.6328983 0.5791986plot(result.LRA, type = "IRP", items = 1:6, nc = 2, nr = 3)
plot(result.LRA, type = "RMP", students = 1:9, nc = 3, nr = 3)
plot(result.LRA, type = "TRP")
plot(result.LRA, type = "LRD")
LRA for ordinal data
LRA can also be applied to ordinal scale data. The sample dataset J15S3810 contains responses to 15 items on a 4-point scale, which we’ll classify into 3 ranks. The mic option enforces monotonic increasing constraints.
result.LRAord <- LRA(J15S3810, nrank = 3, mic = TRUE)
#> Starting EM estimation for Saturation Model...
#> iter 1 logLik -0.715286 iter 2 logLik -0.658925 iter 3 logLik -0.650981 iter 4 logLik -0.648033 iter 5 logLik -0.646591 iter 6 logLik -0.645385 iter 7 logLik -0.644218 iter 8 logLik -0.64299 iter 9 logLik -0.641574 iter 10 logLik -0.639918 iter 11 logLik -0.638005 iter 12 logLik -0.635832 iter 13 logLik -0.63343 iter 14 logLik -0.63086 iter 15 logLik -0.628207 iter 16 logLik -0.625554 iter 17 logLik -0.622973 iter 18 logLik -0.620517 iter 19 logLik -0.618217 iter 20 logLik -0.616089 iter 21 logLik -0.614138 iter 22 logLik -0.612356 iter 23 logLik -0.610732 iter 24 logLik -0.609254 iter 25 logLik -0.607906 iter 26 logLik -0.606672 iter 27 logLik -0.605536 iter 28 logLik -0.604488 iter 29 logLik -0.603515 iter 30 logLik -0.602612 iter 31 logLik -0.601771 iter 32 logLik -0.600989 iter 33 logLik -0.600262 iter 34 logLik -0.599589 iter 35 logLik -0.598965 iter 36 logLik -0.598389 iter 37 logLik -0.597857 iter 38 logLik -0.597367 iter 39 logLik -0.596914 iter 40 logLik -0.596497 iter 41 logLik -0.596113 iter 42 logLik -0.595757 iter 43 logLik -0.595428 iter 44 logLik -0.595121 iter 45 logLik -0.594836 iter 46 logLik -0.594568 iter 47 logLik -0.594316 iter 48 logLik -0.594078 iter 49 logLik -0.59385 iter 50 logLik -0.593631 iter 51 logLik -0.593418 iter 52 logLik -0.593211 iter 53 logLik -0.59301 iter 54 logLik -0.592816 iter 55 logLik -0.59263 iter 56 logLik -0.592452 iter 57 logLik -0.592281 iter 58 logLik -0.592118 iter 59 logLik -0.591962 iter 60 logLik -0.591813 iter 61 logLik -0.591671 iter 62 logLik -0.591534 iter 63 logLik -0.591403 iter 64 logLik -0.591277 iter 65 logLik -0.591156 iter 66 logLik -0.59104 iter 67 logLik -0.590927 iter 68 logLik -0.590819 iter 69 logLik -0.590714 iter 70 logLik -0.590613 iter 71 logLik -0.590515 iter 72 logLik -0.59042 iter 73 logLik -0.590328 iter 74 logLik -0.590238 iter 75 logLik -0.590151 iter 76 logLik -0.590065 iter 77 logLik -0.589982 iter 78 logLik -0.589899 iter 79 logLik -0.589818 iter 80 logLik -0.589738 iter 81 logLik -0.589659 iter 82 logLik -0.589581 iter 83 logLik -0.589504 iter 84 logLik -0.589427 iter 85 logLik -0.589352 iter 86 logLik -0.589277 iter 87 logLik -0.589203 iter 88 logLik -0.58913 iter 89 logLik -0.589059 iter 90 logLik -0.588988 iter 91 logLik -0.588919 iter 92 logLik -0.588851 iter 93 logLik -0.588785 iter 94 logLik -0.588721 iter 95 logLik -0.588658 iter 96 logLik -0.588597 iter 97 logLik -0.588537 iter 98 logLik -0.58848
#> Starting EM estimation for Restricted Model...
#> iter 1 logLik -0.667764 iter 2 logLik -0.670426 iter 3 logLik -0.671721 iter 4 logLik -0.672335 iter 5 logLik -0.672663 iter 6 logLik -0.672853 iter 7 logLik -0.67297 iter 8 logLik -0.673045 iter 9 logLik -0.673092We can visualize the relationship between total scores from the ordinal scale and estimated ranks. ScoreFreq plots a frequency polygon of scores with rank thresholds, while ScoreRank shows the relationship between scores and rank membership probabilities as a heatmap.
plot(result.LRAord, type = "ScoreFreq")
plot(result.LRAord, type = "ScoreRank")
The relationship between items and ranks can be visualized in two complementary ways using ICBR and ICRP plots. These visualizations help understand how items function across different ranks:
- ICBR (Item Category Boundary Reference) shows the cumulative probability curves for each category threshold. For each item, these lines represent the probability of scoring at or above each category boundary across ranks.
- ICRP (Item Category Response Profile) displays the probability of selecting each response category across ranks. These lines show how response patterns change as rank increases.
plot(result.LRAord, type = "ICBR", items = 1:4, nc = 2, nr = 2)
plot(result.LRAord, type = "ICRP", items = 1:4, nc = 2, nr = 2)
Similar to binary data output, we can examine individual examinee characteristics through rank membership probability plots. This visualization shows the probability distribution of rank membership for each examinee, allowing us to understand the certainty of rank classifications. For the first 15 examinees in the dataset:
plot(result.LRAord, type = "RMP", students = 1:9, nc = 3, nr = 3)
Note: The layout parameters nc = 3 and nr = 5 control the arrangement of plots in a 3-column by 5-row grid, making it easier to compare multiple items or examinees simultaneously.
LRA for rated data
If you have data where respondents select the correct answer from multiple choices, like in a multiple-choice test (nominal scale level), you can analyze it using LRA.
result.LRArated <- LRA(J35S5000, nrank = 10, mic = TRUE)
#> Starting EM estimation for Saturation Model...
#> iter 1 logLik -0.394564 iter 2 logLik -0.394427 iter 3 logLik -0.394516 iter 4 logLik -0.394547
#> Starting EM estimation for Restricted Model...
#> iter 1 logLik -1.19393 iter 2 logLik -1.19566 iter 3 logLik -1.1957You can visualize the relationship between scores and ranks, just like with ordinal scale data.
plot(result.LRArated, type = "ScoreFreq")
plot(result.LRArated, type = "ScoreRank")
You can also visualize the relationship between latent ranks and items, or the probability of subjects belonging to certain ranks.
plot(result.LRArated, type = "ICRP", items = 1:4, nc = 2, nr = 2)
plot(result.LRAord, type = "RMP", students = 1:9, nc = 3, nr = 3)
Biclustering/Ranklustering
Biclustering and Ranklustering algorithms are almost the same,
differing only in whether they include a filtering matrix or not. The
difference is specified using the method option in the
Biclustering() function. For more details, please refer to
the help documentation.
## Biclustering
Biclustering(J35S515, nfld = 5, ncls = 6, method = "B")
#> Biclustering is chosen.
#> iter 1 logLik -7966.66 iter 2 logLik -7442.38 iter 3 logLik -7266.35 iter 4
#> logLik -7151.01 iter 5 logLik -7023.94 iter 6 logLik -6984.82 iter 7 logLik
#> -6950.27 iter 8 logLik -6939.34 iter 9 logLik -6930.89 iter 10 logLik -6923.5
#> iter 11 logLik -6914.56 iter 12 logLik -6908.89 iter 13 logLik -6906.84 iter 14
#> logLik -6905.39 iter 15 logLik -6904.24 iter 16 logLik -6903.28 iter 17 logLik
#> -6902.41 iter 18 logLik -6901.58 iter 19 logLik -6900.74 iter 20 logLik
#> -6899.86 iter 21 logLik -6898.9 iter 22 logLik -6897.84 iter 23 logLik -6896.66
#> iter 24 logLik -6895.35 iter 25 logLik -6893.92 iter 26 logLik -6892.4 iter 27
#> logLik -6890.85 iter 28 logLik -6889.32 iter 29 logLik -6887.9 iter 30 logLik
#> -6886.66 iter 31 logLik -6885.67 iter 32 logLik -6884.98 iter 33 logLik
#> -6884.58
#> Bicluster Matrix Profile
#> Class1 Class2 Class3 Class4 Class5 Class6
#> Field1 0.6236 0.8636 0.8718 0.898 0.952 1.000
#> Field2 0.0627 0.3332 0.4255 0.919 0.990 1.000
#> Field3 0.2008 0.5431 0.2281 0.475 0.706 1.000
#> Field4 0.0495 0.2455 0.0782 0.233 0.648 0.983
#> Field5 0.0225 0.0545 0.0284 0.043 0.160 0.983
#>
#> Field Reference Profile Indices
#> Alpha A Beta B Gamma C
#> Field1 1 0.240 1 0.624 0.0 0.0000
#> Field2 3 0.493 3 0.426 0.0 0.0000
#> Field3 1 0.342 4 0.475 0.2 -0.3149
#> Field4 4 0.415 5 0.648 0.2 -0.1673
#> Field5 5 0.823 5 0.160 0.2 -0.0261
#>
#> Class 1 Class 2 Class 3 Class 4 Class 5 Class 6
#> Test Reference Profile 4.431 11.894 8.598 16.002 23.326 34.713
#> Latent Class Ditribution 157.000 64.000 82.000 106.000 89.000 17.000
#> Class Membership Distribution 146.105 73.232 85.753 106.414 86.529 16.968
#>
#> Field Membership Profile
#> CRR LFE Field1 Field2 Field3 Field4 Field5
#> Item01 0.850 1.000 1.000 0.000 0.000 0.000 0.000
#> Item31 0.812 1.000 1.000 0.000 0.000 0.000 0.000
#> Item32 0.808 1.000 1.000 0.000 0.000 0.000 0.000
#> Item21 0.616 2.000 0.000 1.000 0.000 0.000 0.000
#> Item23 0.600 2.000 0.000 1.000 0.000 0.000 0.000
#> Item22 0.586 2.000 0.000 1.000 0.000 0.000 0.000
#> Item24 0.567 2.000 0.000 1.000 0.000 0.000 0.000
#> Item25 0.491 2.000 0.000 1.000 0.000 0.000 0.000
#> Item11 0.476 2.000 0.000 1.000 0.000 0.000 0.000
#> Item26 0.452 2.000 0.000 1.000 0.000 0.000 0.000
#> Item27 0.414 2.000 0.000 1.000 0.000 0.000 0.000
#> Item07 0.573 3.000 0.000 0.000 1.000 0.000 0.000
#> Item03 0.458 3.000 0.000 0.000 1.000 0.000 0.000
#> Item33 0.437 3.000 0.000 0.000 1.000 0.000 0.000
#> Item02 0.392 3.000 0.000 0.000 1.000 0.000 0.000
#> Item09 0.390 3.000 0.000 0.000 1.000 0.000 0.000
#> Item10 0.353 3.000 0.000 0.000 1.000 0.000 0.000
#> Item08 0.350 3.000 0.000 0.000 1.000 0.000 0.000
#> Item12 0.340 4.000 0.000 0.000 0.000 1.000 0.000
#> Item04 0.303 4.000 0.000 0.000 0.000 1.000 0.000
#> Item17 0.276 4.000 0.000 0.000 0.000 1.000 0.000
#> Item05 0.250 4.000 0.000 0.000 0.000 1.000 0.000
#> Item13 0.237 4.000 0.000 0.000 0.000 1.000 0.000
#> Item34 0.229 4.000 0.000 0.000 0.000 1.000 0.000
#> Item29 0.227 4.000 0.000 0.000 0.000 1.000 0.000
#> Item28 0.221 4.000 0.000 0.000 0.000 1.000 0.000
#> Item06 0.216 4.000 0.000 0.000 0.000 1.000 0.000
#> Item16 0.216 4.000 0.000 0.000 0.000 1.000 0.000
#> Item35 0.155 5.000 0.000 0.000 0.000 0.000 1.000
#> Item14 0.126 5.000 0.000 0.000 0.000 0.000 1.000
#> Item15 0.087 5.000 0.000 0.000 0.000 0.000 1.000
#> Item30 0.085 5.000 0.000 0.000 0.000 0.000 1.000
#> Item20 0.054 5.000 0.000 0.000 0.000 0.000 1.000
#> Item19 0.052 5.000 0.000 0.000 0.000 0.000 1.000
#> Item18 0.049 5.000 0.000 0.000 0.000 0.000 1.000
#> Latent Field Distribution
#> Field 1 Field 2 Field 3 Field 4 Field 5
#> N of Items 3 8 7 10 7
#>
#> Model Fit Indices
#> Number of Latent Class : 6
#> Number of Latent Field: 5
#> Number of EM cycle: 33
#> value
#> model_log_like -6884.582
#> bench_log_like -5891.314
#> null_log_like -9862.114
#> model_Chi_sq 1986.535
#> null_Chi_sq 7941.601
#> model_df 1160.000
#> null_df 1155.000
#> NFI 0.750
#> RFI 0.751
#> IFI 0.878
#> TLI 0.879
#> CFI 0.878
#> RMSEA 0.037
#> AIC -333.465
#> CAIC -5258.949
#> BIC -5256.699## Ranklustering
result.Ranklustering <- Biclustering(J35S515, nfld = 5, ncls = 6, method = "R")
#> Ranklustering is chosen.
#> iter 1 logLik -8097.56 iter 2 logLik -7669.21 iter 3 logLik -7586.72 iter 4 logLik -7568.24 iter 5 logLik -7561.02 iter 6 logLik -7557.34 iter 7 logLik -7557.36 Strongly ordinal alignment condition was satisfied.
plot(result.Ranklustering, type = "Array")
plot(result.Ranklustering, type = "FRP", nc = 2, nr = 3)
plot(result.Ranklustering, type = "RRV")
plot(result.Ranklustering, type = "RMP", students = 1:9, nc = 3, nr = 3)
plot(result.Ranklustering, type = "LRD")
To find the optimal number of classes and the optimal number of fields, the Infinite Relational Model is available.
result.IRM <- IRM(J35S515, gamma_c = 1, gamma_f = 1, verbose = TRUE)
#> iter 1 Exact match count of field elements. 0 nfld 15 ncls 30 iter 2 Exact match count of field elements. 0 nfld 12 ncls 27 iter 3 Exact match count of field elements. 1 nfld 12 ncls 24 iter 4 Exact match count of field elements. 2 nfld 12 ncls 23 iter 5 Exact match count of field elements. 3 nfld 12 ncls 23 iter 6 Exact match count of field elements. 0 nfld 12 ncls 23 iter 7 Exact match count of field elements. 1 nfld 12 ncls 23 iter 8 Exact match count of field elements. 2 nfld 12 ncls 23 iter 9 Exact match count of field elements. 3 nfld 12 ncls 21 iter 10 Exact match count of field elements. 4 nfld 12 ncls 21 iter 11 Exact match count of field elements. 5 nfld 12 ncls 21 The minimum class member count is under the setting value.
#> bic -99592.5 nclass 21
#> The minimum class member count is under the setting value.
#> bic -99980.4 nclass 20
#> The minimum class member count is under the setting value.
#> bic -99959.7 nclass 19
#> The minimum class member count is under the setting value.
#> bic -99988.3 nclass 18
#> The minimum class member count is under the setting value.
#> bic -100001 nclass 17
plot(result.IRM, type = "Array")
plot(result.IRM, type = "FRP", nc = 3)
plot(result.IRM, type = "TRP")
Additionally, supplementary notes on the derivation of the Infinite Relational Model with Chinese restaurant process is here.
Bayesian Network Model
The Bayesian network model is a model that represents the conditional probabilities between items in a network format based on the pass rates of the items. By providing a Directed Acyclic Graph (DAG) between items externally, it calculates the conditional probabilities based on the specified graph. The igraph package is used for the analysis and representation of the network.
There are three ways to specify the graph. You can either pass a matrix-type DAG to the argument adj_matrix, pass a DAG described in a CSV file to the argument adj_file, or pass a graph-type object g used in the igraph package to the argument g.
