Estimating true subscores using Wainer's augmentation method
subscore.Wainer.RdThis function estimates subscores using Wainer's augmentation method (Wainer et. al., 2001) <doi:10.4324/9781410604729>. The central idea of this procedure is that, the estimation of subscores will be improved by shrinking the individual observed subscores towards some aggregate values (i.e., group mean subscores). The extent of the shrinkage depends on the closeness of the subscale being estimated with other subscales as well as reliabilities of all the subscales. Wainer's augmentation is a multivariate version of Kelly's formula (Kelly, 1947) <https://www.hup.harvard.edu/catalog.php?isbn=9780674330009>. For details of Wainer's augmentation subscoring method, please refer to Wainer et al. (2001) <doi:10.4324/9781410604729>.
Value
- summary
It contains statistical summary of the augmented subscores (mean, sd, and reliability).
- Augmented.subscores
It contains augmented subscores that are obtained using Wainer's method.
References
Wainer, H., Vevea, J., Camacho, F., Reeve, R., Rosa, K., Nelson, L., Swygert, K., & Thissen, D. (2001). "Augmented scores - "Borrowing strength" to compute scores based on small numbers of items" In Thissen, D. & Wainer, H. (Eds.), Test scoring (pp.343 - 387). Mahwah, NJ: Lawrence Erlbaum Associates, Inc. doi:10.4324/9781410604729.
Kelley, T. L. (1947). Fundamentals of statistics. Harvard University Press. https://www.hup.harvard.edu/catalog.php?isbn=9780674330009.
Examples
test.data<-data.prep(scored.data,c(3,15,15,20),
c("Algebra","Geometry","Measurement", "Math"))
subscore.Wainer(test.data)
#> $summary
#> mean SD Augmented.reliability
#> Algebra 9.773333 3.206071 0.9308608
#> Geometry 9.100000 3.133385 0.9311427
#> Measurement 13.060000 4.010296 0.9310656
#>
#> $subscore.original
#> Algebra Geometry Measurement Math
#> [1,] 15 15 20 50
#> [2,] 11 10 14 35
#> [3,] 13 13 20 46
#> [4,] 10 11 18 39
#> [5,] 9 11 15 35
#> [6,] 10 12 17 39
#> [7,] 10 11 18 39
#> [8,] 10 10 13 33
#> [9,] 13 11 17 41
#> [10,] 14 11 19 44
#> [11,] 12 12 16 40
#> [12,] 10 10 11 31
#> [13,] 11 12 15 38
#> [14,] 11 10 12 33
#> [15,] 12 13 13 38
#> [16,] 9 9 12 30
#> [17,] 9 5 10 24
#> [18,] 10 9 15 34
#> [19,] 10 8 13 31
#> [20,] 11 13 16 40
#> [21,] 13 11 13 37
#> [22,] 14 15 19 48
#> [23,] 14 15 19 48
#> [24,] 11 13 17 41
#> [25,] 12 11 17 40
#> [26,] 11 12 15 38
#> [27,] 13 12 18 43
#> [28,] 14 11 18 43
#> [29,] 15 13 17 45
#> [30,] 15 13 18 46
#> [31,] 14 11 17 42
#> [32,] 12 13 16 41
#> [33,] 12 13 19 44
#> [34,] 8 5 13 26
#> [35,] 13 13 18 44
#> [36,] 11 13 14 38
#> [37,] 14 14 19 47
#> [38,] 8 5 13 26
#> [39,] 14 13 20 47
#> [40,] 12 10 12 34
#> [41,] 14 11 18 43
#> [42,] 11 11 17 39
#> [43,] 15 14 18 47
#> [44,] 15 13 17 45
