Generate multivariate normal data from an explicit factor model for factor analysis parameter recovery testing. The implied covariance matrix \(\Sigma = \Lambda \Phi \Lambda' + \Psi\) is fully specified and stored in the return value for direct comparison with estimated covariances.
Arguments
- loadings
Numeric matrix of dimension
p x m(observed variables x factors). Must be a matrix (not a vector). Zero entries indicate no loading; usematrix(val, nrow = p, ncol = 1)for single-factor models.- phi
Numeric matrix of dimension
m x m. Factor correlation matrix. Must be symmetric and positive-definite. Defaults todiag(m)(orthogonal factors).- psi
Numeric vector of length
p. Unique variances. All values must be positive. Defaults to1 - diag(loadings %*% phi %*% t(loadings))(communality complement). An error is raised if any auto-computed value is \(\le 0\) (over-factored model).- n
Positive integer. Sample size.
- seed
Integer or NULL. Random seed.
Value
A named list with elements:
datadata.frame with columns
y1...yp.paramslist with
loadings,phi,psi, andsigma_implied(the \(p x p\) model-implied covariance matrix).
Examples
# Two orthogonal factors, 6 indicators (3 per factor)
loadings <- matrix(c(0.8, 0.7, 0.6, 0, 0, 0,
0, 0, 0, 0.8, 0.7, 0.6),
nrow = 6, ncol = 2) # byrow = FALSE (default)
r <- simulate_fa(loadings = loadings, n = 500, seed = 1)
r$data # data.frame: y1..y6
#> y1 y2 y3 y4 y5
#> 1 -0.626453811 -0.2867690683 0.6872958302 0.850043472 -0.258144036
#> 2 0.183643324 -0.1431132137 0.9880917992 -0.925312995 -2.110749248
#> 3 -0.835628612 -1.4482594715 -1.3642366102 0.893581214 1.842315572
#> 4 1.595280802 0.9027131581 0.9486193298 -0.941009738 -0.096754176
#> 5 0.329507772 1.0060584798 0.3986777489 0.538952094 0.255541917
#> 6 -0.820468384 0.8611279564 -1.5296039570 -0.181974398 0.475071178
#> 7 0.487429052 -0.8643208703 0.6792079626 0.891767595 0.543727197
#> 8 0.738324705 0.2066610222 -1.3320954007 1.329208205 -0.341203596
#> 9 0.575781352 1.2830122059 -0.5818385116 -0.103466125 -1.816885564
#> 10 -0.305388387 -1.0941424746 0.5065598197 0.615064611 0.172044609
#> 11 1.511781168 -1.2482466734 -0.1999064133 -1.799772395 -1.267013910
#> 12 0.389843236 -0.5570761383 -0.1693443584 -0.262705142 -1.023855341
#> 13 -0.621240581 -1.1492454653 -1.8664615838 -1.210141877 -0.332012787
#> 14 -2.214699887 -1.2008879892 -2.2994260227 0.204477253 -0.146842003
#> 15 1.124930918 0.2954683655 0.7673918993 -0.013994105 0.676124656
#> 16 -0.044933609 0.1666300098 -0.1290945189 -0.235200840 0.938096867
#> 17 -0.016190263 -0.3589989647 -0.5923446707 -0.188318654 -0.623353299
#> 18 0.943836211 0.8385026022 -0.6233781868 0.951283069 -0.192334098
#> 19 0.821221195 0.1566507905 -0.6141749753 -0.154014786 0.032521887
#> 20 0.593901321 1.3185749900 2.2493970825 1.854244505 2.219135998
#> 21 0.918977372 -0.0962430756 0.0263940285 -1.904167354 -2.503560793
#> 22 0.782136301 0.6788114953 1.0926270956 0.359299702 -0.492575726
#> 23 0.074564983 -0.6913349350 0.5479008648 2.114534946 0.591143820
#> 24 -1.989351696 -0.9417051197 -0.2620047813 -1.135748987 -0.325483040
#> 25 0.619825748 0.3075583238 0.5417737951 -0.796255986 1.528666245
#> 26 -0.056128740 -1.4270449959 -1.4089830545 -0.397569450 -0.606987351
#> 27 -0.155795507 -0.2067361197 -0.9673337284 0.091724042 -0.301986967
#> 28 -1.470752384 0.1541769617 0.1541784284 -0.238694074 -1.544670476
#> 29 -0.478150055 0.2967707680 -1.2278226621 -0.461962364 -1.121571783
#> 30 0.417941560 0.3527268769 0.6794239351 1.926340607 1.715641973
#> 31 1.358679552 -0.2269648101 -0.0232489897 0.122054815 -0.549141726
#> 32 -0.102787727 0.9111361675 -0.1777904706 1.092050577 0.208510285
#> 33 0.387671612 0.2827141328 -0.4769735717 -2.972257475 -1.956954756
#> 34 -0.053805041 -0.4044220370 -0.3057536732 1.128489051 0.580338345
#> 35 -1.377059557 0.5892548831 0.3913726527 1.312000129 2.076559217
#> 36 -0.414994563 -0.8699985044 0.0496892797 -0.482607973 0.762216472
#> 37 -0.394289954 0.0305299392 -0.3333675752 -0.224858997 1.538180891
