README.md
In xluo11/Rirt: Data Analysis and Parameter Estimation Using Item Response Theory
Rirt: data analysis and parameter estimation using item response theory
Overview
Rirt provides an interface for conducting data analysis and parameter
estimation using item respone theory (IRT) efficiently in
R. A variety of unidimensional IRT models
is included, such as the 3-parameter-logstic (3PL) model, the
generalized partial credit model (GPCM), the graded response model
(GRM), and a mixture of the aforementioned models.
Installation
Install the stable version from CRAN:
install.packages("Rirt")
Install the most recent version from
github:
devtools::install_github("xluo11/Rirt")
Quickstart
A common interface is created to make the package easy to use:
model_[3pl/gpcm/grm/mixed]_prob: compute the response probability
for the given ability and item parametersmodel_[3pl/gpcm/grm/mixed]_info: compute the information for the
given ability and item parametersmodel_[3pl/gpcm/grm/mixed]_lh: compute the response likelihood for
the given ability and item parametersmodel_[3pl/gpcm/grm/mixed]_gendata: generate response data and
parameters from an IRT modelmodel_[3pl/gpcm/grm]_plot: plot item characteristic curves (ICC)
or item informaiton functions (IIFs)model_[3pl/gpcm/grm]_rescale: rescale ability and item parameters
to a given scalemodel_[3pl/gpcm/grm/mixed]_eap: score response vectors using the
EAP methodmodel_[3pl/gpcm/grm/mixed]_map: score response vectors using the
MAP methodmodel_[3pl/gpcm/grm]_jmle: estimate parameters using the joint
maximum likelihood estimation methodmodel_[3pl/gpcm/grm]_mmle: estimate parameters using the marginal
maximum likelihood estimation method
For example, users can quickly generate data from an IRT model of
choice:
# generate 10 people and 5 items using the 3PL model
# returns a list of 'u' (responses), 't' (ability), 'a' (item discrimination), 'b' (item difficulty), 'c' (pseudo-guessing)
model_3pl_gendata(n_p=10, n_i=5)
## $u
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0 0 0 1 1
## [2,] 1 1 1 0 1
## [3,] 1 0 1 0 1
## [4,] 1 1 1 1 1
## [5,] 0 1 1 1 1
## [6,] 0 0 1 1 0
## [7,] 1 1 1 1 1
## [8,] 1 1 0 1 1
## [9,] 0 0 0 0 0
## [10,] 0 0 0 1 0
##
## $t
## [1] -0.73702846 1.06676908 -0.33373302 0.55168123 0.55682056
## [6] -0.98269231 1.35183104 -0.08724449 -2.06934458 -0.95628154
##
## $a
## [1] 0.9981694 1.0852205 1.1346718 0.7398013 0.7041300
##
## $b
## [1] -0.44867358 -0.23459489 0.56172826 -0.57306103 -0.06587859
##
## $c
## [1] 0.12161099 0.05746704 0.11697387 0.08501213 0.19766003
# generate 10 people and 5 4-category items using the GPCM model
# returns a list of 'u' (responses), 't' (ability), 'a' (item discrimination), 'b' (item difficulty), 'd' (category difficulty)
model_gpcm_gendata(n_p=10, n_i=5, n_c=4)
## $u
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0 0 3 2 2
## [2,] 0 2 3 3 3
## [3,] 0 2 3 2 3
## [4,] 0 3 3 2 2
## [5,] 0 0 2 2 2
## [6,] 0 0 0 1 2
## [7,] 0 0 3 2 3
## [8,] 0 2 3 2 3
## [9,] 2 2 3 3 3
## [10,] 1 2 3 3 2
##
## $t
## [1] 0.71580543 0.16112726 0.64190638 0.07528005 -0.77135585
## [6] -1.50696620 0.67370467 -0.26587520 1.48071264 0.06680653
##
## $a
## [1] 0.8828596 0.7483920 0.7613749 1.0379786 0.7388671
##
## $b
## [1] 1.2525124 0.3984016 -0.3909296 -0.5450076 -0.8616710
##
## $d
## [,1] [,2] [,3] [,4]
## [1,] 0 -0.7388216 0.1009258 0.6378959
## [2,] 0 0.5282723 0.8109394 -1.3392117
## [3,] 0 -1.9253146 1.5360154 0.3892992
## [4,] 0 0.5554536 0.0982999 -0.6537535
## [5,] 0 -0.2729555 1.1965345 -0.9235790
# generate 10 people and 5 4-category items using the GRM model
# returns a list of 'u' (responses), 't' (ability), 'a' (item discrimination), 'b' (item difficulty)
model_grm_gendata(n_p=10, n_i=5, n_c=4)
## $u
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0 0 3 2 0
## [2,] 3 0 0 1 0
## [3,] 1 0 0 0 0
## [4,] 0 1 3 2 3
## [5,] 0 1 3 1 0
## [6,] 0 1 3 0 0
## [7,] 0 1 0 0 0
## [8,] 0 0 0 0 0
## [9,] 3 3 3 3 3
## [10,] 3 3 3 2 3
##
## $t
## [1] 0.51726969 -0.05237931 -1.01519978 0.55510715 0.09494015
## [6] 0.37745550 -1.42838419 -0.98904435 1.59408026 0.84363219
##
## $a
## [1] 0.9589710 0.8230336 0.8624692 1.4175787 0.7818697
##
## $b
## [,1] [,2] [,3]
## [1,] -0.1377936 0.2826489 1.1662412
## [2,] -0.7830236 0.8469811 1.3414404
## [3,] -0.6945578 -0.3015947 0.4116890
## [4,] -0.6068594 0.3140926 0.9510116
## [5,] 0.4490137 0.4704692 0.5825619
Compute the response probability, information, and likelihood using the
3PL model:
# generate data
data <- model_3pl_gendata(n_p=10, n_i=5)
# compute P, returns a 2d array: people, item
with(data, model_3pl_prob(t, a, b, c, D=1.702))
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.3609685 0.1764401 0.1840398 0.3709739 0.4374978
## [2,] 0.5794828 0.3107135 0.4200524 0.6373857 0.7433939
## [3,] 0.8081244 0.5700697 0.8015521 0.9009955 0.9384708
## [4,] 0.8833279 0.7067422 0.9110997 0.9571548 0.9732607
## [5,] 0.6855249 0.4092094 0.5907217 0.7721538 0.8514355
## [6,] 0.7790424 0.5259854 0.7531887 0.8741972 0.9213424
## [7,] 0.8000765 0.5574380 0.7884019 0.8938458 0.9339416
## [8,] 0.4861962 0.2445303 0.2988594 0.5160330 0.6238394
## [9,] 0.9195797 0.7868290 0.9518709 0.9768332 0.9852998
## [10,] 0.8544134 0.6498925 0.8723540 0.9379205 0.9614693
# compute I, returns a 2d array: people, item
with(data, model_3pl_info(t, a, b, c, D=1.702))
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.3403429 0.1186590 0.2665103 0.35258616 0.76268353
## [2,] 0.3983636 0.2893610 0.8506998 0.72454197 0.68922947
## [3,] 0.2654653 0.4251170 0.6801583 0.35364008 0.21789268
## [4,] 0.1781829 0.3793221 0.3552253 0.16773003 0.09877748
## [5,] 0.3614820 0.3742370 0.9570325 0.63653053 0.46936217
## [6,] 0.2934284 0.4223236 0.7840917 0.42886754 0.27263357
## [7,] 0.2735266 0.4251867 0.7108104 0.37455976 0.23261817
## [8,] 0.3955448 0.2116424 0.6049034 0.63398417 0.81388471
## [9,] 0.1283906 0.3139195 0.2025074 0.09348562 0.05508126
## [10,] 0.2142975 0.4084190 0.4843669 0.23572650 0.14034014
# compute L, returns a 2d array: people, item
with(data, model_3pl_lh(u, t, a, b, c, D=1.702, log=FALSE))
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.6390315 0.8235599 0.8159602 0.3709739 0.56250220
## [2,] 0.4205172 0.3107135 0.5799476 0.6373857 0.74339389
## [3,] 0.8081244 0.5700697 0.8015521 0.9009955 0.06152921
## [4,] 0.8833279 0.7067422 0.9110997 0.9571548 0.97326069
## [5,] 0.3144751 0.5907906 0.5907217 0.7721538 0.85143552
## [6,] 0.7790424 0.5259854 0.7531887 0.8741972 0.92134241
## [7,] 0.8000765 0.5574380 0.7884019 0.8938458 0.93394157
## [8,] 0.5138038 0.7554697 0.2988594 0.4839670 0.62383941
## [9,] 0.9195797 0.2131710 0.9518709 0.9768332 0.98529985
## [10,] 0.8544134 0.6498925 0.8723540 0.9379205 0.96146932
Compute the response probability, information, and likelihood using the
GPCM model:
# generate data
data <- model_gpcm_gendata(n_p=10, n_i=5, n_c=3)
# compute P, returns a 3d array: people, item, category
with(data, model_gpcm_prob(t, a, b, d))
## , , 1
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.3354996 0.3652935 0.1803017 0.3065639 0.14992708
## [2,] 0.2985401 0.3361497 0.1492301 0.2814264 0.12651281
## [3,] 0.7177997 0.6750114 0.6914101 0.5907547 0.54048962
## [4,] 0.8003739 0.7517764 0.8176514 0.6711048 0.66582183
## [5,] 0.5839824 0.5610900 0.4797528 0.4808886 0.36894374
## [6,] 0.8704507 0.8228744 0.9076023 0.7523850 0.78122327
## [7,] 0.7230880 0.6797494 0.6998446 0.5955390 0.54809604
## [8,] 0.9408877 0.9033115 0.9716521 0.8562649 0.90096906
## [9,] 0.2431223 0.2914943 0.1085556 0.2430856 0.09502793
## [10,] 0.9820773 0.9598895 0.9940061 0.9402309 0.97016803
##
## , , 2
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.36503156 0.16592013 0.053514130 0.47205052 0.24887815
## [2,] 0.36499113 0.16430294 0.049778759 0.47469939 0.23550982
## [3,] 0.22772488 0.14121662 0.059724451 0.34712244 0.26723547
## [4,] 0.17181532 0.12301385 0.047762412 0.29057559 0.22426027
## [5,] 0.29989428 0.15891988 0.067129908 0.41173896 0.29283343
## [6,] 0.11757454 0.10060877 0.033335356 0.22679338 0.16689362
## [7,] 0.22440175 0.14024981 0.059133189 0.34395623 0.26518954
## [8,] 0.05654954 0.06636208 0.015859993 0.13705369 0.08684845
## [9,] 0.35843131 0.16028193 0.043678344 0.47464258 0.21263025
## [10,] 0.01768260 0.03303744 0.004851627 0.05865468 0.02860468
##
## , , 3
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.2994688236 0.468786388 0.766184179 0.221385533 0.601194776
