Latent Class Analysis (LCA)
LCA classifies examinees into unordered latent classes. Specify the
dataset and the number of classes.
LCA(J15S500, ncls = 5)
#>
#> Item Reference Profile
#> IRP1 IRP2 IRP3 IRP4 IRP5
#> Item01 0.5899 0.6616 0.7723 0.8528 0.884
#> Item02 0.5174 0.7102 0.8282 0.8863 0.871
#> Item03 0.5913 0.6112 0.8905 0.7196 0.839
#> Item04 0.3894 0.8671 0.8214 0.8657 1.000
#> Item05 0.6729 0.6759 1.0000 0.7718 0.920
#> Item06 0.6261 0.8857 0.9478 1.0000 0.903
#> Item07 0.3530 0.7209 0.7978 0.8818 0.891
#> Item08 0.3087 0.5911 0.5724 0.9172 0.609
#> Item09 0.2856 0.5414 0.2028 0.0993 0.692
#> Item10 0.5334 0.4995 0.8109 0.6898 0.800
#> Item11 0.0977 0.0203 0.0437 0.6450 0.678
#> Item12 0.0324 0.2904 0.1232 0.2821 0.684
#> Item13 0.2753 0.5690 0.9059 0.6655 0.811
#> Item14 0.4745 0.7017 0.7876 0.9101 1.000
#> Item15 0.4076 0.6879 0.8160 0.8471 0.824
#>
#> Test Profile
#> Class 1 Class 2 Class 3 Class 4 Class 5
#> Test Reference Profile 6.155 9.034 10.321 11.034 12.405
#> Latent Class Ditribution 119.000 90.000 112.000 86.000 93.000
#> Class Membership Distribution 112.597 99.086 98.336 93.200 96.780
#>
#> Item Fit Indices
#> model_log_like bench_log_like null_log_like model_Chi_sq null_Chi_sq
#> Item01 -265.973 -240.190 -283.343 51.567 86.307
#> Item02 -252.897 -235.436 -278.949 34.922 87.025
#> Item03 -274.316 -260.906 -293.598 26.818 65.383
#> Item04 -197.016 -192.072 -265.962 9.889 147.780
#> Item05 -210.738 -206.537 -247.403 8.402 81.732
#> Item06 -160.728 -153.940 -198.817 13.576 89.755
#> Item07 -248.566 -228.379 -298.345 40.375 139.933
#> Item08 -295.140 -293.225 -338.789 3.829 91.127
#> Item09 -275.187 -300.492 -327.842 -50.611 54.700
#> Item10 -300.309 -288.198 -319.850 24.221 63.303
#> Item11 -184.967 -224.085 -299.265 -78.237 150.360
#> Item12 -228.339 -214.797 -293.598 27.085 157.603
#> Item13 -270.948 -262.031 -328.396 17.834 132.730
#> Item14 -217.326 -204.953 -273.212 24.746 136.519
#> Item15 -269.454 -254.764 -302.847 29.380 96.166
#> model_df null_df NFI RFI IFI TLI CFI RMSEA AIC CAIC
#> Item01 9 13 0.403 0.137 0.449 0.161 0.419 0.097 33.567 -13.365
#> Item02 9 13 0.599 0.420 0.668 0.494 0.650 0.076 16.922 -30.009
#> Item03 9 13 0.590 0.408 0.684 0.509 0.660 0.063 8.818 -38.113
#> Item04 9 13 0.933 0.903 0.994 0.990 0.993 0.014 -8.111 -55.043
#> Item05 9 13 0.897 0.852 1.000 1.000 1.000 0.000 -9.598 -56.529
#> Item06 9 13 0.849 0.782 0.943 0.914 0.940 0.032 -4.424 -51.355