#> 38 -0.059313397 1.0286944755 0.9519773003 1.667765311 -0.602424283
#> 39 1.100025372 1.1149513419 2.0588942806 0.041218532 -0.360273067
#> 40 0.763175748 0.1730127428 0.3048277368 1.038174846 1.445581728
#> 41 -0.164523596 -0.4387895193 0.2483828316 0.587642933 0.077352902
#> 42 -0.253361680 0.1523446203 0.1819312998 -0.158299577 0.174879745
#> 43 0.696963375 0.8157147976 -0.4108618512 -0.292085415 -0.701051590
#> 44 0.556663199 0.3271474849 -0.5244867561 -1.419898963 -2.343516851
#> 45 -0.688755695 0.7066220374 1.5573225210 -2.138318025 -2.197445836
#> 46 -0.707495157 -0.4507386131 0.4045909284 -0.022500105 1.559264216
#> 47 0.364581962 -0.3760242249 0.6839574728 0.269770954 -0.065649968
#> 48 0.768532925 0.8755491653 0.5935870218 -0.502969399 1.199666770
#> 49 -0.112346212 -1.8870742553 -0.0392516434 0.630807390 1.071804204
#> 50 0.881107726 0.8181676757 0.6247709236 0.474565208 0.847839630
#> 51 0.398105880 0.6346677681 0.3178243613 -0.340659062 1.154315383
#> 52 -0.612026393 -0.5290418195 -0.1430991007 0.049601560 -1.289657861
#> 53 0.341119691 -0.7345178263 -0.9071967274 1.295263191 0.219974175
#> 54 -1.129363096 -0.9596934425 1.4462492068 0.314624205 -0.083361548
#> 55 1.433023702 2.0865162537 1.6588394307 0.801572504 -0.315097094
#> 56 1.980399899 0.4930276934 0.5993450538 0.486058896 0.059700694
#> 57 -0.367221476 -2.1374502782 0.4389467047 1.725561057 -0.759315477
#> 58 -1.044134626 0.0882239533 -0.8911845781 0.102432861 -1.116739193
#> 59 0.569719627 -0.0962891820 1.6502795003 -0.989837075 -0.048281211
#> 60 -0.135054604 -0.4988963211 -0.4964653858 -0.460441209 -0.057748882
#> 61 2.401617761 0.3379858229 -1.0610421783 0.237661492 1.213415651
#> 62 -0.039240003 -0.0406640583 1.1475652203 -0.269397264 -0.428016922
#> 63 0.689739362 0.9672256192 1.4305893235 -0.169554070 -0.413538740
#> 64 0.028002159 -0.4710433061 -0.7581964310 0.664868973 0.023391875
#> 65 -0.743273209 -0.9189026110 -1.6777073835 -0.775550915 -1.228298738
#> 66 0.188792300 1.0142819985 -1.2834060492 -1.919261131 -0.868395214
#> 67 -1.804958629 -1.2158367698 -1.1423846360 2.019332540 0.154133380
#> 68 1.465554862 0.6882333351 -0.8993650816 0.435210515 0.654313260
#> 69 0.153253338 -0.4326307850 -0.7395181235 1.050168832 0.440867842
#> 70 2.172611670 1.9626659742 1.3475402749 -0.312114841 0.467077027
#> 71 0.475509529 -0.5573967357 0.7809042728 0.093617617 1.086340213
#> 72 -0.709946431 0.3060276650 -0.8893763279 -2.153187751 -1.550617647
#> 73 0.610726353 1.0095251479 0.3341289036 0.357993071 -0.155542013
#> 74 -0.934097632 -0.9104985923 -0.3149392337 0.467864811 0.223458490
#> 75 -1.253633400 0.0008752535 1.0280614967 -0.906688835 -1.053383970
#> 76 0.291446236 0.9807414288 -0.5025229850 -0.529349916 1.031293107
#> 77 -0.443291873 0.2286537045 -1.4843487909 1.343576717 0.699651140
#> 78 0.001105352 1.6781856621 -1.2136078962 0.469434057 0.982700343
#> 79 0.074341324 -1.5841639699 0.1544186525 -0.253105088 0.171742439
#> 80 -0.589520946 -1.2952601087 -0.5471550737 0.369103715 1.379585262
#> 81 -0.568668733 -1.4588904825 0.1383324927 -1.346707190 -0.927333149
#> 82 -0.135178615 0.0632215616 -0.1551349664 1.171935426 -1.562456308
#> 83 1.178086997 1.9725821951 1.1919810544 -1.908736345 -0.564190837
#> 84 -1.523566800 0.5369211848 -0.2327006456 -1.004390811 -1.418822099
#> 85 0.593946188 0.7371600698 0.3944091786 0.676687514 0.791030185
#> 86 0.332950371 0.9144265100 -0.2143748937 1.152321744 -0.161562424
#> 87 1.063099837 0.4753081468 1.5667476752 -1.732759918 -1.048614831
#> 88 -0.304183924 0.2181966389 -0.4086056793 0.601308857 -0.122232636
#> 89 0.370018810 0.5189188449 0.8184057604 0.341650314 -0.012285816
#> 90 0.267098791 -0.4809409187 0.1305903168 -0.035281009 -0.562242401
#> 91 -0.542520031 -0.5468039017 1.2147466771 -0.626165020 -1.165727781
#> 92 1.207867806 0.5646909807 1.1654433371 -0.855971210 -0.316621748
#> 93 1.160402616 1.8046164189 0.5216057484 0.151777842 -0.922887465
#> 94 0.700213650 -0.4670178193 0.1390014992 1.206372694 0.439412803