The methods to create the matrix-type adj_matrix and the graph object g are as follows:
library(igraph)
DAG <-
matrix(
c(
"Item01", "Item02",
"Item02", "Item03",
"Item02", "Item04",
"Item03", "Item05",
"Item04", "Item05"
),
ncol = 2, byrow = T
)
## graph object
g <- igraph::graph_from_data_frame(DAG)
g
#> IGRAPH 97d24cb DN-- 5 5 --
#> + attr: name (v/c)
#> + edges from 97d24cb (vertex names):
#> [1] Item01->Item02 Item02->Item03 Item02->Item04 Item03->Item05 Item04->Item05
## Adjacency matrix
adj_mat <- as.matrix(igraph::get.adjacency(g))
print(adj_mat)
#> Item01 Item02 Item03 Item04 Item05
#> Item01 0 1 0 0 0
#> Item02 0 0 1 1 0
#> Item03 0 0 0 0 1
#> Item04 0 0 0 0 1
#> Item05 0 0 0 0 0A CSV file with the same information as the graph above in the following format. The first line contains column names (headers) and will not be read as data.
#> From,To
#> Item01,Item02
#> Item02,Item03
#> Item02,Item04
#> Item03,Item05
#> Item04,Item05While only one specification is sufficient, if multiple specifications are provided, they will be prioritized in the order of file, matrix, and graph object.
An example of executing BNM by providing a graph structure (DAG) is as follows:
result.BNM <- BNM(J5S10, adj_matrix = adj_mat)
result.BNM
#> Adjacency Matrix
#> Item01 Item02 Item03 Item04 Item05
#> Item01 0 1 0 0 0
#> Item02 0 0 1 1 0
#> Item03 0 0 0 0 1
#> Item04 0 0 0 0 1
#> Item05 0 0 0 0 0
#> [1] "Your graph is an acyclic graph."
#> [1] "Your graph is connected DAG."
#>
#> Parameter Learning
#> PIRP 1 PIRP 2 PIRP 3 PIRP 4
#> Item01 0.600
#> Item02 0.250 0.5
#> Item03 0.833 1.0
#> Item04 0.167 0.5
#> Item05 0.000 NaN 0.333 0.667
#>
#> Conditional Correct Response Rate
#> Child Item N of Parents Parent Items PIRP Conditional CRR
#> 1 Item01 0 No Parents No Pattern 0.6000000
#> 2 Item02 1 Item01 0 0.2500000
#> 3 Item02 1 Item01 1 0.5000000
#> 4 Item03 1 Item02 0 0.8333333
#> 5 Item03 1 Item02 1 1.0000000
#> 6 Item04 1 Item02 0 0.1666667
#> 7 Item04 1 Item02 1 0.5000000
#> 8 Item05 2 Item03, Item04 00 0.0000000
#> 9 Item05 2 Item03, Item04 01 NaN(0/0)
#> 10 Item05 2 Item03, Item04 10 0.3333333
#> 11 Item05 2 Item03, Item04 11 0.6666667
#>
#> Model Fit Indices
#> value
#> model_log_like -27.046
#> bench_log_like -8.935
#> null_log_like -28.882
#> model_Chi_sq 36.222
#> null_Chi_sq 39.894
#> model_df 20.000
#> null_df 25.000
#> NFI 0.092
#> RFI 0.000
#> IFI 0.185
#> TLI 0.000
#> CFI 0.000
#> RMSEA 0.300
#> AIC -3.778
#> CAIC -11.736
#> BIC -9.829Structure Learning for Bayesian network with GA
The function searches for a DAG suitable for the data using a genetic algorithm. A best DAG is not necessarily identified. Instead of exploring all combinations of nodes and edges, only the space topologically sorted by the pass rate, namely the upper triangular matrix of the adjacency matrix, is explored. For interpretability, the number of parent nodes should be limited. A null model is not proposed. Utilize the content of the items and the experience of the questioner to aid in interpreting the results. For more details, please refer to Section 8.5 of the text(Shojima,2022).
Please note that the GA may take a considerable amount of time, depending on the number of items and the size of the population.
StrLearningGA_BNM(J5S10,
population = 20, Rs = 0.5, Rm = 0.002, maxParents = 2,
maxGeneration = 100, crossover = 2, elitism = 2
)
#> Adjacency Matrix
#> Item01 Item02 Item03 Item04 Item05
#> Item01 0 0 0 1 0
#> Item02 0 0 0 0 0
#> Item03 0 0 0 0 0
#> Item04 0 0 0 0 0
#> Item05 0 0 0 0 0
#> [1] "Your graph is an acyclic graph."
#> [1] "Your graph is connected DAG."
#>
#> Parameter Learning
#> PIRP 1 PIRP 2
#> Item01 0.6
#> Item02 0.4
#> Item03 0.9
#> Item04 0.0 0.5
#> Item05 0.4
#>
#> Conditional Correct Response Rate
#> Child Item N of Parents Parent Items PIRP Conditional CRR
#> 1 Item01 0 No Parents No Pattern 0.6000000
#> 2 Item02 0 No Parents No Pattern 0.4000000
#> 3 Item03 0 No Parents No Pattern 0.9000000
#> 4 Item04 1 Item01 0 0.0000000
#> 5 Item04 1 Item01 1 0.5000000
#> 6 Item05 0 No Parents No Pattern 0.4000000
#>
#> Model Fit Indices
#> value
#> model_log_like -27.600
#> bench_log_like -8.935
#> null_log_like -28.882
#> model_Chi_sq 37.330
#> null_Chi_sq 39.894
#> model_df 24.000
#> null_df 25.000
#> NFI 0.064
#> RFI 0.025
#> IFI 0.161
#> TLI 0.068
#> CFI 0.105
#> RMSEA 0.248
#> AIC -10.670
#> CAIC -20.220
#> BIC -17.932The method of Population-Based incremental learning proposed by Fukuda (2014) can also be used for learning. This method has several variations for estimating the optimal adjacency matrix at the end, which can be specified as options. See help or text Section 8.5.2.
StrLearningPBIL_BNM(J5S10,
population = 20, Rs = 0.5, Rm = 0.005, maxParents = 2,
alpha = 0.05, estimate = 4
)
#> Adjacency Matrix
#> Item01 Item02 Item03 Item04 Item05
#> Item01 0 0 0 1 0
#> Item02 0 0 0 0 0
#> Item03 1 0 0 0 0
#> Item04 0 0 0 0 0
#> Item05 0 0 0 0 0
#> [1] "Your graph is an acyclic graph."
#> [1] "Your graph is connected DAG."
#>
#> Parameter Learning
#> PIRP 1 PIRP 2
#> Item01 0.0 0.667
#> Item02 0.4
#> Item03 0.9
#> Item04 0.0 0.500
#> Item05 0.4
#>
#> Conditional Correct Response Rate
#> Child Item N of Parents Parent Items PIRP Conditional CRR
#> 1 Item01 1 Item03 0 0.0000000
#> 2 Item01 1 Item03 1 0.6666667
#> 3 Item02 0 No Parents No Pattern 0.4000000
#> 4 Item03 0 No Parents No Pattern 0.9000000
#> 5 Item04 1 Item01 0 0.0000000
#> 6 Item04 1 Item01 1 0.5000000
#> 7 Item05 0 No Parents No Pattern 0.4000000
#>
#> Model Fit Indices
#> value
#> model_log_like -26.599
#> bench_log_like -8.935
#> null_log_like -28.882
#> model_Chi_sq 35.327
#> null_Chi_sq 39.894
#> model_df 23.000
#> null_df 25.000
#> NFI 0.114
#> RFI 0.037
#> IFI 0.270
#> TLI 0.100
#> CFI 0.172
#> RMSEA 0.244
#> AIC -10.673
#> CAIC -19.825
#> BIC -17.633Local Dependence Latent Rank Analysis
LD-LRA is an analysis that combines LRA and BNM, and it is used to analyze the network structure among items in the latent rank. In this function, structural learning is not performed, so you need to provide item graphs for each rank as separate files.
For each class, it is necessary to specify a graph, and there are three ways to do so. You can either pass a matrix-type DAG for each class or a list of graph-type objects used in the igraph package to the arguments adj_list or g_list, respectively, or you can provide a DAG described in a CSV file. The way to specify it in a CSV file is as follows.
DAG_dat <- matrix(c(
"From", "To", "Rank",
"Item01", "Item02", 1,
"Item04", "Item05", 1,
"Item01", "Item02", 2,
"Item02", "Item03", 2,
"Item04", "Item05", 2,
"Item08", "Item09", 2,
"Item08", "Item10", 2,
"Item09", "Item10", 2,
"Item08", "Item11", 2,
"Item01", "Item02", 3,
"Item02", "Item03", 3,
"Item04", "Item05", 3,
"Item08", "Item09", 3,
"Item08", "Item10", 3,
"Item09", "Item10", 3,
"Item08", "Item11", 3,
"Item02", "Item03", 4,
"Item04", "Item06", 4,
"Item04", "Item07", 4,
"Item05", "Item06", 4,
"Item05", "Item07", 4,
"Item08", "Item10", 4,
"Item08", "Item11", 4,
"Item09", "Item11", 4,
"Item02", "Item03", 5,
"Item04", "Item06", 5,
"Item04", "Item07", 5,
"Item05", "Item06", 5,
"Item05", "Item07", 5,
"Item09", "Item11", 5,
"Item10", "Item11", 5,
"Item10", "Item12", 5
), ncol = 3, byrow = TRUE)
# save csv file
edgeFile <- tempfile(fileext = ".csv")
write.csv(DAG_dat, edgeFile, row.names = FALSE, quote = TRUE)Here, it is shown an example of specifying with matrix-type and graph objects using the aforementioned CSV file. While only one specification is sufficient, if multiple specifications are provided, they will be prioritized in the order of file, matrix, and graph object.
g_csv <- read.csv(edgeFile)
colnames(g_csv) <- c("From", "To", "Rank")
adj_list <- list()
g_list <- list()
for (i in 1:5) {
adj_R <- g_csv[g_csv$Rank == i, 1:2]
g_tmp <- igraph::graph_from_data_frame(adj_R)
adj_tmp <- igraph::get.adjacency(g_tmp)
g_list[[i]] <- g_tmp
adj_list[[i]] <- adj_tmp
}
## Example of graph list
g_list
#> [[1]]
#> IGRAPH edfe830 DN-- 4 2 --
#> + attr: name (v/c)
#> + edges from edfe830 (vertex names):
#> [1] Item01->Item02 Item04->Item05
#>
#> [[2]]
#> IGRAPH 1c18926 DN-- 9 7 --
#> + attr: name (v/c)
#> + edges from 1c18926 (vertex names):
#> [1] Item01->Item02 Item02->Item03 Item04->Item05 Item08->Item09 Item08->Item10
#> [6] Item09->Item10 Item08->Item11
#>
#> [[3]]
#> IGRAPH ab56615 DN-- 9 7 --
#> + attr: name (v/c)
#> + edges from ab56615 (vertex names):
#> [1] Item01->Item02 Item02->Item03 Item04->Item05 Item08->Item09 Item08->Item10
#> [6] Item09->Item10 Item08->Item11
#>
#> [[4]]
#> IGRAPH 232031e DN-- 10 8 --
#> + attr: name (v/c)
#> + edges from 232031e (vertex names):
#> [1] Item02->Item03 Item04->Item06 Item04->Item07 Item05->Item06 Item05->Item07
#> [6] Item08->Item10 Item08->Item11 Item09->Item11
#>
#> [[5]]
#> IGRAPH 40b9704 DN-- 10 8 --
#> + attr: name (v/c)
#> + edges from 40b9704 (vertex names):
#> [1] Item02->Item03 Item04->Item06 Item04->Item07 Item05->Item06 Item05->Item07
#> [6] Item09->Item11 Item10->Item11 Item10->Item12### Example of adjacency list
adj_list
#> [[1]]
#> 4 x 4 sparse Matrix of class "dgCMatrix"
#> Item01 Item04 Item02 Item05
#> Item01 . . 1 .
#> Item04 . . . 1
#> Item02 . . . .
#> Item05 . . . .
#>
#> [[2]]
#> 9 x 9 sparse Matrix of class "dgCMatrix"
#> Item01 Item02 Item04 Item08 Item09 Item03 Item05 Item10 Item11
#> Item01 . 1 . . . . . . .
#> Item02 . . . . . 1 . . .
#> Item04 . . . . . . 1 . .
#> Item08 . . . . 1 . . 1 1
#> Item09 . . . . . . . 1 .
#> Item03 . . . . . . . . .
#> Item05 . . . . . . . . .
#> Item10 . . . . . . . . .
#> Item11 . . . . . . . . .
#>
#> [[3]]
#> 9 x 9 sparse Matrix of class "dgCMatrix"
#> Item01 Item02 Item04 Item08 Item09 Item03 Item05 Item10 Item11
#> Item01 . 1 . . . . . . .
#> Item02 . . . . . 1 . . .
#> Item04 . . . . . . 1 . .
#> Item08 . . . . 1 . . 1 1
#> Item09 . . . . . . . 1 .
#> Item03 . . . . . . . . .
#> Item05 . . . . . . . . .
#> Item10 . . . . . . . . .
#> Item11 . . . . . . . . .
#>
#> [[4]]
#> 10 x 10 sparse Matrix of class "dgCMatrix"
#>
#> Item02 . . . . . 1 . . . .
#> Item04 . . . . . . 1 1 . .
#> Item05 . . . . . . 1 1 . .
#> Item08 . . . . . . . . 1 1
#> Item09 . . . . . . . . . 1
#> Item03 . . . . . . . . . .
#> Item06 . . . . . . . . . .
#> Item07 . . . . . . . . . .
#> Item10 . . . . . . . . . .
#> Item11 . . . . . . . . . .
#>
#> [[5]]
#> 10 x 10 sparse Matrix of class "dgCMatrix"
#>
#> Item02 . . . . . 1 . . . .
#> Item04 . . . . . . 1 1 . .
#> Item05 . . . . . . 1 1 . .
#> Item09 . . . . . . . . 1 .
#> Item10 . . . . . . . . 1 1
#> Item03 . . . . . . . . . .
#> Item06 . . . . . . . . . .
#> Item07 . . . . . . . . . .
#> Item11 . . . . . . . . . .