## [2,] 0.3364687688 0.499547357 0.800991128 0.243874197 0.637977374
## [3,] 0.0544754115 0.183771980 0.248865490 0.062122880 0.192274905
## [4,] 0.0278108151 0.125209724 0.134586139 0.038319630 0.109917899
## [5,] 0.1161233128 0.279990130 0.453117273 0.107372439 0.338222827
## [6,] 0.0119747163 0.076516814 0.059062341 0.020821593 0.051883114
## [7,] 0.0525102576 0.180000821 0.241022178 0.060504756 0.186714417
## [8,] 0.0025627319 0.030326423 0.012487953 0.006681387 0.012182488
## [9,] 0.3984463764 0.548223813 0.847766024 0.282271808 0.692341828
## [10,] 0.0002400653 0.007073108 0.001142304 0.001114459 0.001227295
# compute I, returns a 3d array: people, item, category
with(data, model_gpcm_info(t, a, b, d))
## , , 1
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.62510798 0.34994060 0.32078354 0.2868053 0.2330967
## [2,] 0.55615673 0.31640124 0.23126049 0.2649041 0.1806756
## [3,] 0.70129697 0.48493425 1.51806565 0.3963648 0.9385987
## [4,] 0.54441048 0.42369269 1.17119999 0.3725997 0.8824578
## [5,] 0.82629926 0.49748643 1.31898354 0.3877298 0.7399292
## [6,] 0.37225526 0.32775629 0.66023454 0.3217658 0.6681504
## [7,] 0.69295794 0.48243369 1.50751977 0.3956758 0.9404498
## [8,] 0.17428301 0.18028022 0.18382284 0.2171626 0.3152420
## [9,] 0.44138899 0.26243458 0.13123522 0.2287843 0.1161982
## [10,] 0.05288157 0.06600743 0.02746983 0.1003566 0.0896758
##
## , , 2
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.6801323432 0.158946690 0.095209599 0.441625912 0.386939248
## [2,] 0.6799497618 0.154650300 0.077141670 0.446830226 0.336336444
## [3,] 0.2224893232 0.101451290 0.131131498 0.232900605 0.464073437
## [4,] 0.1168679643 0.069329213 0.068414648 0.161328598 0.297227007
## [5,] 0.4243319942 0.140905180 0.184560133 0.331975968 0.587287409
## [6,] 0.0502816994 0.040073135 0.024249777 0.096990693 0.142737723
## [7,] 0.2150512472 0.099538497 0.127377488 0.228524347 0.455025061
## [8,] 0.0104748141 0.013244346 0.003000487 0.034759033 0.030387588
## [9,] 0.6507326752 0.144303091 0.052803681 0.446718267 0.259999855
## [10,] 0.0009521487 0.002271841 0.000134077 0.006260574 0.002644024
##
## , , 3
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 5.579749e-01 0.4490838132 1.363156e+00 0.2071167904 0.9346977953
## [2,] 6.268148e-01 0.4701994429 1.241288e+00 0.2295565686 0.9111086937
## [3,] 5.322298e-02 0.1320234405 5.464111e-01 0.0416811321 0.3338990689
## [4,] 1.891678e-02 0.0705667840 1.927805e-01 0.0212751941 0.1456814771
## [5,] 1.643074e-01 0.2482512482 1.245754e+00 0.0865720095 0.6783173855
## [6,] 5.121084e-03 0.0304771504 4.296485e-02 0.0089045844 0.0443736418
## [7,] 5.032223e-02 0.1277507019 5.191805e-01 0.0401993293 0.3203736455
## [8,] 4.747013e-04 0.0060524573 2.362544e-03 0.0016945080 0.0042625565
## [9,] 7.233801e-01 0.4935702366 1.024882e+00 0.2656651093 0.8465812309
## [10,] 1.292671e-05 0.0004863869 3.156812e-05 0.0001189531 0.0001134429
# compute L, returns a 2d array: people, item
with(data, model_gpcm_lh(u, t, a, b, d, D=1.702, log=FALSE))
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.3354996 0.4687864 0.7661842 0.47205052 0.1499271
## [2,] 0.3364688 0.1643029 0.1492301 0.28142641 0.6379774
## [3,] 0.7177997 0.1412166 0.6914101 0.06212288 0.5404896
## [4,] 0.8003739 0.1230138 0.8176514 0.67110478 0.2242603
## [5,] 0.5839824 0.5610900 0.4797528 0.41173896 0.2928334
## [6,] 0.8704507 0.1006088 0.9076023 0.75238503 0.7812233
## [7,] 0.2244018 0.1402498 0.6998446 0.34395623 0.5480960
## [8,] 0.9408877 0.9033115 0.9716521 0.85626493 0.9009691
## [9,] 0.3584313 0.5482238 0.8477660 0.24308561 0.6923418
## [10,] 0.9820773 0.9598895 0.9940061 0.94023086 0.9701680
Compute the response probability, information, and likelihood using the
GRM model:
# generate data
data <- model_grm_gendata(n_p=10, n_i=5, n_c=3)
# compute P, returns a 3d array: people, item, category
with(data, model_grm_prob(t, a, b))
## , , 1
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.46874907 0.34455110 0.49171940 0.26674793 0.56119361
## [2,] 0.44431842 0.31904938 0.45589862 0.24242890 0.54044458
## [3,] 0.21262551 0.11645956 0.14664585 0.07224665 0.31813989
## [4,] 0.53704511 0.41984769 0.59054431 0.34187464 0.61751922
## [5,] 0.17317187 0.08921036 0.10601893 0.05297255 0.27313616
## [6,] 0.48843782 0.36565918 0.52049897 0.28731713 0.57766864
## [7,] 0.04084875 0.01500415 0.01146629 0.00698243 0.08824085
## [8,] 0.62687904 0.52731058 0.71234580 0.45692768 0.68879170
## [9,] 0.29473514 0.18002034 0.24530039 0.12087628 0.40361019
## [10,] 0.93566990 0.93283454 0.98299202 0.93322727 0.93293255
##
## , , 2
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.16802383 0.48707488 0.24493278 0.39807532 0.007197214
## [2,] 0.16938295 0.49584456 0.25193807 0.39324577 0.007263255
## [3,] 0.13656033 0.43680752 0.18530197 0.22579698 0.006385403
## [4,] 0.16036688 0.45194177 0.21604232 0.39718704 0.006891671
## [5,] 0.12067764 0.39004110 0.14932686 0.18073281 0.005851710
## [6,] 0.16638226 0.47848451 0.23787904 0.39999835 0.007126919
## [7,] 0.03716675 0.11020081 0.02098437 0.02994043 0.002384370
## [8,] 0.14260545 0.38559376 0.16511164 0.36410082 0.006241677
## [9,] 0.15891218 0.49348080 0.23918693 0.30759048 0.007067665
## [10,] 0.03088351 0.05956066 0.01105968 0.05381989 0.001809038
##
## , , 3
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.36322711 0.168374024 0.263347818 0.33517674 0.43160917
## [2,] 0.38629863 0.185106064 0.292163309 0.36432533 0.45229217
## [3,] 0.65081416 0.446732922 0.668052180 0.70195637 0.67547470
## [4,] 0.30258801 0.128210534 0.193413368 0.26093833 0.37558911
## [5,] 0.70615049 0.520748534 0.744654216 0.76629464 0.72101213
## [6,] 0.34517992 0.155856310 0.241621991 0.31268452 0.41520444
## [7,] 0.92198450 0.874795032 0.967549343 0.96307714 0.90937478
## [8,] 0.23051552 0.087095660 0.122542565 0.17897150 0.30496662
## [9,] 0.54635268 0.326498859 0.515512674 0.57153325 0.58932215
## [10,] 0.03344659 0.007604802 0.005948298 0.01295284 0.06525841
# compute I, returns a 3d array: people, item, category
with(data, model_grm_info(t, a, b))
## , , 1
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.165584696 0.1506680434 0.3713410462 0.1254504679 0.142182475
## [2,] 0.155616017 0.1342166699 0.3417048307 0.1070555131 0.138098167
## [3,] 0.050495390 0.0232034068 0.0554502641 0.0116435337 0.071003415
## [4,] 0.189408674 0.1980161457 0.4314660501 0.1849520926 0.150057693
## [5,] 0.035173014 0.0140353661 0.0303619659 0.0063897111 0.055790359
## [6,] 0.173123751 0.1642293077 0.3925220227 0.1414608340 0.144996863
## [7,] 0.002270306 0.0004293707 0.0003927101 0.0001164091 0.007304108
## [8,] 0.207996871 0.2544977139 0.4410493588 0.2726273728 0.151905911
## [9,] 0.086907196 0.0514542540 0.1372161895 0.0308851001 0.099954454
## [10,] 0.079891248 0.1131697336 0.0496577925 0.1398269392 0.060056492
##
## , , 2
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.114482231 0.345052773 0.3285438862 0.400499126 0.0036400270
## [2,] 0.116286471 0.353414896 0.3462623828 0.392635941 0.0037071305
## [3,] 0.076453862 0.296078467 0.1942308473 0.149976306 0.0028652165
## [4,] 0.104565262 0.310965094 0.2596840628 0.399050679 0.0033375419
## [5,] 0.060032177 0.249849780 0.1285584175 0.099625397 0.0024063009
## [6,] 0.112320672 0.336797173 0.3110991622 0.403638147 0.0035692736
## [7,] 0.005857932 0.026525154 0.0027094939 0.003059107 0.0003995339
## [8,] 0.083198055 0.245464944 0.1558735079 0.345883292 0.0027376904
## [9,] 0.102728942 0.351167970 0.3143033996 0.259704926 0.0035101722
## [10,] 0.004053222 0.008096164 0.0007562981 0.009718405 0.0002299882
##
## , , 3
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.119173784 0.0456513124 0.1543677829 0.179585314 0.10893717