#> Item07 9 13 0.711 0.583 0.760 0.643 0.753 0.084 22.375 -24.556
#> Item08 9 13 0.958 0.939 1.000 1.000 1.000 0.000 -14.171 -61.102
#> Item09 9 13 1.000 1.000 1.000 1.000 1.000 0.000 -68.611 -115.542
#> Item10 9 13 0.617 0.447 0.720 0.563 0.697 0.058 6.221 -40.710
#> Item11 9 13 1.000 1.000 1.000 1.000 1.000 0.000 -96.237 -143.169
#> Item12 9 13 0.828 0.752 0.878 0.819 0.875 0.063 9.085 -37.846
#> Item13 9 13 0.866 0.806 0.929 0.893 0.926 0.044 -0.166 -47.098
#> Item14 9 13 0.819 0.738 0.877 0.816 0.873 0.059 6.746 -40.186
#> Item15 9 13 0.694 0.559 0.766 0.646 0.755 0.067 11.380 -35.552
#> BIC
#> Item01 -4.365
#> Item02 -21.009
#> Item03 -29.113
#> Item04 -46.043
#> Item05 -47.529
#> Item06 -42.355
#> Item07 -15.556
#> Item08 -52.102
#> Item09 -106.542
#> Item10 -31.710
#> Item11 -134.169
#> Item12 -28.846
#> Item13 -38.098
#> Item14 -31.186
#> Item15 -26.552
#>
#> Model Fit Indices
#> Number of Latent class: 5
#> Number of EM cycle: 337
#> value
#> model_log_like -3651.904
#> bench_log_like -3560.005
#> null_log_like -4350.217
#> model_Chi_sq 183.797
#> null_Chi_sq 1580.424
#> model_df 135.000
#> null_df 195.000
#> NFI 0.884
#> RFI 0.832
#> IFI 0.966
#> TLI 0.949
#> CFI 0.965
#> RMSEA 0.027
#> AIC -86.203
#> CAIC -790.175
#> BIC -655.175
The Class Membership Matrix indicates which latent class each
examinee belongs to:
result.LCA <- LCA(J15S500, ncls = 5)
head(result.LCA$Students)
#> Membership 1 Membership 2 Membership 3 Membership 4 Membership 5
#> Student001 0.7285244374 0.012211535 0.226232540 3.303149e-02 3.055593e-12
#> Student002 0.0220645036 0.086986302 0.830839343 6.010974e-02 1.074954e-07
#> Student003 0.0170578933 0.054109896 0.879752304 2.100872e-02 2.807118e-02
#> Student004 0.0010508039 0.223175413 0.203820488 3.286491e-01 2.433042e-01
#> Student005 0.9407961670 0.053321705 0.004873703 1.808344e-08 1.008407e-03
#> Student006 0.0002372397 0.002528968 0.029747250 8.551046e-01 1.123819e-01
#> Estimate
#> Student001 1
#> Student002 3
#> Student003 3
#> Student004 4
#> Student005 1
#> Student006 4
LCA Plot Types
- IRP: Item Reference Profile
- CMP: Class Membership Profile
- TRP: Test Reference Profile
- LCD: Latent Class Distribution
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")
Latent Rank Analysis (LRA)
LRA is similar to LCA but assumes an ordering among the latent
classes (ranks). Specify the dataset and the number of ranks.