#> Item12 . . . . . . . . . .The example of running the LDLRA function using this CSV file would look like this.
result.LDLRA <- LDLRA(J12S5000,
ncls = 5,
adj_file = edgeFile
)
result.LDLRA
#> Adjacency Matrix
#> [[1]]
#> Item01 Item02 Item03 Item04 Item05 Item06 Item07 Item08 Item09 Item10
#> Item01 0 1 0 0 0 0 0 0 0 0
#> Item02 0 0 0 0 0 0 0 0 0 0
#> Item03 0 0 0 0 0 0 0 0 0 0
#> Item04 0 0 0 0 1 0 0 0 0 0
#> Item05 0 0 0 0 0 0 0 0 0 0
#> Item06 0 0 0 0 0 0 0 0 0 0
#> Item07 0 0 0 0 0 0 0 0 0 0
#> Item08 0 0 0 0 0 0 0 0 0 0
#> Item09 0 0 0 0 0 0 0 0 0 0
#> Item10 0 0 0 0 0 0 0 0 0 0
#> Item11 0 0 0 0 0 0 0 0 0 0
#> Item12 0 0 0 0 0 0 0 0 0 0
#> Item11 Item12
#> Item01 0 0
#> Item02 0 0
#> Item03 0 0
#> Item04 0 0
#> Item05 0 0
#> Item06 0 0
#> Item07 0 0
#> Item08 0 0
#> Item09 0 0
#> Item10 0 0
#> Item11 0 0
#> Item12 0 0
#>
#> [[2]]
#> Item01 Item02 Item03 Item04 Item05 Item06 Item07 Item08 Item09 Item10
#> Item01 0 1 0 0 0 0 0 0 0 0
#> Item02 0 0 1 0 0 0 0 0 0 0
#> Item03 0 0 0 0 0 0 0 0 0 0
#> Item04 0 0 0 0 1 0 0 0 0 0
#> Item05 0 0 0 0 0 0 0 0 0 0
#> Item06 0 0 0 0 0 0 0 0 0 0
#> Item07 0 0 0 0 0 0 0 0 0 0
#> Item08 0 0 0 0 0 0 0 0 1 1
#> Item09 0 0 0 0 0 0 0 0 0 1
#> Item10 0 0 0 0 0 0 0 0 0 0
#> Item11 0 0 0 0 0 0 0 0 0 0
#> Item12 0 0 0 0 0 0 0 0 0 0
#> Item11 Item12
#> Item01 0 0
#> Item02 0 0
#> Item03 0 0
#> Item04 0 0
#> Item05 0 0
#> Item06 0 0
#> Item07 0 0
#> Item08 1 0
#> Item09 0 0
#> Item10 0 0
#> Item11 0 0
#> Item12 0 0
#>
#> [[3]]
#> Item01 Item02 Item03 Item04 Item05 Item06 Item07 Item08 Item09 Item10
#> Item01 0 1 0 0 0 0 0 0 0 0
#> Item02 0 0 1 0 0 0 0 0 0 0
#> Item03 0 0 0 0 0 0 0 0 0 0
#> Item04 0 0 0 0 1 0 0 0 0 0
#> Item05 0 0 0 0 0 0 0 0 0 0
#> Item06 0 0 0 0 0 0 0 0 0 0
#> Item07 0 0 0 0 0 0 0 0 0 0
#> Item08 0 0 0 0 0 0 0 0 1 1
#> Item09 0 0 0 0 0 0 0 0 0 1
#> Item10 0 0 0 0 0 0 0 0 0 0
#> Item11 0 0 0 0 0 0 0 0 0 0
#> Item12 0 0 0 0 0 0 0 0 0 0
#> Item11 Item12
#> Item01 0 0
#> Item02 0 0
#> Item03 0 0
#> Item04 0 0
#> Item05 0 0
#> Item06 0 0
#> Item07 0 0
#> Item08 1 0
#> Item09 0 0
#> Item10 0 0
#> Item11 0 0
#> Item12 0 0
#>
#> [[4]]
#> Item01 Item02 Item03 Item04 Item05 Item06 Item07 Item08 Item09 Item10
#> Item01 0 0 0 0 0 0 0 0 0 0
#> Item02 0 0 1 0 0 0 0 0 0 0
#> Item03 0 0 0 0 0 0 0 0 0 0
#> Item04 0 0 0 0 0 1 1 0 0 0
#> Item05 0 0 0 0 0 1 1 0 0 0
#> Item06 0 0 0 0 0 0 0 0 0 0
#> Item07 0 0 0 0 0 0 0 0 0 0
#> Item08 0 0 0 0 0 0 0 0 0 1
#> Item09 0 0 0 0 0 0 0 0 0 0
#> Item10 0 0 0 0 0 0 0 0 0 0
#> Item11 0 0 0 0 0 0 0 0 0 0
#> Item12 0 0 0 0 0 0 0 0 0 0
#> Item11 Item12
#> Item01 0 0
#> Item02 0 0
#> Item03 0 0
#> Item04 0 0
#> Item05 0 0
#> Item06 0 0
#> Item07 0 0
#> Item08 1 0
#> Item09 1 0
#> Item10 0 0
#> Item11 0 0
#> Item12 0 0
#>
#> [[5]]
#> Item01 Item02 Item03 Item04 Item05 Item06 Item07 Item08 Item09 Item10
#> Item01 0 0 0 0 0 0 0 0 0 0
#> Item02 0 0 1 0 0 0 0 0 0 0
#> Item03 0 0 0 0 0 0 0 0 0 0
#> Item04 0 0 0 0 0 1 1 0 0 0
#> Item05 0 0 0 0 0 1 1 0 0 0
#> Item06 0 0 0 0 0 0 0 0 0 0
#> Item07 0 0 0 0 0 0 0 0 0 0
#> Item08 0 0 0 0 0 0 0 0 0 0
#> Item09 0 0 0 0 0 0 0 0 0 0
#> Item10 0 0 0 0 0 0 0 0 0 0
#> Item11 0 0 0 0 0 0 0 0 0 0
#> Item12 0 0 0 0 0 0 0 0 0 0
#> Item11 Item12
#> Item01 0 0
#> Item02 0 0
#> Item03 0 0
#> Item04 0 0
#> Item05 0 0
#> Item06 0 0
#> Item07 0 0
#> Item08 0 0
#> Item09 1 0
#> Item10 1 1
#> Item11 0 0
#> Item12 0 0
#>
#> Parameter Learning
#> Item Rank PIRP 1 PIRP 2 PIRP 3 PIRP 4
#> 1 Item01 1 0.456
#> 2 Item02 1 0.030 0.444
#> 3 Item03 1 0.083
#> 4 Item04 1 0.421
#> 5 Item05 1 0.101 0.240
#> 6 Item06 1 0.025
#> 7 Item07 1 0.016
#> 8 Item08 1 0.286
#> 9 Item09 1 0.326
#> 10 Item10 1 0.181
#> 11 Item11 1 0.106
#> 12 Item12 1 0.055
#> 13 Item01 2 0.549
#> 14 Item02 2 0.035 0.568
#> 15 Item03 2 0.020 0.459
#> 16 Item04 2 0.495
#> 17 Item05 2 0.148 0.351
#> 18 Item06 2 0.066
#> 19 Item07 2 0.045
#> 20 Item08 2 0.407
#> 21 Item09 2 0.264 0.734
#> 22 Item10 2 0.081 0.133 0.159 0.745
#> 23 Item11 2 0.041 0.445
#> 24 Item12 2 0.086
#> 25 Item01 3 0.683
#> 26 Item02 3 0.040 0.728
#> 27 Item03 3 0.032 0.617
#> 28 Item04 3 0.612
#> 29 Item05 3 0.227 0.556
#> 30 Item06 3 0.205
#> 31 Item07 3 0.156
#> 32 Item08 3 0.581
#> 33 Item09 3 0.330 0.845
#> 34 Item10 3 0.092 0.160 0.211 0.843
#> 35 Item11 3 0.056 0.636
#> 36 Item12 3 0.152
#> 37 Item01 4 0.836
#> 38 Item02 4 0.720
#> 39 Item03 4 0.058 0.713
#> 40 Item04 4 0.740
#> 41 Item05 4 0.635
#> 42 Item06 4 0.008 0.105 0.023 0.684
#> 43 Item07 4 0.010 0.031 0.039 0.542
#> 44 Item08 4 0.760
#> 45 Item09 4 0.805
#> 46 Item10 4 0.150 0.844
#> 47 Item11 4 0.064 0.124 0.105 0.825
#> 48 Item12 4 0.227
#> 49 Item01 5 0.931
#> 50 Item02 5 0.869
#> 51 Item03 5 0.099 0.789
#> 52 Item04 5 0.846
#> 53 Item05 5 0.811
#> 54 Item06 5 0.015 0.125 0.040 0.788
#> 55 Item07 5 0.016 0.034 0.064 0.650
#> 56 Item08 5 0.880
#> 57 Item09 5 0.912
#> 58 Item10 5 0.825
#> 59 Item11 5 0.082 0.190 0.216 0.915
#> 60 Item12 5 0.153 0.341
#>
#> Conditional Correct Response Rate
#> Child Item Rank N of Parents Parent Items PIRP Conditional CRR
#> 1 Item01 1 0 No Parents No Pattern 0.45558
#> 2 Item02 1 1 Item01 0 0.03025
#> 3 Item02 1 1 Item01 1 0.44394
#> 4 Item03 1 0 No Parents No Pattern 0.08278
#> 5 Item04 1 0 No Parents No Pattern 0.42148
#> 6 Item05 1 1 Item04 0 0.10127
#> 7 Item05 1 1 Item04 1 0.24025
#> 8 Item06 1 0 No Parents No Pattern 0.02499
#> 9 Item07 1 0 No Parents No Pattern 0.01574
#> 10 Item08 1 0 No Parents No Pattern 0.28642
#> 11 Item09 1 0 No Parents No Pattern 0.32630
#> 12 Item10 1 0 No Parents No Pattern 0.18092
#> 13 Item11 1 0 No Parents No Pattern 0.10575
#> 14 Item12 1 0 No Parents No Pattern 0.05523
#> 15 Item01 2 0 No Parents No Pattern 0.54940
#> 16 Item02 2 1 Item01 0 0.03471
#> 17 Item02 2 1 Item01 1 0.56821
#> 18 Item03 2 1 Item02 0 0.02016
#> 19 Item03 2 1 Item02 1 0.45853
#> 20 Item04 2 0 No Parents No Pattern 0.49508
#> 21 Item05 2 1 Item04 0 0.14771
#> 22 Item05 2 1 Item04 1 0.35073
#> 23 Item06 2 0 No Parents No Pattern 0.06647
#> 24 Item07 2 0 No Parents No Pattern 0.04491
#> 25 Item08 2 0 No Parents No Pattern 0.40721
#> 26 Item09 2 1 Item08 0 0.26431
#> 27 Item09 2 1 Item08 1 0.73427
#> 28 Item10 2 2 Item08, Item09 00 0.08098
#> 29 Item10 2 2 Item08, Item09 01 0.13279
#> 30 Item10 2 2 Item08, Item09 10 0.15937
#> 31 Item10 2 2 Item08, Item09 11 0.74499
#> 32 Item11 2 1 Item08 0 0.04094
#> 33 Item11 2 1 Item08 1 0.44457
#> 34 Item12 2 0 No Parents No Pattern 0.08574
#> 35 Item01 3 0 No Parents No Pattern 0.68342
#> 36 Item02 3 1 Item01 0 0.04020
#> 37 Item02 3 1 Item01 1 0.72757
#> 38 Item03 3 1 Item02 0 0.03175
#> 39 Item03 3 1 Item02 1 0.61691
#> 40 Item04 3 0 No Parents No Pattern 0.61195
#> 41 Item05 3 1 Item04 0 0.22705
#> 42 Item05 3 1 Item04 1 0.55588
#> 43 Item06 3 0 No Parents No Pattern 0.20488
#> 44 Item07 3 0 No Parents No Pattern 0.15633
#> 45 Item08 3 0 No Parents No Pattern 0.58065
#> 46 Item09 3 1 Item08 0 0.32967
#> 47 Item09 3 1 Item08 1 0.84549
#> 48 Item10 3 2 Item08, Item09 00 0.09192
#> 49 Item10 3 2 Item08, Item09 01 0.15977
#> 50 Item10 3 2 Item08, Item09 10 0.21087
#> 51 Item10 3 2 Item08, Item09 11 0.84330
#> 52 Item11 3 1 Item08 0 0.05581
#> 53 Item11 3 1 Item08 1 0.63598
#> 54 Item12 3 0 No Parents No Pattern 0.15169
#> 55 Item01 4 0 No Parents No Pattern 0.83557
#> 56 Item02 4 0 No Parents No Pattern 0.71950
#> 57 Item03 4 1 Item02 0 0.05808
#> 58 Item03 4 1 Item02 1 0.71297
#> 59 Item04 4 0 No Parents No Pattern 0.73957
#> 60 Item05 4 0 No Parents No Pattern 0.63526
#> 61 Item06 4 2 Item04, Item05 00 0.00816
#> 62 Item06 4 2 Item04, Item05 01 0.10474
#> 63 Item06 4 2 Item04, Item05 10 0.02265
#> 64 Item06 4 2 Item04, Item05 11 0.68419
#> 65 Item07 4 2 Item04, Item05 00 0.00984
#> 66 Item07 4 2 Item04, Item05 01 0.03091
#> 67 Item07 4 2 Item04, Item05 10 0.03850
#> 68 Item07 4 2 Item04, Item05 11 0.54195
#> 69 Item08 4 0 No Parents No Pattern 0.75976
#> 70 Item09 4 0 No Parents No Pattern 0.80490
#> 71 Item10 4 1 Item08 0 0.14956
#> 72 Item10 4 1 Item08 1 0.84430
#> 73 Item11 4 2 Item08, Item09 00 0.06376
#> 74 Item11 4 2 Item08, Item09 01 0.12384
#> 75 Item11 4 2 Item08, Item09 10 0.10494
#> 76 Item11 4 2 Item08, Item09 11 0.82451
#> 77 Item12 4 0 No Parents No Pattern 0.22688
#> 78 Item01 5 0 No Parents No Pattern 0.93131
#> 79 Item02 5 0 No Parents No Pattern 0.86923
#> 80 Item03 5 1 Item02 0 0.09865
#> 81 Item03 5 1 Item02 1 0.78854
#> 82 Item04 5 0 No Parents No Pattern 0.84621
#> 83 Item05 5 0 No Parents No Pattern 0.81118
#> 84 Item06 5 2 Item04, Item05 00 0.01452
#> 85 Item06 5 2 Item04, Item05 01 0.12528
#> 86 Item06 5 2 Item04, Item05 10 0.04000
#> 87 Item06 5 2 Item04, Item05 11 0.78805
#> 88 Item07 5 2 Item04, Item05 00 0.01570
#> 89 Item07 5 2 Item04, Item05 01 0.03361
#> 90 Item07 5 2 Item04, Item05 10 0.06363
#> 91 Item07 5 2 Item04, Item05 11 0.65039
#> 92 Item08 5 0 No Parents No Pattern 0.88028
#> 93 Item09 5 0 No Parents No Pattern 0.91209
#> 94 Item10 5 0 No Parents No Pattern 0.82476
#> 95 Item11 5 2 Item09, Item10 00 0.08248
#> 96 Item11 5 2 Item09, Item10 01 0.18951
#> 97 Item11 5 2 Item09, Item10 10 0.21590
#> 98 Item11 5 2 Item09, Item10 11 0.91466
#> 99 Item12 5 1 Item10 0 0.15301
#> 100 Item12 5 1 Item10 1 0.34114
#>
#> Marginal Item Reference Profile
#> Rank 1 Rank 2 Rank 3 Rank 4 Rank 5
#> Item01 0.4556 0.5494 0.683 0.836 0.931
#> Item02 0.2099 0.2964 0.474 0.720 0.869
#> Item03 0.0828 0.1397 0.316 0.554 0.741
#> Item04 0.4215 0.4951 0.612 0.740 0.846
#> Item05 0.1555 0.2393 0.432 0.635 0.811
#> Item06 0.0250 0.0665 0.205 0.385 0.631
#> Item07 0.0157 0.0449 0.156 0.304 0.517
#> Item08 0.2864 0.4072 0.581 0.760 0.880
#> Item09 0.3263 0.4409 0.624 0.805 0.912
#> Item10 0.1809 0.2977 0.498 0.650 0.825
#> Item11 0.1057 0.1926 0.387 0.565 0.808
#> Item12 0.0552 0.0857 0.152 0.227 0.317
#>
#> IRP Indices
#> Alpha A Beta B Gamma C
#> Item01 3 0.15215133 1 0.4555806 0 0
#> Item02 3 0.24578705 3 0.4737140 0 0
#> Item03 3 0.23808314 4 0.5544465 0 0
#> Item04 3 0.12762155 2 0.4950757 0 0
#> Item05 3 0.20322441 3 0.4320364 0 0
#> Item06 4 0.24595102 4 0.3851075 0 0
#> Item07 4 0.21361675 5 0.5173874 0 0
#> Item08 3 0.17910918 3 0.5806476 0 0
#> Item09 2 0.18320368 2 0.4408936 0 0
#> Item10 2 0.20070396 3 0.4984108 0 0
#> Item11 4 0.24332189 4 0.5650492 0 0
#> Item12 4 0.09047482 5 0.3173548 0 0
#> [1] "Strongly ordinal alignment condition was satisfied."
#>
#> Test reference Profile and Latent Rank Distribution
#> Rank 1 Rank 2 Rank 3 Rank 4 Rank 5
#> Test Reference Profile 2.321 3.255 5.121 7.179 9.090
#> Latent Rank Ditribution 1829.000 593.000 759.000 569.000 1250.000
#> Rank Membership Distribution 1121.838 1087.855 873.796 835.528 1080.983
#> [1] "Weakly ordinal alignment condition was satisfied."
