mlFA: Maximum likelihood factor analysis
In FMradio: Factor Modeling for Radiomics Data
Description
Usage
Arguments
Details
Value
Note
Author(s)
References
See Also
Examples
Description
mlFA is a function that performs a maximum likelihood factor analysis.
Usage
1
mlFA(R, m)
Arguments
R
(Regularized) correlation matrix.
m
A numeric integer or integer indicating the latent dimension of the factor solution (i.e., the number of factors).
Details
This function is basically a wrapper around the
factanal function from the stats package.
Its purpose is to produce a factor solution of the chosen dimension (argument m) by a maximum likelihood estimation procedure (Joreskog, 1967).
The wrapper ensures that the model is fitted under the same circumstances under which latent dimensionality is assessed with functions such as dimLRT and dimIC.
The function produces a Varimax rotated (Kaiser, 1958) factor solution.
The output can be used to produce factor scores by the facScore function.
Value
The function returns an object of class list:
$Loadings
A matrix of class loadings representing the loadings matrix in which in which each element λ_{jk} is the loading of the jth feature on the kth latent factor.
$Uniqueness
A matrix representing the diagonal matrix carrying the unique variances.
$rotmatrix
A matrix representing the Varimax rotation matrix.
Note
Note that the order of the features in the $Loadings and $Uniqueness slots of the output is determined by the order of the features for the input argument R. As the $Loadings slot gives an object of class "loadings" it can be subjected to the print function, which sorts the output to emphasize the loadings structure when calling sort = TRUE.
Note that the maximum likelihood procedure is stable when a regularized correlation matrix is used as the input for argument R.
In high-dimensional situations usage of dimGB on the regularized correlation matrix is recommended to determine the value for argument m.
Author(s)
Carel F.W. Peeters <cf.peeters@vumc.nl>
References
Joreskog, K.G (1967). Some contributions to maximum likelihood factor analysis. Psychometrika,
32:443–482.
Kaiser, H.F. (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika,
23:187–200.
Peeters, C.F.W. et al. (2019). Stable prediction with radiomics data.
arXiv:1903.11696 [stat.ML].
See Also
Examples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
## Simulate some data according to a factor model with 5 latent factors
## Simulate high-dimensional situation in the sense that p > n
## $cormatrix gives the correlation matrix on the generated data
simDAT <- FAsim(p = 50, m = 5, n = 40, loadingvalue = .9)
simDAT$cormatrix
## Regularize the correlation matrix
RegR <- regcor(simDAT$data)
## Evaluate the Guttman bounds
## First Guttman bound indicates to retain 5 latent factors
GB <- dimGB(RegR$optCor)
print(GB)
## Produce ML factor solution under 5 factors
## Print loadings structure of this solution
fit <- mlFA(RegR$optCor, 5)
print(fit$Loadings, digits = 2, cutoff = .3, sort = TRUE)
Example output
feature.1 feature.2 feature.3 feature.4 feature.5
feature.1 1.000000000 -0.866876912 0.822617429 0.867990517 -0.8755472089
feature.2 -0.866876912 1.000000000 -0.828803457 -0.862851706 0.9001987725
feature.3 0.822617429 -0.828803457 1.000000000 0.840953403 -0.8383367736
feature.4 0.867990517 -0.862851706 0.840953403 1.000000000 -0.8683060803
feature.5 -0.875547209 0.900198773 -0.838336774 -0.868306080 1.0000000000
feature.6 0.884006681 -0.890518596 0.825456386 0.882890679 -0.8988627174
feature.7 0.870240536 -0.898367604 0.853122853 0.868617858 -0.8885788931
feature.8 -0.899910199 0.878842797 -0.852656247 -0.891742696 0.9077208268
feature.9 0.803373650 -0.872952144 0.816165592 0.860933188 -0.8696890740
feature.10 0.853537655 -0.856735845 0.853759445 0.846444084 -0.8685343941
feature.11 -0.060315059 0.018369055 0.037976022 0.008140202 0.0173243842
feature.12 -0.034342173 0.019341793 -0.058904141 -0.026050338 -0.0108526694
feature.13 0.099645941 -0.042257788 0.010385601 0.038185837 -0.0793784683
feature.14 -0.005610550 0.024892390 -0.087245616 -0.067804030 0.0266973053
feature.15 -0.085333517 0.071274916 0.021702427 -0.021379961 0.0379620328
feature.16 0.011541465 -0.002664394 0.098211099 0.009294626 -0.0211330595
feature.17 -0.027283249 0.013303721 -0.012024852 0.053793130 0.0108726636
feature.18 0.100594095 -0.120004464 0.125533854 0.127894679 -0.0874556962
feature.19 0.014030594 0.010510570 -0.070634602 -0.009865705 0.0208757570
