¿Cómo simular una regresión logística?

Author

Andrés Gutiérrez

Published

August 15, 2013

Durante el semestre que acaba de pasar, dirigí dos trabajos de grados que estaban relacionados con algunas propiedades de la regresión logística. De paso diré que la regresión logística me ha sorprendido por sus inesperadas propiedades (como por ejemplo ser robusta ante los diseños de muestreo complejos). En este periplo por las propiedades de este tipo de modelos, nos encontramos con un problema nada inocuo. La correcta simulación del modelo.

Suponga que se tiene una variable auxiliar, que para este caso, seguirá una distribución normal de media 20 y varianza 4. Ahora, queremos generar una variable dicotómica, tal que el LOGIT de la esperanza está inducida por un modelo lineal de la forma -5+0.3X (intercepto -5, pendiente 0.3). Luego, la manera INCORRECTA de generar la respuesta dicotómica es la siguiente:

x <- rnorm(1000, 20, 4)
p <- exp(-5+0.3*x)/(1+exp(-5+0.3*x))
p

   [1] 0.45010999 0.82318259 0.54604367 0.80391176 0.07450941 0.92751295
   [7] 0.73048589 0.96751949 0.78513839 0.94298815 0.87299941 0.47170476
  [13] 0.53294610 0.48823760 0.79356654 0.43764058 0.87124767 0.07607519
  [19] 0.68016993 0.41823526 0.85477745 0.57493712 0.54675472 0.91873715
  [25] 0.93724703 0.72714170 0.94002956 0.94984157 0.81503308 0.82541238
  [31] 0.66825148 0.58328678 0.62401426 0.94572702 0.59303916 0.89092618
  [37] 0.50577062 0.65056995 0.64703436 0.42779545 0.63363937 0.67388262
  [43] 0.91053169 0.97337239 0.86217832 0.70834575 0.96008006 0.91106031
  [49] 0.80934716 0.98421929 0.71348300 0.56410929 0.95284931 0.94656120
  [55] 0.45083672 0.78269901 0.64389011 0.90676522 0.54917120 0.64511490
  [61] 0.82773910 0.83521434 0.63043120 0.66565078 0.54464983 0.64358775
  [67] 0.68292117 0.78623127 0.45542999 0.80832905 0.59903209 0.94948767
  [73] 0.40909672 0.85683727 0.81146698 0.63031510 0.89370446 0.49258438
  [79] 0.40114989 0.85560094 0.46860782 0.84918922 0.69446745 0.90254510
  [85] 0.95961459 0.67018800 0.85197342 0.19440311 0.37476978 0.57441744
  [91] 0.63150755 0.70621920 0.68438316 0.87520160 0.50184460 0.76751530
  [97] 0.56457210 0.87347576 0.94398026 0.95919634 0.73488658 0.85435452
 [103] 0.70603774 0.88047463 0.83417287 0.89963502 0.87795147 0.89872289
 [109] 0.91726297 0.60739917 0.69113175 0.42069842 0.78081554 0.90285310
 [115] 0.89928892 0.58333544 0.53073368 0.42056257 0.85308351 0.59863142
 [121] 0.80743096 0.84223564 0.58663409 0.28922972 0.67683621 0.58301310
 [127] 0.67708644 0.88228991 0.77882584 0.62395135 0.52810745 0.39408461
 [133] 0.80757102 0.72532537 0.72467939 0.66443355 0.87409749 0.73272924
 [139] 0.25643088 0.82916690 0.89200374 0.51059507 0.15352160 0.55632506
 [145] 0.81026279 0.69539812 0.91800526 0.80871772 0.95191543 0.72552067
 [151] 0.78527859 0.45145743 0.94328496 0.88943800 0.57715693 0.76217384
 [157] 0.93043593 0.93238712 0.45015113 0.47690410 0.41909392 0.64199317
 [163] 0.54777961 0.37461541 0.93782884 0.30784459 0.44086555 0.32566565
 [169] 0.27356040 0.58536542 0.46827093 0.47621964 0.79672546 0.83781579
 [175] 0.85507064 0.86542786 0.29887100 0.67612669 0.42292033 0.62167388
 [181] 0.88603073 0.45628944 0.84519283 0.50968653 0.61596582 0.46612897
 [187] 0.49592057 0.93995298 0.70021536 0.92694753 0.37672303 0.89361636