## [2,] 0.129910150 0.0540651505 0.1825656883 0.202875917 0.11527491
## [3,] 0.209802311 0.2137990556 0.4476363341 0.353115631 0.15234017
## [4,] 0.090580038 0.0277481523 0.0911714229 0.120996634 0.09062419
## [5,] 0.207854369 0.2516486462 0.4278314788 0.329972014 0.14921695
## [6,] 0.110675808 0.0397045426 0.1337806767 0.161579307 0.10372316
## [7,] 0.094073610 0.1855278935 0.0917919261 0.082344447 0.07710501
## [8,] 0.058001583 0.0134089058 0.0398139009 0.063232865 0.06650556
## [9,] 0.192089736 0.1390198154 0.3890414302 0.336524730 0.14674202
## [10,] 0.001533802 0.0001111311 0.0001062745 0.000398185 0.00409555
# compute L, returns a 2d array: people, item
with(data, model_grm_lh(u, t, a, b, D=1.702, log=FALSE))
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.36322711 0.16837402 0.2449328 0.3351767 0.4316092
## [2,] 0.38629863 0.49584456 0.4558986 0.3643253 0.5404446
## [3,] 0.65081416 0.11645956 0.1466459 0.7019564 0.6754747
## [4,] 0.16036688 0.45194177 0.5905443 0.2609383 0.3755891
## [5,] 0.70615049 0.39004110 0.1493269 0.7662946 0.7210121
## [6,] 0.48843782 0.47848451 0.5204990 0.3999984 0.4152044
## [7,] 0.04084875 0.87479503 0.9675493 0.9630771 0.9093748
## [8,] 0.62687904 0.08709566 0.7123458 0.4569277 0.3049666
## [9,] 0.54635268 0.32649886 0.5155127 0.5715332 0.4036102
## [10,] 0.93566990 0.93283454 0.9829920 0.9332273 0.9329326
Compute the response probability, information, and likelihood using a
mixture of IRT models:
# generate data, returns response, theta, and a list of items
data <- model_mixed_gendata(n_p=10, n_3pl=5, n_gpcm=3, n_grm=3, n_c=3)
# compute P, returns a list
with(data, model_mixed_prob(t, items))
## $`3pl`
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.4638027 0.7876843 0.7775934 0.6172807 0.8715012
## [2,] 0.1005899 0.1675718 0.2532055 0.1756530 0.3408528
## [3,] 0.1485265 0.3676959 0.4545735 0.3166527 0.5898376
## [4,] 0.1362926 0.3276309 0.4197982 0.2908427 0.5514164
## [5,] 0.3948003 0.7384522 0.7392978 0.5731808 0.8443237
## [6,] 0.1612890 0.4046603 0.4853123 0.3401780 0.6223262
## [7,] 0.1798952 0.4516805 0.5229803 0.3700406 0.6603308
## [8,] 0.6311038 0.8746184 0.8501477 0.7114999 0.9190191
## [9,] 0.7605168 0.9238484 0.8966344 0.7818173 0.9467997
## [10,] 0.8717470 0.9594914 0.9354970 0.8493599 0.9683955
##
## $gpcm
## , , 1
##
## [,1] [,2] [,3]
## [1,] 0.036972214 0.6297609 0.40827616
## [2,] 0.853443457 0.9746159 0.98030083
## [3,] 0.475656048 0.9184826 0.90858017
## [4,] 0.549143806 0.9311171 0.92682974
## [5,] 0.058338965 0.6950175 0.51501079
## [6,] 0.411639188 0.9058485 0.88938454
## [7,] 0.336747012 0.8881083 0.86104800
## [8,] 0.012381074 0.4548380 0.19240789
## [9,] 0.004628683 0.2990495 0.08053891
## [10,] 0.001388117 0.1517540 0.02392218
##
## , , 2
##
## [,1] [,2] [,3]
## [1,] 0.11272074 0.24151671 0.27665133
## [2,] 0.11090037 0.02452634 0.01907532
## [3,] 0.23560227 0.07337181 0.07967640
## [4,] 0.22425853 0.06296540 0.06541128
## [5,] 0.13768281 0.21368175 0.26161976
## [6,] 0.24055113 0.08346395 0.09393907
## [7,] 0.24000812 0.09712894 0.11371220
## [8,] 0.06784200 0.28934900 0.25216058
## [9,] 0.04222662 0.29540788 0.18730724
## [10,] 0.02339068 0.25460374 0.11096530
##
## , , 3
##
## [,1] [,2] [,3]
## [1,] 0.85030705 0.1287223905 0.3150725099
## [2,] 0.03565618 0.0008577634 0.0006238544
## [3,] 0.28874169 0.0081455977 0.0117434351
## [4,] 0.22659767 0.0059174548 0.0077589757
## [5,] 0.80397823 0.0913007937 0.2233694490
## [6,] 0.34780968 0.0106875375 0.0166763875
## [7,] 0.42324487 0.0147627315 0.0252397932
## [8,] 0.91977693 0.2558130434 0.5554315244
## [9,] 0.95314470 0.4055426308 0.7321538489
## [10,] 0.97522120 0.5936422645 0.8651125217
##
##
## $grm
## , , 1
##
## [,1] [,2] [,3]
## [1,] 0.16198717 0.22075570 0.4532402
## [2,] 0.93053209 0.83939376 0.8625801
## [3,] 0.68944467 0.60290244 0.7267857
## [4,] 0.74207504 0.64471299 0.7506738
## [5,] 0.21423844 0.26411050 0.4941757
## [6,] 0.64001783 0.56582435 0.7052235
## [7,] 0.57657689 0.52034042 0.6780677
## [8,] 0.08083886 0.14150236 0.3626848
## [9,] 0.04246285 0.09331656 0.2909464
## [10,] 0.01907808 0.05516628 0.2167887
##
## , , 2
##
## [,1] [,2] [,3]
## [1,] 0.40192168 0.4646233 0.04412764
## [2,] 0.05843141 0.1363280 0.01967340
## [3,] 0.24746870 0.3182017 0.03371633
## [4,] 0.20853418 0.2884143 0.03165130
## [5,] 0.43164520 0.4699181 0.04418884
## [6,] 0.28242422 0.3434417 0.03542770
## [7,] 0.32450416 0.3726137 0.03737451
## [8,] 0.28957399 0.4174728 0.04183400
## [9,] 0.18632342 0.3484676 0.03782801
## [10,] 0.09605023 0.2546912 0.03156335
##
## , , 3
##
## [,1] [,2] [,3]
## [1,] 0.43609115 0.31462096 0.5026321
## [2,] 0.01103650 0.02427827 0.1177465
## [3,] 0.06308663 0.07889583 0.2394980
## [4,] 0.04939078 0.06687271 0.2176749
## [5,] 0.35411635 0.26597145 0.4616355
## [6,] 0.07755795 0.09073397 0.2593488
## [7,] 0.09891895 0.10704585 0.2845578
## [8,] 0.62958715 0.44102483 0.5954812
## [9,] 0.77121373 0.55821587 0.6712256
## [10,] 0.88487169 0.69014253 0.7516480
# compute I, returns a list when combine=FALSE or a 2d array when combine=TRUE
with(data, model_mixed_info(t, items, combine=TRUE))
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.775452241 0.4726678 0.2872265 0.30086135 0.21143290 0.55698184
## [2,] 0.005666158 0.1326179 0.1607771 0.09282275 0.26558509 0.54355573
## [3,] 0.096797338 0.4977745 0.3353109 0.21842316 0.40015026 1.79963732
## [4,] 0.067497120 0.4399661 0.3153444 0.19895303 0.39714881 1.65715044
## [5,] 0.684412242 0.5376474 0.3155627 0.30604296 0.24602566 0.75576322
## [6,] 0.129542690 0.5424508 0.3490453 0.23457042 0.39738759 1.86356898
## [7,] 0.179626123 0.5867696 0.3607629 0.25279992 0.38793540 1.85650163
## [8,] 0.820309938 0.3169700 0.2165739 0.26947236 0.14246668 0.26839541
## [9,] 0.678128375 0.2055471 0.1595932 0.22792844 0.09713821 0.14331448
## [10,] 0.430491908 0.1143631 0.1049195 0.17339460 0.05934461 0.06971581
## [,7] [,8] [,9] [,10] [,11]
## [1,] 0.93291961 2.2320363 1.2580533 0.6020251 0.2639822
## [2,] 0.05013278 0.0660799 0.2879373 0.2849627 0.1207019
## [3,] 0.18001324 0.3623153 0.9746404 0.5250644 0.2038961
## [4,] 0.14898978 0.2807689 0.8637916 0.4967046 0.1918171
## [5,] 0.77554800 2.0404707 1.2839438 0.6028639 0.2643314
## [6,] 0.21183370 0.4511204 1.0606431 0.5457333 0.2138661
## [7,] 0.25764036 0.5862647 1.1472089 0.5657805 0.2251630
## [8,] 1.23368651 1.9240616 1.0766585 0.5885706 0.2508616
## [9,] 1.27452120 1.2120876 0.7934486 0.5494808 0.2277877
## [10,] 1.01139893 0.5666544 0.4544041 0.4594141 0.1913015
# compute L, returns a list when combine=FALSE or a 2d array when combine=TRUE
with(data, model_mixed_lh(u, t, items, combine=TRUE))
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.5361973 0.78768427 0.7775934 0.6172807 0.8715012 0.8503070
## [2,] 0.8994101 0.83242816 0.7467945 0.8243470 0.6591472 0.8534435
## [3,] 0.8514735 0.36769585 0.5454265 0.3166527 0.5898376 0.2887417
## [4,] 0.1362926 0.67236911 0.4197982 0.7091573 0.4485836 0.5491438
## [5,] 0.3948003 0.73845224 0.7392978 0.5731808 0.8443237 0.8039782
## [6,] 0.8387110 0.59533969 0.5146877 0.3401780 0.6223262 0.2405511
## [7,] 0.8201048 0.54831954 0.5229803 0.6299594 0.6603308 0.4232449
## [8,] 0.6311038 0.87461842 0.8501477 0.7114999 0.9190191 0.9197769
## [9,] 0.2394832 0.92384844 0.8966344 0.7818173 0.9467997 0.9531447
## [10,] 0.8717470 0.04050863 0.9354970 0.8493599 0.9683955 0.9752212
## [,7] [,8] [,9] [,10] [,11]
## [1,] 0.6297609 0.4082762 0.1619872 0.2207557 0.4532402
## [2,] 0.9746159 0.9803008 0.9305321 0.8393938 0.8625801
## [3,] 0.9184826 0.9085802 0.6894447 0.6029024 0.7267857
## [4,] 0.0629654 0.9268297 0.7420750 0.6447130 0.2176749
## [5,] 0.2136817 0.5150108 0.4316452 0.2641105 0.4941757
## [6,] 0.9058485 0.8893845 0.6400178 0.5658244 0.2593488
## [7,] 0.8881083 0.8610480 0.3245042 0.1070459 0.6780677
## [8,] 0.2893490 0.5554315 0.6295871 0.4410248 0.3626848
## [9,] 0.2954079 0.1873072 0.7712137 0.3484676 0.2909464
## [10,] 0.5936423 0.8651125 0.8848717 0.6901425 0.2167887
Visualization
Visualization is essential in IRT data analysis, and this package makes
it really easy to produce high-quality characteristic curves and
information function curves at both item and test level.