#>
#> Model Fit Indices
#> value
#> model_log_like -26657.783
#> bench_log_like -21318.465
#> null_log_like -37736.228
#> model_Chi_sq 10678.636
#> null_Chi_sq 32835.527
#> model_df 56.000
#> null_df 144.000
#> NFI 0.675
#> RFI 0.164
#> IFI 0.676
#> TLI 0.164
#> CFI 0.675
#> RMSEA 0.195
#> AIC 10566.636
#> CAIC 10201.662
#> BIC 10201.673Of course, it also supports various types of plots.
plot(result.LDLRA, type = "IRP", nc = 4, nr = 3)
plot(result.LDLRA, type = "TRP")
plot(result.LDLRA, type = "LRD")
Structure Learning for LDLRA with GA(PBIL)
You can learn item-interaction graphs for each rank using the PBIL algorithm. In addition to various options, the learning process requires a very long computation time. It’s also important to note that the result is merely one of the feasible solutions, and it’s not necessarily the optimal solution.
result.LDLRA.PBIL <- StrLearningPBIL_LDLRA(J35S515,
seed = 123,
ncls = 5,
method = "R",
elitism = 1,
successiveLimit = 15
)
result.LDLRA.PBIL
#> Adjacency Matrix
#> [[1]]
#> Item01 Item02 Item03 Item04 Item05 Item06 Item07 Item08 Item09 Item10
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#> Item33 0 0 0 0 0
#> Item34 0 0 0 0 0
#> Item35 0 0 0 0 0
#>
#> [[2]]
#> Item01 Item02 Item03 Item04 Item05 Item06 Item07 Item08 Item09 Item10
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#> Item21 Item22 Item23 Item24 Item25 Item26 Item27 Item28 Item29 Item30
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#>
#> [[3]]
#> Item01 Item02 Item03 Item04 Item05 Item06 Item07 Item08 Item09 Item10
#> Item01 0 0 0 0 1 0 0 0 0 0
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#> [[4]]
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#> Item24 0 0 0 0 0 0 0 0 0 0
#> Item25 0 0 0 0 0 0 0 0 0 0
#> Item26 0 0 0 0 0 0 0 0 0 0
#> Item27 0 0 0 0 0 0 0 0 0 0
#> Item28 0 0 0 0 0 0 0 0 0 0
#> Item29 0 0 0 0 0 0 0 0 0 0
#> Item30 0 0 0 0 0 0 0 0 0 0
#> Item31 0 0 0 0 0 0 0 0 0 0
#> Item32 0 0 0 0 0 0 0 0 0 0
#> Item33 0 0 0 0 0 0 0 0 0 0
#> Item34 0 0 0 0 0 0 0 0 0 0
#> Item35 0 0 0 0 0 0 0 0 0 0
#> Item11 Item12 Item13 Item14 Item15 Item16 Item17 Item18 Item19 Item20
#> Item01 0 0 0 0 0 0 0 0 0 0
#> Item02 0 0 0 0 0 0 0 0 0 0
#> Item03 0 0 0 0 0 0 0 0 0 0
#> Item04 0 0 0 0 0 0 0 0 0 0
#> Item05 0 0 0 0 0 1 0 0 0 0
#> Item06 0 0 0 0 0 0 0 0 0 0
#> Item07 0 0 0 0 0 0 0 0 0 0
#> Item08 0 0 0 0 0 0 1 0 0 0
#> Item09 0 0 0 0 0 0 0 0 0 0
#> Item10 0 0 0 0 0 0 0 0 0 1
#> Item11 0 0 0 0 0 0 0 0 0 0
#> Item12 0 0 0 0 0 0 0 0 0 0
#> Item13 0 0 0 0 0 0 0 1 0 0
#> Item14 0 0 0 0 0 0 0 0 0 0
#> Item15 0 0 0 0 0 0 0 0 0 0
#> Item16 0 0 0 0 0 0 0 0 0 0
#> Item17 0 0 0 0 0 0 0 0 0 0
#> Item18 0 0 0 0 0 0 0 0 0 0
#> Item19 0 0 0 0 0 0 0 0 0 0
#> Item20 0 0 0 0 0 0 0 0 0 0
#> Item21 0 0 1 0 0 0 0 0 0 0
#> Item22 0 0 0 0 0 0 0 0 0 0
#> Item23 0 0 0 0 0 0 0 0 0 0
#> Item24 0 0 0 0 0 0 0 0 0 0
#> Item25 0 0 0 0 0 0 0 0 0 0
#> Item26 0 0 0 0 0 0 0 0 0 0
#> Item27 0 0 0 0 0 0 0 0 0 0
#> Item28 0 0 0 0 0 0 0 0 0 0
#> Item29 0 0 0 0 0 0 0 0 0 0
#> Item30 0 0 0 0 0 0 0 0 0 0
#> Item31 0 0 0 0 0 0 0 0 0 0
#> Item32 0 0 0 0 0 0 0 0 0 0
#> Item33 0 0 0 0 0 0 0 0 0 0
#> Item34 0 0 0 0 0 0 0 0 0 0
#> Item35 0 0 0 0 0 0 0 0 0 0
#> Item21 Item22 Item23 Item24 Item25 Item26 Item27 Item28 Item29 Item30
#> Item01 0 0 0 0 0 0 0 0 0 0
#> Item02 0 0 0 0 0 0 0 0 0 0
#> Item03 0 0 0 0 0 0 0 0 0 0
#> Item04 0 0 0 0 0 0 0 0 0 0
#> Item05 0 0 0 0 0 0 0 0 0 0
#> Item06 0 0 0 0 0 0 0 0 0 0
#> Item07 0 0 0 0 0 0 0 0 0 0
#> Item08 0 0 0 0 0 0 0 0 0 0
#> Item09 0 0 0 0 0 0 0 0 0 0
#> Item10 0 0 0 0 0 0 0 0 0 0
#> Item11 0 0 0 0 0 0 0 0 0 0
#> Item12 0 0 0 0 0 0 0 0 0 0
#> Item13 0 0 0 0 0 0 0 0 0 0
#> Item14 0 0 0 0 0 0 0 0 0 0
#> Item15 0 0 0 0 0 0 0 0 0 0
#> Item16 0 0 0 0 0 0 0 0 0 0
#> Item17 0 0 0 0 0 0 0 0 0 0
#> Item18 0 0 0 0 0 0 0 0 0 0
#> Item19 0 0 0 0 0 0 0 0 0 0
#> Item20 0 0 0 0 0 0 0 0 0 0
#> Item21 0 0 0 0 0 0 0 0 0 0
#> Item22 0 0 0 0 0 0 0 0 0 0
#> Item23 0 1 0 0 0 0 0 0 0 0
#> Item24 0 0 0 0 1 0 0 0 0 0
#> Item25 0 0 0 0 0 0 0 0 0 0
#> Item26 0 0 0 0 0 0 0 0 0 0
#> Item27 0 0 0 0 0 0 0 0 0 0
#> Item28 0 0 0 0 0 0 0 0 0 0
#> Item29 0 0 0 0 0 0 0 0 0 0
#> Item30 0 0 0 0 0 0 0 0 0 0
#> Item31 0 0 1 0 0 0 0 0 0 0
#> Item32 1 0 0 1 0 0 0 0 0 0
#> Item33 0 0 0 0 0 0 0 0 0 0
#> Item34 0 0 0 0 0 0 0 0 0 0
#> Item35 0 0 0 0 0 0 0 0 0 0
#> Item31 Item32 Item33 Item34 Item35
#> Item01 1 1 0 0 0
#> Item02 0 0 0 0 0
#> Item03 0 0 0 0 0
#> Item04 0 0 0 0 0
#> Item05 0 0 0 0 0
#> Item06 0 0 0 0 0
#> Item07 0 0 0 0 0
#> Item08 0 0 0 0 0
#> Item09 0 0 0 0 0
#> Item10 0 0 0 0 0
#> Item11 0 0 0 0 1
#> Item12 0 0 0 0 0
#> Item13 0 0 0 0 0
#> Item14 0 0 0 0 0
#> Item15 0 0 0 0 0
#> Item16 0 0 0 0 1
#> Item17 0 0 0 1 0
#> Item18 0 0 0 0 0
#> Item19 0 0 0 0 0
#> Item20 0 0 0 0 0
#> Item21 0 0 0 0 0
#> Item22 0 0 0 0 0
#> Item23 0 0 0 0 0
#> Item24 0 0 0 0 0
#> Item25 0 0 0 0 0
#> Item26 0 0 0 0 0
#> Item27 0 0 0 0 0
#> Item28 0 0 0 0 0
#> Item29 0 0 0 0 0
#> Item30 0 0 0 0 0
#> Item31 0 0 0 0 0
#> Item32 0 0 0 0 0
#> Item33 0 0 0 0 0
#> Item34 0 0 0 0 0
#> Item35 0 0 0 0 0
#>
#> [[5]]
#> Item01 Item02 Item03 Item04 Item05 Item06 Item07 Item08 Item09 Item10
#> Item01 0 0 0 0 0 0 0 0 0 0
#> Item02 0 0 0 0 0 0 0 0 0 0
#> Item03 0 0 0 0 0 0 0 0 0 0
#> Item04 0 0 0 0 0 0 0 0 0 0
#> Item05 0 0 0 0 0 0 0 0 0 0
#> Item06 0 0 0 0 0 0 0 0 0 0
#> Item07 0 0 0 0 0 0 0 1 0 0
#> Item08 0 0 0 0 0 0 0 0 0 0
#> Item09 0 0 0 0 0 0 0 0 0 1
#> Item10 0 0 0 0 0 0 0 0 0 0
#> Item11 0 0 0 0 0 1 0 0 0 0
#> Item12 0 0 0 0 0 0 0 0 0 0
#> Item13 0 0 0 0 0 0 0 0 0 0
#> Item14 0 0 0 0 0 0 0 0 0 0
#> Item15 0 0 0 0 0 0 0 0 0 0
#> Item16 0 0 0 0 0 0 0 0 0 0
#> Item17 0 0 0 0 1 0 0 0 0 0
#> Item18 0 0 0 0 0 0 0 0 0 0
#> Item19 0 0 0 0 0 0 0 0 0 0
#> Item20 0 0 0 0 0 0 0 0 0 0
#> Item21 0 0 0 0 0 0 0 0 0 0
#> Item22 0 0 0 0 0 0 0 0 0 0
#> Item23 0 0 0 0 0 0 0 0 0 0
#> Item24 0 0 0 0 0 0 0 0 0 0
#> Item25 0 0 0 0 0 0 0 0 0 0
#> Item26 0 0 0 0 0 0 0 0 0 0
#> Item27 0 0 0 0 0 0 0 0 0 0
#> Item28 0 0 0 0 0 0 0 0 0 0
#> Item29 0 0 0 0 0 0 0 0 0 0
#> Item30 0 0 0 0 0 0 0 0 0 0
#> Item31 0 0 0 0 0 0 0 0 0 0
#> Item32 0 0 0 0 0 0 0 0 0 0
#> Item33 0 0 0 0 0 0 0 0 0 0
#> Item34 0 0 0 0 0 0 0 0 0 0
#> Item35 0 0 0 0 0 0 0 0 0 0
#> Item11 Item12 Item13 Item14 Item15 Item16 Item17 Item18 Item19 Item20
#> Item01 0 0 0 0 0 0 0 0 0 0
#> Item02 0 0 0 0 0 0 0 0 0 0
#> Item03 0 0 0 0 0 0 0 0 0 0
#> Item04 0 0 0 0 0 0 0 0 1 0
#> Item05 0 0 0 0 0 0 0 1 0 0
#> Item06 0 0 0 0 0 0 0 0 0 0
#> Item07 0 0 0 0 0 0 0 0 0 0
#> Item08 0 1 0 0 0 0 0 0 0 0
#> Item09 0 0 0 0 0 0 0 0 0 0
#> Item10 0 0 0 0 0 0 0 0 0 0
#> Item11 0 0 0 0 0 0 0 0 1 0
#> Item12 0 0 0 0 0 0 0 0 0 0
#> Item13 0 0 0 0 0 0 0 0 0 0
#> Item14 0 0 0 0 0 0 0 0 0 0
#> Item15 0 0 0 0 0 0 0 0 0 0
#> Item16 0 0 0 0 0 0 0 0 0 0
#> Item17 0 0 0 0 0 0 0 0 0 0
#> Item18 0 0 0 0 0 0 0 0 0 0
#> Item19 0 0 0 0 0 0 0 0 0 0
#> Item20 0 0 0 0 0 0 0 0 0 0
#> Item21 0 0 0 0 0 0 0 0 0 0
#> Item22 0 0 0 0 0 0 0 0 0 0
#> Item23 0 0 0 0 0 0 0 0 0 0
#> Item24 0 0 0 0 0 0 0 0 0 0
#> Item25 0 0 0 0 0 0 0 1 0 0
#> Item26 0 0 0 0 0 0 0 0 0 0
#> Item27 0 0 0 0 0 0 0 0 0 0
#> Item28 0 0 0 0 0 0 0 0 0 0
#> Item29 0 0 0 0 0 0 0 0 0 0
#> Item30 0 0 0 0 0 0 0 0 0 0
#> Item31 0 0 0 0 0 0 0 0 0 0
#> Item32 0 0 0 0 0 0 0 0 0 0
#> Item33 0 0 0 0 0 0 0 0 0 0
#> Item34 0 0 0 0 0 0 0 0 0 0
#> Item35 0 0 0 0 0 0 0 0 0 0
#> Item21 Item22 Item23 Item24 Item25 Item26 Item27 Item28 Item29 Item30
#> Item01 0 0 0 0 0 0 0 0 0 0
#> Item02 0 0 0 0 0 0 0 0 0 0
#> Item03 0 0 0 0 0 0 0 0 0 0
#> Item04 0 0 0 0 0 0 0 0 0 0
#> Item05 0 0 0 0 0 0 0 0 0 0
#> Item06 0 0 0 0 0 0 0 0 0 0
#> Item07 0 0 0 0 0 0 0 0 0 0
#> Item08 0 0 0 0 0 0 0 1 0 0
#> Item09 0 0 0 0 0 0 0 0 0 0
#> Item10 0 0 0 0 0 0 0 0 0 0
#> Item11 0 0 0 0 0 0 0 0 0 0
#> Item12 0 0 0 0 0 0 0 0 0 0
#> Item13 0 0 0 0 0 0 0 0 0 0
#> Item14 0 0 0 0 0 0 0 0 0 0
#> Item15 0 0 0 0 0 0 0 0 0 0
#> Item16 0 0 0 0 0 0 0 0 0 1
#> Item17 0 0 0 0 0 0 0 0 0 0
#> Item18 0 0 0 0 0 0 0 0 0 0
#> Item19 0 0 0 0 0 0 0 0 0 0
#> Item20 0 0 0 0 0 0 0 0 0 0
#> Item21 0 0 1 0 0 0 0 0 0 0
#> Item22 0 0 0 0 0 0 0 0 0 0
#> Item23 0 1 0 0 0 0 0 0 0 0
#> Item24 0 0 0 0 0 0 0 0 0 0
#> Item25 0 0 0 0 0 0 0 0 0 0
#> Item26 0 0 0 0 0 0 0 0 0 0
#> Item27 0 0 0 0 0 0 0 0 0 0
#> Item28 0 0 0 0 0 0 0 0 0 0
#> Item29 0 0 0 0 0 0 0 0 0 0
#> Item30 0 0 0 0 0 0 0 0 0 0
#> Item31 1 0 0 0 0 0 0 0 0 0
#> Item32 1 0 0 0 0 0 0 0 0 0
#> Item33 0 0 0 0 0 0 0 0 0 0
#> Item34 0 0 0 0 0 0 0 0 0 0
#> Item35 0 0 0 0 0 0 0 0 0 0
#> Item31 Item32 Item33 Item34 Item35
#> Item01 0 0 0 0 0
#> Item02 0 0 0 0 0
#> Item03 0 0 0 0 0
#> Item04 0 0 0 0 0
#> Item05 0 0 0 0 0
#> Item06 0 0 0 0 0
#> Item07 0 0 0 0 0
#> Item08 0 0 0 0 0
#> Item09 0 0 0 0 0
#> Item10 0 0 0 0 0
#> Item11 0 0 0 0 0
#> Item12 0 0 0 0 0
#> Item13 0 0 0 0 0
#> Item14 0 0 0 0 0
#> Item15 0 0 0 0 0
#> Item16 0 0 0 0 0
#> Item17 0 0 0 0 0
#> Item18 0 0 0 0 0
#> Item19 0 0 0 0 0
#> Item20 0 0 0 0 0
#> Item21 0 0 0 0 0
#> Item22 0 0 0 0 0
#> Item23 0 0 0 0 0
#> Item24 0 0 0 0 0
#> Item25 0 0 0 0 0
#> Item26 0 0 0 0 0
#> Item27 0 0 0 0 0
#> Item28 0 0 0 0 0
#> Item29 0 0 0 0 0
#> Item30 0 0 0 0 0
#> Item31 0 1 0 0 0
#> Item32 0 0 0 0 0
#> Item33 0 0 0 0 0
#> Item34 0 0 0 0 0
#> Item35 0 0 0 0 0
#>
#> Parameter Learning
#> Item Rank PIRP 1 PIRP 2 PIRP 3 PIRP 4
#> 1 Item01 1 0.710
#> 2 Item02 1 0.073 0.256
#> 3 Item03 1 0.236
#> 4 Item04 1 0.079
#> 5 Item05 1 0.061
#> 6 Item06 1 0.040
#> 7 Item07 1 0.398 0.429
#> 8 Item08 1 0.258
#> 9 Item09 1 0.227 0.246