#> [45,] 14 10 17 41
#> [46,] 13 13 18 44
#> [47,] 12 10 19 41
#> [48,] 11 13 18 42
#> [49,] 12 8 14 34
#> [50,] 10 13 17 40
#> [51,] 13 14 16 43
#> [52,] 14 12 16 42
#> [53,] 15 13 19 47
#> [54,] 13 13 16 42
#> [55,] 12 13 16 41
#> [56,] 14 12 15 41
#> [57,] 15 8 18 41
#> [58,] 14 14 20 48
#> [59,] 13 13 17 43
#> [60,] 15 13 19 47
#> [61,] 14 13 19 46
#> [62,] 11 10 14 35
#> [63,] 11 5 11 27
#> [64,] 8 8 12 28
#> [65,] 2 2 4 8
#> [66,] 7 6 5 18
#> [67,] 10 7 8 25
#> [68,] 15 15 18 48
#> [69,] 6 7 10 23
#> [70,] 13 11 15 39
#> [71,] 7 2 7 16
#> [72,] 8 5 8 21
#> [73,] 12 10 15 37
#> [74,] 9 7 11 27
#> [75,] 8 9 16 33
#> [76,] 8 2 8 18
#> [77,] 10 9 12 31
#> [78,] 5 7 10 22
#> [79,] 5 4 4 13
#> [80,] 12 11 18 41
#> [81,] 4 7 11 22
#> [82,] 13 12 12 37
#> [83,] 8 5 8 21
#> [84,] 7 9 15 31
#> [85,] 7 8 10 25
#> [86,] 1 2 6 9
#> [87,] 8 7 10 25
#> [88,] 3 2 9 14
#> [89,] 7 10 14 31
#> [90,] 5 8 6 19
#> [91,] 2 4 2 8
#> [92,] 3 4 9 16
#> [93,] 6 4 5 15
#> [94,] 5 5 8 18
#> [95,] 9 10 8 27
#> [96,] 13 11 17 41
#> [97,] 7 1 5 13
#> [98,] 2 3 6 11
#> [99,] 7 8 11 26
#> [100,] 10 10 14 34
#> [101,] 10 9 13 32
#> [102,] 4 2 8 14
#> [103,] 5 1 9 15
#> [104,] 10 10 9 29
#> [105,] 14 13 17 44
#> [106,] 8 8 10 26
#> [107,] 10 11 16 37
#> [108,] 6 3 9 18
#> [109,] 9 9 13 31
#> [110,] 13 12 17 42
#> [111,] 9 6 16 31
#> [112,] 7 4 11 22
#> [113,] 12 11 13 36
#> [114,] 13 11 14 38
#> [115,] 13 10 13 36
#> [116,] 2 5 7 14
#> [117,] 5 4 8 17
#> [118,] 9 5 11 25
#> [119,] 12 12 14 38
#> [120,] 3 4 4 11
#> [121,] 6 5 14 25
#> [122,] 3 4 4 11
#> [123,] 7 8 7 22
#> [124,] 5 4 6 15
#> [125,] 10 7 8 25
#> [126,] 14 10 18 42
#> [127,] 14 12 19 45
#> [128,] 10 5 8 23
#> [129,] 13 12 15 40
#> [130,] 14 11 20 45
#> [131,] 8 11 16 35
#> [132,] 4 2 7 13
#> [133,] 12 11 17 40
#> [134,] 9 9 14 32
#> [135,] 13 9 13 35
#> [136,] 7 10 14 31
#> [137,] 11 10 9 30
#> [138,] 10 10 13 33
#> [139,] 3 6 7 16
#> [140,] 5 7 5 17
#> [141,] 5 9 10 24
#> [142,] 5 5 5 15
#> [143,] 3 6 7 16
#> [144,] 6 8 5 19
#> [145,] 9 12 14 35
#> [146,] 4 4 9 17
#> [147,] 11 14 13 38
#> [148,] 5 7 11 23
#> [149,] 9 2 10 21
#> [150,] 14 12 17 43
#>
#> $subscore.augmented
#> Algebra Geometry Measurement
#> [1,] 14.990343 14.172447 19.559098
#> [2,] 10.652423 9.976406 14.188895
#> [3,] 13.855137 13.024940 18.032893
#> [4,] 11.852344 11.015454 15.440777
#> [5,] 10.688819 9.879820 14.047290
#> [6,] 11.848295 11.000886 15.465543
#> [7,] 11.852344 11.015454 15.440777
#> [8,] 10.080772 9.388084 13.450559
#> [9,] 12.383551 11.685794 16.345484
#> [10,] 13.251138 12.547772 17.411396
#> [11,] 12.103788 11.356561 15.959489
#> [12,] 9.488899 8.840772 12.795408
#> [13,] 11.532137 10.768239 15.221153
#> [14,] 10.060550 9.429093 13.533744
#> [15,] 11.507867 10.794680 15.329105
#> [16,] 9.217232 8.540674 12.359881