 [193] 0.78534448 0.73381639 0.64904930 0.59712444 0.86422034 0.82898686
 [199] 0.77433238 0.87427048 0.55758940 0.85932477 0.97181144 0.93375371
 [205] 0.80785040 0.77362420 0.92958036 0.83107448 0.41066599 0.58138257
 [211] 0.92356622 0.45338211 0.69427631 0.62741238 0.93654453 0.97379099
 [217] 0.90074534 0.63152258 0.44283236 0.51478243 0.69453867 0.89878423
 [223] 0.95013740 0.65401300 0.76047286 0.94291713 0.73854739 0.76087859
 [229] 0.53433439 0.66484977 0.78081365 0.48081325 0.94168846 0.88378100
 [235] 0.49217084 0.47137741 0.93567433 0.29856254 0.50019941 0.49360870
 [241] 0.62680060 0.86130584 0.32969027 0.48207241 0.78421358 0.79868330
 [247] 0.46986114 0.70189770 0.50387301 0.88634082 0.48462627 0.50816621
 [253] 0.95907260 0.97515814 0.83892976 0.94780476 0.64779170 0.69168797
 [259] 0.52117698 0.95843851 0.91397722 0.67214891 0.74234573 0.41240654
 [265] 0.54655368 0.82030569 0.72468515 0.44425721 0.66974831 0.89483560
 [271] 0.32784895 0.86378936 0.75588933 0.51847194 0.94697694 0.73346932
 [277] 0.73529520 0.71253819 0.76540276 0.60754201 0.36286275 0.70077005
 [283] 0.60547962 0.59826311 0.80848854 0.78367180 0.68523384 0.63322618
 [289] 0.31663037 0.78863580 0.88771188 0.58873112 0.34291498 0.45477274
 [295] 0.92197542 0.69736681 0.42557904 0.86041373 0.91624472 0.57127651
 [301] 0.53454154 0.59469081 0.53662513 0.80515138 0.69332291 0.88896269
 [307] 0.45779025 0.80250657 0.93644776 0.91645914 0.73811532 0.77290660
 [313] 0.56276717 0.87958967 0.90048494 0.92140731 0.57848697 0.45622951
 [319] 0.65894808 0.52440900 0.59333622 0.42540211 0.37112133 0.86327701
 [325] 0.66640271 0.62317660 0.92973663 0.81763369 0.69316548 0.51512447
 [331] 0.88295012 0.93449259 0.94785643 0.81283962 0.89831170 0.53930420
 [337] 0.86980965 0.81370865 0.80600488 0.65117513 0.41722548 0.37906203
 [343] 0.43626664 0.67549814 0.93497810 0.76803812 0.75215769 0.90719938
 [349] 0.46415497 0.89044192 0.92772863 0.57708335 0.46008320 0.46087404
 [355] 0.89995355 0.23683902 0.42253590 0.76639414 0.83743328 0.69001759
 [361] 0.89183743 0.76181680 0.95112934 0.38776147 0.51805270 0.71167193
 [367] 0.47319907 0.73922499 0.43590365 0.76633156 0.41247598 0.89179469
 [373] 0.75843619 0.82510392 0.37938536 0.38793436 0.48591818 0.79522178
 [379] 0.90253675 0.99583776 0.75159398 0.61481648 0.56071620 0.71145787
 [385] 0.14429858 0.83058274 0.53192808 0.61251710 0.64500035 0.38353130
 [391] 0.85763195 0.85581606 0.28314921 0.81779673 0.58195776 0.52506053
 [397] 0.41242059 0.53775947 0.69748103 0.51737079 0.40809884 0.71364386
 [403] 0.23752043 0.88098601 0.78757537 0.13153703 0.27675556 0.57024565
 [409] 0.90893798 0.65119185 0.29021869 0.92784545 0.78385274 0.63631218
 [415] 0.70489511 0.81025585 0.60310826 0.69455599 0.86676020 0.48363268
 [421] 0.39728115 0.33412553 0.68645860 0.43656733 0.91719218 0.81859816
 [427] 0.89590405 0.75611218 0.68759806 0.81660724 0.21403783 0.71875865
 [433] 0.54569378 0.53753658 0.81027123 0.96776669 0.90881224 0.52634091
 [439] 0.08845972 0.32924708 0.97849461 0.90090783 0.25928220 0.52514663
 [445] 0.92377738 0.93226497 0.94581675 0.94530292 0.77911471 0.74451680
 [451] 0.24539952 0.98315356 0.90404065 0.84988935 0.62898459 0.22401682
 [457] 0.68695573 0.67708615 0.89339605 0.89692330 0.64748227 0.71542166