#> 10 Item10 1 0.192
#> 11 Item11 1 0.133
#> 12 Item12 1 0.111
#> 13 Item13 1 0.088
#> 14 Item14 1 0.013
#> 15 Item15 1 0.014
#> 16 Item16 1 0.058
#> 17 Item17 1 0.125
#> 18 Item18 1 0.030
#> 19 Item19 1 0.035 0.079
#> 20 Item20 1 0.028
#> 21 Item21 1 0.174 0.298
#> 22 Item22 1 0.226
#> 23 Item23 1 0.301 0.304
#> 24 Item24 1 0.231
#> 25 Item25 1 0.133
#> 26 Item26 1 0.092
#> 27 Item27 1 0.106
#> 28 Item28 1 0.017 0.112
#> 29 Item29 1 0.061 0.069
#> 30 Item30 1 0.027
#> 31 Item31 1 0.645 0.706
#> 32 Item32 1 0.484 0.801 0.543 0.809
#> 33 Item33 1 0.312
#> 34 Item34 1 0.183 0.239
#> 35 Item35 1 0.098
#> 36 Item01 2 0.802
#> 37 Item02 2 0.231
#> 38 Item03 2 0.331
#> 39 Item04 2 0.157
#> 40 Item05 2 0.124
#> 41 Item06 2 0.094
#> 42 Item07 2 0.480
#> 43 Item08 2 0.276 0.285
#> 44 Item09 2 0.239 0.302 0.348 0.436
#> 45 Item10 2 0.258
#> 46 Item11 2 0.282
#> 47 Item12 2 0.173
#> 48 Item13 2 0.114
#> 49 Item14 2 0.030
#> 50 Item15 2 0.020
#> 51 Item16 2 0.081
#> 52 Item17 2 0.143 0.216
#> 53 Item18 2 0.026
#> 54 Item19 2 0.029
#> 55 Item20 2 0.050 0.036
#> 56 Item21 2 0.307 0.522
#> 57 Item22 2 0.317 0.586
#> 58 Item23 2 0.361 0.456
#> 59 Item24 2 0.386
#> 60 Item25 2 0.133 0.520
#> 61 Item26 2 0.167 0.242
#> 62 Item27 2 0.158 0.331
#> 63 Item28 2 0.046 0.149
#> 64 Item29 2 0.100
#> 65 Item30 2 0.040
#> 66 Item31 2 0.659 0.773
#> 67 Item32 2 0.497 0.782 0.564 0.839
#> 68 Item33 2 0.354
#> 69 Item34 2 0.196
#> 70 Item35 2 0.131 0.106
#> 71 Item01 3 0.877
#> 72 Item02 3 0.417
#> 73 Item03 3 0.434 0.523
#> 74 Item04 3 0.308
#> 75 Item05 3 0.097 0.284
#> 76 Item06 3 0.047 0.024 0.775 0.778
#> 77 Item07 3 0.577
#> 78 Item08 3 0.327 0.354
#> 79 Item09 3 0.301 0.311 0.440 0.503
#> 80 Item10 3 0.366
#> 81 Item11 3 0.501
#> 82 Item12 3 0.316
#> 83 Item13 3 0.201
#> 84 Item14 3 0.041 0.072 0.271 0.437
#> 85 Item15 3 0.024 0.133
#> 86 Item16 3 0.185
#> 87 Item17 3 0.247
#> 88 Item18 3 0.041
#> 89 Item19 3 0.045
#> 90 Item20 3 0.050
#> 91 Item21 3 0.366 0.390 0.502 0.781
#> 92 Item22 3 0.416 0.787
#> 93 Item23 3 0.436 0.669
#> 94 Item24 3 0.598
#> 95 Item25 3 0.354 0.548
#> 96 Item26 3 0.456
#> 97 Item27 3 0.098 0.761
#> 98 Item28 3 0.163 0.295
#> 99 Item29 3 0.171 0.301
#> 100 Item30 3 0.082
#> 101 Item31 3 0.662 0.833
#> 102 Item32 3 0.814
#> 103 Item33 3 0.366 0.480
#> 104 Item34 3 0.232
#> 105 Item35 3 0.155
#> 106 Item01 4 0.950
#> 107 Item02 4 0.595
#> 108 Item03 4 0.618
#> 109 Item04 4 0.157 0.082 0.677 0.695
#> 110 Item05 4 0.168 0.449
#> 111 Item06 4 0.329
#> 112 Item07 4 0.688
#> 113 Item08 4 0.408
#> 114 Item09 4 0.499
#> 115 Item10 4 0.470
#> 116 Item11 4 0.740
#> 117 Item12 4 0.496
#> 118 Item13 4 0.198 0.334
#> 119 Item14 4 0.194
#> 120 Item15 4 0.128
#> 121 Item16 4 0.248 0.417
#> 122 Item17 4 0.335 0.410
#> 123 Item18 4 0.019 0.182
#> 124 Item19 4 0.066
#> 125 Item20 4 0.038 0.128
#> 126 Item21 4 0.802 0.912
#> 127 Item22 4 0.636 0.912
#> 128 Item23 4 0.672 0.858
#> 129 Item24 4 0.757 0.849
#> 130 Item25 4 0.253 0.883
#> 131 Item26 4 0.751
#> 132 Item27 4 0.656
#> 133 Item28 4 0.363
#> 134 Item29 4 0.340
#> 135 Item30 4 0.131
#> 136 Item31 4 0.685 0.900
#> 137 Item32 4 0.640 0.860
#> 138 Item33 4 0.520
#> 139 Item34 4 0.207 0.344
#> 140 Item35 4 0.157 0.269 0.166 0.327
#> 141 Item01 5 0.967
#> 142 Item02 5 0.739
#> 143 Item03 5 0.732
#> 144 Item04 5 0.614
#> 145 Item05 5 0.457 0.591
#> 146 Item06 5 0.157 0.527
#> 147 Item07 5 0.759
#> 148 Item08 5 0.454 0.511
#> 149 Item09 5 0.627
#> 150 Item10 5 0.319 0.710
#> 151 Item11 5 0.885
#> 152 Item12 5 0.628 0.723
#> 153 Item13 5 0.502
#> 154 Item14 5 0.335
#> 155 Item15 5 0.244
#> 156 Item16 5 0.492
#> 157 Item17 5 0.533
#> 158 Item18 5 0.171 0.048 0.181 0.211
#> 159 Item19 5 0.104 0.031 0.229 0.206
#> 160 Item20 5 0.131
#> 161 Item21 5 0.631 0.799 0.901 0.971
#> 162 Item22 5 0.727 0.959
#> 163 Item23 5 0.622 0.941
#> 164 Item24 5 0.941
#> 165 Item25 5 0.915
#> 166 Item26 5 0.902
#> 167 Item27 5 0.824
#> 168 Item28 5 0.488 0.614
#> 169 Item29 5 0.496
#> 170 Item30 5 0.101 0.309
#> 171 Item31 5 0.930
#> 172 Item32 5 0.628 0.899
#> 173 Item33 5 0.616
#> 174 Item34 5 0.318
#> 175 Item35 5 0.260
#>
#> Conditional Correct Response Rate
#> Child Item Rank N of Parents Parent Items PIRP Conditional CRR
#> 1 Item01 1 0 No Parents No Pattern 0.7104
#> 2 Item02 1 1 Item21 0 0.0725
#> 3 Item02 1 1 Item21 1 0.2563
#> 4 Item03 1 0 No Parents No Pattern 0.2360
#> 5 Item04 1 0 No Parents No Pattern 0.0789
#> 6 Item05 1 0 No Parents No Pattern 0.0608
#> 7 Item06 1 0 No Parents No Pattern 0.0400
#> 8 Item07 1 1 Item01 0 0.3979
#> 9 Item07 1 1 Item01 1 0.4292
#> 10 Item08 1 0 No Parents No Pattern 0.2581
#> 11 Item09 1 1 Item03 0 0.2275
#> 12 Item09 1 1 Item03 1 0.2465
#> 13 Item10 1 0 No Parents No Pattern 0.1916
#> 14 Item11 1 0 No Parents No Pattern 0.1325
#> 15 Item12 1 0 No Parents No Pattern 0.1111
#> 16 Item13 1 0 No Parents No Pattern 0.0884
#> 17 Item14 1 0 No Parents No Pattern 0.0134
#> 18 Item15 1 0 No Parents No Pattern 0.0139
#> 19 Item16 1 0 No Parents No Pattern 0.0578
#> 20 Item17 1 0 No Parents No Pattern 0.1253
#> 21 Item18 1 0 No Parents No Pattern 0.0303
#> 22 Item19 1 1 Item13 0 0.0354
#> 23 Item19 1 1 Item13 1 0.0795
#> 24 Item20 1 0 No Parents No Pattern 0.0283
#> 25 Item21 1 1 Item31 0 0.1737
#> 26 Item21 1 1 Item31 1 0.2978
#> 27 Item22 1 0 No Parents No Pattern 0.2256
#> 28 Item23 1 1 Item01 0 0.3009
#> 29 Item23 1 1 Item01 1 0.3036
#> 30 Item24 1 0 No Parents No Pattern 0.2312
#> 31 Item25 1 0 No Parents No Pattern 0.1329
#> 32 Item26 1 0 No Parents No Pattern 0.0922
#> 33 Item27 1 0 No Parents No Pattern 0.1058
#> 34 Item28 1 1 Item05 0 0.0166
#> 35 Item28 1 1 Item05 1 0.1120
#> 36 Item29 1 1 Item07 0 0.0611
#> 37 Item29 1 1 Item07 1 0.0693
#> 38 Item30 1 0 No Parents No Pattern 0.0275
#> 39 Item31 1 1 Item01 0 0.6446
#> 40 Item31 1 1 Item01 1 0.7056
#> 41 Item32 1 2 Item01, Item31 00 0.4841
#> 42 Item32 1 2 Item01, Item31 01 0.8011
#> 43 Item32 1 2 Item01, Item31 10 0.5430
#> 44 Item32 1 2 Item01, Item31 11 0.8090
#> 45 Item33 1 0 No Parents No Pattern 0.3122
#> 46 Item34 1 1 Item13 0 0.1826
#> 47 Item34 1 1 Item13 1 0.2390
#> 48 Item35 1 0 No Parents No Pattern 0.0985
#> 49 Item01 2 0 No Parents No Pattern 0.8019
#> 50 Item02 2 0 No Parents No Pattern 0.2314
#> 51 Item03 2 0 No Parents No Pattern 0.3315
#> 52 Item04 2 0 No Parents No Pattern 0.1574
#> 53 Item05 2 0 No Parents No Pattern 0.1245
#> 54 Item06 2 0 No Parents No Pattern 0.0938
#> 55 Item07 2 0 No Parents No Pattern 0.4805
#> 56 Item08 2 1 Item32 0 0.2758
#> 57 Item08 2 1 Item32 1 0.2853
#> 58 Item09 2 2 Item11, Item26 00 0.2390
#> 59 Item09 2 2 Item11, Item26 01 0.3025
#> 60 Item09 2 2 Item11, Item26 10 0.3484
#> 61 Item09 2 2 Item11, Item26 11 0.4357
#> 62 Item10 2 0 No Parents No Pattern 0.2584
#> 63 Item11 2 0 No Parents No Pattern 0.2817
#> 64 Item12 2 0 No Parents No Pattern 0.1729
#> 65 Item13 2 0 No Parents No Pattern 0.1141
#> 66 Item14 2 0 No Parents No Pattern 0.0304
#> 67 Item15 2 0 No Parents No Pattern 0.0204
#> 68 Item16 2 0 No Parents No Pattern 0.0814
#> 69 Item17 2 1 Item26 0 0.1429
#> 70 Item17 2 1 Item26 1 0.2164
#> 71 Item18 2 0 No Parents No Pattern 0.0261
#> 72 Item19 2 0 No Parents No Pattern 0.0287
#> 73 Item20 2 1 Item31 0 0.0498
#> 74 Item20 2 1 Item31 1 0.0362
#> 75 Item21 2 1 Item31 0 0.3073
#> 76 Item21 2 1 Item31 1 0.5221
#> 77 Item22 2 1 Item23 0 0.3171
#> 78 Item22 2 1 Item23 1 0.5862
#> 79 Item23 2 1 Item32 0 0.3612
#> 80 Item23 2 1 Item32 1 0.4558
#> 81 Item24 2 0 No Parents No Pattern 0.3861
#> 82 Item25 2 1 Item24 0 0.1334
#> 83 Item25 2 1 Item24 1 0.5204
#> 84 Item26 2 1 Item31 0 0.1670
#> 85 Item26 2 1 Item31 1 0.2420
#> 86 Item27 2 1 Item11 0 0.1581
#> 87 Item27 2 1 Item11 1 0.3310
#> 88 Item28 2 1 Item12 0 0.0458
#> 89 Item28 2 1 Item12 1 0.1493
#> 90 Item29 2 0 No Parents No Pattern 0.1001
#> 91 Item30 2 0 No Parents No Pattern 0.0402
#> 92 Item31 2 1 Item01 0 0.6591
#> 93 Item31 2 1 Item01 1 0.7730
#> 94 Item32 2 2 Item01, Item31 00 0.4973
#> 95 Item32 2 2 Item01, Item31 01 0.7824
#> 96 Item32 2 2 Item01, Item31 10 0.5641
#> 97 Item32 2 2 Item01, Item31 11 0.8388
#> 98 Item33 2 0 No Parents No Pattern 0.3538
#> 99 Item34 2 0 No Parents No Pattern 0.1957
#> 100 Item35 2 1 Item09 0 0.1307
#> 101 Item35 2 1 Item09 1 0.1056
#> 102 Item01 3 0 No Parents No Pattern 0.8766
#> 103 Item02 3 0 No Parents No Pattern 0.4172
#> 104 Item03 3 1 Item25 0 0.4344
#> 105 Item03 3 1 Item25 1 0.5228
#> 106 Item04 3 0 No Parents No Pattern 0.3078
#> 107 Item05 3 1 Item01 0 0.0972
#> 108 Item05 3 1 Item01 1 0.2836
#> 109 Item06 3 2 Item05, Item23 00 0.0471
#> 110 Item06 3 2 Item05, Item23 01 0.0239
#> 111 Item06 3 2 Item05, Item23 10 0.7752
#> 112 Item06 3 2 Item05, Item23 11 0.7779
#> 113 Item07 3 0 No Parents No Pattern 0.5768
#> 114 Item08 3 1 Item23 0 0.3271
#> 115 Item08 3 1 Item23 1 0.3542
#> 116 Item09 3 2 Item26, Item27 00 0.3008
#> 117 Item09 3 2 Item26, Item27 01 0.3107
#> 118 Item09 3 2 Item26, Item27 10 0.4403
#> 119 Item09 3 2 Item26, Item27 11 0.5028
#> 120 Item10 3 0 No Parents No Pattern 0.3665
#> 121 Item11 3 0 No Parents No Pattern 0.5008
#> 122 Item12 3 0 No Parents No Pattern 0.3162
#> 123 Item13 3 0 No Parents No Pattern 0.2012
#> 124 Item14 3 2 Item13, Item24 00 0.0414
#> 125 Item14 3 2 Item13, Item24 01 0.0721
#> 126 Item14 3 2 Item13, Item24 10 0.2709
#> 127 Item14 3 2 Item13, Item24 11 0.4372
#> 128 Item15 3 1 Item11 0 0.0244
#> 129 Item15 3 1 Item11 1 0.1329
#> 130 Item16 3 0 No Parents No Pattern 0.1848