LRA(J15S500, nrank = 6)
#> estimating method is isotonic
#> Item Reference Profile
#> IRP1 IRP2 IRP3 IRP4 IRP5 IRP6
#> Item01 0.4582 0.7484 0.7484 0.7484 0.839 0.914
#> Item02 0.5601 0.5601 0.8048 0.8048 0.883 0.883
#> Item03 0.5996 0.5996 0.7590 0.7590 0.759 0.866
#> Item04 0.4668 0.4668 0.8949 0.8949 0.895 0.995
#> Item05 0.5593 0.8284 0.8284 0.8284 0.828 0.936
#> Item06 0.6196 0.7691 0.9416 0.9416 0.942 0.942
#> Item07 0.4112 0.4112 0.7402 0.8959 0.896 0.896
#> Item08 0.3485 0.3485 0.6006 0.7315 0.732 0.732
#> Item09 0.3149 0.3149 0.3149 0.3149 0.315 0.619
#> Item10 0.4438 0.6344 0.6940 0.6940 0.722 0.765
#> Item11 0.0816 0.0816 0.0816 0.0816 0.688 0.688
#> Item12 0.0653 0.0653 0.2193 0.2193 0.219 0.859
#> Item13 0.2228 0.4902 0.7527 0.7527 0.753 0.788
#> Item14 0.2894 0.7760 0.7760 0.7760 0.933 1.000
#> Item15 0.3830 0.5188 0.8164 0.8164 0.816 0.845
#>
#> Item Reference Profile Indices
#> Alpha A Beta B Gamma C
#> Item01 1 0.290 1 0.458 0 0
#> Item02 2 0.245 1 0.560 0 0
#> Item03 2 0.159 1 0.600 0 0
#> Item04 2 0.428 1 0.467 0 0
#> Item05 1 0.269 1 0.559 0 0
#> Item06 2 0.173 1 0.620 0 0
#> Item07 2 0.329 1 0.411 0 0
#> Item08 2 0.252 3 0.601 0 0
#> Item09 5 0.304 6 0.619 0 0
#> Item10 1 0.191 1 0.444 0 0
#> Item11 4 0.607 5 0.688 0 0
#> Item12 5 0.640 3 0.219 0 0
#> Item13 1 0.267 2 0.490 0 0
#> Item14 1 0.487 1 0.289 0 0
#> Item15 2 0.298 2 0.519 0 0
#>
#> Test Profile
#> Rank 1 Rank 2 Rank 3 Rank 4 Rank 5 Rank 6
#> Test Reference Profile 5.824 7.613 9.973 10.259 11.219 12.727
#> Latent Rank Ditribution 75.000 82.000 71.000 107.000 74.000 91.000
#> Rank Membership Distribution 77.702 79.964 86.849 87.007 87.796 80.682
#>
#> Item Fit Indices
#> model_log_like bench_log_like null_log_like model_Chi_sq null_Chi_sq
#> Item01 -259.078 -240.190 -283.343 37.776 86.307
#> Item02 -254.771 -235.436 -278.949 38.669 87.025
#> Item03 -282.423 -260.906 -293.598 43.033 65.383
#> Item04 -199.586 -192.072 -265.962 15.029 147.780
#> Item05 -229.023 -206.537 -247.403 44.972 81.732
#> Item06 -170.972 -153.940 -198.817 34.064 89.755
#> Item07 -241.895 -228.379 -298.345 27.033 139.933
#> Item08 -308.979 -293.225 -338.789 31.508 91.127
#> Item09 -314.833 -300.492 -327.842 28.681 54.700
#> Item10 -308.797 -288.198 -319.850 41.198 63.303
#> Item11 -198.273 -224.085 -299.265 -51.625 150.360
#> Item12 -208.480 -214.797 -293.598 -12.633 157.603
#> Item13 -284.705 -262.031 -328.396 45.349 132.730
#> Item14 -203.455 -204.953 -273.212 -2.995 136.519
#> Item15 -266.694 -254.764 -302.847 23.862 96.166
#> model_df null_df NFI RFI IFI TLI CFI RMSEA AIC CAIC
#> Item01 10 13 0.562 0.431 0.636 0.507 0.621 0.075 17.776 -34.370
#> Item02 11 13 0.556 0.475 0.636 0.558 0.626 0.071 16.669 -40.692
#> Item03 11 13 0.342 0.222 0.411 0.277 0.388 0.076 21.033 -36.328
#> Item04 11 13 0.898 0.880 0.971 0.965 0.970 0.027 -6.971 -64.332
#> Item05 11 13 0.450 0.350 0.520 0.416 0.506 0.079 22.972 -34.389
#> Item06 11 13 0.620 0.551 0.707 0.645 0.700 0.065 12.064 -45.297
#> Item07 11 13 0.807 0.772 0.876 0.851 0.874 0.054 5.033 -52.328
#> Item08 11 13 0.654 0.591 0.744 0.690 0.738 0.061 9.508 -47.853