 [463] 0.77324005 0.58946259 0.82636432 0.64861768 0.90519755 0.51102770
 [469] 0.89655865 0.69998894 0.34490260 0.30253275 0.64810615 0.64886985
 [475] 0.94748442 0.75949976 0.59781629 0.48214149 0.58129534 0.72623841
 [481] 0.91595320 0.70859062 0.70037201 0.84338353 0.87316041 0.80326655
 [487] 0.13221658 0.75008392 0.95314778 0.60796310 0.95229195 0.92257900
 [493] 0.91172002 0.89919306 0.49259206 0.77057971 0.89697843 0.62776470
 [499] 0.90710946 0.67245645 0.46786056 0.84160435 0.80017741 0.66609863
 [505] 0.68394364 0.83770706 0.76308731 0.91354227 0.40667890 0.20805640
 [511] 0.32361917 0.48777973 0.93478843 0.91433380 0.73281501 0.88513890
 [517] 0.87452873 0.84018003 0.90850537 0.85113716 0.27775619 0.72651702
 [523] 0.38649400 0.60663954 0.55990030 0.89319363 0.75742423 0.97696029
 [529] 0.87173490 0.27771892 0.73358960 0.60074847 0.12634684 0.86144025
 [535] 0.32893283 0.90896180 0.30940379 0.81081787 0.75133418 0.56225421
 [541] 0.76066392 0.64695441 0.70184268 0.72770953 0.77062529 0.18590811
 [547] 0.58315383 0.69695932 0.81792694 0.96023926 0.82652904 0.78316600
 [553] 0.98906578 0.43615533 0.55504225 0.51256948 0.81253776 0.42432362
 [559] 0.68195957 0.49801849 0.86280114 0.93914090 0.66963536 0.65997921
 [565] 0.82191355 0.90163970 0.36957168 0.70958667 0.70958321 0.78847847
 [571] 0.14832650 0.91288471 0.76284263 0.83206729 0.78650043 0.56508758
 [577] 0.24521800 0.07105085 0.62476028 0.42452690 0.58926417 0.90220517
 [583] 0.37545039 0.40052803 0.92786176 0.87441329 0.37801088 0.85723280
 [589] 0.80821541 0.42744419 0.92492257 0.72977252 0.79058812 0.24461007
 [595] 0.90235813 0.93629827 0.24434826 0.75925372 0.21097143 0.85873939
 [601] 0.87131277 0.63163282 0.74242331 0.73285816 0.66859113 0.56691012
 [607] 0.90757434 0.71375612 0.88287681 0.16346698 0.39670572 0.39238229
 [613] 0.74386650 0.93310027 0.64786263 0.81540686 0.23764291 0.76777628
 [619] 0.74620921 0.84025769 0.72683570 0.88918766 0.44775041 0.69380375
 [625] 0.65588961 0.68450278 0.33019351 0.88275727 0.82054787 0.61971169
 [631] 0.15782266 0.41637543 0.68488526 0.94998803 0.79650191 0.66404285
 [637] 0.52320165 0.64016756 0.95622537 0.66439872 0.74369660 0.82481866
 [643] 0.75447749 0.93854438 0.59406191 0.74255615 0.93743469 0.82014358
 [649] 0.62249159 0.49087486 0.91605824 0.65755010 0.77609708 0.96093360
 [655] 0.20685282 0.95765225 0.90507583 0.59115105 0.50681190 0.38742666
 [661] 0.34741863 0.49981720 0.81905512 0.61201187 0.51256805 0.68639698
 [667] 0.54795772 0.49586544 0.86348031 0.94601306 0.85311239 0.75694711
 [673] 0.67382683 0.84682856 0.61076848 0.96619317 0.95998892 0.97646664
 [679] 0.57434012 0.97522272 0.72023978 0.18056512 0.31807534 0.71353640
 [685] 0.62667241 0.59118390 0.86601519 0.26594444 0.53438401 0.91789165
 [691] 0.67732736 0.69517841 0.40281273 0.53856276 0.27797953 0.55158739
 [697] 0.71251007 0.47694185 0.71447433 0.91694289 0.87871450 0.90577828
 [703] 0.69855960 0.57049775 0.65204124 0.82323723 0.61351425 0.89554349
 [709] 0.62780927 0.66050783 0.69940145 0.91209776 0.69320857 0.74764918
 [715] 0.73632038 0.78063194 0.74647780 0.46030886 0.95712766 0.67352154
 [721] 0.92315822 0.24983224 0.37523555 0.82320493 0.43617373 0.48069226
 [727] 0.85359816 0.64204894 0.65719180 0.79706442 0.78183606 0.85825141