#> 95 1.586833455 -0.8630843043 0.2682888685 -1.488577883 -1.082438109
#> 96 0.558486426 0.9492652223 -1.3858196327 -1.050374123 -1.117301872
#> 97 -1.276592208 -1.3910746772 0.6947785850 1.637874117 2.361146103
#> 98 -0.573265414 -0.6823399937 0.5581334774 -1.207334181 -2.224891715
#> 99 -1.224612615 0.0637583657 -1.0805202957 -2.626155085 -1.444810726
#> 100 -0.473400636 -0.8973156852 -0.0387042811 0.017463266 -0.991962688
#> 101 -0.620366677 -0.6299768104 -0.1524558012 0.344194027 -0.918302141
#> 102 0.042115873 1.2683325083 -0.6775849156 0.012719837 -0.030744990
#> 103 -0.910921649 -0.0724170893 0.9376270022 -0.873450133 1.323602389
#> 104 0.158028772 0.5376976422 -0.0382109813 0.342800284 1.370164012
#> 105 -0.654584644 -0.4798002654 -0.1816148356 -0.177387754 0.052481660
#> 106 1.767287269 0.0479052772 0.9882537525 0.921433255 -0.024947407
#> 107 0.716707476 -0.8385883464 -0.7821489801 0.300944402 1.860774158
#> 108 0.910174229 0.3246242143 -0.5352646814 0.693386753 0.194022360
#> 109 0.384185358 1.8735533821 0.5080697897 0.325471906 -0.556274561
#> 110 1.682176081 1.1256985778 1.3422304792 0.408051023 -0.896145540
#> 111 -0.635736454 -0.2198306741 -0.2118032456 0.759842256 -0.058351207
#> 112 -0.461644730 0.0170551519 0.4448690356 -2.287153520 0.101198501
#> 113 1.432282239 0.4829360741 0.9046332161 0.517838262 1.189519770
#> 114 -0.650696353 -1.5232473280 -0.0996230673 -1.347236258 -0.292544206
#> 115 -0.207380744 2.1006982514 -0.8150190445 0.363159859 -1.152734368
#> 116 -0.392807929 -0.5709932327 0.3150928442 1.314752565 0.511294186
#> 117 -0.319992869 -0.4265851515 -0.8857998669 -0.448305911 0.653495720
#> 118 -0.279113303 -1.6412453197 -0.8607075536 -0.808164468 -0.712514359
#> 119 0.494188331 0.0712724845 0.1019778712 -0.086268499 -0.613362116
#> 120 -0.177330482 -0.3041945381 -1.5442324088 1.585424273 0.930986056
#> 121 -0.505957462 -0.4950246384 -0.3749004846 0.101057448 -0.531054709
#> 122 1.343038825 -0.7283632632 -0.0588888794 0.148762212 -0.210566378
#> 123 -0.214579409 1.3584154804 -0.3938340716 -0.546434668 1.678264003
#> 124 -0.179556530 1.3605665942 -0.2402190706 -0.527675587 0.161009330
#> 125 -0.100190741 0.5152218706 1.1464844189 -0.139871559 0.956281809
#> 126 0.712666307 -0.5128623543 -0.9390551443 -1.589411414 -0.849084737
#> 127 -0.073564404 0.5507696536 -0.0897290353 1.234104166 0.313679507
#> 128 -0.037634171 -0.2252739715 -2.1841752504 -0.335454848 0.357201791
#> 129 -0.681660479 -0.6466703424 -0.9640462758 -2.001002507 -2.351260173
#> 130 -0.324270272 0.9473493703 -0.3371178587 -0.017458403 1.207431832
#> 131 0.060160440 -0.9836018015 1.1282203915 -0.240966896 -0.318668948
#> 132 -0.588894486 -0.7533222672 -1.2014511369 -1.791989399 -3.507469932
#> 133 0.531496193 -0.3081517832 -0.6267209667 2.049691511 0.427788017
#> 134 -1.518394082 -0.8339362941 -1.9244962650 -1.117217612 -0.968547956
#> 135 0.306557861 -1.1314337233 0.0143863438 -1.353549545 0.890064881
#> 136 -1.536449824 -1.4431183156 0.1244438341 0.451302311 0.187286856
#> 137 -0.300976127 0.4245978118 -1.0422432753 1.033227425 -0.650733955
#> 138 -0.528279904 0.0895903642 -0.5339762920 -0.195885695 -0.347366534
#> 139 -0.652094781 -1.1720439381 0.6816532740 0.227310387 -0.619197715
#> 140 -0.056896778 0.4314455599 -0.5124320365 -1.102199750 -1.921293401
#> 141 -1.914359426 -3.0874653232 -0.7746282812 0.955277369 -0.018738654
#> 142 1.176583312 0.3767974101 0.8368449295 0.312830446 -1.362633528
#> 143 -1.664972436 -0.3416418337 -1.7891864944 0.101005713 2.280173495
#> 144 -0.463530401 -0.8055846273 0.7014128950 0.748115414 0.649469995
#> 145 -1.115920105 -0.6550745625 0.1400342760 0.998960895 1.077283657
#> 146 -0.750819001 -1.7404845625 -1.0700827143 -0.732515309 -1.364161889
#> 147 2.087166546 1.8712033304 0.3808675182 0.480895240 1.342752120
#> 148 0.017395620 -1.5232919343 -0.6643548007 -0.989450308 -0.121512592
#> 149 -1.286300530 -0.9884704527 0.0220374060 2.229559048 0.470907951
#> 150 -1.640605534 -1.1302102735 -1.3182419977 0.456736600 0.166512473