feature.20 -0.046723575 0.091107075 -0.158110960 -0.072124954 0.1237978109
feature.21 0.056022857 0.015014541 -0.048503850 0.070370859 -0.0261872709
feature.22 -0.003206638 -0.008792304 -0.062060842 -0.008654589 0.0228862639
feature.23 -0.072371112 -0.032811166 -0.080705051 0.026971337 -0.0270707836
feature.24 0.084297672 -0.098972304 0.108549972 0.036647315 -0.0657151830
feature.25 0.024753281 -0.003876501 -0.017120428 0.037239775 -0.0253195447
feature.26 -0.096231064 0.119241670 -0.006885731 -0.187046834 0.1562887566
feature.27 0.044385767 -0.105769833 0.154783234 0.026156942 -0.0157741420
feature.28 -0.085933070 0.085967763 -0.152278193 0.025672480 0.0790826889
feature.29 -0.001455415 -0.036872347 0.031394181 -0.016345184 -0.0001874017
feature.30 -0.079292307 0.065054907 -0.142590390 -0.048471342 0.0635472462
feature.31 -0.258307076 0.252543892 -0.291642972 -0.323519787 0.3049831440
feature.32 -0.160111511 0.148682514 -0.177754135 -0.226746897 0.1823124677
feature.33 0.102584676 -0.061736644 0.071966851 0.173126493 -0.1483715596
feature.34 0.049368620 -0.151132817 0.143095692 0.185879010 -0.2065184272
feature.35 -0.110521554 0.094244128 -0.146276555 -0.214314943 0.1676240614
feature.36 0.115768247 -0.140225761 0.123765027 0.192881603 -0.2084009312
feature.37 0.188947503 -0.205247487 0.188223570 0.243959451 -0.3078798640
feature.38 0.175405982 -0.221364995 0.192675105 0.264427009 -0.2752817760
feature.39 0.184278499 -0.181292809 0.142183375 0.208352037 -0.2172891359
feature.40 -0.152001375 0.202593871 -0.165545630 -0.269147977 0.2572767643
feature.41 -0.019624475 0.047792400 -0.139446465 -0.091283674 0.0416284091
feature.42 0.056078293 -0.014988705 0.132675451 0.051869841 -0.0558966236
feature.43 -0.093486694 0.101682470 -0.049450485 -0.068928506 0.1248897903
feature.44 -0.074324771 0.072057690 -0.119679091 -0.063448947 0.0235394682
feature.45 0.044104561 -0.105394480 -0.010307326 0.032451789 -0.1020489606
feature.46 0.126982161 -0.117275239 0.033845721 0.080741546 -0.0970135614
feature.47 -0.001587552 0.003958769 0.079403015 0.074013684 -0.0036018210
feature.48 -0.037355162 0.025830861 -0.134200267 -0.102054486 0.0323808632
feature.49 -0.050346196 0.022468396 -0.127762899 -0.028368584 0.0243404530
feature.50 0.121794645 -0.095976297 -0.013182335 0.007631305 -0.0765969123
feature.6 feature.7 feature.8 feature.9 feature.10
feature.1 0.88400668 0.870240536 -0.8999101988 0.803373650 0.853537655
feature.2 -0.89051860 -0.898367604 0.8788427970 -0.872952144 -0.856735845
feature.3 0.82545639 0.853122853 -0.8526562469 0.816165592 0.853759445
feature.4 0.88289068 0.868617858 -0.8917426960 0.860933188 0.846444084
feature.5 -0.89886272 -0.888578893 0.9077208268 -0.869689074 -0.868534394
feature.6 1.00000000 0.910619458 -0.8984002587 0.861302979 0.897581507
feature.7 0.91061946 1.000000000 -0.8726673747 0.841535837 0.853120802
feature.8 -0.89840026 -0.872667375 1.0000000000 -0.859086969 -0.919794439
feature.9 0.86130298 0.841535837 -0.8590869686 1.000000000 0.870813026
feature.10 0.89758151 0.853120802 -0.9197944390 0.870813026 1.000000000
feature.11 -0.17900865 -0.030309214 0.0383411535 -0.085185546 -0.148287657
feature.12 0.13195365 0.014734911 0.0253688501 0.114097270 0.100067348
feature.13 0.19909814 0.064134888 -0.1021523489 0.132075924 0.149575506
feature.14 0.05356802 -0.068064454 0.0275835901 0.045912298 0.074153704
feature.15 -0.17819484 -0.074937082 0.0479961444 -0.097205869 -0.160030994
feature.16 -0.12695514 0.021442133 0.0001212702 -0.035954512 -0.067674212
feature.17 -0.11787460 -0.023224641 0.0360390615 -0.091191304 -0.092489533
feature.18 -0.02235377 0.100234392 -0.1181754433 -0.038978881 -0.034530695
feature.19 0.16004001 0.010920544 -0.0408059335 0.052400257 0.151017240
feature.20 0.07103616 -0.089860280 0.0147846861 -0.056750553 0.037375124
feature.21 0.07710228 -0.013835882 0.0095848651 0.048076456 -0.029997098
feature.22 0.10346198 0.008962608 0.0712924720 0.106250868 -0.014987578
feature.23 0.06693758 0.021229088 0.0460493201 0.100389062 -0.057796753
feature.24 -0.03252537 0.084215666 -0.0911883905 0.009029140 0.075614884
feature.25 0.08389840 0.036595521 -0.0134930840 0.062678969 -0.010967067
feature.26 -0.21143629 -0.141818241 0.1077196222 -0.192027817 -0.132310808
feature.27 -0.03322237 0.094084649 -0.0742200302 -0.013821363 0.029887096
feature.28 0.01901324 -0.060291979 0.0587453734 0.054599325 -0.020561485
feature.29 -0.09611312 0.009486258 -0.0427436209 -0.034121134 0.017595689