#> 131 Item17 3 0 No Parents No Pattern 0.2474
#> 132 Item18 3 0 No Parents No Pattern 0.0408
#> 133 Item19 3 0 No Parents No Pattern 0.0450
#> 134 Item20 3 0 No Parents No Pattern 0.0495
#> 135 Item21 3 2 Item01, Item31 00 0.3659
#> 136 Item21 3 2 Item01, Item31 01 0.3898
#> 137 Item21 3 2 Item01, Item31 10 0.5020
#> 138 Item21 3 2 Item01, Item31 11 0.7812
#> 139 Item22 3 1 Item23 0 0.4160
#> 140 Item22 3 1 Item23 1 0.7868
#> 141 Item23 3 1 Item31 0 0.4360
#> 142 Item23 3 1 Item31 1 0.6688
#> 143 Item24 3 0 No Parents No Pattern 0.5984
#> 144 Item25 3 1 Item31 0 0.3543
#> 145 Item25 3 1 Item31 1 0.5483
#> 146 Item26 3 0 No Parents No Pattern 0.4555
#> 147 Item27 3 1 Item26 0 0.0977
#> 148 Item27 3 1 Item26 1 0.7606
#> 149 Item28 3 1 Item34 0 0.1633
#> 150 Item28 3 1 Item34 1 0.2946
#> 151 Item29 3 1 Item04 0 0.1713
#> 152 Item29 3 1 Item04 1 0.3011
#> 153 Item30 3 0 No Parents No Pattern 0.0821
#> 154 Item31 3 1 Item01 0 0.6618
#> 155 Item31 3 1 Item01 1 0.8333
#> 156 Item32 3 0 No Parents No Pattern 0.8141
#> 157 Item33 3 1 Item25 0 0.3665
#> 158 Item33 3 1 Item25 1 0.4800
#> 159 Item34 3 0 No Parents No Pattern 0.2321
#> 160 Item35 3 0 No Parents No Pattern 0.1546
#> 161 Item01 4 0 No Parents No Pattern 0.9497
#> 162 Item02 4 0 No Parents No Pattern 0.5947
#> 163 Item03 4 0 No Parents No Pattern 0.6182
#> 164 Item04 4 2 Item03, Item23 00 0.1572
#> 165 Item04 4 2 Item03, Item23 01 0.0821
#> 166 Item04 4 2 Item03, Item23 10 0.6769
#> 167 Item04 4 2 Item03, Item23 11 0.6946
#> 168 Item05 4 1 Item11 0 0.1677
#> 169 Item05 4 1 Item11 1 0.4492
#> 170 Item06 4 0 No Parents No Pattern 0.3285
#> 171 Item07 4 0 No Parents No Pattern 0.6876
#> 172 Item08 4 0 No Parents No Pattern 0.4083
#> 173 Item09 4 0 No Parents No Pattern 0.4991
#> 174 Item10 4 0 No Parents No Pattern 0.4701
#> 175 Item11 4 0 No Parents No Pattern 0.7402
#> 176 Item12 4 0 No Parents No Pattern 0.4962
#> 177 Item13 4 1 Item21 0 0.1981
#> 178 Item13 4 1 Item21 1 0.3335
#> 179 Item14 4 0 No Parents No Pattern 0.1940
#> 180 Item15 4 0 No Parents No Pattern 0.1281
#> 181 Item16 4 1 Item05 0 0.2483
#> 182 Item16 4 1 Item05 1 0.4170
#> 183 Item17 4 1 Item08 0 0.3349
#> 184 Item17 4 1 Item08 1 0.4103
#> 185 Item18 4 1 Item13 0 0.0188
#> 186 Item18 4 1 Item13 1 0.1821
#> 187 Item19 4 0 No Parents No Pattern 0.0657
#> 188 Item20 4 1 Item10 0 0.0383
#> 189 Item20 4 1 Item10 1 0.1284
#> 190 Item21 4 1 Item32 0 0.8020
#> 191 Item21 4 1 Item32 1 0.9118
#> 192 Item22 4 1 Item23 0 0.6357
#> 193 Item22 4 1 Item23 1 0.9125
#> 194 Item23 4 1 Item31 0 0.6720
#> 195 Item23 4 1 Item31 1 0.8585
#> 196 Item24 4 1 Item32 0 0.7567
#> 197 Item24 4 1 Item32 1 0.8491
#> 198 Item25 4 1 Item24 0 0.2533
#> 199 Item25 4 1 Item24 1 0.8835
#> 200 Item26 4 0 No Parents No Pattern 0.7507
#> 201 Item27 4 0 No Parents No Pattern 0.6559
#> 202 Item28 4 0 No Parents No Pattern 0.3633
#> 203 Item29 4 0 No Parents No Pattern 0.3401
#> 204 Item30 4 0 No Parents No Pattern 0.1310
#> 205 Item31 4 1 Item01 0 0.6849
#> 206 Item31 4 1 Item01 1 0.9001
#> 207 Item32 4 1 Item01 0 0.6398
#> 208 Item32 4 1 Item01 1 0.8596
#> 209 Item33 4 0 No Parents No Pattern 0.5199
#> 210 Item34 4 1 Item17 0 0.2075
#> 211 Item34 4 1 Item17 1 0.3444
#> 212 Item35 4 2 Item11, Item16 00 0.1565
#> 213 Item35 4 2 Item11, Item16 01 0.2690
#> 214 Item35 4 2 Item11, Item16 10 0.1658
#> 215 Item35 4 2 Item11, Item16 11 0.3272
#> 216 Item01 5 0 No Parents No Pattern 0.9674
#> 217 Item02 5 0 No Parents No Pattern 0.7392
#> 218 Item03 5 0 No Parents No Pattern 0.7320
#> 219 Item04 5 0 No Parents No Pattern 0.6143
#> 220 Item05 5 1 Item17 0 0.4570
#> 221 Item05 5 1 Item17 1 0.5908
#> 222 Item06 5 1 Item11 0 0.1572
#> 223 Item06 5 1 Item11 1 0.5268
#> 224 Item07 5 0 No Parents No Pattern 0.7591
#> 225 Item08 5 1 Item07 0 0.4539
#> 226 Item08 5 1 Item07 1 0.5110
#> 227 Item09 5 0 No Parents No Pattern 0.6270
#> 228 Item10 5 1 Item09 0 0.3187
#> 229 Item10 5 1 Item09 1 0.7098
#> 230 Item11 5 0 No Parents No Pattern 0.8853
#> 231 Item12 5 1 Item08 0 0.6280
#> 232 Item12 5 1 Item08 1 0.7234
#> 233 Item13 5 0 No Parents No Pattern 0.5022
#> 234 Item14 5 0 No Parents No Pattern 0.3348
#> 235 Item15 5 0 No Parents No Pattern 0.2438
#> 236 Item16 5 0 No Parents No Pattern 0.4923
#> 237 Item17 5 0 No Parents No Pattern 0.5326
#> 238 Item18 5 2 Item05, Item25 00 0.1711
#> 239 Item18 5 2 Item05, Item25 01 0.0482
#> 240 Item18 5 2 Item05, Item25 10 0.1809
#> 241 Item18 5 2 Item05, Item25 11 0.2111
#> 242 Item19 5 2 Item04, Item11 00 0.1040
#> 243 Item19 5 2 Item04, Item11 01 0.0311
#> 244 Item19 5 2 Item04, Item11 10 0.2288
#> 245 Item19 5 2 Item04, Item11 11 0.2065
#> 246 Item20 5 0 No Parents No Pattern 0.1309
#> 247 Item21 5 2 Item31, Item32 00 0.6312
#> 248 Item21 5 2 Item31, Item32 01 0.7986
#> 249 Item21 5 2 Item31, Item32 10 0.9008
#> 250 Item21 5 2 Item31, Item32 11 0.9715
#> 251 Item22 5 1 Item23 0 0.7273
#> 252 Item22 5 1 Item23 1 0.9585
#> 253 Item23 5 1 Item21 0 0.6220
#> 254 Item23 5 1 Item21 1 0.9412
#> 255 Item24 5 0 No Parents No Pattern 0.9410
#> 256 Item25 5 0 No Parents No Pattern 0.9148
#> 257 Item26 5 0 No Parents No Pattern 0.9019
#> 258 Item27 5 0 No Parents No Pattern 0.8242
#> 259 Item28 5 1 Item08 0 0.4880
#> 260 Item28 5 1 Item08 1 0.6142
#> 261 Item29 5 0 No Parents No Pattern 0.4960
#> 262 Item30 5 1 Item16 0 0.1008
#> 263 Item30 5 1 Item16 1 0.3090
#> 264 Item31 5 0 No Parents No Pattern 0.9299
#> 265 Item32 5 1 Item31 0 0.6278
#> 266 Item32 5 1 Item31 1 0.8989
#> 267 Item33 5 0 No Parents No Pattern 0.6160
#> 268 Item34 5 0 No Parents No Pattern 0.3181
#> 269 Item35 5 0 No Parents No Pattern 0.2602
#>
#> Marginal Item Reference Profile
#> Rank 1 Rank 2 Rank 3 Rank 4 Rank 5
#> Item01 0.7104 0.8019 0.8766 0.9497 0.967
#> Item02 0.0957 0.2314 0.4172 0.5947 0.739
#> Item03 0.2360 0.3315 0.4774 0.6182 0.732
#> Item04 0.0789 0.1574 0.3078 0.4316 0.614
#> Item05 0.0608 0.1245 0.2751 0.3886 0.550
#> Item06 0.0400 0.0938 0.2827 0.3285 0.520
#> Item07 0.4183 0.4805 0.5768 0.6876 0.759
#> Item08 0.2581 0.2834 0.3445 0.4083 0.501
#> Item09 0.2308 0.2753 0.3750 0.4991 0.627
#> Item10 0.1916 0.2584 0.3665 0.4701 0.606
#> Item11 0.1325 0.2817 0.5008 0.7402 0.885
#> Item12 0.1111 0.1729 0.3162 0.4962 0.682
#> Item13 0.0884 0.1141 0.2012 0.3305 0.502
#> Item14 0.0134 0.0304 0.1007 0.1940 0.335
#> Item15 0.0139 0.0204 0.0888 0.1281 0.244
#> Item16 0.0578 0.0814 0.1848 0.2941 0.492
#> Item17 0.1253 0.1548 0.2474 0.3624 0.533
#> Item18 0.0303 0.0261 0.0408 0.0487 0.155
#> Item19 0.0384 0.0287 0.0450 0.0657 0.161
#> Item20 0.0283 0.0391 0.0495 0.0758 0.131
#> Item21 0.2576 0.4755 0.7174 0.8948 0.955
#> Item22 0.2256 0.4260 0.6539 0.8968 0.955
#> Item23 0.3027 0.4369 0.6292 0.8418 0.939
#> Item24 0.2312 0.3861 0.5984 0.8347 0.941
#> Item25 0.1329 0.2833 0.5152 0.8528 0.915
#> Item26 0.0922 0.2257 0.4555 0.7507 0.902
#> Item27 0.1058 0.1985 0.3572 0.6559 0.824
#> Item28 0.0180 0.0589 0.1966 0.3633 0.559
#> Item29 0.0643 0.1001 0.2182 0.3401 0.496
#> Item30 0.0275 0.0402 0.0821 0.1310 0.239
#> Item31 0.6844 0.7566 0.8254 0.8934 0.930
#> Item32 0.7139 0.7705 0.8141 0.8527 0.886
#> Item33 0.3122 0.3538 0.4217 0.5199 0.616
#> Item34 0.1866 0.1957 0.2321 0.2429 0.318
#> Item35 0.0985 0.1239 0.1546 0.1967 0.260
#>
#> IRP Indices
#> Alpha A Beta B Gamma C
#> Item01 1 0.09147897 1 0.7104169 0.00 0.000000000
#> Item02 2 0.18575838 3 0.4171962 0.00 0.000000000
#> Item03 2 0.14596474 3 0.4774193 0.00 0.000000000
#> Item04 4 0.18270703 4 0.4316332 0.00 0.000000000
#> Item05 4 0.16182210 5 0.5504280 0.00 0.000000000
#> Item06 4 0.19192621 5 0.5204344 0.00 0.000000000
#> Item07 3 0.11078674 2 0.4804693 0.00 0.000000000
#> Item08 4 0.09271153 5 0.5009651 0.00 0.000000000
#> Item09 4 0.12790164 4 0.4991280 0.00 0.000000000
#> Item10 4 0.13606029 4 0.4700913 0.00 0.000000000
#> Item11 3 0.23945895 3 0.5007893 0.00 0.000000000
#> Item12 4 0.18559983 4 0.4961763 0.00 0.000000000
#> Item13 4 0.17168271 5 0.5022082 0.00 0.000000000
#> Item14 4 0.14087675 5 0.3348446 0.00 0.000000000
#> Item15 4 0.11578699 5 0.2438481 0.00 0.000000000
#> Item16 4 0.19823273 5 0.4923369 0.00 0.000000000
#> Item17 4 0.17017469 5 0.5325965 0.00 0.000000000
#> Item18 4 0.10679453 5 0.1554454 0.25 -0.004248974
#> Item19 4 0.09483464 5 0.1605088 0.25 -0.009740845
#> Item20 4 0.05504400 5 0.1308940 0.00 0.000000000
#> Item21 2 0.24190430 2 0.4755097 0.00 0.000000000
#> Item22 3 0.24291837 2 0.4259656 0.00 0.000000000
#> Item23 3 0.21261028 2 0.4369185 0.00 0.000000000
#> Item24 3 0.23639488 3 0.5983545 0.00 0.000000000
#> Item25 3 0.33752441 3 0.5152274 0.00 0.000000000
#> Item26 3 0.29514977 3 0.4555486 0.00 0.000000000
#> Item27 3 0.29864648 3 0.3572087 0.00 0.000000000
#> Item28 4 0.19588834 5 0.5591385 0.00 0.000000000
#> Item29 4 0.15593083 5 0.4960204 0.00 0.000000000
#> Item30 4 0.10842632 5 0.2394654 0.00 0.000000000
#> Item31 1 0.07218040 1 0.6843920 0.00 0.000000000
#> Item32 1 0.05661584 1 0.7139340 0.00 0.000000000
#> Item33 3 0.09815971 4 0.5199007 0.00 0.000000000
#> Item34 4 0.07524326 5 0.3181130 0.00 0.000000000
#> Item35 4 0.06353034 5 0.2601808 0.00 0.000000000
#>
#> Test reference Profile and Latent Rank Distribution
#> Rank 1 Rank 2 Rank 3 Rank 4 Rank 5
#> Test Reference Profile 6.413 8.819 12.947 17.380 21.472
#> Latent Rank Ditribution 181.000 60.000 83.000 82.000 109.000
#> Rank Membership Distribution 165.388 78.163 81.015 80.658 109.777
#> [1] "Weakly ordinal alignment condition was satisfied."