#> 151 0.450187101 0.3025775632 -1.3810018818 -0.035815444 -0.351148399
#> 152 -0.018559833 -0.6926502330 -0.2818190838 -0.490442706 0.791374651
#> 153 -0.318068375 1.3377998899 0.8436051960 -0.220566611 -0.259099371
#> 154 -0.929362147 -1.7051178161 -1.7016645052 -1.101209361 0.282427655
#> 155 -1.487460310 -0.6224270295 -0.8717503950 0.063470471 0.640203835
#> 156 -1.075192297 -3.0376884592 -2.1013356047 0.583525933 0.078909755
#> 157 1.000028804 0.5620176099 0.5806832705 -0.513358846 -0.328838156
#> 158 -0.621266695 0.0743448088 -0.3188886813 -0.812552122 -1.169529751
#> 159 -1.384426847 -1.6739615362 -0.8913905007 -1.629319906 -1.325006890
#> 160 1.869290622 1.6307516591 0.4918884973 0.568905707 -0.837920727
#> 161 0.425100377 0.5122676704 -0.3442360612 0.333783307 -0.164674729
#> 162 -0.238647101 0.6752377169 0.8113953044 -0.211322559 -0.420647089
#> 163 1.058483049 -0.1059503489 0.4443786012 -0.551097930 -0.489293087
#> 164 0.886422651 -0.3077215910 0.0879473351 0.258361129 2.215376711
#> 165 -0.619243048 -1.8144767326 -3.3776999699 -1.375210446 -0.649842356
#> 166 2.206102465 0.9682530861 -0.0944989555 -1.472754149 -2.049091551
#> 167 -0.255027030 -1.2520016543 0.2878870133 -1.445209673 -0.039446058
#> 168 -1.424494650 -0.2275848223 -0.9072470160 -0.361288425 -0.440944501
#> 169 -0.144399602 -0.0218082837 -1.0931310683 0.790709965 1.610452241
#> 170 0.207538339 1.9304151110 0.1222764256 0.714657846 0.444754785
#> 171 2.307978399 0.3333157163 0.5742711885 -0.565153517 1.014709256
#> 172 0.105802368 1.0382691510 -0.1638321258 1.418014303 0.779815465
#> 173 0.456998805 -0.1810014578 0.5492457630 -1.145756781 -1.091196239
#> 174 -0.077152935 -1.2500111824 1.0021626024 -0.492817823 1.363354259
#> 175 -0.334000842 0.2876587588 0.7921036807 -0.733052274 -1.542323500
#> 176 -0.034726028 -1.2069894073 0.7910072236 0.159232080 0.441193459
#> 177 0.787639606 -0.4331348078 -0.0975554809 -1.751192959 -1.789532864
#> 178 2.075245009 0.5547594342 1.5868361815 1.184830867 1.651873506
#> 179 1.027392439 0.7500748749 -0.3101981792 1.207031601 1.041269490
#> 180 1.207908398 -0.1511697343 0.2780303905 0.977649327 0.191979467
#> 181 -1.231323422 0.2034499083 1.4809434766 -0.081991744 -0.358807429
#> 182 0.983895570 -0.4423598928 0.2424495032 1.156263423 1.383788944
#> 183 0.219924804 0.3026400721 -0.1168779441 0.486796337 -0.609743741
#> 184 -1.467250029 -0.7031134699 -1.1794659198 0.958848663 -0.107008190
#> 185 0.521022743 -0.5911933356 0.7173980667 0.177792371 0.437474905
#> 186 -0.158754605 -0.4440139056 1.6528436306 0.292250881 0.785255028
#> 187 1.464587312 0.2765290493 0.9369109664 -2.113340685 -2.604338233
#> 188 -0.766082000 0.3658451873 -0.4168147929 -0.538047911 -0.142025243
#> 189 -0.430211754 1.0482595231 1.8569412618 1.014025585 -0.394633072
#> 190 -0.926109497 -1.3809134694 -1.3027732306 0.101269641 -0.475039304
#> 191 -0.177103961 0.6717941424 1.0557600956 -1.218707372 0.404870036
#> 192 0.402011779 0.1626202004 -0.1823492038 -0.958376793 0.293077560
#> 193 -0.731748173 -2.0395857803 -1.5869889802 -0.555797485 -0.471582976
#> 194 0.830373168 -0.1612516055 1.3060247136 2.121875912 0.415120414
#> 195 -1.208082786 -0.2944676901 -2.0400506509 -0.179589988 0.156535258
#> 196 -1.047984413 -0.4666515001 -0.7410522862 -0.644735408 0.950718506
#> 197 1.441157707 -1.2163885579 0.2593468422 -2.092583772 -0.421109020
#> 198 -1.015847465 -0.0880848399 -0.7612405881 -1.901870838 -0.732703799
#> 199 0.411974712 0.7734115769 -1.0882395104 1.222250276 0.747174925
#> 200 -0.381076051 -0.4656859091 -0.7888600749 -0.851220441 0.401095496
#> 201 0.409401840 -0.3569500195 -1.2686727585 1.621202883 1.667550000
#> 202 1.688873286 2.5792018786 2.8206910790 -0.329102802 0.486179192
#> 203 1.586588433 0.8139262812 -0.8481624880 -2.326409527 -1.892363132
#> 204 -0.330907801 -0.1969215580 -1.9686026992 2.192998000 -1.000222557
#> 205 -2.285235535 -2.2105076665 -0.7033089159 -1.082479976 -1.075983055
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#> 207 0.667066167 -0.8883662424 -0.1259570386 0.095918957 -0.103940522