For example, users can draw ICCs and IIFs for a set of 3PL items:
data <- model_3pl_gendata(1, 5)
# item characteristic curves
with(data, model_3pl_plot(a, b, c, type='prob', total=FALSE))
# test characteristic curves
with(data, model_3pl_plot(a, b, c, type='prob', total=TRUE))
# item information funciton curves
with(data, model_3pl_plot(a, b, c, type='info', total=FALSE))
# test information function curves
with(data, model_3pl_plot(a, b, c, type='info', total=TRUE))
Draw ICCs and IIFs for a set of GPCM items:
# Reproduce Figure 1 in Muraki's 1992 APM article
b <- matrix(c(-2, 0, 2, -.5, 0, 2, -.5, 0, 2), nrow=3, byrow=TRUE)
model_gpcm_plot(a=c(1, 1, .7), b=rowMeans(b), d=rowMeans(b)-b, D=1.0, d0=0)
# Reproduce Figure 2 in Muraki's 1992 APM article
b <- matrix(c(.5, 0, NA, 0, 0, 0), nrow=2, byrow=TRUE)
model_gpcm_plot(a=.7, b=rowMeans(b, na.rm=TRUE), d=rowMeans(b, na.rm=TRUE)-b, D=1.0, d0=0)
# Reproduce Figure 3 in Muraki's 1992 APM article
b <- matrix(c(1.759, -1.643, 3.970, -2.764), nrow=2, byrow=TRUE)
model_gpcm_plot(a=c(.778,.946), b=rowMeans(b), d=rowMeans(b)-b, D=1.0, d0=0)
# Reproduce Figure 1 in Muraki's 1993 APM article
b <- matrix(c(0, -2, 4, 0, -2, 2, 0, -2, 0, 0, -2, -2, 0, -2, -4), nrow=5, byrow=TRUE)
model_gpcm_plot(a=1, b=rowMeans(b), d=rowMeans(b)-b, D=1.0)
# Reproduce Figure 2 in Muraki's 1993 APM article
b <- matrix(c(0, -2, 4, 0, -2, 2, 0, -2, 0, 0, -2, -2, 0, -2, -4), nrow=5, byrow=TRUE)
model_gpcm_plot(a=1, b=rowMeans(b), d=rowMeans(b)-b, D=1.0, type='info', item_level=TRUE)
Draw ICCs and IIFs for a set of GRM items:
data <- model_grm_gendata(1, 3, n_c=3)
# item category characteristic curves
with(data, model_grm_plot(a, b, type='prob', total=FALSE))
# item category information funciton curves
with(data, model_grm_plot(a, b, type='info', total=FALSE))
Estimation
IRT models describe the interaction between people and items using
probabilistic functions. However, neither people nor item parameters are
naturally known in reality and they have to be estimated based on the
observed response data. Thus, parameter estimation, which calibrates
items and scores response vectors, is the centerpiece of data analysis
in IRT.
Here are some examples of estimating model parameters using the 3PL
model in this package.
# generate data
data <- model_3pl_gendata(2000, 40)
# score response vectors using EAP
scores <- with(data, model_3pl_eap(u, a, b, c))
round(c(corr=cor(data$t, scores$t), rmse=rmse(data$t, scores$t)), 2)
## corr rmse
## 0.96 0.28
# score response vectors using MAP
scores <- with(data, model_3pl_map(u, a, b, c))
round(c(corr=cor(data$t, scores$t), rmse=rmse(data$t, scores$t)), 2)
## corr rmse
## 0.96 0.28
# calibrate items using joint MLE
calib <- with(data, model_3pl_jmle(u, true_params=data))
## t: corr = 0.959, rmse = 0.287
## a: corr = 0.905, rmse = 0.129
## b: corr = 0.98, rmse = 0.127
## c: corr = 0.489, rmse = 0.06
# calibrate items using marginal MLE
calib <- with(data, model_3pl_mmle(u, true_params=data))
## t: corr = 0.958, rmse = 0.287
## a: corr = 0.932, rmse = 0.089
## b: corr = 0.98, rmse = 0.118
## c: corr = 0.424, rmse = 0.056
# calibrate items using joint MLE and fix c=0
calib <- with(data, model_3pl_jmle(u, c=0, true_params=data))
## t: corr = 0.955, rmse = 0.298
## a: corr = 0.81, rmse = 0.185
## b: corr = 0.976, rmse = 0.23
# calibrate items using joint MLE and fix a=1 and c=0
calib <- with(data, model_3pl_jmle(u, a=1, c=0, true_params=data))
## t: corr = 0.955, rmse = 0.3
## b: corr = 0.976, rmse = 0.232
Here are examples of estimating parameters using the GPCM model.
# generate data
data <- model_gpcm_gendata(1000, 20, n_c=3)
# score response vectors using EAP
scores <- with(data, model_gpcm_eap(u, a, b, d))
round(c(corr=cor(data$t, scores$t), rmse=rmse(data$t, scores$t)), 2)
## corr rmse
## 0.96 0.28
# score response vectors using MAP
scores <- with(data, model_gpcm_map(u, a, b, d))
round(c(corr=cor(data$t, scores$t), rmse=rmse(data$t, scores$t)), 2)
## corr rmse
## 0.96 0.27
# calibrate items using joint MLE
calib <- with(data, model_gpcm_jmle(u, true_params=data))
## t: corr = 0.962, rmse = 0.276
## a: corr = 0.963, rmse = 0.092
## b: corr = 0.998, rmse = 0.07
## d_2: corr = 0.993, rmse = 0.089
## d_3: corr = 0.993, rmse = 0.089
# calibrate items using marginal MLE
calib <- with(data, model_gpcm_mmle(u, true_params=data))
## t: corr = 0.96, rmse = 0.283
## a: corr = 0.959, rmse = 0.076
## b: corr = 0.999, rmse = 0.085
## d_2: corr = 0.994, rmse = 0.065
## d_3: corr = 0.994, rmse = 0.065
Here are examples of estimating parameters using the GRM model.
# generate data
data <- model_grm_gendata(1000, 20, n_c=3)
# score response vectors using EAP
scores <- with(data, model_grm_eap(u, a, b))
round(c(corr=cor(data$t, scores$t), rmse=rmse(data$t, scores$t)), 2)
## corr rmse
## 0.95 0.30
# score response vectors using MAP
scores <- with(data, model_grm_map(u, a, b))
round(c(corr=cor(data$t, scores$t), rmse=rmse(data$t, scores$t)), 2)
## corr rmse
## 0.95 0.30
# calibrate items using joint MLE
calib <- with(data, model_grm_jmle(u, true_params=data))
## t: corr = 0.953, rmse = 0.308
## a: corr = 0.95, rmse = 0.11
## b_1: corr = 0.997, rmse = 0.082
## b_2: corr = 0.997, rmse = 0.061
# calibrate items using marginal MLE
calib <- with(data, model_grm_mmle(u, true_params=data))
## t: corr = 0.953, rmse = 0.3
## a: corr = 0.952, rmse = 0.058
## b_1: corr = 0.997, rmse = 0.057
## b_2: corr = 0.997, rmse = 0.053
Here are examples of estimating parameters using the mixed model.
# generate data
data <- model_mixed_gendata(2000, n_3pl=40, n_gpcm=10, n_grm=10, n_c=3)
# score response vectors using EAP
scores <- with(data, model_mixed_eap(u, items))
round(c(corr=cor(data$t, scores$t), rmse=rmse(data$t, scores$t)), 2)
## corr rmse
## 0.97 0.24
# score response vectors using MAP
scores <- with(data, model_mixed_map(u, items))
round(c(corr=cor(data$t, scores$t), rmse=rmse(data$t, scores$t)), 2)
## corr rmse
## 0.98 0.22
Getting help
If you encounter a bug, please post a code example that exposes the bug
on github. You can post your
questions and feature requests on
github or contact the
author.
xluo11/Rirt documentation built on Nov. 5, 2019, 12:29 p.m.