feature.30 0.01618877 -0.003377282 0.0690835707 0.002075876 -0.079914797
feature.31 -0.26622854 -0.296557985 0.2993029633 -0.232820742 -0.267769254
feature.32 -0.15787834 -0.209149888 0.1978626953 -0.204645361 -0.151721141
feature.33 0.13653652 0.188549776 -0.1424215656 0.111451807 0.083749366
feature.34 0.10721465 0.172708226 -0.1719371999 0.211031362 0.124690485
feature.35 -0.09344387 -0.201205206 0.1416188179 -0.187430754 -0.137472333
feature.36 0.14045941 0.218170758 -0.1485916252 0.226422833 0.102349196
feature.37 0.23368405 0.265652356 -0.2855153349 0.221847823 0.206196959
feature.38 0.22116852 0.293432257 -0.2196043180 0.255627164 0.155223752
feature.39 0.10082701 0.198646728 -0.2226727379 0.143784970 0.103159311
feature.40 -0.20101004 -0.213505007 0.2312938680 -0.244071500 -0.150211052
feature.41 -0.10110568 -0.079524359 0.0516001891 -0.069745550 -0.149199111
feature.42 0.06086419 0.084592502 -0.0769636696 0.050739116 0.128287865
feature.43 -0.09536982 -0.071791692 0.0925802059 -0.114881601 0.000741006
feature.44 -0.07182215 -0.045069492 0.0874113913 -0.140172141 -0.125388085
feature.45 0.04659212 0.074001705 -0.0941255057 0.056324480 -0.003729613
feature.46 0.09926548 0.097909352 -0.1504770707 0.074162317 0.065939079
feature.47 0.03435570 0.014411566 -0.0236168636 0.054115046 0.104582369
feature.48 -0.10848049 -0.051495689 0.0890098417 -0.069121909 -0.161991722
feature.49 -0.04525542 -0.020113254 0.0016315999 -0.064896611 -0.120801875
feature.50 0.01374113 0.061915056 -0.0984185155 0.068363433 0.002981323
feature.11 feature.12 feature.13 feature.14 feature.15
feature.1 -6.031506e-02 -3.434217e-02 0.099645941 -0.005610550 -0.085333517
feature.2 1.836905e-02 1.934179e-02 -0.042257788 0.024892390 0.071274916
feature.3 3.797602e-02 -5.890414e-02 0.010385601 -0.087245616 0.021702427
feature.4 8.140202e-03 -2.605034e-02 0.038185837 -0.067804030 -0.021379961
feature.5 1.732438e-02 -1.085267e-02 -0.079378468 0.026697305 0.037962033
feature.6 -1.790087e-01 1.319536e-01 0.199098142 0.053568016 -0.178194836
feature.7 -3.030921e-02 1.473491e-02 0.064134888 -0.068064454 -0.074937082
feature.8 3.834115e-02 2.536885e-02 -0.102152349 0.027583590 0.047996144
feature.9 -8.518555e-02 1.140973e-01 0.132075924 0.045912298 -0.097205869
feature.10 -1.482877e-01 1.000673e-01 0.149575506 0.074153704 -0.160030994
feature.11 1.000000e+00 -8.591021e-01 -0.838226974 -0.792378428 0.799684529
feature.12 -8.591021e-01 1.000000e+00 0.841902942 0.873161037 -0.831186896
feature.13 -8.382270e-01 8.419029e-01 1.000000000 0.808839126 -0.825949664
feature.14 -7.923784e-01 8.731610e-01 0.808839126 1.000000000 -0.796698844
feature.15 7.996845e-01 -8.311869e-01 -0.825949664 -0.796698844 1.000000000
feature.16 8.146275e-01 -7.641024e-01 -0.764732698 -0.798966421 0.810145402
feature.17 8.258615e-01 -8.103206e-01 -0.763081501 -0.771112783 0.774792940
feature.18 8.360744e-01 -8.467215e-01 -0.796828873 -0.806176956 0.822422636
feature.19 -8.588877e-01 7.903354e-01 0.845715405 0.758402753 -0.789609372
feature.20 -8.193790e-01 7.993939e-01 0.767094208 0.845853561 -0.745623784
feature.21 -4.944730e-02 1.906110e-01 0.167096264 0.266569850 -0.143618782
feature.22 -1.474106e-01 2.800012e-01 0.113846941 0.266097942 -0.232309212
feature.23 -4.497573e-02 2.339956e-01 0.079471102 0.229503138 -0.115270092
feature.24 1.396139e-01 -2.567089e-01 -0.193099986 -0.234558990 0.284823173
feature.25 8.114234e-03 8.884429e-02 -0.028918735 0.098185534 -0.047018748
feature.26 1.648934e-01 -2.644116e-01 -0.144091652 -0.200219130 0.221869846
feature.27 1.099778e-01 -1.608390e-01 -0.118094091 -0.125144182 0.180305699
feature.28 -1.449608e-01 3.457819e-01 0.193821914 0.330675271 -0.302776823
feature.29 1.380381e-01 -2.648549e-01 -0.171529730 -0.251237498 0.241172158
feature.30 -1.113508e-01 2.868764e-01 0.183156530 0.283726776 -0.229031540
feature.31 -9.584336e-02 1.351236e-01 0.093860368 0.133499918 -0.124760965
feature.32 2.844011e-02 2.396207e-02 -0.005961998 0.023806632 -0.048539711
feature.33 3.831486e-02 -9.768068e-02 -0.067835114 -0.158915693 0.069000706
feature.34 5.854769e-02 -2.404956e-02 -0.042055281 -0.055478932 0.089166425
feature.35 -8.454697e-04 -1.437246e-02 -0.066426803 -0.008564893 0.002815495
feature.36 8.126408e-02 -7.113974e-02 -0.070867509 -0.062203550 0.104820107
feature.37 3.837504e-02 -2.504854e-02 0.039005979 -0.052815616 0.010674264
feature.38 -2.852667e-03 -2.449694e-02 -0.033458077 -0.072918988 -0.002681395