#> [17,] 7.457805 6.957008 10.295364
#> [18,] 10.380757 9.676308 13.753368
#> [19,] 9.496995 8.869908 12.745876
#> [20,] 12.119962 11.300984 15.901070
#> [21,] 11.199804 10.591169 15.035183
#> [22,] 14.418692 13.584126 18.820762
#> [23,] 14.418692 13.584126 18.820762
#> [24,] 12.415898 11.574640 16.228645
#> [25,] 12.107836 11.371129 15.934723
#> [26,] 11.532137 10.768239 15.221153
#> [27,] 12.971376 12.218539 17.025401
#> [28,] 12.955202 12.274116 17.083820
#> [29,] 13.518757 12.833302 17.871689
#> [30,] 13.814693 13.106958 18.199264
#> [31,] 12.659265 12.000459 16.756245
#> [32,] 12.395676 11.615649 16.311831
#> [33,] 13.283486 12.436618 17.294557
#> [34,] 8.069901 7.463311 10.867329
#> [35,] 13.263264 12.477627 17.377742
#> [36,] 11.528089 10.753671 15.245919
#> [37,] 14.126804 13.325037 18.468420
#> [38,] 8.069901 7.463311 10.867329
#> [39,] 14.130852 13.339605 18.443654
#> [40,] 10.336265 9.743759 13.944505
#> [41,] 12.955202 12.274116 17.083820
#> [42,] 11.832122 11.056463 15.523962
#> [43,] 14.106582 13.366046 18.551606
#> [44,] 13.518757 12.833302 17.871689
#> [45,] 12.367377 11.741371 16.403904
#> [46,] 13.263264 12.477627 17.377742
#> [47,] 12.407821 11.659353 16.237532
#> [48,] 12.711835 11.848296 16.556221
#> [49,] 10.344361 9.772895 13.894973
#> [50,] 12.140184 11.259975 15.817884
#> [51,] 12.963279 12.189403 17.074933
#> [52,] 12.655217 11.985892 16.781011
#> [53,] 14.110630 13.380614 18.526840
#> [54,] 12.671391 11.930315 16.722592
#> [55,] 12.395676 11.615649 16.311831
#> [56,] 12.359280 11.712235 16.453436
#> [57,] 12.355251 11.811516 16.437557
#> [58,] 14.422740 13.598694 18.795995
#> [59,] 12.967328 12.203971 17.050167
#> [60,] 14.110630 13.380614 18.526840
#> [61,] 13.834915 13.065949 18.116079
#> [62,] 10.652423 9.976406 14.188895
#> [63,] 8.305171 7.859995 11.444461
#> [64,] 8.649629 7.966920 11.596779
#> [65,] 2.876518 2.335147 4.397562
#> [66,] 5.718582 5.218484 8.188307
#> [67,] 7.725424 7.242538 10.755658
#> [68,] 14.398470 13.625135 18.903947
#> [69,] 7.214439 6.531189 9.767765
#> [70,] 11.791677 11.138482 15.690333
#> [71,] 5.142901 4.729443 7.434092
#> [72,] 6.590218 6.095030 9.229453
#> [73,] 11.224074 10.564728 14.927231
#> [74,] 8.337519 7.748841 11.327623
#> [75,] 10.125264 9.320634 13.259421
#> [76,] 5.714552 5.317765 8.172428
#> [77,] 9.492947 8.855340 12.770642
#> [78,] 6.938724 6.216523 9.357004
#> [79,] 4.287439 3.797320 6.334527
#> [80,] 12.403773 11.644785 16.262299
#> [81,] 6.958946 6.175514 9.273818
#> [82,] 11.195756 10.576601 15.059949
#> [83,] 6.590218 6.095030 9.229453
#> [84,] 9.553613 8.732312 12.521085
#> [85,] 7.782042 7.104942 10.530867
#> [86,] 3.192677 2.567794 4.641951
#> [87,] 7.765868 7.160519 10.589286
#> [88,] 4.631916 4.018094 6.446199
#> [89,] 9.549565 8.717744 12.545851