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#> 209 -0.013399523 1.1200121243 0.8640789478 -1.518667630 -2.536754371
#> 210 0.510108423 1.7386667806 1.7651263956 -1.510585063 -1.346645664
#> 211 -0.164375832 1.2073281218 -1.1574901847 1.395604928 0.111982609
#> 212 0.420694643 1.3099414031 1.5579485652 -0.708421944 -0.899428352
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#> 218 -1.253289756 0.6733549723 -0.5484884304 1.899876533 0.154962769
#> 219 0.642241306 0.4681488069 0.9981894106 -0.756206936 -0.447590610
#> 220 -0.044709137 0.8828985551 0.4322943753 -0.377955780 1.589226649
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#> 223 -0.630300334 1.0915578469 0.1544294648 1.610975996 0.256962185
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#> 225 -1.156572363 -1.4559196280 -0.2792958033 -0.400515674 0.030897813
#> 226 1.803141908 1.0683006108 1.2128252189 -1.669580400 -1.281690580
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#> 228 -1.605513412 -0.1834529680 -0.6586378120 0.691487433 1.370901047
#> 229 0.197193439 0.5259740249 2.4227155155 -0.040977477 -0.110484005
#> 230 0.263175646 -0.1465553468 -0.0007136956 1.204344853 0.842154530
#> 231 -0.985826700 -0.9567226747 -0.8165588227 1.298514964 1.373731031
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#> 234 0.570507636 -0.4956040428 -0.3676067714 0.963266754 0.521474296
#> 235 -0.059723276 0.3180411951 -0.8802998914 0.693211228 1.400850390
#> 236 -0.098178744 -0.4288899788 0.5979871257 -0.501393166 -0.424674121
#> 237 0.560820729 1.0804855230 0.5400000415 3.153971417 2.316354014
#> 238 -1.186458639 -0.8289727101 -0.1919439012 -2.549106895 -1.881379758
#> 239 1.096777044 1.6041203789 0.4945347332 -0.008138085 -0.764651246
#> 240 -0.005344028 0.4075624689 -1.3297636501 1.138780641 0.999856044
#> 241 0.707310667 -1.4639983357 -0.0109431056 1.482346468 0.511170415
#> 242 1.034107735 -0.5272449684 2.3039617752 0.975845009 1.244374917
#> 243 0.223480415 1.1879188072 -0.9129333642 0.148350293 0.872643127
#> 244 -0.878707613 0.0802430364 -1.1407475528 0.788908757 0.101583481
#> 245 1.162964556 -0.1499441196 -0.7647011617 0.557581220 0.395764395
#> 246 -2.000164945 -2.2350724878 -1.5189972028 -0.027394393 0.747498425
#> 247 -0.544790740 0.5513013189 1.1125366615 -0.877074959 0.718101868
#> 248 -0.255670709 -0.8157263926 -0.4024290351 1.216243471 0.352315238
#> 249 -0.166121037 1.3996884584 0.0297772389 -0.351249812 -0.596931276
#> 250 1.020463909 2.0391692062 -0.2926101572 -0.294393117 -0.426845359
#> 251 0.136221893 -1.1289167853 -0.1440913217 0.253137809 1.144236499
#> 252 0.407167603 -0.4726044882 0.9294967591 0.011497986 -0.617400028
#> 253 -0.069654813 -1.0750198484 1.6777889857 -1.975783962 -1.644352130
#> 254 -0.247664342 0.4141511998 1.1310100413 -0.426326724 -0.449382615
#> 255 0.695550807 -0.6798846920 -0.6482470250 0.612302560 -0.731143161
#> 256 1.146228357 -1.0440975070 -0.2529223765 1.230814109 0.337430641
#> 257 -2.403096215 0.3289371610 -0.2633473348 -1.270807808 -1.042349943
#> 258 0.572739555 1.1541757484 0.4498889744 -0.650555664 -0.314705252
#> 259 0.374724407 0.8868265305 0.5652470777 -2.443166237 -0.804236514
#> 260 -0.425267722 -0.7882593082 -1.4513579913 0.996548595 1.858321539
#> 261 0.951012808 0.5232207556 0.0884324570 0.005113528 0.675027103
#> 262 -0.389237182 0.2954252038 -0.8686675388 0.791024764 -0.610262210
#> 263 -0.284330662 -1.2207219962 -1.1670778241 -0.563191537 0.931225586
#> 264 0.857409778 0.3771998855 -0.2279920669 0.213826203 0.788325521
#> 265 1.719627299 1.1085919490 0.5015723168 0.977082664 0.600667843
#> 266 0.270054901 1.5536813187 0.4336942919 -0.234161103 0.982408345
#> 267 -0.422184010 0.2955791635 0.4580880240 0.848660554 0.924208208
#> 268 -1.189113295 0.3964171427 -0.7736822888 0.653970891 0.238581951
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#> 270 -0.939829327 -1.4480046712 0.7326439862 0.309423133 0.907809083
#> 271 -0.258932583 -0.4264221066 0.6665337686 -2.485982190 -3.224077227