feature.39 1.008755e-01 -1.504300e-01 -0.093005495 -0.111269037 0.116114477
feature.40 -1.161297e-06 -8.374821e-06 -0.069157731 -0.007551128 -0.032829297
feature.41 2.201525e-01 -1.153037e-01 -0.140808505 -0.089397690 0.156455052
feature.42 -1.847217e-01 9.942845e-02 0.155291951 0.083989160 -0.140802406
feature.43 -1.272810e-01 3.657528e-02 0.105466158 0.042108822 -0.120351712
feature.44 2.208698e-01 -9.376837e-02 -0.222288646 -0.101714913 0.149761104
feature.45 2.259095e-01 -1.465916e-01 -0.168919847 -0.170506183 0.204096383
feature.46 2.231763e-01 -1.705681e-01 -0.175644795 -0.138701851 0.126447323
feature.47 -1.147745e-01 8.828150e-02 0.170316709 0.088019031 -0.072719896
feature.48 1.801741e-01 -1.074437e-01 -0.197594274 -0.117816840 0.124673336
feature.49 2.057407e-01 -1.376146e-01 -0.233671523 -0.166003912 0.222477016
feature.50 2.227431e-01 -1.748305e-01 -0.218781370 -0.135182971 0.186524913
feature.16 feature.17 feature.18 feature.19 feature.20
feature.1 0.0115414654 -0.02728325 0.10059410 0.014030594 -0.046723575
feature.2 -0.0026643941 0.01330372 -0.12000446 0.010510570 0.091107075
feature.3 0.0982110993 -0.01202485 0.12553385 -0.070634602 -0.158110960
feature.4 0.0092946261 0.05379313 0.12789468 -0.009865705 -0.072124954
feature.5 -0.0211330595 0.01087266 -0.08745570 0.020875757 0.123797811
feature.6 -0.1269551412 -0.11787460 -0.02235377 0.160040013 0.071036161
feature.7 0.0214421326 -0.02322464 0.10023439 0.010920544 -0.089860280
feature.8 0.0001212702 0.03603906 -0.11817544 -0.040805934 0.014784686
feature.9 -0.0359545122 -0.09119130 -0.03897888 0.052400257 -0.056750553
feature.10 -0.0676742120 -0.09248953 -0.03453070 0.151017240 0.037375124
feature.11 0.8146275416 0.82586151 0.83607439 -0.858887724 -0.819379018
feature.12 -0.7641023582 -0.81032064 -0.84672147 0.790335413 0.799393859
feature.13 -0.7647326977 -0.76308150 -0.79682887 0.845715405 0.767094208
feature.14 -0.7989664211 -0.77111278 -0.80617696 0.758402753 0.845853561
feature.15 0.8101454017 0.77479294 0.82242264 -0.789609372 -0.745623784
feature.16 1.0000000000 0.77443604 0.77554653 -0.833681610 -0.803152391
feature.17 0.7744360390 1.00000000 0.86175374 -0.790393567 -0.755872420
feature.18 0.7755465288 0.86175374 1.00000000 -0.794968387 -0.753728698
feature.19 -0.8336816100 -0.79039357 -0.79496839 1.000000000 0.814100012
feature.20 -0.8031523911 -0.75587242 -0.75372870 0.814100012 1.000000000
feature.21 -0.0731768454 -0.10465351 -0.12026925 0.099244779 0.213780245
feature.22 -0.1952090738 -0.22057950 -0.20654069 0.146003686 0.272435396
feature.23 -0.0986491704 -0.06899574 -0.07244922 0.041479991 0.190898707
feature.24 0.1492455368 0.14521009 0.20140650 -0.125137881 -0.222828993
feature.25 0.0077195054 -0.06078008 -0.03628613 0.021553197 0.112045259
feature.26 0.1305707430 0.12534322 0.19287220 -0.144917327 -0.248935080
feature.27 0.1153160015 0.04672113 0.12264462 -0.130666691 -0.150570419
feature.28 -0.1867168081 -0.16192382 -0.23826586 0.170919261 0.327138498
feature.29 0.1541286762 0.11797300 0.13238661 -0.124573426 -0.256042363
feature.30 -0.1411700104 -0.19667020 -0.18763997 0.128742265 0.278145805
feature.31 -0.0292987382 -0.25388446 -0.22417332 -0.005742177 0.092534716
feature.32 0.0768395829 -0.08502154 -0.08179205 -0.109410118 0.003174347
feature.33 -0.0145273124 0.17223786 0.13438777 0.034210147 -0.069561571
feature.34 -0.0159472004 0.14520988 0.15078512 0.028138397 -0.086045352
feature.35 0.0834077391 -0.10411360 -0.01787777 -0.093723575 0.060066845
feature.36 0.0263728092 0.20814504 0.11447755 -0.034013342 -0.121057268
feature.37 -0.0179877548 0.13951387 0.11260735 0.037329176 -0.054498159
feature.38 -0.0416858455 0.08236783 0.07336681 -0.010182640 -0.084211399
feature.39 0.0214112422 0.16736902 0.22668616 0.002182691 -0.104777137
feature.40 0.0694999781 -0.13682678 -0.11257100 -0.073749062 0.049843329
feature.41 0.1513878401 0.17137997 0.13097110 -0.203874319 -0.082078292
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feature.10 -0.1491991109 0.128287865 0.000741006 -0.125388085 -0.003729613
feature.11 0.2201525042 -0.184721732 -0.127281044 0.220869813 0.225909547
feature.12 -0.1153036658 0.099428450 0.036575280 -0.093768369 -0.146591618
feature.13 -0.1408085046 0.155291951 0.105466158 -0.222288646 -0.168919847
feature.14 -0.0893976895 0.083989160 0.042108822 -0.101714913 -0.170506183
feature.15 0.1564550518 -0.140802406 -0.120351712 0.149761104 0.204096383
feature.16 0.1513878401 -0.108358339 -0.032653095 0.143084347 0.192719111