#> [90,] 6.046866 5.380987 8.399044
#> [91,] 2.868422 2.306012 4.447094
#> [92,] 5.215693 4.536271 7.150882
#> [93,] 4.859090 4.385642 7.072863
#> [94,] 5.763074 5.151034 7.997170
#> [95,] 8.325374 7.705138 11.401921
#> [96,] 12.383551 11.685794 16.345484
#> [97,] 4.259139 3.923042 6.426600
#> [98,] 3.760280 3.141548 5.405054
#> [99,] 8.077978 7.378599 10.858442
#> [100,] 10.376709 9.661741 13.778134
#> [101,] 9.788884 9.128996 13.098217
#> [102,] 4.611694 4.059103 6.529385
#> [103,] 4.891457 4.388336 6.915379
#> [104,] 8.897026 8.293459 12.140257
#> [105,] 13.243042 12.518636 17.460928
#> [106,] 8.057756 7.419608 10.941628
#> [107,] 11.260470 10.468142 14.785626
#> [108,] 5.750948 5.221179 8.030823
#> [109,] 9.513169 8.814331 12.687456
#> [110,] 12.675439 11.944882 16.697826
#> [111,] 9.525313 8.858034 12.613158
#> [112,] 6.910424 6.342245 9.449076
#> [113,] 10.924090 10.276504 14.624422
#> [114,] 11.495741 10.864825 15.362758
#> [115,] 10.907916 10.332081 14.682841
#> [116,] 4.639994 3.933381 6.437312
#> [117,] 5.471185 4.891945 7.644829
#> [118,] 7.753742 7.230664 10.622940
#> [119,] 11.511915 10.809248 15.304339
#> [120,] 3.736010 3.167990 5.513005
#> [121,] 7.814408 7.107637 10.373383
#> [122,] 3.736010 3.167990 5.513005
#> [123,] 6.894232 6.283974 9.548141
#> [124,] 4.879312 4.344633 6.989678
#> [125,] 7.725424 7.242538 10.755658
#> [126,] 12.663313 12.015027 16.731479
#> [127,] 13.543027 12.806860 17.763737
#> [128,] 7.141647 6.724361 10.050975
#> [129,] 12.083566 11.397570 16.042675
#> [130,] 13.547075 12.821428 17.738971
#> [131,] 10.709041 9.838811 13.964104
#> [132,] 4.315757 3.785447 6.201809
#> [133,] 12.107836 11.371129 15.934723
#> [134,] 9.809106 9.087987 13.015032
#> [135,] 10.616027 10.072992 14.330500
#> [136,] 9.549565 8.717744 12.545851
#> [137,] 9.172740 8.608125 12.551018
#> [138,] 10.080772 9.388084 13.450559
#> [139,] 5.207597 4.507135 7.200414
#> [140,] 5.459041 4.848242 7.719127
#> [141,] 7.522501 6.734700 10.061687
#> [142,] 4.875264 4.330065 7.014444
#> [143,] 5.207597 4.507135 7.200414
#> [144,] 6.026644 5.421996 8.482229
#> [145,] 10.684771 9.865252 14.072056
#> [146,] 5.491407 4.850936 7.561643
#> [147,] 11.524040 10.739103 15.270685
#> [148,] 7.234661 6.490180 9.684579
#> [149,] 6.582140 6.179743 9.238340
#> [150,] 12.951154 12.259548 17.108587
#>
subscore.Wainer(test.data)$summary
#> mean SD Augmented.reliability
#> Algebra 9.773333 3.206071 0.9308608
#> Geometry 9.100000 3.133385 0.9311427
#> Measurement 13.060000 4.010296 0.9310656
subscore.Wainer(test.data)$subscore.augmented
#> Algebra Geometry Measurement
#> [1,] 14.990343 14.172447 19.559098
#> [2,] 10.652423 9.976406 14.188895
#> [3,] 13.855137 13.024940 18.032893
#> [4,] 11.852344 11.015454 15.440777