#>
#> Model Fit Indices
#> value
#> model_log_like -7796.306
#> bench_log_like -5891.314
#> null_log_like -9862.114
#> model_Chi_sq 3809.985
#> null_Chi_sq 7941.601
#> model_df 921.000
#> null_df 1155.000
#> NFI 0.520
#> RFI 0.398
#> IFI 0.588
#> TLI 0.466
#> CFI 0.574
#> RMSEA 0.078
#> AIC 1967.985
#> CAIC -1942.680
#> BIC -1940.893Local Dependence Biclustering
Local Dependence Biclustering combines biclustering and Bayesian network models. The model requires three main components:
- Number of latent classes/ranks
- Field assignments for items
- Network structure between fields at each rank
Here’s an example implementation:
# Create field configuration vector (assign items to fields)
conf <- c(1, 6, 6, 8, 9, 9, 4, 7, 7, 7, 5, 8, 9, 10, 10, 9, 9, 10, 10, 10, 2, 2, 3, 3, 5, 5, 6, 9, 9, 10, 1, 1, 7, 9, 10)
# Create edge data for network structure between fields
edges_data <- data.frame(
"From Field (Parent) >>>" = c(
6, 4, 5, 1, 1, 4, # Class/Rank 2
3, 4, 6, 2, 4, 4, # Class/Rank 3
3, 6, 4, 1, # Class/Rank 4
7, 9, 6, 7 # Class/Rank 5
),
">>> To Field (Child)" = c(
8, 7, 8, 7, 2, 5, # Class/Rank 2
5, 8, 8, 4, 6, 7, # Class/Rank 3
5, 8, 5, 8, # Class/Rank 4
10, 10, 8, 9 # Class/Rank 5
),
"At Class/Rank (Locus)" = c(
2, 2, 2, 2, 2, 2, # Class/Rank 2
3, 3, 3, 3, 3, 3, # Class/Rank 3
4, 4, 4, 4, # Class/Rank 4
5, 5, 5, 5 # Class/Rank 5
)
)
# Save edge data to temporary file
edgeFile <- tempfile(fileext = ".csv")
write.csv(edges_data, file = edgeFile, row.names = FALSE)Additionally, as mentioned in the text (Shojima, 2022), it is often the case that seeking the network structure exploratively does not yield appropriate results, so it has not been implemented.
result.LDB <- LDB(U = J35S515, ncls = 5, conf = conf, adj_file = edgeFile)
result.LDB
#> Adjacency Matrix
#> [[1]]
#> Field01 Field02 Field03 Field04 Field05 Field06 Field07 Field08 Field09
#> Field01 0 0 0 0 0 0 0 0 0
#> Field02 0 0 0 0 0 0 0 0 0
#> Field03 0 0 0 0 0 0 0 0 0
#> Field04 0 0 0 0 0 0 0 0 0
#> Field05 0 0 0 0 0 0 0 0 0
#> Field06 0 0 0 0 0 0 0 0 0
#> Field07 0 0 0 0 0 0 0 0 0
#> Field08 0 0 0 0 0 0 0 0 0
#> Field09 0 0 0 0 0 0 0 0 0
#> Field10 0 0 0 0 0 0 0 0 0
#> Field10
#> Field01 0
#> Field02 0
#> Field03 0
#> Field04 0
#> Field05 0
#> Field06 0
#> Field07 0
#> Field08 0
#> Field09 0
#> Field10 0
#>
#> [[2]]
#> Field01 Field02 Field03 Field04 Field05 Field06 Field07 Field08 Field09
#> Field01 0 1 0 0 0 0 1 0 0
#> Field02 0 0 0 0 0 0 0 0 0
#> Field03 0 0 0 0 0 0 0 0 0
#> Field04 0 0 0 0 1 0 1 0 0
#> Field05 0 0 0 0 0 0 0 1 0
#> Field06 0 0 0 0 0 0 0 1 0
#> Field07 0 0 0 0 0 0 0 0 0
#> Field08 0 0 0 0 0 0 0 0 0
#> Field09 0 0 0 0 0 0 0 0 0
#> Field10 0 0 0 0 0 0 0 0 0
#> Field10
#> Field01 0
#> Field02 0
#> Field03 0
#> Field04 0
#> Field05 0
#> Field06 0
#> Field07 0
#> Field08 0
#> Field09 0
#> Field10 0
#>
#> [[3]]
#> Field01 Field02 Field03 Field04 Field05 Field06 Field07 Field08 Field09
#> Field01 0 0 0 0 0 0 0 0 0
#> Field02 0 0 0 1 0 0 0 0 0
#> Field03 0 0 0 0 1 0 0 0 0
#> Field04 0 0 0 0 0 1 1 1 0
#> Field05 0 0 0 0 0 0 0 0 0
#> Field06 0 0 0 0 0 0 0 1 0
#> Field07 0 0 0 0 0 0 0 0 0
#> Field08 0 0 0 0 0 0 0 0 0
#> Field09 0 0 0 0 0 0 0 0 0
#> Field10 0 0 0 0 0 0 0 0 0
#> Field10
#> Field01 0
#> Field02 0
#> Field03 0
#> Field04 0
#> Field05 0
#> Field06 0
#> Field07 0
#> Field08 0
#> Field09 0
#> Field10 0
#>
#> [[4]]
#> Field01 Field02 Field03 Field04 Field05 Field06 Field07 Field08 Field09
#> Field01 0 0 0 0 0 0 0 1 0
#> Field02 0 0 0 0 0 0 0 0 0
#> Field03 0 0 0 0 1 0 0 0 0
#> Field04 0 0 0 0 1 0 0 0 0
#> Field05 0 0 0 0 0 0 0 0 0
#> Field06 0 0 0 0 0 0 0 1 0
#> Field07 0 0 0 0 0 0 0 0 0
#> Field08 0 0 0 0 0 0 0 0 0
#> Field09 0 0 0 0 0 0 0 0 0
#> Field10 0 0 0 0 0 0 0 0 0
#> Field10
#> Field01 0
#> Field02 0
#> Field03 0
#> Field04 0
#> Field05 0
#> Field06 0
#> Field07 0
#> Field08 0
#> Field09 0
#> Field10 0
#>
#> [[5]]
#> Field01 Field02 Field03 Field04 Field05 Field06 Field07 Field08 Field09
#> Field01 0 0 0 0 0 0 0 0 0
#> Field02 0 0 0 0 0 0 0 0 0
#> Field03 0 0 0 0 0 0 0 0 0
#> Field04 0 0 0 0 0 0 0 0 0
#> Field05 0 0 0 0 0 0 0 0 0
#> Field06 0 0 0 0 0 0 0 1 0
#> Field07 0 0 0 0 0 0 0 0 1
#> Field08 0 0 0 0 0 0 0 0 0
#> Field09 0 0 0 0 0 0 0 0 0
#> Field10 0 0 0 0 0 0 0 0 0
#> Field10
#> Field01 0
#> Field02 0
#> Field03 0
#> Field04 0
#> Field05 0
#> Field06 0
#> Field07 1
#> Field08 0
#> Field09 1
#> Field10 0
#>
#> Parameter Learning
#> Rank 1
#> PIRP 0 PIRP 1 PIRP 2 PIRP 3 PIRP 4 PIRP 5 PIRP 6 PIRP 7 PIRP 8 PIRP 9
#> Field01 0.6538
#> Field02 0.0756
#> Field03 0.1835
#> Field04 0.3819
#> Field05 0.0500
#> Field06 0.0985
#> Field07 0.2176
#> Field08 0.0608
#> Field09 0.0563
#> Field10 0.0237
#> PIRP 10 PIRP 11 PIRP 12
#> Field01
#> Field02
#> Field03
#> Field04
#> Field05
#> Field06
#> Field07
#> Field08
#> Field09
#> Field10
#> Rank 2
#> PIRP 0 PIRP 1 PIRP 2 PIRP 3 PIRP 4 PIRP 5 PIRP 6 PIRP 7 PIRP 8 PIRP 9
#> Field01 0.8216
#> Field02 0.1463 0.3181 0.383 0.597
#> Field03 0.3320
#> Field04 0.4931
#> Field05 0.1596 0.2552
#> Field06 0.2541
#> Field07 0.1232 0.2926 0.217 0.306 0.376
#> Field08 0.0648 0.0887 0.236 0.443 0.196 0.285 0.624
#> Field09 0.1101
#> Field10 0.0359
#> PIRP 10 PIRP 11 PIRP 12
#> Field01
#> Field02
#> Field03
#> Field04
#> Field05
#> Field06
#> Field07
#> Field08
#> Field09
#> Field10
#> Rank 3
#> PIRP 0 PIRP 1 PIRP 2 PIRP 3 PIRP 4 PIRP 5 PIRP 6 PIRP 7 PIRP 8 PIRP 9
#> Field01 0.8923
#> Field02 0.8736
#> Field03 0.8030
#> Field04 0.4730 0.492 0.650
#> Field05 0.2732 0.319 0.714
#> Field06 0.4025 0.486
#> Field07 0.3162 0.408
#> Field08 0.1028 0.166 0.177 0.439 0.59
#> Field09 0.1799
#> Field10 0.0431
#> PIRP 10 PIRP 11 PIRP 12
#> Field01
#> Field02
#> Field03
#> Field04
#> Field05
#> Field06
#> Field07
#> Field08
#> Field09
#> Field10
#> Rank 4
#> PIRP 0 PIRP 1 PIRP 2 PIRP 3 PIRP 4 PIRP 5 PIRP 6 PIRP 7 PIRP 8
#> Field01 0.91975
#> Field02 0.97126
#> Field03 0.96955
#> Field04 0.70098
#> Field05 0.28691 0.476702 0.911 0.952
#> Field06 0.72620
#> Field07 0.48152
#> Field08 0.00353 0.000122 0.370 0.370 0.401 0.532 0.779
#> Field09 0.36220
#> Field10 0.08630
#> PIRP 9 PIRP 10 PIRP 11 PIRP 12
#> Field01
#> Field02
#> Field03
#> Field04
#> Field05
#> Field06
#> Field07
#> Field08
#> Field09
#> Field10
#> Rank 5
#> PIRP 0 PIRP 1 PIRP 2 PIRP 3 PIRP 4 PIRP 5 PIRP 6 PIRP 7 PIRP 8 PIRP 9
#> Field01 0.9627
#> Field02 0.9959
#> Field03 0.9947
#> Field04 0.8654
#> Field05 0.9939
#> Field06 0.9178
#> Field07 0.7334
#> Field08 0.5109 0.4442 0.5939 0.9174
#> Field09 0.4062 0.5193 0.6496 0.6786 0.851
#> Field10 0.0874 0.0278 0.0652 0.0429 0.110 0.117 0.118 0.163 0.217 0.275
#> PIRP 10 PIRP 11 PIRP 12
#> Field01
#> Field02
#> Field03
#> Field04
#> Field05
#> Field06
#> Field07
#> Field08
#> Field09
#> Field10 0.262 0.257 0.95
#>
#> Marginal Rankluster Reference Matrix
#> Rank 1 Rank 2 Rank 3 Rank 4 Rank 5
#> Field01 0.6538 0.8216 0.8923 0.9198 0.963
#> Field02 0.0756 0.5069 0.8736 0.9713 0.996
#> Field03 0.1835 0.3320 0.8030 0.9696 0.995
#> Field04 0.3819 0.4931 0.6271 0.7010 0.865
#> Field05 0.0500 0.2072 0.6182 0.9263 0.994
#> Field06 0.0985 0.2541 0.4550 0.7262 0.918
#> Field07 0.2176 0.3119 0.3738 0.4815 0.733
#> Field08 0.0608 0.1723 0.2718 0.5700 0.863
#> Field09 0.0563 0.1101 0.1799 0.3622 0.715
#> Field10 0.0237 0.0359 0.0431 0.0863 0.377
#>
#> IRP Indices
#> Alpha A Beta B Gamma C
#> Field01 1 0.1677977 1 0.6538429 0 0
#> Field02 1 0.4312713 2 0.5068824 0 0
#> Field03 2 0.4710088 2 0.3320336 0 0
#> Field04 4 0.1643891 2 0.4930958 0 0
#> Field05 2 0.4110466 3 0.6182062 0 0
#> Field06 3 0.2712108 3 0.4549879 0 0
#> Field07 4 0.2518684 4 0.4815211 0 0
#> Field08 3 0.2982121 4 0.5699954 0 0
#> Field09 4 0.3528379 4 0.3621986 0 0
#> Field10 4 0.2906998 5 0.3769977 0 0
#> Rank 1 Rank 2 Rank 3 Rank 4 Rank 5
#> Test Reference Profile 4.915 8.744 13.657 18.867 26.488
#> Latent Rank Ditribution 163.000 91.000 102.000 91.000 68.000
#> Rank Membership Dsitribution 148.275 103.002 105.606 86.100 72.017
#>
#> Latent Field Distribution
#> Field 1 Field 2 Field 3 Field 4 Field 5 Field 6 Field 7 Field 8
#> N of Items 3 2 2 1 3 3 4 2
#> Field 9 Field 10
#> N of Items 8 7
#>
#> Model Fit Indices
#> value
#> model_log_like -6804.899
#> bench_log_like -5891.314
#> null_log_like -9862.114
#> model_Chi_sq 1827.169
#> null_Chi_sq 7941.601
#> model_df 1088.000
#> null_df 1155.000
#> NFI 0.770
#> RFI 0.756
#> IFI 0.892
#> TLI 0.884
#> CFI 0.891
#> RMSEA 0.036
#> AIC -348.831
#> CAIC -4968.595
#> BIC -4966.485
#> Strongly ordinal alignment condition was satisfied.Of course, it also supports various types of plots.
# Show bicluster structure
plot(result.LDB, type = "Array")
# Test Response Profile
plot(result.LDB, type = "TRP")
# Latent Rank Distribution
plot(result.LDB, type = "LRD")
# Rank Membership Profiles for first 9 students
plot(result.LDB, type = "RMP", students = 1:9, nc = 3, nr = 3)
# Field Reference Profiles
plot(result.LDB, type = "FRP", nc = 3, nr = 2)
In this model, you can draw a Field PIRP Profile that visualizes the correct answer count for each rank and each field.
plot(result.LDB, type = "FieldPIRP")
Bicluster Network Model
Bicluster Network Model: BINET is a model that combines the Bayesian network model and Biclustering. BINET is very similar to LDB and LDR.
The most significant difference is that in LDB, the nodes represent the fields, whereas in BINET, they represent the class. BINET explores the local dependency structure among latent classes at each latent field, where each field is a locus.
To execute this analysis, in addition to the dataset, the same field correspondence file used during exploratory Biclustering is required, as well as an adjacency matrix between classes.