feature.17 0.1713799671 -0.137237222 -0.074875021 0.204461534 0.166791158
feature.18 0.1309710968 -0.086966621 -0.084880797 0.192026736 0.183649231
feature.19 -0.2038743188 0.139193873 0.059847701 -0.165982495 -0.159904624
feature.20 -0.0820782917 0.076340270 0.032893029 -0.108645557 -0.151534835
feature.21 -0.0312021848 0.047930926 0.100272126 -0.117190361 -0.155068148
feature.22 -0.0321455915 0.059562995 0.033218224 -0.143238302 -0.170194937
feature.23 -0.0305689478 0.007353475 0.026105561 -0.050243214 -0.087596795
feature.24 0.0479269165 -0.133153157 -0.114945645 0.125024101 0.200003019
feature.25 -0.0307812271 0.095080477 0.081387335 -0.115970569 -0.112885210
feature.26 0.0008476133 -0.050366716 -0.042236593 0.100444139 0.076735632
feature.27 0.0250783413 -0.059570916 -0.064494508 0.078939251 0.121310151
feature.28 -0.0441089392 0.116494636 0.155193841 -0.127458338 -0.197432615
feature.29 0.0920326172 -0.168114455 -0.114951662 0.121436323 0.236265788
feature.30 0.0012261736 0.029997150 0.076048358 -0.065413800 -0.122240823
feature.31 -0.0175503833 -0.112838052 0.177286680 -0.000278452 0.030129282
feature.32 0.1119974120 -0.101081046 0.121296519 0.108003481 0.044591912
feature.33 -0.1083652995 0.222072643 -0.045378203 -0.143868838 -0.103021678
feature.34 -0.1737893957 0.161722413 -0.044436869 -0.159202986 -0.102024858
feature.35 0.0725207567 -0.175201560 0.044635368 0.125893966 0.095528128
feature.36 -0.0674120987 0.150555350 -0.125952851 -0.131294776 -0.053882386
feature.37 -0.0714771288 0.130176640 -0.071206208 -0.072763785 0.031044575
feature.38 -0.0951760217 0.185913773 -0.091596467 -0.139218513 -0.043850027
feature.39 0.0189342354 0.081940676 -0.169498560 -0.051416158 0.032231885
feature.40 0.1994578088 -0.273920693 0.005984937 0.222121237 0.129938378
feature.41 1.0000000000 -0.806924163 -0.834131808 0.790440005 0.831490330
feature.42 -0.8069241630 1.000000000 0.827164570 -0.832659581 -0.898701976
feature.43 -0.8341318077 0.827164570 1.000000000 -0.803541030 -0.844514121
feature.44 0.7904400052 -0.832659581 -0.803541030 1.000000000 0.847889073
feature.45 0.8314903296 -0.898701976 -0.844514121 0.847889073 1.000000000
feature.46 0.8155844847 -0.857621541 -0.811240654 0.800323958 0.887624055
feature.47 -0.8395366602 0.785260825 0.822019125 -0.785177610 -0.834221804
feature.48 0.8761741234 -0.800122309 -0.822252555 0.815468151 0.862850042
feature.49 0.7946581556 -0.854337218 -0.849793522 0.881236354 0.916853197
feature.50 0.8546739064 -0.849753569 -0.811320120 0.805021908 0.880348655
feature.46 feature.47 feature.48 feature.49 feature.50
feature.1 0.126982161 -0.001587552 -0.037355162 -0.05034620 0.121794645
feature.2 -0.117275239 0.003958769 0.025830861 0.02246840 -0.095976297
feature.3 0.033845721 0.079403015 -0.134200267 -0.12776290 -0.013182335
feature.4 0.080741546 0.074013684 -0.102054486 -0.02836858 0.007631305
feature.5 -0.097013561 -0.003601821 0.032380863 0.02434045 -0.076596912
feature.6 0.099265480 0.034355699 -0.108480486 -0.04525542 0.013741133
feature.7 0.097909352 0.014411566 -0.051495689 -0.02011325 0.061915056
feature.8 -0.150477071 -0.023616864 0.089009842 0.00163160 -0.098418515
feature.9 0.074162317 0.054115046 -0.069121909 -0.06489661 0.068363433
feature.10 0.065939079 0.104582369 -0.161991722 -0.12080188 0.002981323
feature.11 0.223176305 -0.114774451 0.180174073 0.20574067 0.222743095
feature.12 -0.170568054 0.088281500 -0.107443736 -0.13761460 -0.174830478
feature.13 -0.175644795 0.170316709 -0.197594274 -0.23367152 -0.218781370
feature.14 -0.138701851 0.088019031 -0.117816840 -0.16600391 -0.135182971
feature.15 0.126447323 -0.072719896 0.124673336 0.22247702 0.186524913
feature.16 0.158549168 -0.024382380 0.094650343 0.15104405 0.206505315
feature.17 0.175514792 -0.057183613 0.128253998 0.21658163 0.192504001
feature.18 0.180049172 -0.050855349 0.123223023 0.22061832 0.228200316
feature.19 -0.135278719 0.103895720 -0.149897140 -0.14788195 -0.230846206
feature.20 -0.056262767 0.011834108 -0.176124739 -0.08641088 -0.162848168
feature.21 -0.118965444 0.255636766 -0.147272989 -0.15661016 -0.159231864
feature.22 -0.116023626 0.116566524 -0.077279806 -0.16128527 -0.170371896
feature.23 -0.113988116 0.180378050 -0.091984735 -0.06486588 -0.136863975
feature.24 0.147708047 -0.197765916 0.165869350 0.19262372 0.218899097
feature.25 -0.101557089 0.110202661 -0.060289957 -0.09167976 -0.138601096
feature.26 0.065860346 -0.101905291 0.094402187 0.09637209 0.132061251
feature.27 0.121991032 -0.133934186 0.070139638 0.12524848 0.161197783