 [733] 0.93366202 0.52532756 0.83724469 0.93548748 0.70603221 0.53820875
 [739] 0.81966099 0.87376971 0.73161796 0.98025672 0.81458300 0.17005217
 [745] 0.78407451 0.63679641 0.41289618 0.91641362 0.68506247 0.61871251
 [751] 0.86570699 0.55001924 0.16791417 0.69872225 0.76127238 0.86567833
 [757] 0.70870413 0.91201959 0.74275717 0.63542711 0.45814522 0.85433099
 [763] 0.89725203 0.74367477 0.89552871 0.89942593 0.84389645 0.73106147
 [769] 0.97948957 0.51706382 0.89203146 0.80446000 0.48458086 0.82040265
 [775] 0.71072712 0.56714064 0.58550555 0.85590978 0.66038817 0.83060360
 [781] 0.96668474 0.67874413 0.62974007 0.96808348 0.87428081 0.79351740
 [787] 0.78085353 0.90027629 0.72689194 0.86261363 0.96802763 0.65898493
 [793] 0.96115273 0.74148245 0.89745805 0.70307640 0.95304897 0.77737174
 [799] 0.95469966 0.40429175 0.88068562 0.79414113 0.89484258 0.85434576
 [805] 0.58520032 0.90442024 0.91913056 0.75887957 0.54132032 0.67491837
 [811] 0.76516880 0.49293945 0.87560422 0.11708032 0.97625558 0.54140703
 [817] 0.55886235 0.78073088 0.46194385 0.88017895 0.81123735 0.79579761
 [823] 0.55493176 0.90999467 0.96716383 0.40433688 0.89349044 0.69581266
 [829] 0.47609445 0.51561163 0.46722349 0.73381428 0.36130466 0.97196711
 [835] 0.54653115 0.43305134 0.53516849 0.14414233 0.31390520 0.86668269
 [841] 0.84567998 0.92373683 0.34170069 0.84325961 0.91052444 0.47357342
 [847] 0.90283433 0.76803086 0.98684521 0.77034903 0.65856530 0.89423601
 [853] 0.94107097 0.57435548 0.75057165 0.41443026 0.82735650 0.75805515
 [859] 0.86676235 0.81643102 0.22631132 0.73738431 0.60901844 0.42893818
 [865] 0.96694377 0.68600250 0.90003581 0.70280159 0.40200390 0.95663393
 [871] 0.49880131 0.72752142 0.29982741 0.52824339 0.88284201 0.23823458
 [877] 0.72942379 0.20774354 0.78883895 0.86799181 0.54508523 0.70060067
 [883] 0.79462743 0.38253385 0.88085429 0.56388185 0.81135741 0.32427171
 [889] 0.30407175 0.64750192 0.75348386 0.63154105 0.52065995 0.59161942
 [895] 0.70341936 0.56173863 0.80798764 0.75997696 0.53891864 0.46410711
 [901] 0.97082929 0.52450560 0.79611763 0.61853425 0.61107146 0.74956332
 [907] 0.68192998 0.81258369 0.93871910 0.87376701 0.79964256 0.37603484
 [913] 0.94121536 0.51992971 0.98998898 0.91024806 0.90145442 0.77323921
 [919] 0.83178850 0.54567015 0.33885266 0.76256904 0.91878883 0.81928071
 [925] 0.64619664 0.97009769 0.20441581 0.83385861 0.51039441 0.88980444
 [931] 0.60914431 0.54090013 0.60545950 0.78743558 0.90675603 0.81939327
 [937] 0.81210703 0.73398365 0.50054148 0.90899406 0.39051331 0.56567806
 [943] 0.86751488 0.82610630 0.67451786 0.93622909 0.85049646 0.54578103
 [949] 0.74019059 0.47430336 0.94549413 0.76148500 0.88940215 0.74834396
 [955] 0.73445101 0.97624180 0.73872050 0.88030052 0.28644410 0.90245543
 [961] 0.52892813 0.82681410 0.94282738 0.73806640 0.80260461 0.85219906
 [967] 0.92014972 0.82986727 0.85484194 0.47924152 0.35278989 0.91696398
 [973] 0.92814134 0.98996959 0.75252095 0.20089237 0.72887514 0.90857539
 [979] 0.73841014 0.96429926 0.86062542 0.81685330 0.93195887 0.64335784
 [985] 0.76108780 0.87255416 0.65133162 0.89177626 0.79788427 0.35256629
 [991] 0.81832037 0.95351192 0.45495734 0.87796228 0.76131049 0.91693544
 [997] 0.91846698 0.73369477 0.52643756 0.43209023