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Rirt: data analysis and parameter estimation using item response theory
Overview
Rirt provides an interface for conducting data analysis and parameter estimation using item respone theory (IRT) efficiently in R. A variety of unidimensional IRT models is included, such as the 3-parameter-logstic (3PL) model, the generalized partial credit model (GPCM), the graded response model (GRM), and a mixture of the aforementioned models.
Installation
Install the stable version from CRAN:
install.packages("Rirt")
Install the most recent version from github:
devtools::install_github("xluo11/Rirt")
Quickstart
A common interface is created to make the package easy to use:
model_[3pl/gpcm/grm/mixed]_prob: compute the response probability for the given ability and item parametersmodel_[3pl/gpcm/grm/mixed]_info: compute the information for the given ability and item parametersmodel_[3pl/gpcm/grm/mixed]_lh: compute the response likelihood for the given ability and item parametersmodel_[3pl/gpcm/grm/mixed]_gendata: generate response data and parameters from an IRT modelmodel_[3pl/gpcm/grm]_plot: plot item characteristic curves (ICC) or item informaiton functions (IIFs)model_[3pl/gpcm/grm]_rescale: rescale ability and item parameters to a given scalemodel_[3pl/gpcm/grm/mixed]_eap: score response vectors using the EAP methodmodel_[3pl/gpcm/grm/mixed]_map: score response vectors using the MAP methodmodel_[3pl/gpcm/grm]_jmle: estimate parameters using the joint maximum likelihood estimation methodmodel_[3pl/gpcm/grm]_mmle: estimate parameters using the marginal maximum likelihood estimation method
For example, users can quickly generate data from an IRT model of choice:
# generate 10 people and 5 items using the 3PL model
# returns a list of 'u' (responses), 't' (ability), 'a' (item discrimination), 'b' (item difficulty), 'c' (pseudo-guessing)
model_3pl_gendata(n_p=10, n_i=5)
## $u
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0 0 0 1 1
## [2,] 1 1 1 0 1
## [3,] 1 0 1 0 1
## [4,] 1 1 1 1 1
## [5,] 0 1 1 1 1
## [6,] 0 0 1 1 0
## [7,] 1 1 1 1 1
## [8,] 1 1 0 1 1
## [9,] 0 0 0 0 0
## [10,] 0 0 0 1 0
##
## $t
## [1] -0.73702846 1.06676908 -0.33373302 0.55168123 0.55682056
## [6] -0.98269231 1.35183104 -0.08724449 -2.06934458 -0.95628154
##
## $a
## [1] 0.9981694 1.0852205 1.1346718 0.7398013 0.7041300
##
## $b
## [1] -0.44867358 -0.23459489 0.56172826 -0.57306103 -0.06587859
##
## $c
## [1] 0.12161099 0.05746704 0.11697387 0.08501213 0.19766003
# generate 10 people and 5 4-category items using the GPCM model
# returns a list of 'u' (responses), 't' (ability), 'a' (item discrimination), 'b' (item difficulty), 'd' (category difficulty)
model_gpcm_gendata(n_p=10, n_i=5, n_c=4)
## $u
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0 0 3 2 2
## [2,] 0 2 3 3 3
## [3,] 0 2 3 2 3
## [4,] 0 3 3 2 2
## [5,] 0 0 2 2 2
## [6,] 0 0 0 1 2
## [7,] 0 0 3 2 3
## [8,] 0 2 3 2 3
## [9,] 2 2 3 3 3
## [10,] 1 2 3 3 2
##
## $t
## [1] 0.71580543 0.16112726 0.64190638 0.07528005 -0.77135585
## [6] -1.50696620 0.67370467 -0.26587520 1.48071264 0.06680653
##
## $a
## [1] 0.8828596 0.7483920 0.7613749 1.0379786 0.7388671
##
## $b
## [1] 1.2525124 0.3984016 -0.3909296 -0.5450076 -0.8616710
##
## $d
## [,1] [,2] [,3] [,4]
## [1,] 0 -0.7388216 0.1009258 0.6378959
## [2,] 0 0.5282723 0.8109394 -1.3392117
## [3,] 0 -1.9253146 1.5360154 0.3892992
## [4,] 0 0.5554536 0.0982999 -0.6537535
## [5,] 0 -0.2729555 1.1965345 -0.9235790
# generate 10 people and 5 4-category items using the GRM model
# returns a list of 'u' (responses), 't' (ability), 'a' (item discrimination), 'b' (item difficulty)
model_grm_gendata(n_p=10, n_i=5, n_c=4)
## $u
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0 0 3 2 0
## [2,] 3 0 0 1 0
## [3,] 1 0 0 0 0
## [4,] 0 1 3 2 3
## [5,] 0 1 3 1 0
## [6,] 0 1 3 0 0
## [7,] 0 1 0 0 0
## [8,] 0 0 0 0 0
## [9,] 3 3 3 3 3
## [10,] 3 3 3 2 3
##
## $t
## [1] 0.51726969 -0.05237931 -1.01519978 0.55510715 0.09494015
## [6] 0.37745550 -1.42838419 -0.98904435 1.59408026 0.84363219
##
## $a
## [1] 0.9589710 0.8230336 0.8624692 1.4175787 0.7818697
##
## $b
## [,1] [,2] [,3]
## [1,] -0.1377936 0.2826489 1.1662412
## [2,] -0.7830236 0.8469811 1.3414404
## [3,] -0.6945578 -0.3015947 0.4116890
## [4,] -0.6068594 0.3140926 0.9510116
## [5,] 0.4490137 0.4704692 0.5825619
Compute the response probability, information, and likelihood using the 3PL model:
# generate data
data <- model_3pl_gendata(n_p=10, n_i=5)
# compute P, returns a 2d array: people, item
with(data, model_3pl_prob(t, a, b, c, D=1.702))
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.3609685 0.1764401 0.1840398 0.3709739 0.4374978
## [2,] 0.5794828 0.3107135 0.4200524 0.6373857 0.7433939
## [3,] 0.8081244 0.5700697 0.8015521 0.9009955 0.9384708
## [4,] 0.8833279 0.7067422 0.9110997 0.9571548 0.9732607
## [5,] 0.6855249 0.4092094 0.5907217 0.7721538 0.8514355
## [6,] 0.7790424 0.5259854 0.7531887 0.8741972 0.9213424
## [7,] 0.8000765 0.5574380 0.7884019 0.8938458 0.9339416
## [8,] 0.4861962 0.2445303 0.2988594 0.5160330 0.6238394
## [9,] 0.9195797 0.7868290 0.9518709 0.9768332 0.9852998
## [10,] 0.8544134 0.6498925 0.8723540 0.9379205 0.9614693
# compute I, returns a 2d array: people, item
with(data, model_3pl_info(t, a, b, c, D=1.702))
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.3403429 0.1186590 0.2665103 0.35258616 0.76268353
## [2,] 0.3983636 0.2893610 0.8506998 0.72454197 0.68922947
## [3,] 0.2654653 0.4251170 0.6801583 0.35364008 0.21789268
## [4,] 0.1781829 0.3793221 0.3552253 0.16773003 0.09877748
## [5,] 0.3614820 0.3742370 0.9570325 0.63653053 0.46936217
## [6,] 0.2934284 0.4223236 0.7840917 0.42886754 0.27263357
## [7,] 0.2735266 0.4251867 0.7108104 0.37455976 0.23261817
## [8,] 0.3955448 0.2116424 0.6049034 0.63398417 0.81388471
## [9,] 0.1283906 0.3139195 0.2025074 0.09348562 0.05508126
## [10,] 0.2142975 0.4084190 0.4843669 0.23572650 0.14034014
# compute L, returns a 2d array: people, item
with(data, model_3pl_lh(u, t, a, b, c, D=1.702, log=FALSE))
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.6390315 0.8235599 0.8159602 0.3709739 0.56250220
## [2,] 0.4205172 0.3107135 0.5799476 0.6373857 0.74339389
## [3,] 0.8081244 0.5700697 0.8015521 0.9009955 0.06152921
## [4,] 0.8833279 0.7067422 0.9110997 0.9571548 0.97326069
## [5,] 0.3144751 0.5907906 0.5907217 0.7721538 0.85143552
## [6,] 0.7790424 0.5259854 0.7531887 0.8741972 0.92134241
## [7,] 0.8000765 0.5574380 0.7884019 0.8938458 0.93394157
## [8,] 0.5138038 0.7554697 0.2988594 0.4839670 0.62383941
## [9,] 0.9195797 0.2131710 0.9518709 0.9768332 0.98529985
## [10,] 0.8544134 0.6498925 0.8723540 0.9379205 0.96146932
Compute the response probability, information, and likelihood using the GPCM model:
# generate data
data <- model_gpcm_gendata(n_p=10, n_i=5, n_c=3)
# compute P, returns a 3d array: people, item, category
with(data, model_gpcm_prob(t, a, b, d))
## , , 1
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.3354996 0.3652935 0.1803017 0.3065639 0.14992708
## [2,] 0.2985401 0.3361497 0.1492301 0.2814264 0.12651281
## [3,] 0.7177997 0.6750114 0.6914101 0.5907547 0.54048962
## [4,] 0.8003739 0.7517764 0.8176514 0.6711048 0.66582183
## [5,] 0.5839824 0.5610900 0.4797528 0.4808886 0.36894374
## [6,] 0.8704507 0.8228744 0.9076023 0.7523850 0.78122327
## [7,] 0.7230880 0.6797494 0.6998446 0.5955390 0.54809604
## [8,] 0.9408877 0.9033115 0.9716521 0.8562649 0.90096906
## [9,] 0.2431223 0.2914943 0.1085556 0.2430856 0.09502793
## [10,] 0.9820773 0.9598895 0.9940061 0.9402309 0.97016803
##
## , , 2
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.36503156 0.16592013 0.053514130 0.47205052 0.24887815
## [2,] 0.36499113 0.16430294 0.049778759 0.47469939 0.23550982
## [3,] 0.22772488 0.14121662 0.059724451 0.34712244 0.26723547
## [4,] 0.17181532 0.12301385 0.047762412 0.29057559 0.22426027
## [5,] 0.29989428 0.15891988 0.067129908 0.41173896 0.29283343
## [6,] 0.11757454 0.10060877 0.033335356 0.22679338 0.16689362
## [7,] 0.22440175 0.14024981 0.059133189 0.34395623 0.26518954
## [8,] 0.05654954 0.06636208 0.015859993 0.13705369 0.08684845
## [9,] 0.35843131 0.16028193 0.043678344 0.47464258 0.21263025
## [10,] 0.01768260 0.03303744 0.004851627 0.05865468 0.02860468
##
## , , 3
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.2994688236 0.468786388 0.766184179 0.221385533 0.601194776