feature.28 -0.116551013 0.223391495 -0.175289550 -0.16671282 -0.182313219
feature.29 0.204626688 -0.236683810 0.205729686 0.22064687 0.240099201
feature.30 -0.133922152 0.172882282 -0.093704234 -0.10770841 -0.138438348
feature.31 -0.060842234 0.010762561 -0.041952623 0.02259649 0.061642341
feature.32 0.008347155 -0.054705867 0.019952603 0.03795343 0.099344246
feature.33 -0.089869843 0.065323222 -0.039979950 -0.07176696 -0.184402857
feature.34 -0.104685751 0.171769872 -0.100107225 -0.11879993 -0.179335697
feature.35 0.077821508 -0.103156087 -0.013843120 0.13219997 0.173368626
feature.36 -0.037808528 0.039450334 0.029187796 -0.05329783 -0.112453540
feature.37 0.015400739 0.037048627 0.016358451 -0.02615746 -0.113369933
feature.38 -0.044730261 0.045186527 -0.001281237 -0.07318082 -0.119790207
feature.39 0.076862785 -0.057421353 0.080345557 0.03174024 0.034609455
feature.40 0.186021177 -0.203558043 0.113601626 0.15298887 0.266986734
feature.41 0.815584485 -0.839536660 0.876174123 0.79465816 0.854673906
feature.42 -0.857621541 0.785260825 -0.800122309 -0.85433722 -0.849753569
feature.43 -0.811240654 0.822019125 -0.822252555 -0.84979352 -0.811320120
feature.44 0.800323958 -0.785177610 0.815468151 0.88123635 0.805021908
feature.45 0.887624055 -0.834221804 0.862850042 0.91685320 0.880348655
feature.46 1.000000000 -0.830958040 0.758282935 0.84819192 0.905661797
feature.47 -0.830958040 1.000000000 -0.881366494 -0.85117726 -0.819824812
feature.48 0.758282935 -0.881366494 1.000000000 0.84179045 0.812897212
feature.49 0.848191920 -0.851177257 0.841790454 1.00000000 0.859495540
feature.50 0.905661797 -0.819824812 0.812897212 0.85949554 1.000000000
Determining folds...
Determining optimal penalty value...
Calculating Guttman bounds...
Visualizing...
First.lower-bound Second.lower-bound Third.lower-bound
5 11 23
Loadings:
Factor1 Factor2 Factor3 Factor4 Factor5
feature.1 0.82
feature.2 -0.84
feature.3 0.80
feature.4 0.82
feature.5 -0.84
feature.6 0.85
feature.7 0.83
feature.8 -0.85
feature.9 0.81
feature.10 0.83
feature.41 0.80
feature.42 -0.81
feature.43 -0.82
feature.44 0.79
feature.45 0.84
feature.46 0.81
feature.47 -0.80
feature.48 0.81
feature.49 0.83
feature.50 0.81
feature.31 -0.76
feature.32 -0.82
feature.33 0.83
feature.34 0.81
feature.35 -0.79
feature.36 0.82
feature.37 0.81
feature.38 0.78
feature.39 0.81
feature.40 -0.82
feature.21 0.81
feature.22 0.81
feature.23 0.80
feature.24 -0.82
feature.25 0.82
feature.26 -0.79
feature.27 -0.78
feature.28 0.81
feature.29 -0.83
feature.30 0.80
feature.11 0.82
feature.12 -0.81
feature.13 -0.80
feature.14 -0.79
feature.15 0.78
feature.16 0.79
feature.17 0.78
feature.18 0.80
feature.19 -0.80
feature.20 -0.78
Factor1 Factor2 Factor3 Factor4 Factor5
SS loadings 7.05 6.75 6.72 6.69 6.54
Proportion Var 0.14 0.13 0.13 0.13 0.13
Cumulative Var 0.14 0.28 0.41 0.54 0.67
FMradio documentation built on Dec. 16, 2019, 5:43 p.m.
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Description Usage Arguments Details Value Note Author(s) References See Also Examples
Description
mlFA is a function that performs a maximum likelihood factor analysis.
Usage
1 | mlFA(R, m)
|
Arguments
R |
(Regularized) correlation |
m |
A |
Details
This function is basically a wrapper around the
factanal function from the stats package.
Its purpose is to produce a factor solution of the chosen dimension (argument m) by a maximum likelihood estimation procedure (Joreskog, 1967).
The wrapper ensures that the model is fitted under the same circumstances under which latent dimensionality is assessed with functions such as dimLRT and dimIC.
The function produces a Varimax rotated (Kaiser, 1958) factor solution.
The output can be used to produce factor scores by the facScore function.
Value
The function returns an object of class list:
$Loadings |
A matrix of class |
$Uniqueness |
A |
$rotmatrix |
A |
Note
Note that the order of the features in the
$Loadingsand$Uniquenessslots of the output is determined by the order of the features for the input argumentR. As the$Loadingsslot gives an object of class "loadings" it can be subjected to theprintfunction, which sorts the output to emphasize the loadings structure when callingsort = TRUE.Note that the maximum likelihood procedure is stable when a regularized correlation matrix is used as the input for argument
R.In high-dimensional situations usage of
dimGBon the regularized correlation matrix is recommended to determine the value for argumentm.