#> Item09 12 13 0.476 0.432 0.609 0.567 0.600 0.053 4.681 -57.895
#> Item10 9 13 0.349 0.060 0.407 0.075 0.360 0.085 23.198 -23.734
#> Item11 12 13 1.000 1.000 1.000 1.000 1.000 0.000 -75.625 -138.201
#> Item12 11 13 1.000 1.000 1.000 1.000 1.000 0.000 -34.633 -91.994
#> Item13 10 13 0.658 0.556 0.712 0.616 0.705 0.084 25.349 -26.797
#> Item14 10 13 1.000 1.000 1.000 1.000 1.000 0.000 -22.995 -75.141
#> Item15 10 13 0.752 0.677 0.839 0.783 0.833 0.053 3.862 -48.284
#> BIC
#> Item01 -24.370
#> Item02 -29.692
#> Item03 -25.328
#> Item04 -53.332
#> Item05 -23.389
#> Item06 -34.297
#> Item07 -41.328
#> Item08 -36.853
#> Item09 -45.895
#> Item10 -14.734
#> Item11 -126.201
#> Item12 -80.994
#> Item13 -16.797
#> Item14 -65.141
#> Item15 -38.284
#>
#> Model Fit Indices
#> Number of Latent rank: 6
#> Number of EM cycle: 65
#> value
#> model_log_like -3731.964
#> bench_log_like -3560.005
#> null_log_like -4350.217
#> model_Chi_sq 343.918
#> null_Chi_sq 1580.424
#> model_df 161.000
#> null_df 195.000
#> NFI 0.782
#> RFI 0.736
#> IFI 0.871
#> TLI 0.840
#> CFI 0.868
#> RMSEA 0.048
#> AIC 21.918
#> CAIC -817.634
#> BIC -656.634
Rank membership probabilities and rank-up/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 0.3732015798 0.428451221 0.11219848 0.030244910 0.055903805
#> Student002 0.0200254309 0.080848295 0.57928413 0.282874219 0.036967734
#> Student003 0.0062126694 0.213151690 0.54304200 0.146385733 0.076483260
#> Student004 0.0010041013 0.009115691 0.19880904 0.293017561 0.133718573
#> Student005 0.2529143262 0.727293364 0.01394516 0.003759143 0.001964068
#> Student006 0.0001300928 0.002053329 0.04593162 0.067696979 0.766760085
#> Membership 6 Estimate Rank-Up Odds Rank-Down Odds
#> Student001 6.653884e-09 2 0.26186990 0.87104800
#> Student002 1.950328e-07 3 0.48831688 0.13956587
#> Student003 1.472465e-02 3 0.26956613 0.39251419
#> Student004 3.643350e-01 6 NA 0.36702090
#> Student005 1.239411e-04 2 0.01917405 0.34774733
#> Student006 1.174279e-01 5 0.15314816 0.08828965
plot(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 handle ordinal scale data. The mic option
enforces monotonic increasing constraints.
result.LRAord <- LRA(J15S3810, nrank = 3, mic = TRUE)
Score-rank relationship visualizations:
plot(result.LRAord, type = "ScoreFreq")
plot(result.LRAord, type = "ScoreRank")
Item-rank relationship plots:
- ICBR: Item Category Boundary Reference – cumulative
probability curves for each category threshold
- ICRP: Item Category Response Profile – probability
of each response category across ranks
plot(result.LRAord, type = "ICBR", items = 1:4, nc = 2, nr = 2)
plot(result.LRAord, type = "ICRP", items = 1:4, nc = 2, nr = 2)
Rank membership profiles for individual examinees:
plot(result.LRAord, type = "RMP", students = 1:9, nc = 3, nr = 3)
LRA for Rated/Nominal Data
For multiple-choice tests (nominal scale), LRA can analyze response
patterns including distractor choices.
result.LRArated <- LRA(J35S5000, nrank = 10, mic = TRUE)
plot(result.LRArated, type = "ScoreFreq")
plot(result.LRArated, type = "ScoreRank")
plot(result.LRArated, type = "ICRP", items = 1:4, nc = 2, nr = 2)