## [2,] 0.3364687688 0.499547357 0.800991128 0.243874197 0.637977374
## [3,] 0.0544754115 0.183771980 0.248865490 0.062122880 0.192274905
## [4,] 0.0278108151 0.125209724 0.134586139 0.038319630 0.109917899
## [5,] 0.1161233128 0.279990130 0.453117273 0.107372439 0.338222827
## [6,] 0.0119747163 0.076516814 0.059062341 0.020821593 0.051883114
## [7,] 0.0525102576 0.180000821 0.241022178 0.060504756 0.186714417
## [8,] 0.0025627319 0.030326423 0.012487953 0.006681387 0.012182488
## [9,] 0.3984463764 0.548223813 0.847766024 0.282271808 0.692341828
## [10,] 0.0002400653 0.007073108 0.001142304 0.001114459 0.001227295
# compute I, returns a 3d array: people, item, category
with(data, model_gpcm_info(t, a, b, d))
## , , 1
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.62510798 0.34994060 0.32078354 0.2868053 0.2330967
## [2,] 0.55615673 0.31640124 0.23126049 0.2649041 0.1806756
## [3,] 0.70129697 0.48493425 1.51806565 0.3963648 0.9385987
## [4,] 0.54441048 0.42369269 1.17119999 0.3725997 0.8824578
## [5,] 0.82629926 0.49748643 1.31898354 0.3877298 0.7399292
## [6,] 0.37225526 0.32775629 0.66023454 0.3217658 0.6681504
## [7,] 0.69295794 0.48243369 1.50751977 0.3956758 0.9404498
## [8,] 0.17428301 0.18028022 0.18382284 0.2171626 0.3152420
## [9,] 0.44138899 0.26243458 0.13123522 0.2287843 0.1161982
## [10,] 0.05288157 0.06600743 0.02746983 0.1003566 0.0896758
##
## , , 2
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.6801323432 0.158946690 0.095209599 0.441625912 0.386939248
## [2,] 0.6799497618 0.154650300 0.077141670 0.446830226 0.336336444
## [3,] 0.2224893232 0.101451290 0.131131498 0.232900605 0.464073437
## [4,] 0.1168679643 0.069329213 0.068414648 0.161328598 0.297227007
## [5,] 0.4243319942 0.140905180 0.184560133 0.331975968 0.587287409
## [6,] 0.0502816994 0.040073135 0.024249777 0.096990693 0.142737723
## [7,] 0.2150512472 0.099538497 0.127377488 0.228524347 0.455025061
## [8,] 0.0104748141 0.013244346 0.003000487 0.034759033 0.030387588
## [9,] 0.6507326752 0.144303091 0.052803681 0.446718267 0.259999855
## [10,] 0.0009521487 0.002271841 0.000134077 0.006260574 0.002644024
##
## , , 3
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 5.579749e-01 0.4490838132 1.363156e+00 0.2071167904 0.9346977953
## [2,] 6.268148e-01 0.4701994429 1.241288e+00 0.2295565686 0.9111086937
## [3,] 5.322298e-02 0.1320234405 5.464111e-01 0.0416811321 0.3338990689
## [4,] 1.891678e-02 0.0705667840 1.927805e-01 0.0212751941 0.1456814771
## [5,] 1.643074e-01 0.2482512482 1.245754e+00 0.0865720095 0.6783173855
## [6,] 5.121084e-03 0.0304771504 4.296485e-02 0.0089045844 0.0443736418
## [7,] 5.032223e-02 0.1277507019 5.191805e-01 0.0401993293 0.3203736455
## [8,] 4.747013e-04 0.0060524573 2.362544e-03 0.0016945080 0.0042625565
## [9,] 7.233801e-01 0.4935702366 1.024882e+00 0.2656651093 0.8465812309
## [10,] 1.292671e-05 0.0004863869 3.156812e-05 0.0001189531 0.0001134429
# compute L, returns a 2d array: people, item
with(data, model_gpcm_lh(u, t, a, b, d, D=1.702, log=FALSE))
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.3354996 0.4687864 0.7661842 0.47205052 0.1499271
## [2,] 0.3364688 0.1643029 0.1492301 0.28142641 0.6379774
## [3,] 0.7177997 0.1412166 0.6914101 0.06212288 0.5404896
## [4,] 0.8003739 0.1230138 0.8176514 0.67110478 0.2242603
## [5,] 0.5839824 0.5610900 0.4797528 0.41173896 0.2928334
## [6,] 0.8704507 0.1006088 0.9076023 0.75238503 0.7812233
## [7,] 0.2244018 0.1402498 0.6998446 0.34395623 0.5480960
## [8,] 0.9408877 0.9033115 0.9716521 0.85626493 0.9009691
## [9,] 0.3584313 0.5482238 0.8477660 0.24308561 0.6923418
## [10,] 0.9820773 0.9598895 0.9940061 0.94023086 0.9701680
Compute the response probability, information, and likelihood using the GRM model:
# generate data
data <- model_grm_gendata(n_p=10, n_i=5, n_c=3)
# compute P, returns a 3d array: people, item, category
with(data, model_grm_prob(t, a, b))
## , , 1
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.46874907 0.34455110 0.49171940 0.26674793 0.56119361
## [2,] 0.44431842 0.31904938 0.45589862 0.24242890 0.54044458
## [3,] 0.21262551 0.11645956 0.14664585 0.07224665 0.31813989
## [4,] 0.53704511 0.41984769 0.59054431 0.34187464 0.61751922
## [5,] 0.17317187 0.08921036 0.10601893 0.05297255 0.27313616
## [6,] 0.48843782 0.36565918 0.52049897 0.28731713 0.57766864
## [7,] 0.04084875 0.01500415 0.01146629 0.00698243 0.08824085
## [8,] 0.62687904 0.52731058 0.71234580 0.45692768 0.68879170
## [9,] 0.29473514 0.18002034 0.24530039 0.12087628 0.40361019
## [10,] 0.93566990 0.93283454 0.98299202 0.93322727 0.93293255
##
## , , 2
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.16802383 0.48707488 0.24493278 0.39807532 0.007197214
## [2,] 0.16938295 0.49584456 0.25193807 0.39324577 0.007263255
## [3,] 0.13656033 0.43680752 0.18530197 0.22579698 0.006385403
## [4,] 0.16036688 0.45194177 0.21604232 0.39718704 0.006891671
## [5,] 0.12067764 0.39004110 0.14932686 0.18073281 0.005851710
## [6,] 0.16638226 0.47848451 0.23787904 0.39999835 0.007126919
## [7,] 0.03716675 0.11020081 0.02098437 0.02994043 0.002384370
## [8,] 0.14260545 0.38559376 0.16511164 0.36410082 0.006241677
## [9,] 0.15891218 0.49348080 0.23918693 0.30759048 0.007067665
## [10,] 0.03088351 0.05956066 0.01105968 0.05381989 0.001809038
##
## , , 3
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.36322711 0.168374024 0.263347818 0.33517674 0.43160917
## [2,] 0.38629863 0.185106064 0.292163309 0.36432533 0.45229217
## [3,] 0.65081416 0.446732922 0.668052180 0.70195637 0.67547470
## [4,] 0.30258801 0.128210534 0.193413368 0.26093833 0.37558911
## [5,] 0.70615049 0.520748534 0.744654216 0.76629464 0.72101213
## [6,] 0.34517992 0.155856310 0.241621991 0.31268452 0.41520444
## [7,] 0.92198450 0.874795032 0.967549343 0.96307714 0.90937478
## [8,] 0.23051552 0.087095660 0.122542565 0.17897150 0.30496662
## [9,] 0.54635268 0.326498859 0.515512674 0.57153325 0.58932215
## [10,] 0.03344659 0.007604802 0.005948298 0.01295284 0.06525841
# compute I, returns a 3d array: people, item, category
with(data, model_grm_info(t, a, b))
## , , 1
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.165584696 0.1506680434 0.3713410462 0.1254504679 0.142182475
## [2,] 0.155616017 0.1342166699 0.3417048307 0.1070555131 0.138098167
## [3,] 0.050495390 0.0232034068 0.0554502641 0.0116435337 0.071003415
## [4,] 0.189408674 0.1980161457 0.4314660501 0.1849520926 0.150057693
## [5,] 0.035173014 0.0140353661 0.0303619659 0.0063897111 0.055790359
## [6,] 0.173123751 0.1642293077 0.3925220227 0.1414608340 0.144996863
## [7,] 0.002270306 0.0004293707 0.0003927101 0.0001164091 0.007304108
## [8,] 0.207996871 0.2544977139 0.4410493588 0.2726273728 0.151905911
## [9,] 0.086907196 0.0514542540 0.1372161895 0.0308851001 0.099954454
## [10,] 0.079891248 0.1131697336 0.0496577925 0.1398269392 0.060056492
##
## , , 2
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.114482231 0.345052773 0.3285438862 0.400499126 0.0036400270
## [2,] 0.116286471 0.353414896 0.3462623828 0.392635941 0.0037071305
## [3,] 0.076453862 0.296078467 0.1942308473 0.149976306 0.0028652165
## [4,] 0.104565262 0.310965094 0.2596840628 0.399050679 0.0033375419
## [5,] 0.060032177 0.249849780 0.1285584175 0.099625397 0.0024063009
## [6,] 0.112320672 0.336797173 0.3110991622 0.403638147 0.0035692736
## [7,] 0.005857932 0.026525154 0.0027094939 0.003059107 0.0003995339
## [8,] 0.083198055 0.245464944 0.1558735079 0.345883292 0.0027376904
## [9,] 0.102728942 0.351167970 0.3143033996 0.259704926 0.0035101722
## [10,] 0.004053222 0.008096164 0.0007562981 0.009718405 0.0002299882
##
## , , 3
##
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.119173784 0.0456513124 0.1543677829 0.179585314 0.10893717