#> [5,] 10.688819 9.879820 14.047290
#> [6,] 11.848295 11.000886 15.465543
#> [7,] 11.852344 11.015454 15.440777
#> [8,] 10.080772 9.388084 13.450559
#> [9,] 12.383551 11.685794 16.345484
#> [10,] 13.251138 12.547772 17.411396
#> [11,] 12.103788 11.356561 15.959489
#> [12,] 9.488899 8.840772 12.795408
#> [13,] 11.532137 10.768239 15.221153
#> [14,] 10.060550 9.429093 13.533744
#> [15,] 11.507867 10.794680 15.329105
#> [16,] 9.217232 8.540674 12.359881
#> [17,] 7.457805 6.957008 10.295364
#> [18,] 10.380757 9.676308 13.753368
#> [19,] 9.496995 8.869908 12.745876
#> [20,] 12.119962 11.300984 15.901070
#> [21,] 11.199804 10.591169 15.035183
#> [22,] 14.418692 13.584126 18.820762
#> [23,] 14.418692 13.584126 18.820762
#> [24,] 12.415898 11.574640 16.228645
#> [25,] 12.107836 11.371129 15.934723
#> [26,] 11.532137 10.768239 15.221153
#> [27,] 12.971376 12.218539 17.025401
#> [28,] 12.955202 12.274116 17.083820
#> [29,] 13.518757 12.833302 17.871689
#> [30,] 13.814693 13.106958 18.199264
#> [31,] 12.659265 12.000459 16.756245
#> [32,] 12.395676 11.615649 16.311831
#> [33,] 13.283486 12.436618 17.294557
#> [34,] 8.069901 7.463311 10.867329
#> [35,] 13.263264 12.477627 17.377742
#> [36,] 11.528089 10.753671 15.245919
#> [37,] 14.126804 13.325037 18.468420
#> [38,] 8.069901 7.463311 10.867329
#> [39,] 14.130852 13.339605 18.443654
#> [40,] 10.336265 9.743759 13.944505
#> [41,] 12.955202 12.274116 17.083820
#> [42,] 11.832122 11.056463 15.523962
#> [43,] 14.106582 13.366046 18.551606
#> [44,] 13.518757 12.833302 17.871689
#> [45,] 12.367377 11.741371 16.403904
#> [46,] 13.263264 12.477627 17.377742
#> [47,] 12.407821 11.659353 16.237532
#> [48,] 12.711835 11.848296 16.556221
#> [49,] 10.344361 9.772895 13.894973
#> [50,] 12.140184 11.259975 15.817884
#> [51,] 12.963279 12.189403 17.074933
#> [52,] 12.655217 11.985892 16.781011
#> [53,] 14.110630 13.380614 18.526840
#> [54,] 12.671391 11.930315 16.722592
#> [55,] 12.395676 11.615649 16.311831
#> [56,] 12.359280 11.712235 16.453436
#> [57,] 12.355251 11.811516 16.437557
#> [58,] 14.422740 13.598694 18.795995
#> [59,] 12.967328 12.203971 17.050167
#> [60,] 14.110630 13.380614 18.526840
#> [61,] 13.834915 13.065949 18.116079
#> [62,] 10.652423 9.976406 14.188895
#> [63,] 8.305171 7.859995 11.444461
#> [64,] 8.649629 7.966920 11.596779
#> [65,] 2.876518 2.335147 4.397562
#> [66,] 5.718582 5.218484 8.188307
#> [67,] 7.725424 7.242538 10.755658
#> [68,] 14.398470 13.625135 18.903947
#> [69,] 7.214439 6.531189 9.767765
#> [70,] 11.791677 11.138482 15.690333
#> [71,] 5.142901 4.729443 7.434092
#> [72,] 6.590218 6.095030 9.229453
#> [73,] 11.224074 10.564728 14.927231
#> [74,] 8.337519 7.748841 11.327623
#> [75,] 10.125264 9.320634 13.259421
#> [76,] 5.714552 5.317765 8.172428
#> [77,] 9.492947 8.855340 12.770642