#> 272 0.394379168 -1.1583928620 0.4834286719 -0.278439107 -0.284249696
#> 273 -0.851857092 0.2925068461 1.2193274800 -0.552433759 -1.243168065
#> 274 2.649166881 2.6573640115 0.6911946801 0.876042073 -1.183068469
#> 275 0.156011676 0.0354028378 1.8214793962 0.293483694 0.348795695
#> 276 1.130207267 -0.1797542640 1.4158709135 -0.597222047 -0.440411137
#> 277 -2.289123980 -0.3812134828 -0.6369255093 -0.157203603 -1.764940452
#> 278 0.741001157 0.5303920595 0.9168326437 0.184185060 -0.128568250
#> 279 -1.316245160 -1.0571208671 -0.6496567871 -1.013032014 -0.218808536
#> 280 0.919803678 1.4459722760 0.5514633339 -0.679843405 0.151913111
#> 281 0.398130155 -0.4065736226 -1.3000969800 -1.242693614 -0.452015625
#> 282 -0.407528579 0.7236881185 0.6512696708 -1.535018736 -0.263806884
#> 283 1.324258630 0.0435994807 2.3201675233 -0.524540013 -0.134253687
#> 284 -0.701231669 -0.0684066759 1.5280249286 -0.018527385 0.294045686
#> 285 -0.580614304 0.4133554450 0.0296779537 -1.568147171 -0.972112584
#> 286 -1.001072181 -1.6668523933 -0.8167668185 0.467851674 -0.596813209
#> 287 -0.668178607 -0.0443376617 -0.0919031168 -0.667828864 -0.164714252
#> 288 0.945184953 0.4368547326 1.4166171625 0.311082360 0.938528184
#> 289 0.433702150 0.8027220430 1.5991344261 0.142551004 1.964596175
#> 290 1.005159218 -0.0904583975 0.6886041234 -0.374034721 -0.575188117
#> 291 -0.390118664 -0.2905339866 0.7530034003 -1.944926376 -1.079422298
#> 292 0.376370292 1.3559794447 0.4096916164 -1.597681314 -2.656899577
#> 293 0.244164924 0.2763252128 0.5750319087 -2.540657861 -2.079029464
#> 294 -1.426257342 -0.1166964969 0.4829976683 1.362047250 1.830522714
#> 295 1.778429287 0.8129108599 3.9363620294 -0.360666003 -0.412688895
#> 296 0.134447661 -0.7775526856 -0.1696675317 1.936397311 1.555700120
#> 297 0.765598999 0.4196835854 -0.2078273188 -1.455883798 -0.262317262
#> 298 0.955136677 -0.4800195770 0.6174414954 1.481905708 0.043110491
#> 299 -0.050565701 -2.1791758853 -0.8462909109 1.076119636 0.615475817
#> 300 -0.305815420 0.7973527892 0.6116945662 -0.757488401 -0.231542127
#> 301 0.893673702 -0.4000386461 0.5234193070 0.714085544 0.141655740
#> 302 -1.047298149 -2.0993832394 0.2934074258 0.581384629 0.496036966
#> 303 1.971337386 1.9285325556 0.3043202285 -0.146723902 -0.623988718
#> 304 -0.383632106 -0.2246613777 -1.2207828400 1.506981777 -0.111555689
#> 305 1.654145302 0.4295336772 2.0296536572 -0.279532643 0.280484046
#> 306 1.512212694 0.6994105173 0.8866988083 2.027738733 2.290385368
#> 307 0.082965734 -0.3064615816 -0.8294161565 -1.195740196 -0.519134937
#> 308 0.567220915 1.1433681404 -0.4672215020 1.312317892 0.677482219
#> 309 -1.024548480 0.0291143689 -0.8329502617 -0.524007513 -0.537406603
#> 310 0.323006503 -1.2496170422 0.3465933420 0.354249509 1.369893404
#> 311 1.043612458 0.8772110177 0.1129870923 -0.071674488 0.871202274
#> 312 0.099078487 0.6576437016 1.3316066842 -0.133144306 1.088983520
#> 313 -0.454136909 0.2993326091 0.1132091135 -0.088077384 0.466937150
#> 314 -0.655781852 -2.3757668054 -0.5891329658 0.917788931 0.740921466
#> 315 -0.035922423 -0.2151084433 -0.0735126213 0.031439264 0.431558834
#> 316 1.069161461 2.2388419891 0.6191369799 1.358930582 0.979177598
#> 317 -0.483974930 0.3891124237 -0.2307742497 0.113494845 -0.930732007
#> 318 -0.121010111 -1.4838883415 0.6223180707 0.174306546 1.880347097
#> 319 -1.294140004 -2.1030556306 -2.0735676488 -0.053480405 -0.978962123
#> 320 0.494312836 0.6836958215 1.2594274639 -0.424758034 0.551113728
#> 321 1.307901520 0.5882211682 0.2951140361 0.207911552 -0.907523454
#> 322 1.497041009 1.6347650907 1.5244634040 0.992922918 -0.184508264
#> 323 0.814702731 0.6996667463 0.3945566695 0.813857888 -0.131567091
#> 324 -1.869788790 -0.9799743487 -2.0375254183 0.996005584 -0.434486899
#> 325 0.482029504 0.4220990013 0.5625634115 1.732744271 0.912267461
#> 326 0.456135603 0.3931770056 -0.2985127863 0.357042034 -0.503636765
#> 327 -0.353400286 -1.2497576004 0.5634795064 0.155726335 0.217166880