## [2,] 0.129910150 0.0540651505 0.1825656883 0.202875917 0.11527491
## [3,] 0.209802311 0.2137990556 0.4476363341 0.353115631 0.15234017
## [4,] 0.090580038 0.0277481523 0.0911714229 0.120996634 0.09062419
## [5,] 0.207854369 0.2516486462 0.4278314788 0.329972014 0.14921695
## [6,] 0.110675808 0.0397045426 0.1337806767 0.161579307 0.10372316
## [7,] 0.094073610 0.1855278935 0.0917919261 0.082344447 0.07710501
## [8,] 0.058001583 0.0134089058 0.0398139009 0.063232865 0.06650556
## [9,] 0.192089736 0.1390198154 0.3890414302 0.336524730 0.14674202
## [10,] 0.001533802 0.0001111311 0.0001062745 0.000398185 0.00409555
# compute L, returns a 2d array: people, item
with(data, model_grm_lh(u, t, a, b, D=1.702, log=FALSE))
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.36322711 0.16837402 0.2449328 0.3351767 0.4316092
## [2,] 0.38629863 0.49584456 0.4558986 0.3643253 0.5404446
## [3,] 0.65081416 0.11645956 0.1466459 0.7019564 0.6754747
## [4,] 0.16036688 0.45194177 0.5905443 0.2609383 0.3755891
## [5,] 0.70615049 0.39004110 0.1493269 0.7662946 0.7210121
## [6,] 0.48843782 0.47848451 0.5204990 0.3999984 0.4152044
## [7,] 0.04084875 0.87479503 0.9675493 0.9630771 0.9093748
## [8,] 0.62687904 0.08709566 0.7123458 0.4569277 0.3049666
## [9,] 0.54635268 0.32649886 0.5155127 0.5715332 0.4036102
## [10,] 0.93566990 0.93283454 0.9829920 0.9332273 0.9329326
Compute the response probability, information, and likelihood using a mixture of IRT models:
# generate data, returns response, theta, and a list of items
data <- model_mixed_gendata(n_p=10, n_3pl=5, n_gpcm=3, n_grm=3, n_c=3)
# compute P, returns a list
with(data, model_mixed_prob(t, items))
## $`3pl`
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.4638027 0.7876843 0.7775934 0.6172807 0.8715012
## [2,] 0.1005899 0.1675718 0.2532055 0.1756530 0.3408528
## [3,] 0.1485265 0.3676959 0.4545735 0.3166527 0.5898376
## [4,] 0.1362926 0.3276309 0.4197982 0.2908427 0.5514164
## [5,] 0.3948003 0.7384522 0.7392978 0.5731808 0.8443237
## [6,] 0.1612890 0.4046603 0.4853123 0.3401780 0.6223262
## [7,] 0.1798952 0.4516805 0.5229803 0.3700406 0.6603308
## [8,] 0.6311038 0.8746184 0.8501477 0.7114999 0.9190191
## [9,] 0.7605168 0.9238484 0.8966344 0.7818173 0.9467997
## [10,] 0.8717470 0.9594914 0.9354970 0.8493599 0.9683955
##
## $gpcm
## , , 1
##
## [,1] [,2] [,3]
## [1,] 0.036972214 0.6297609 0.40827616
## [2,] 0.853443457 0.9746159 0.98030083
## [3,] 0.475656048 0.9184826 0.90858017
## [4,] 0.549143806 0.9311171 0.92682974
## [5,] 0.058338965 0.6950175 0.51501079
## [6,] 0.411639188 0.9058485 0.88938454
## [7,] 0.336747012 0.8881083 0.86104800
## [8,] 0.012381074 0.4548380 0.19240789
## [9,] 0.004628683 0.2990495 0.08053891
## [10,] 0.001388117 0.1517540 0.02392218
##
## , , 2
##
## [,1] [,2] [,3]
## [1,] 0.11272074 0.24151671 0.27665133
## [2,] 0.11090037 0.02452634 0.01907532
## [3,] 0.23560227 0.07337181 0.07967640
## [4,] 0.22425853 0.06296540 0.06541128
## [5,] 0.13768281 0.21368175 0.26161976
## [6,] 0.24055113 0.08346395 0.09393907
## [7,] 0.24000812 0.09712894 0.11371220
## [8,] 0.06784200 0.28934900 0.25216058
## [9,] 0.04222662 0.29540788 0.18730724
## [10,] 0.02339068 0.25460374 0.11096530
##
## , , 3
##
## [,1] [,2] [,3]
## [1,] 0.85030705 0.1287223905 0.3150725099
## [2,] 0.03565618 0.0008577634 0.0006238544
## [3,] 0.28874169 0.0081455977 0.0117434351
## [4,] 0.22659767 0.0059174548 0.0077589757
## [5,] 0.80397823 0.0913007937 0.2233694490
## [6,] 0.34780968 0.0106875375 0.0166763875
## [7,] 0.42324487 0.0147627315 0.0252397932
## [8,] 0.91977693 0.2558130434 0.5554315244
## [9,] 0.95314470 0.4055426308 0.7321538489
## [10,] 0.97522120 0.5936422645 0.8651125217
##
##
## $grm
## , , 1
##
## [,1] [,2] [,3]
## [1,] 0.16198717 0.22075570 0.4532402
## [2,] 0.93053209 0.83939376 0.8625801
## [3,] 0.68944467 0.60290244 0.7267857
## [4,] 0.74207504 0.64471299 0.7506738
## [5,] 0.21423844 0.26411050 0.4941757
## [6,] 0.64001783 0.56582435 0.7052235
## [7,] 0.57657689 0.52034042 0.6780677
## [8,] 0.08083886 0.14150236 0.3626848
## [9,] 0.04246285 0.09331656 0.2909464
## [10,] 0.01907808 0.05516628 0.2167887
##
## , , 2
##
## [,1] [,2] [,3]
## [1,] 0.40192168 0.4646233 0.04412764
## [2,] 0.05843141 0.1363280 0.01967340
## [3,] 0.24746870 0.3182017 0.03371633
## [4,] 0.20853418 0.2884143 0.03165130
## [5,] 0.43164520 0.4699181 0.04418884
## [6,] 0.28242422 0.3434417 0.03542770
## [7,] 0.32450416 0.3726137 0.03737451
## [8,] 0.28957399 0.4174728 0.04183400
## [9,] 0.18632342 0.3484676 0.03782801
## [10,] 0.09605023 0.2546912 0.03156335
##
## , , 3
##
## [,1] [,2] [,3]
## [1,] 0.43609115 0.31462096 0.5026321
## [2,] 0.01103650 0.02427827 0.1177465
## [3,] 0.06308663 0.07889583 0.2394980
## [4,] 0.04939078 0.06687271 0.2176749
## [5,] 0.35411635 0.26597145 0.4616355
## [6,] 0.07755795 0.09073397 0.2593488
## [7,] 0.09891895 0.10704585 0.2845578
## [8,] 0.62958715 0.44102483 0.5954812
## [9,] 0.77121373 0.55821587 0.6712256
## [10,] 0.88487169 0.69014253 0.7516480
# compute I, returns a list when combine=FALSE or a 2d array when combine=TRUE
with(data, model_mixed_info(t, items, combine=TRUE))
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.775452241 0.4726678 0.2872265 0.30086135 0.21143290 0.55698184
## [2,] 0.005666158 0.1326179 0.1607771 0.09282275 0.26558509 0.54355573
## [3,] 0.096797338 0.4977745 0.3353109 0.21842316 0.40015026 1.79963732
## [4,] 0.067497120 0.4399661 0.3153444 0.19895303 0.39714881 1.65715044
## [5,] 0.684412242 0.5376474 0.3155627 0.30604296 0.24602566 0.75576322
## [6,] 0.129542690 0.5424508 0.3490453 0.23457042 0.39738759 1.86356898
## [7,] 0.179626123 0.5867696 0.3607629 0.25279992 0.38793540 1.85650163
## [8,] 0.820309938 0.3169700 0.2165739 0.26947236 0.14246668 0.26839541
## [9,] 0.678128375 0.2055471 0.1595932 0.22792844 0.09713821 0.14331448
## [10,] 0.430491908 0.1143631 0.1049195 0.17339460 0.05934461 0.06971581
## [,7] [,8] [,9] [,10] [,11]
## [1,] 0.93291961 2.2320363 1.2580533 0.6020251 0.2639822
## [2,] 0.05013278 0.0660799 0.2879373 0.2849627 0.1207019
## [3,] 0.18001324 0.3623153 0.9746404 0.5250644 0.2038961
## [4,] 0.14898978 0.2807689 0.8637916 0.4967046 0.1918171
## [5,] 0.77554800 2.0404707 1.2839438 0.6028639 0.2643314
## [6,] 0.21183370 0.4511204 1.0606431 0.5457333 0.2138661
## [7,] 0.25764036 0.5862647 1.1472089 0.5657805 0.2251630
## [8,] 1.23368651 1.9240616 1.0766585 0.5885706 0.2508616
## [9,] 1.27452120 1.2120876 0.7934486 0.5494808 0.2277877
## [10,] 1.01139893 0.5666544 0.4544041 0.4594141 0.1913015
# compute L, returns a list when combine=FALSE or a 2d array when combine=TRUE
with(data, model_mixed_lh(u, t, items, combine=TRUE))
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.5361973 0.78768427 0.7775934 0.6172807 0.8715012 0.8503070
## [2,] 0.8994101 0.83242816 0.7467945 0.8243470 0.6591472 0.8534435
## [3,] 0.8514735 0.36769585 0.5454265 0.3166527 0.5898376 0.2887417
## [4,] 0.1362926 0.67236911 0.4197982 0.7091573 0.4485836 0.5491438
## [5,] 0.3948003 0.73845224 0.7392978 0.5731808 0.8443237 0.8039782
## [6,] 0.8387110 0.59533969 0.5146877 0.3401780 0.6223262 0.2405511
## [7,] 0.8201048 0.54831954 0.5229803 0.6299594 0.6603308 0.4232449
## [8,] 0.6311038 0.87461842 0.8501477 0.7114999 0.9190191 0.9197769
## [9,] 0.2394832 0.92384844 0.8966344 0.7818173 0.9467997 0.9531447
## [10,] 0.8717470 0.04050863 0.9354970 0.8493599 0.9683955 0.9752212
## [,7] [,8] [,9] [,10] [,11]
## [1,] 0.6297609 0.4082762 0.1619872 0.2207557 0.4532402
## [2,] 0.9746159 0.9803008 0.9305321 0.8393938 0.8625801
## [3,] 0.9184826 0.9085802 0.6894447 0.6029024 0.7267857
## [4,] 0.0629654 0.9268297 0.7420750 0.6447130 0.2176749
## [5,] 0.2136817 0.5150108 0.4316452 0.2641105 0.4941757
## [6,] 0.9058485 0.8893845 0.6400178 0.5658244 0.2593488
## [7,] 0.8881083 0.8610480 0.3245042 0.1070459 0.6780677
## [8,] 0.2893490 0.5554315 0.6295871 0.4410248 0.3626848
## [9,] 0.2954079 0.1873072 0.7712137 0.3484676 0.2909464
## [10,] 0.5936423 0.8651125 0.8848717 0.6901425 0.2167887
Visualization
Visualization is essential in IRT data analysis, and this package makes it really easy to produce high-quality characteristic curves and information function curves at both item and test level.