Author(s)
Carel F.W. Peeters <cf.peeters@vumc.nl>
References
Joreskog, K.G (1967). Some contributions to maximum likelihood factor analysis. Psychometrika, 32:443–482.
Kaiser, H.F. (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika, 23:187–200.
Peeters, C.F.W. et al. (2019). Stable prediction with radiomics data. arXiv:1903.11696 [stat.ML].
See Also
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | ## Simulate some data according to a factor model with 5 latent factors
## Simulate high-dimensional situation in the sense that p > n
## $cormatrix gives the correlation matrix on the generated data
simDAT <- FAsim(p = 50, m = 5, n = 40, loadingvalue = .9)
simDAT$cormatrix
## Regularize the correlation matrix
RegR <- regcor(simDAT$data)
## Evaluate the Guttman bounds
## First Guttman bound indicates to retain 5 latent factors
GB <- dimGB(RegR$optCor)
print(GB)
## Produce ML factor solution under 5 factors
## Print loadings structure of this solution
fit <- mlFA(RegR$optCor, 5)
print(fit$Loadings, digits = 2, cutoff = .3, sort = TRUE)
|
Example output
feature.1 feature.2 feature.3 feature.4 feature.5
feature.1 1.000000000 -0.866876912 0.822617429 0.867990517 -0.8755472089
feature.2 -0.866876912 1.000000000 -0.828803457 -0.862851706 0.9001987725
feature.3 0.822617429 -0.828803457 1.000000000 0.840953403 -0.8383367736
feature.4 0.867990517 -0.862851706 0.840953403 1.000000000 -0.8683060803
feature.5 -0.875547209 0.900198773 -0.838336774 -0.868306080 1.0000000000
feature.6 0.884006681 -0.890518596 0.825456386 0.882890679 -0.8988627174
feature.7 0.870240536 -0.898367604 0.853122853 0.868617858 -0.8885788931
feature.8 -0.899910199 0.878842797 -0.852656247 -0.891742696 0.9077208268
feature.9 0.803373650 -0.872952144 0.816165592 0.860933188 -0.8696890740
feature.10 0.853537655 -0.856735845 0.853759445 0.846444084 -0.8685343941
feature.11 -0.060315059 0.018369055 0.037976022 0.008140202 0.0173243842
feature.12 -0.034342173 0.019341793 -0.058904141 -0.026050338 -0.0108526694
feature.13 0.099645941 -0.042257788 0.010385601 0.038185837 -0.0793784683
feature.14 -0.005610550 0.024892390 -0.087245616 -0.067804030 0.0266973053
feature.15 -0.085333517 0.071274916 0.021702427 -0.021379961 0.0379620328
feature.16 0.011541465 -0.002664394 0.098211099 0.009294626 -0.0211330595
feature.17 -0.027283249 0.013303721 -0.012024852 0.053793130 0.0108726636
feature.18 0.100594095 -0.120004464 0.125533854 0.127894679 -0.0874556962
feature.19 0.014030594 0.010510570 -0.070634602 -0.009865705 0.0208757570
feature.20 -0.046723575 0.091107075 -0.158110960 -0.072124954 0.1237978109
feature.21 0.056022857 0.015014541 -0.048503850 0.070370859 -0.0261872709
feature.22 -0.003206638 -0.008792304 -0.062060842 -0.008654589 0.0228862639
feature.23 -0.072371112 -0.032811166 -0.080705051 0.026971337 -0.0270707836
feature.24 0.084297672 -0.098972304 0.108549972 0.036647315 -0.0657151830
feature.25 0.024753281 -0.003876501 -0.017120428 0.037239775 -0.0253195447
feature.26 -0.096231064 0.119241670 -0.006885731 -0.187046834 0.1562887566
feature.27 0.044385767 -0.105769833 0.154783234 0.026156942 -0.0157741420
feature.28 -0.085933070 0.085967763 -0.152278193 0.025672480 0.0790826889
feature.29 -0.001455415 -0.036872347 0.031394181 -0.016345184 -0.0001874017
feature.30 -0.079292307 0.065054907 -0.142590390 -0.048471342 0.0635472462
feature.31 -0.258307076 0.252543892 -0.291642972 -0.323519787 0.3049831440
feature.32 -0.160111511 0.148682514 -0.177754135 -0.226746897 0.1823124677
feature.33 0.102584676 -0.061736644 0.071966851 0.173126493 -0.1483715596
feature.34 0.049368620 -0.151132817 0.143095692 0.185879010 -0.2065184272
feature.35 -0.110521554 0.094244128 -0.146276555 -0.214314943 0.1676240614
feature.36 0.115768247 -0.140225761 0.123765027 0.192881603 -0.2084009312
feature.37 0.188947503 -0.205247487 0.188223570 0.243959451 -0.3078798640
feature.38 0.175405982 -0.221364995 0.192675105 0.264427009 -0.2752817760
feature.39 0.184278499 -0.181292809 0.142183375 0.208352037 -0.2172891359
feature.40 -0.152001375 0.202593871 -0.165545630 -0.269147977 0.2572767643
feature.41 -0.019624475 0.047792400 -0.139446465 -0.091283674 0.0416284091
feature.42 0.056078293 -0.014988705 0.132675451 0.051869841 -0.0558966236
feature.43 -0.093486694 0.101682470 -0.049450485 -0.068928506 0.1248897903
feature.44 -0.074324771 0.072057690 -0.119679091 -0.063448947 0.0235394682
feature.45 0.044104561 -0.105394480 -0.010307326 0.032451789 -0.1020489606