#> [78,] 6.938724 6.216523 9.357004
#> [79,] 4.287439 3.797320 6.334527
#> [80,] 12.403773 11.644785 16.262299
#> [81,] 6.958946 6.175514 9.273818
#> [82,] 11.195756 10.576601 15.059949
#> [83,] 6.590218 6.095030 9.229453
#> [84,] 9.553613 8.732312 12.521085
#> [85,] 7.782042 7.104942 10.530867
#> [86,] 3.192677 2.567794 4.641951
#> [87,] 7.765868 7.160519 10.589286
#> [88,] 4.631916 4.018094 6.446199
#> [89,] 9.549565 8.717744 12.545851
#> [90,] 6.046866 5.380987 8.399044
#> [91,] 2.868422 2.306012 4.447094
#> [92,] 5.215693 4.536271 7.150882
#> [93,] 4.859090 4.385642 7.072863
#> [94,] 5.763074 5.151034 7.997170
#> [95,] 8.325374 7.705138 11.401921
#> [96,] 12.383551 11.685794 16.345484
#> [97,] 4.259139 3.923042 6.426600
#> [98,] 3.760280 3.141548 5.405054
#> [99,] 8.077978 7.378599 10.858442
#> [100,] 10.376709 9.661741 13.778134
#> [101,] 9.788884 9.128996 13.098217
#> [102,] 4.611694 4.059103 6.529385
#> [103,] 4.891457 4.388336 6.915379
#> [104,] 8.897026 8.293459 12.140257
#> [105,] 13.243042 12.518636 17.460928
#> [106,] 8.057756 7.419608 10.941628
#> [107,] 11.260470 10.468142 14.785626
#> [108,] 5.750948 5.221179 8.030823
#> [109,] 9.513169 8.814331 12.687456
#> [110,] 12.675439 11.944882 16.697826
#> [111,] 9.525313 8.858034 12.613158
#> [112,] 6.910424 6.342245 9.449076
#> [113,] 10.924090 10.276504 14.624422
#> [114,] 11.495741 10.864825 15.362758
#> [115,] 10.907916 10.332081 14.682841
#> [116,] 4.639994 3.933381 6.437312
#> [117,] 5.471185 4.891945 7.644829
#> [118,] 7.753742 7.230664 10.622940
#> [119,] 11.511915 10.809248 15.304339
#> [120,] 3.736010 3.167990 5.513005
#> [121,] 7.814408 7.107637 10.373383
#> [122,] 3.736010 3.167990 5.513005
#> [123,] 6.894232 6.283974 9.548141
#> [124,] 4.879312 4.344633 6.989678
#> [125,] 7.725424 7.242538 10.755658
#> [126,] 12.663313 12.015027 16.731479
#> [127,] 13.543027 12.806860 17.763737
#> [128,] 7.141647 6.724361 10.050975
#> [129,] 12.083566 11.397570 16.042675
#> [130,] 13.547075 12.821428 17.738971
#> [131,] 10.709041 9.838811 13.964104
#> [132,] 4.315757 3.785447 6.201809
#> [133,] 12.107836 11.371129 15.934723
#> [134,] 9.809106 9.087987 13.015032
#> [135,] 10.616027 10.072992 14.330500
#> [136,] 9.549565 8.717744 12.545851
#> [137,] 9.172740 8.608125 12.551018
#> [138,] 10.080772 9.388084 13.450559
#> [139,] 5.207597 4.507135 7.200414
#> [140,] 5.459041 4.848242 7.719127
#> [141,] 7.522501 6.734700 10.061687
#> [142,] 4.875264 4.330065 7.014444
#> [143,] 5.207597 4.507135 7.200414
#> [144,] 6.026644 5.421996 8.482229
#> [145,] 10.684771 9.865252 14.072056
#> [146,] 5.491407 4.850936 7.561643
#> [147,] 11.524040 10.739103 15.270685
#> [148,] 7.234661 6.490180 9.684579
#> [149,] 6.582140 6.179743 9.238340
#> [150,] 12.951154 12.259548 17.108587