For example, users can draw ICCs and IIFs for a set of 3PL items:
data <- model_3pl_gendata(1, 5)
# item characteristic curves
with(data, model_3pl_plot(a, b, c, type='prob', total=FALSE))
# test characteristic curves
with(data, model_3pl_plot(a, b, c, type='prob', total=TRUE))
# item information funciton curves
with(data, model_3pl_plot(a, b, c, type='info', total=FALSE))
# test information function curves
with(data, model_3pl_plot(a, b, c, type='info', total=TRUE))
Draw ICCs and IIFs for a set of GPCM items:
# Reproduce Figure 1 in Muraki's 1992 APM article
b <- matrix(c(-2, 0, 2, -.5, 0, 2, -.5, 0, 2), nrow=3, byrow=TRUE)
model_gpcm_plot(a=c(1, 1, .7), b=rowMeans(b), d=rowMeans(b)-b, D=1.0, d0=0)
# Reproduce Figure 2 in Muraki's 1992 APM article
b <- matrix(c(.5, 0, NA, 0, 0, 0), nrow=2, byrow=TRUE)
model_gpcm_plot(a=.7, b=rowMeans(b, na.rm=TRUE), d=rowMeans(b, na.rm=TRUE)-b, D=1.0, d0=0)
# Reproduce Figure 3 in Muraki's 1992 APM article
b <- matrix(c(1.759, -1.643, 3.970, -2.764), nrow=2, byrow=TRUE)
model_gpcm_plot(a=c(.778,.946), b=rowMeans(b), d=rowMeans(b)-b, D=1.0, d0=0)
# Reproduce Figure 1 in Muraki's 1993 APM article
b <- matrix(c(0, -2, 4, 0, -2, 2, 0, -2, 0, 0, -2, -2, 0, -2, -4), nrow=5, byrow=TRUE)
model_gpcm_plot(a=1, b=rowMeans(b), d=rowMeans(b)-b, D=1.0)
# Reproduce Figure 2 in Muraki's 1993 APM article
b <- matrix(c(0, -2, 4, 0, -2, 2, 0, -2, 0, 0, -2, -2, 0, -2, -4), nrow=5, byrow=TRUE)
model_gpcm_plot(a=1, b=rowMeans(b), d=rowMeans(b)-b, D=1.0, type='info', item_level=TRUE)
Draw ICCs and IIFs for a set of GRM items:
data <- model_grm_gendata(1, 3, n_c=3)
# item category characteristic curves
with(data, model_grm_plot(a, b, type='prob', total=FALSE))
# item category information funciton curves
with(data, model_grm_plot(a, b, type='info', total=FALSE))
Estimation
IRT models describe the interaction between people and items using probabilistic functions. However, neither people nor item parameters are naturally known in reality and they have to be estimated based on the observed response data. Thus, parameter estimation, which calibrates items and scores response vectors, is the centerpiece of data analysis in IRT.
Here are some examples of estimating model parameters using the 3PL model in this package.
# generate data
data <- model_3pl_gendata(2000, 40)
# score response vectors using EAP
scores <- with(data, model_3pl_eap(u, a, b, c))
round(c(corr=cor(data$t, scores$t), rmse=rmse(data$t, scores$t)), 2)
## corr rmse
## 0.96 0.28
# score response vectors using MAP
scores <- with(data, model_3pl_map(u, a, b, c))
round(c(corr=cor(data$t, scores$t), rmse=rmse(data$t, scores$t)), 2)
## corr rmse
## 0.96 0.28
# calibrate items using joint MLE
calib <- with(data, model_3pl_jmle(u, true_params=data))
## t: corr = 0.959, rmse = 0.287
## a: corr = 0.905, rmse = 0.129
## b: corr = 0.98, rmse = 0.127
## c: corr = 0.489, rmse = 0.06
# calibrate items using marginal MLE
calib <- with(data, model_3pl_mmle(u, true_params=data))
## t: corr = 0.958, rmse = 0.287
## a: corr = 0.932, rmse = 0.089
## b: corr = 0.98, rmse = 0.118
## c: corr = 0.424, rmse = 0.056
# calibrate items using joint MLE and fix c=0
calib <- with(data, model_3pl_jmle(u, c=0, true_params=data))
## t: corr = 0.955, rmse = 0.298
## a: corr = 0.81, rmse = 0.185
## b: corr = 0.976, rmse = 0.23
# calibrate items using joint MLE and fix a=1 and c=0
calib <- with(data, model_3pl_jmle(u, a=1, c=0, true_params=data))
## t: corr = 0.955, rmse = 0.3
## b: corr = 0.976, rmse = 0.232
Here are examples of estimating parameters using the GPCM model.
# generate data
data <- model_gpcm_gendata(1000, 20, n_c=3)
# score response vectors using EAP
scores <- with(data, model_gpcm_eap(u, a, b, d))
round(c(corr=cor(data$t, scores$t), rmse=rmse(data$t, scores$t)), 2)
## corr rmse
## 0.96 0.28
# score response vectors using MAP
scores <- with(data, model_gpcm_map(u, a, b, d))
round(c(corr=cor(data$t, scores$t), rmse=rmse(data$t, scores$t)), 2)
## corr rmse
## 0.96 0.27
# calibrate items using joint MLE
calib <- with(data, model_gpcm_jmle(u, true_params=data))
## t: corr = 0.962, rmse = 0.276
## a: corr = 0.963, rmse = 0.092
## b: corr = 0.998, rmse = 0.07
## d_2: corr = 0.993, rmse = 0.089
## d_3: corr = 0.993, rmse = 0.089
# calibrate items using marginal MLE
calib <- with(data, model_gpcm_mmle(u, true_params=data))
## t: corr = 0.96, rmse = 0.283
## a: corr = 0.959, rmse = 0.076
## b: corr = 0.999, rmse = 0.085
## d_2: corr = 0.994, rmse = 0.065
## d_3: corr = 0.994, rmse = 0.065
Here are examples of estimating parameters using the GRM model.
# generate data
data <- model_grm_gendata(1000, 20, n_c=3)
# score response vectors using EAP
scores <- with(data, model_grm_eap(u, a, b))
round(c(corr=cor(data$t, scores$t), rmse=rmse(data$t, scores$t)), 2)
## corr rmse
## 0.95 0.30
# score response vectors using MAP
scores <- with(data, model_grm_map(u, a, b))
round(c(corr=cor(data$t, scores$t), rmse=rmse(data$t, scores$t)), 2)
## corr rmse
## 0.95 0.30
# calibrate items using joint MLE
calib <- with(data, model_grm_jmle(u, true_params=data))
## t: corr = 0.953, rmse = 0.308
## a: corr = 0.95, rmse = 0.11
## b_1: corr = 0.997, rmse = 0.082
## b_2: corr = 0.997, rmse = 0.061
# calibrate items using marginal MLE
calib <- with(data, model_grm_mmle(u, true_params=data))
## t: corr = 0.953, rmse = 0.3
## a: corr = 0.952, rmse = 0.058
## b_1: corr = 0.997, rmse = 0.057
## b_2: corr = 0.997, rmse = 0.053
Here are examples of estimating parameters using the mixed model.
# generate data
data <- model_mixed_gendata(2000, n_3pl=40, n_gpcm=10, n_grm=10, n_c=3)
# score response vectors using EAP
scores <- with(data, model_mixed_eap(u, items))
round(c(corr=cor(data$t, scores$t), rmse=rmse(data$t, scores$t)), 2)
## corr rmse
## 0.97 0.24
# score response vectors using MAP
scores <- with(data, model_mixed_map(u, items))
round(c(corr=cor(data$t, scores$t), rmse=rmse(data$t, scores$t)), 2)
## corr rmse
## 0.98 0.22
Getting help
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