# Create field configuration vector for item assignment
conf <- c(1, 5, 5, 5, 9, 9, 6, 6, 6, 6, 2, 7, 7, 11, 11, 7, 7, 12, 12, 12, 2, 2, 3, 3, 4, 4, 4, 8, 8, 12, 1, 1, 6, 10, 10)
# Create edge data for network structure between classes
edges_data <- data.frame(
"From Class (Parent) >>>" = c(
1, 2, 3, 4, 5, 7, # Dependencies in various fields
2, 4, 6, 8, 10,
6, 6, 11, 8, 9, 12
),
">>> To Class (Child)" = c(
2, 4, 5, 5, 6, 11, # Target classes
3, 7, 9, 12, 12,
10, 8, 12, 12, 11, 13
),
"At Field (Locus)" = c(
1, 2, 2, 3, 4, 4, # Field locations
5, 5, 5, 5, 5,
7, 8, 8, 9, 9, 12
)
)
# Save edge data to temporary file
edgeFile <- tempfile(fileext = ".csv")
write.csv(edges_data, file = edgeFile, row.names = FALSE)The model requires three components:
- Field assignments for items (vector from configuration file)
- Network structure between classes for each field
- Number of classes and fields
# Fit Bicluster Network Model
result.BINET <- BINET(
U = J35S515,
ncls = 13, # Maximum class number from edges (13)
nfld = 12, # Maximum field number from conf (12)
conf = conf, # Field configuration vector
adj_file = edgeFile # Network structure file
)
# Display model results
print(result.BINET)
#> Total Graph
#> Class01 Class02 Class03 Class04 Class05 Class06 Class07 Class08 Class09
#> Class01 0 1 0 0 0 0 0 0 0
#> Class02 0 0 1 1 0 0 0 0 0
#> Class03 0 0 0 0 1 0 0 0 0
#> Class04 0 0 0 0 1 0 1 0 0
#> Class05 0 0 0 0 0 1 0 0 0
#> Class06 0 0 0 0 0 0 0 1 1
#> Class07 0 0 0 0 0 0 0 0 0
#> Class08 0 0 0 0 0 0 0 0 0
#> Class09 0 0 0 0 0 0 0 0 0
#> Class10 0 0 0 0 0 0 0 0 0
#> Class11 0 0 0 0 0 0 0 0 0
#> Class12 0 0 0 0 0 0 0 0 0
#> Class13 0 0 0 0 0 0 0 0 0
#> Class10 Class11 Class12 Class13
#> Class01 0 0 0 0
#> Class02 0 0 0 0
#> Class03 0 0 0 0
#> Class04 0 0 0 0
#> Class05 0 0 0 0
#> Class06 1 0 0 0
#> Class07 0 1 0 0
#> Class08 0 0 1 0
#> Class09 0 1 0 0
#> Class10 0 0 1 0
#> Class11 0 0 1 0
#> Class12 0 0 0 1
#> Class13 0 0 0 0
#> Estimation of Parameter set
#> Field 1
#> PSRP 1 PSRP 2 PSRP 3 PSRP 4
#> Class 1 0.000
#> Class 2 0.554 0.558 0.649
#> Class 3 0.740
#> Class 4 0.859
#> Class 5 0.875
#> Class 6 0.910
#> Class 7 0.868
#> Class 8 0.889
#> Class 9 0.961
#> Class 10 0.932
#> Class 11 0.898
#> Class 12 0.975
#> Class 13 1.000
#> Field 2
#> PSRP 1 PSRP 2 PSRP 3 PSRP 4
#> Class 1 0.0000
#> Class 2 0.0090
#> Class 3 0.0396
#> Class 4 0.6813 0.785 0.637
#> Class 5 0.4040 0.728 0.696
#> Class 6 0.6877
#> Class 7 0.8316
#> Class 8 0.8218
#> Class 9 1.0000
#> Class 10 0.9836
#> Class 11 1.0000
#> Class 12 1.0000
#> Class 13 1.0000
#> Field 3
#> PSRP 1 PSRP 2 PSRP 3 PSRP 4
#> Class 1 0.000
#> Class 2 0.177
#> Class 3 0.219
#> Class 4 0.206
#> Class 5 0.189 0.253
#> Class 6 1.000
#> Class 7 1.000
#> Class 8 1.000
#> Class 9 0.986
#> Class 10 1.000
#> Class 11 0.973
#> Class 12 1.000
#> Class 13 1.000
#> Field 4
#> PSRP 1 PSRP 2 PSRP 3 PSRP 4
#> Class 1 0.0000
#> Class 2 0.0127
#> Class 3 0.1228
#> Class 4 0.0468
#> Class 5 0.1131
#> Class 6 0.6131 0.436 0.179
#> Class 7 0.9775
#> Class 8 0.9539
#> Class 9 0.9751
#> Class 10 0.9660
#> Class 11 0.9411 0.925 0.757
#> Class 12 1.0000
#> Class 13 1.0000
#> Field 5
#> PSRP 1 PSRP 2 PSRP 3 PSRP 4
#> Class 1 0.0000
#> Class 2 0.0157
#> Class 3 0.0731 0.330 0.06789
#> Class 4 0.9626
#> Class 5 0.1028
#> Class 6 0.2199
#> Class 7 0.1446 0.265 0.00602
#> Class 8 0.9403
#> Class 9 0.2936 0.298 0.12080
#> Class 10 0.8255
#> Class 11 0.9123
#> Class 12 1.0000 1.000 1.00000
#> Class 13 1.0000
#> Field 6
#> PSRP 1 PSRP 2 PSRP 3 PSRP 4
#> Class 1 0.000
#> Class 2 0.236
#> Class 3 0.275
#> Class 4 0.449
#> Class 5 0.414
#> Class 6 0.302
#> Class 7 0.415
#> Class 8 0.469
#> Class 9 0.560
#> Class 10 0.564
#> Class 11 0.614
#> Class 12 0.764
#> Class 13 1.000
#> Field 7
#> PSRP 1 PSRP 2 PSRP 3 PSRP 4
#> Class 1 0.0000
#> Class 2 0.0731
#> Class 3 0.0810
#> Class 4 0.1924
#> Class 5 0.1596
#> Class 6 0.1316
#> Class 7 0.1263
#> Class 8 0.1792
#> Class 9 0.7542
#> Class 10 0.9818 0.883 0.933 0.975
#> Class 11 0.3047
#> Class 12 0.7862
#> Class 13 1.0000
#> Field 8
#> PSRP 1 PSRP 2 PSRP 3 PSRP 4
#> Class 1 0.00e+00
#> Class 2 9.83e-05
#> Class 3 3.70e-02
#> Class 4 3.91e-02
#> Class 5 4.21e-02
#> Class 6 6.88e-02
#> Class 7 4.56e-01
#> Class 8 1.65e-01 0.192
#> Class 9 6.15e-01
#> Class 10 3.88e-01
#> Class 11 3.16e-01
#> Class 12 1.00e+00 1.000
#> Class 13 1.00e+00
#> Field 9
#> PSRP 1 PSRP 2 PSRP 3 PSRP 4
#> Class 1 0.00e+00
#> Class 2 3.13e-16
#> Class 3 1.61e-02
#> Class 4 6.15e-01
#> Class 5 3.46e-02
#> Class 6 5.26e-02
#> Class 7 1.44e-11
#> Class 8 2.09e-01
#> Class 9 1.90e-17
#> Class 10 8.09e-01
#> Class 11 1.00e+00 1.000
#> Class 12 7.81e-01 0.703
#> Class 13 1.00e+00
#> Field 10
#> PSRP 1 PSRP 2 PSRP 3 PSRP 4
#> Class 1 0.0000
#> Class 2 0.0952
#> Class 3 0.1798
#> Class 4 0.1741
#> Class 5 0.1594
#> Class 6 0.1789
#> Class 7 0.1208
#> Class 8 0.1550
#> Class 9 0.2228
#> Class 10 0.2602
#> Class 11 0.1724
#> Class 12 0.3109
#> Class 13 1.0000
#> Field 11
#> PSRP 1 PSRP 2 PSRP 3 PSRP 4
#> Class 1 0.00e+00
#> Class 2 6.13e-14
#> Class 3 8.84e-07
#> Class 4 8.14e-02
#> Class 5 2.46e-02
#> Class 6 2.13e-02
#> Class 7 2.56e-02
#> Class 8 3.84e-16
#> Class 9 2.44e-01
#> Class 10 4.30e-01
#> Class 11 3.84e-02
#> Class 12 5.86e-01
#> Class 13 1.00e+00
#> Field 12
#> PSRP 1 PSRP 2 PSRP 3 PSRP 4
#> Class 1 0.00e+00
#> Class 2 2.35e-03
#> Class 3 5.57e-02
#> Class 4 1.50e-18
#> Class 5 2.02e-02
#> Class 6 1.67e-02
#> Class 7 1.93e-02
#> Class 8 4.62e-02
#> Class 9 1.85e-02
#> Class 10 2.54e-02
#> Class 11 5.76e-15
#> Class 12 2.26e-01
#> Class 13 1.00e+00 1 1 1
#> Local Dependence Passing Student Rate
#> Field Field Item 1 Field Item 2 Field Item 3 Field Item 4 Parent Class
#> 1 1.000 Item01 Item31 Item32 1.000
#> 2 2.000 Item11 Item21 Item22 2.000
#> 3 2.000 Item11 Item21 Item22 3.000
#> 4 3.000 Item23 Item24 4.000
#> 5 4.000 Item25 Item26 Item27 5.000
#> 6 4.000 Item25 Item26 Item27 7.000
#> 7 5.000 Item02 Item03 Item04 2.000
#> 8 5.000 Item02 Item03 Item04 4.000
#> 9 5.000 Item02 Item03 Item04 6.000
#> 10 5.000 Item02 Item03 Item04 8.000
#> 11 5.000 Item02 Item03 Item04 10.000
#> 12 7.000 Item12 Item13 Item16 Item17 6.000
#> 13 8.000 Item28 Item29 6.000
#> 14 8.000 Item28 Item29 11.000
#> 15 9.000 Item05 Item06 8.000
#> 16 9.000 Item05 Item06 9.000
#> 17 12.000 Item18 Item19 Item20 Item30 12.000
#> Parent CCR 1 Parent CCR 2 Parent CCR 3 Parent CCR 4 Child Class Child CCR 1
#> 1 0.000 0.000 0.000 2.000 0.554
#> 2 0.005 0.018 0.003 4.000 0.681
#> 3 0.034 0.068 0.016 5.000 0.404
#> 4 0.221 0.190 5.000 0.189
#> 5 0.147 0.050 0.142 6.000 0.613
#> 6 0.999 0.991 0.943 11.000 0.941
#> 7 0.005 0.040 0.002 3.000 0.073
#> 8 0.996 0.998 0.893 7.000 0.145
#> 9 0.263 0.334 0.063 9.000 0.294
#> 10 0.980 0.958 0.882 12.000 1.000
#> 11 0.943 0.800 0.733 12.000 1.000
#> 12 0.181 0.146 0.037 0.162 10.000 0.982
#> 13 0.009 0.129 8.000 0.165
#> 14 0.359 0.273 12.000 1.000
#> 15 0.266 0.152 12.000 0.781
#> 16 0.000 0.000 11.000 1.000
#> 17 0.158 0.178 0.217 0.352 13.000 1.000
#> Child CCR 2 Child CCR 3 Child CCR 4
#> 1 0.558 0.649
#> 2 0.785 0.637
#> 3 0.728 0.696
#> 4 0.253
#> 5 0.436 0.179
#> 6 0.925 0.757
#> 7 0.330 0.068
#> 8 0.265 0.006
#> 9 0.298 0.121
#> 10 1.000 1.000
#> 11 1.000 1.000
#> 12 0.883 0.933 0.975
#> 13 0.192
#> 14 1.000
#> 15 0.703
#> 16 1.000
#> 17 1.000 1.000 1.000
#> Marginal Bicluster Reference Matrix
#> Class1 Class2 Class3 Class4 Class5 Class6 Class7 Class8 Class9 Class10
#> Field1 0 0.587 0.740 0.859 0.875 0.910 0.868 0.889 0.961 0.932
#> Field2 0 0.009 0.040 0.701 0.609 0.688 0.832 0.822 1.000 0.984
#> Field3 0 0.177 0.219 0.206 0.221 1.000 1.000 1.000 0.986 1.000
#> Field4 0 0.013 0.123 0.047 0.113 0.410 0.978 0.954 0.975 0.966
#> Field5 0 0.016 0.157 0.963 0.103 0.220 0.138 0.940 0.237 0.825
#> Field6 0 0.236 0.275 0.449 0.414 0.302 0.415 0.469 0.560 0.564
#> Field7 0 0.073 0.081 0.192 0.160 0.132 0.126 0.179 0.754 0.943
#> Field8 0 0.000 0.037 0.039 0.042 0.069 0.456 0.179 0.615 0.388
#> Field9 0 0.000 0.016 0.615 0.035 0.053 0.000 0.209 0.000 0.809
#> Field10 0 0.095 0.180 0.174 0.159 0.179 0.121 0.155 0.223 0.260
#> Field11 0 0.000 0.000 0.081 0.025 0.021 0.026 0.000 0.244 0.430
#> Field12 0 0.002 0.056 0.000 0.020 0.017 0.019 0.046 0.019 0.025
#> Class11 Class12 Class13
#> Field1 0.898 0.975 1
#> Field2 1.000 1.000 1
#> Field3 0.973 1.000 1
#> Field4 0.874 1.000 1
#> Field5 0.912 1.000 1
#> Field6 0.614 0.764 1
#> Field7 0.305 0.786 1
#> Field8 0.316 1.000 1
#> Field9 1.000 0.742 1
#> Field10 0.172 0.311 1
#> Field11 0.038 0.586 1
#> Field12 0.000 0.226 1
#> Class 1 Class 2 Class 3 Class 4 Class 5 Class 6
#> Test Reference Profile 0.000 3.900 6.001 12.951 8.853 11.428
#> Latent Class Ditribution 2.000 95.000 73.000 37.000 60.000 44.000
#> Class Membership Dsitribution 1.987 82.567 86.281 37.258 60.781 43.222
#> Class 7 Class 8 Class 9 Class 10 Class 11
#> Test Reference Profile 14.305 17.148 19.544 23.589 20.343
#> Latent Class Ditribution 43.000 30.000 34.000 18.000 37.000
#> Class Membership Dsitribution 43.062 30.087 34.435 20.063 34.811
#> Class 12 Class 13
#> Test Reference Profile 27.076 35
#> Latent Class Ditribution 27.000 15
#> Class Membership Dsitribution 25.445 15
#>
#> Model Fit Indices
#> Multigroup Model Saturated Moodel
#> model_log_like -5786.942 -5786.942
#> bench_log_like -5891.314 0
#> null_log_like -9862.114 -9862.114
#> model_Chi_sq -208.744 11573.88
#> null_Chi_sq 7941.601 19724.23
#> model_df 1005 16895
#> null_df 1155 17045
#> NFI 1 0.4132149
#> RFI 1 0.4080052
#> IFI 1 1
#> TLI 1 1
#> CFI 1 1
#> RMSEA 0 0
#> AIC -2218.744 -22216.12
#> CAIC -6486.081 -93954.09
#> BIC -6484.132 -93921.32Of course, it also supports various types of plots.
# Show bicluster structure
plot(result.BINET, type = "Array")
# Test Response Profile
plot(result.BINET, type = "TRP")
# Latent Rank Distribution
plot(result.BINET, type = "LRD")
# Rank Membership Profiles for first 9 students
plot(result.BINET, type = "RMP", students = 1:9, nc = 3, nr = 3)
# Field Reference Profiles
plot(result.BINET, type = "FRP", nc = 3, nr = 2)
LDPSR plot shows all Passing Student Rates for all locally dependent classes compared with their respective parents.
# Locally Dependent Passing Student Rates
plot(result.BINET, type = "LDPSR", nc = 3, nr = 2)
Available Output Types by Model
Pattern Analysis
| Model | IRP | FRP | TRP | ICRP | C/R RV |
|---|---|---|---|---|---|
| IRT | |||||
| LCA | ✓ | ✓ | ✓ | ||
| LRA | ✓ | ✓ | ✓ | ||
| LRAordinal | ✓ | ||||
| LRArated | ✓ | ||||
| Biclustering | ✓ | ✓ | ✓ | ||
| IRM | ✓ | ✓ | |||
| LDLRA | ✓ | ||||
| LDB | ✓ | ✓ | |||
| BINET | ✓ | ✓ |
Diagnostics & Visualization
| Model | LCD/LRD | CMP/RMP | Array | Other |
|---|---|---|---|---|
| IRT | IIC, ICC, TIC | |||
| LCA | ✓ | ✓ | ||
| LRA | ✓ | ✓ | ||
| LRAordinal | ✓ | |||
| LRArated | ✓ | |||
| Biclustering | ✓ | ✓ | ✓ | |
| IRM | ✓ | |||
| LDLRA | ✓ | ✓ | ||
| LDB | ✓ | ✓ | ✓ | FieldPIRP |
| BINET | ✓ | ✓ | ✓ | LDPSR |
Note: ✓ indicates available output type for the model.
Community and Support
We welcome community involvement and feedback to improve
exametrika. Here’s how you can participate and get
support:
Reporting Issues
If you encounter bugs or have suggestions for improvements:
- Open an issue on GitHub Issues
- Provide a minimal reproducible example
- Include your R session information (
sessionInfo())
Discussions and Community
Join our GitHub Discussions:
- Ask questions
- Share your use cases
- Discuss feature requests
- Exchange tips and tricks
- Get updates about package development
Contributing
We appreciate contributions from the community:
- Bug reports and feature requests through Issues
- Usage examples and tips through Discussions
- Code improvements through Pull Requests
Please check our existing Issues and Discussions before posting to avoid duplicates.
Reference
- Shojima, Kojiro (2022) Test Data Engineering: Latent Rank Analysis, Biclustering, and Bayesian Network (Behaviormetrics: Quantitative Approaches to Human Behavior, 13),Springer.
- Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika, 34(S1), 1-97.
Future Updates
Upcoming Features
Polytomous Data Support
- Item Response Theory
- Partial Credit Model (PCM)
- Latent Structure Analysis
- Extended Biclustering for polytomous data
Current Development Status
- Binary response models: ✅ Implemented
- Polytomous response models: 🚧 Under development
- CRAN submission: ✅ Now on CRAN!
Follow our GitHub repository and join the Discussions to stay updated on development progress and provide feedback on desired features.
Citation