#> 328 0.170489471 2.0420119503 -0.1996397447 1.552788656 2.540975151
#> 329 -0.864035954 -1.6536961755 -2.2064788978 -0.064069618 0.575211047
#> 330 0.679230774 0.3663167621 -0.4098698339 -1.008218758 -1.966071671
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#> 332 -1.569082185 0.6126937364 -1.1333496812 -0.654514621 -0.304654278
#> 333 -0.367450756 0.6321982850 -0.1956535197 -1.433782829 0.806371621
#> 334 1.364434929 0.2970485612 0.0412152967 1.418184353 1.212694110
#> 335 -0.334281365 -0.0170079539 -0.4754135836 -0.593885665 -0.546709478
#> 336 0.732750042 1.3759166332 0.8458584105 1.118971798 -0.419324134
#> 337 0.946585640 2.3828648981 -0.1360064515 2.384081563 1.373850545
#> 338 0.004398704 0.2528876554 0.7095646838 -0.403286494 0.760041545
#> 339 -0.352322306 -1.0610095039 -0.3615653869 0.370108558 -0.533437418
#> 340 -0.529695509 -1.1114870206 -0.4665527512 -0.575293014 -0.956202277
#> 341 0.739589226 2.0758929821 -1.0261989021 2.919228807 2.450439843
#> 342 -1.063457415 -2.3109893195 -1.1468164212 -1.078914790 0.155346413
#> 343 0.246210844 2.6695380342 0.1990681981 1.178491329 0.420069538
#> 344 -0.289499367 -0.3786466803 0.5403872972 0.869602773 0.083078535
#> 345 -2.264889356 -1.6447994955 0.0233554057 0.931166310 0.296241810
#> 346 -1.408850456 -0.6584185000 0.4090089154 1.024681228 1.490314541
#> 347 0.916019329 1.2862764847 -0.3452948596 0.612975056 0.391834960
#> 348 -0.191278951 0.1437747754 -0.7196221078 0.608190081 0.554881207
#> 349 0.803283216 -1.1708173966 -0.0482365269 -0.892891710 0.851588107
#> 350 1.887474463 1.3498882496 0.0612425323 0.452035606 0.973405395
#> 351 1.473881181 1.1985468411 -0.5869556267 -0.374712307 0.798473410
#> 352 0.677268492 0.9257033759 1.1244956733 2.840185145 3.015840127
#> 353 0.379962687 -0.6417453431 0.0526311358 0.732393453 0.151006548
#> 354 -0.192798426 -2.0723420473 0.0019430013 0.840518928 0.386031329
#> 355 1.577891795 0.6147103337 -0.3341702170 -2.219652117 -1.104477517
#> 356 0.596234109 -0.4484534328 -0.1782782440 -1.327463782 -0.365083835
#> 357 -1.173576941 -1.2917372884 -1.7081118030 -0.976660698 -0.962480965
#> 358 -0.155642535 -0.8773588908 -0.9604738075 -0.587075872 -0.140744398
#> 359 -1.918909820 -1.4043332436 -0.7970313171 -0.568012815 -0.905184479
#> 360 -0.195258846 -0.3671859897 -0.8498469210 0.643640957 0.653300630
#> 361 -2.592327670 -0.7921465677 -1.9514816053 -0.237307108 -0.917638137
#> 362 1.314002167 1.5530898040 -0.3100469309 -0.298578026 -0.359706441
#> 363 -0.635543001 -1.0141676890 -1.6807334469 1.189562559 1.819048627
#> 364 -0.429978839 -0.4966415560 -1.0031379028 -2.108941204 -1.466193517
#> 365 -0.169318332 0.2046360504 1.4466794089 -0.073282004 -0.618846881
#> 366 0.612218174 1.5017302667 -0.0134380549 0.863402869 0.968516315
#> 367 0.678340177 0.3334165711 -0.6398123907 1.074707461 -1.879584507
#> 368 0.567951972 -1.0894510235 -1.2434575743 -0.205852630 -0.030783039
#> 369 -0.572542604 -0.1285357386 -0.3617710020 -0.433613832 -0.071525277
#> 370 -1.363291256 -0.8621088440 -1.0142015758 -0.182223784 1.840294479
#> 371 -0.388722244 1.2508127684 0.3531880092 -0.431078673 -0.706088231
#> 372 0.277914132 0.4401546394 -0.0465101215 -0.424609610 -0.902415859
#> 373 -0.823081122 -0.9771175666 0.2357387943 -1.353819676 1.135325183
#> 374 -0.068840934 -0.4026918944 0.5046360890 -0.196594949 0.194165231
#> 375 -1.167662326 -1.0725255572 0.2534928979 0.234116137 0.126245134
#> 376 -0.008309014 0.1494756878 0.2389062804 -0.557206294 0.112216659
#> 377 0.128855402 0.2183198660 1.0502542530 -1.113325275 -0.994116677
#> 378 -0.145875628 0.6770831166 0.0314340913 -1.630635426 -1.916487264
#> 379 -0.163910957 0.1734739148 0.7802752413 -0.387823881 0.421548167
#> 380 1.763552003 0.6837914068 0.6440242132 1.722632017 1.038417120
#> 381 0.762586512 -0.3522410572 -0.3904068223 -0.558500300 -0.394712663
#> 382 1.111431081 1.1482480916 -0.3976186757 -0.636660846 -1.446598311
#> 383 -0.923206953 -0.5687671392 -1.0628444215 2.781387134 0.898414330
#> 384 0.164341838 0.2435112512 -0.5301716673 -1.957255876 -0.696414399