feature.46 0.126982161 -0.117275239 0.033845721 0.080741546 -0.0970135614
feature.47 -0.001587552 0.003958769 0.079403015 0.074013684 -0.0036018210
feature.48 -0.037355162 0.025830861 -0.134200267 -0.102054486 0.0323808632
feature.49 -0.050346196 0.022468396 -0.127762899 -0.028368584 0.0243404530
feature.50 0.121794645 -0.095976297 -0.013182335 0.007631305 -0.0765969123
feature.6 feature.7 feature.8 feature.9 feature.10
feature.1 0.88400668 0.870240536 -0.8999101988 0.803373650 0.853537655
feature.2 -0.89051860 -0.898367604 0.8788427970 -0.872952144 -0.856735845
feature.3 0.82545639 0.853122853 -0.8526562469 0.816165592 0.853759445
feature.4 0.88289068 0.868617858 -0.8917426960 0.860933188 0.846444084
feature.5 -0.89886272 -0.888578893 0.9077208268 -0.869689074 -0.868534394
feature.6 1.00000000 0.910619458 -0.8984002587 0.861302979 0.897581507
feature.7 0.91061946 1.000000000 -0.8726673747 0.841535837 0.853120802
feature.8 -0.89840026 -0.872667375 1.0000000000 -0.859086969 -0.919794439
feature.9 0.86130298 0.841535837 -0.8590869686 1.000000000 0.870813026
feature.10 0.89758151 0.853120802 -0.9197944390 0.870813026 1.000000000
feature.11 -0.17900865 -0.030309214 0.0383411535 -0.085185546 -0.148287657
feature.12 0.13195365 0.014734911 0.0253688501 0.114097270 0.100067348
feature.13 0.19909814 0.064134888 -0.1021523489 0.132075924 0.149575506
feature.14 0.05356802 -0.068064454 0.0275835901 0.045912298 0.074153704
feature.15 -0.17819484 -0.074937082 0.0479961444 -0.097205869 -0.160030994
feature.16 -0.12695514 0.021442133 0.0001212702 -0.035954512 -0.067674212
feature.17 -0.11787460 -0.023224641 0.0360390615 -0.091191304 -0.092489533
feature.18 -0.02235377 0.100234392 -0.1181754433 -0.038978881 -0.034530695
feature.19 0.16004001 0.010920544 -0.0408059335 0.052400257 0.151017240
feature.20 0.07103616 -0.089860280 0.0147846861 -0.056750553 0.037375124
feature.21 0.07710228 -0.013835882 0.0095848651 0.048076456 -0.029997098
feature.22 0.10346198 0.008962608 0.0712924720 0.106250868 -0.014987578
feature.23 0.06693758 0.021229088 0.0460493201 0.100389062 -0.057796753
feature.24 -0.03252537 0.084215666 -0.0911883905 0.009029140 0.075614884
feature.25 0.08389840 0.036595521 -0.0134930840 0.062678969 -0.010967067
feature.26 -0.21143629 -0.141818241 0.1077196222 -0.192027817 -0.132310808
feature.27 -0.03322237 0.094084649 -0.0742200302 -0.013821363 0.029887096
feature.28 0.01901324 -0.060291979 0.0587453734 0.054599325 -0.020561485
feature.29 -0.09611312 0.009486258 -0.0427436209 -0.034121134 0.017595689
feature.30 0.01618877 -0.003377282 0.0690835707 0.002075876 -0.079914797
feature.31 -0.26622854 -0.296557985 0.2993029633 -0.232820742 -0.267769254
feature.32 -0.15787834 -0.209149888 0.1978626953 -0.204645361 -0.151721141
feature.33 0.13653652 0.188549776 -0.1424215656 0.111451807 0.083749366
feature.34 0.10721465 0.172708226 -0.1719371999 0.211031362 0.124690485
feature.35 -0.09344387 -0.201205206 0.1416188179 -0.187430754 -0.137472333
feature.36 0.14045941 0.218170758 -0.1485916252 0.226422833 0.102349196
feature.37 0.23368405 0.265652356 -0.2855153349 0.221847823 0.206196959
feature.38 0.22116852 0.293432257 -0.2196043180 0.255627164 0.155223752
feature.39 0.10082701 0.198646728 -0.2226727379 0.143784970 0.103159311
feature.40 -0.20101004 -0.213505007 0.2312938680 -0.244071500 -0.150211052
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Determining folds...
Determining optimal penalty value...
Calculating Guttman bounds...
Visualizing...
First.lower-bound Second.lower-bound Third.lower-bound
5 11 23
Loadings:
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feature.1 0.82
feature.2 -0.84
feature.3 0.80
feature.4 0.82
feature.5 -0.84
feature.6 0.85
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feature.34 0.81
feature.35 -0.79
feature.36 0.82
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feature.38 0.78
feature.39 0.81
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feature.21 0.81
feature.22 0.81
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feature.24 -0.82
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feature.15 0.78
feature.16 0.79
feature.17 0.78
feature.18 0.80
feature.19 -0.80
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Factor1 Factor2 Factor3 Factor4 Factor5
SS loadings 7.05 6.75 6.72 6.69 6.54
Proportion Var 0.14 0.13 0.13 0.13 0.13
Cumulative Var 0.14 0.28 0.41 0.54 0.67
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