#> 385 1.154825187 1.5610605294 1.2500487670 0.142527896 -0.708602973
#> 386 -0.056521425 1.4198965597 0.2479111485 0.664498799 0.785981703
#> 387 -2.129360648 -1.9826718488 -1.7370416671 -0.478627492 0.761083572
#> 388 0.344845762 1.5552221307 -0.3052693972 1.183689583 1.447680501
#> 389 -1.904955446 -1.7848573710 0.2254412279 0.805437291 -0.649813676
#> 390 -0.811170153 -0.2335844636 -0.5439011540 0.053317758 0.135040704
#> 391 1.324004321 0.9256747310 -0.3954630080 -1.167346415 -0.645499171
#> 392 0.615636849 0.1153399823 -0.1240106606 0.323350156 0.930830346
#> 393 1.091668956 1.7664629520 0.8585104139 -0.320783022 0.137853534
#> 394 0.306604862 -0.3742049981 0.7417310326 -0.885950264 -1.580156765
#> 395 -0.110158762 0.4855563757 -0.0052076539 0.151278172 -0.195612470
#> 396 -0.924312773 -0.5285974025 -0.3501109363 0.579363472 1.258310857
#> 397 1.592913754 0.1203071609 -0.2973112670 -0.079166826 0.730083409
#> 398 0.045010598 1.0315669102 0.9266769713 0.848767635 1.991945053
#> 399 -0.715128401 -2.1306464583 -0.6386593513 -2.511033487 -2.218708511
#> 400 0.865223100 0.0486115047 0.9516513691 0.738477114 -0.142160106
#> 401 1.074440958 -0.6753536946 1.5617666099 -0.570994285 -0.629885545
#> 402 1.895654774 1.2225602434 1.0989056274 0.286539016 0.985105137
#> 403 -0.602997304 -0.1186063627 -0.4836278902 1.147619856 0.727292261
#> 404 -0.390867821 -1.1457497962 -1.0501862030 0.139558702 1.306760788
#> 405 -0.416222032 0.3062253741 -0.5755859872 0.088926609 0.513795094
#> 406 -0.375657423 -1.0661183683 -1.1538354385 -2.630159317 -1.354869651
#> 407 -0.366630946 0.3408302393 0.2617330624 -1.217220832 0.834225838
#> 408 -0.295677453 0.0314605258 -1.3579053992 0.749152579 0.849014189
#> 409 1.441820410 1.4000202639 -0.6750407544 -1.415932924 -0.219709726
#> 410 -0.697538292 -1.1681477033 -1.0787556777 -0.112382662 -0.039728183
#> 411 -0.388167506 -0.1383737112 0.7614750561 0.304032006 0.033519236
#> 412 0.652536452 -0.0180220546 0.3341517298 -0.354012805 -0.795619380
#> 413 1.124772447 -0.5870854369 -0.1522837326 -0.851242490 -1.538682657
#> 414 -0.772110803 -0.3058824087 0.2318093914 -1.461173075 -1.002885523
#> 415 -0.508086216 1.1850209292 -0.2217649168 0.264251846 0.170614441
#> 416 0.523620590 -0.2436944466 0.0344029859 -1.907357944 0.446055700
#> 417 1.017754227 0.4043950655 0.5146836281 -0.925231266 0.458176050
#> 418 -0.251164588 0.4303811679 0.9394789720 -0.111571293 -0.357681073
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#> 446 -0.143320524
#> 447 -0.338717146
#> 448 -1.019502455
#> 449 -0.564628420
#> 450 1.419964151
#> 451 -0.455663939
#> 452 0.194340069
#> 453 -3.080063265
#> 454 1.180386436
#> 455 -0.846740998
#> 456 0.187221358
#> 457 1.052486528
#> 458 0.437110820
#> 459 0.400636475
#> 460 1.620881774
#> 461 0.179217851
#> 462 0.082431587
#> 463 0.663066476
#> 464 0.068360763
#> 465 -1.238691073
#> 466 -1.306197686
#> 467 -1.164491687
#> 468 0.026234636
#> 469 -0.262302823
#> 470 0.485095687
#> 471 1.837058011
#> 472 0.102234549
#> 473 -0.212199140
#> 474 1.447509995
#> 475 -0.533763655
#> 476 0.112339027
#> 477 2.177340316
#> 478 -0.824079949
#> 479 -1.350292628
#> 480 -0.278783882
#> 481 1.361629211
#> 482 -0.944265283
#> 483 0.537436771
#> 484 -0.021406887
#> 485 -0.509549309
#> 486 -1.619875103
#> 487 0.686323545
#> 488 0.536991870
#> 489 -0.368985995
#> 490 0.458823736
#> 491 1.810228459
#> 492 -0.773543026
#> 493 -2.112870550
#> 494 0.094154247
#> 495 1.325119464
#> 496 -0.195478976
#> 497 0.055960572
#> 498 1.041076386
#> 499 -1.611102531
#> 500 -1.317739585
r$params$sigma_implied # implied covariance matrix
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 1.00 0.56 0.48 0.00 0.00 0.00
#> [2,] 0.56 1.00 0.42 0.00 0.00 0.00
#> [3,] 0.48 0.42 1.00 0.00 0.00 0.00
#> [4,] 0.00 0.00 0.00 1.00 0.56 0.48
#> [5,] 0.00 0.00 0.00 0.56 1.00 0.42
#> [6,] 0.00 0.00 0.00 0.48 0.42 1.00
# Oblique model
phi <- matrix(c(1, 0.4, 0.4, 1), nrow = 2)
r2 <- simulate_fa(loadings = loadings, phi = phi, n = 500, seed = 1)