y <- as.numeric(p > 0.5)
y

   [1] 0 1 1 1 0 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
  [38] 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1
  [75] 1 1 1 0 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [112] 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1
 [149] 1 1 1 0 1 1 1 1 1 1 0 0 0 1 1 0 1 0 0 0 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1
 [186] 0 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 0 1 1 1
 [223] 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 0 1 1 0 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1
 [260] 1 1 1 1 0 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 0 0 1 1
 [297] 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1
 [334] 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 1 1 1 0 0 1 0 0 1 1 1 1 1 1 0 1 1 0 1 0 1
 [371] 0 1 1 1 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 1 1 1 0 1 1 1 0 1 0 1 1 0 0
 [408] 1 1 1 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 0 1
 [445] 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 0 1 1 1
 [482] 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1
 [519] 1 1 0 1 0 1 1 1 1 1 1 0 1 1 0 1 0 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1
 [556] 1 1 0 1 0 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 0 0 1 0 1 1 0 0 1 1 0 1 1 0 1 1
 [593] 1 0 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 1
 [630] 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 0 0 1 1 1 1
 [667] 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 1
 [704] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [741] 1 1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1
 [778] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 0
 [815] 1 1 1 1 0 1 1 1 1 1 1 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 1 1 0 1 1 0 1 1 1 1 1
 [852] 1 1 1 1 0 1 1 1 1 0 1 1 0 1 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 1 1 0 1 1 1 0
 [889] 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1
 [926] 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1
 [963] 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1
[1000] 0

Por alguna razón, denominada como el problema de la separación completa, cuando se quiere realizar la estimación de los parámetros del modelo logístico, se genera una serie de errores.

glm(y~x, family=binomial(link="logit"))


Call:  glm(formula = y ~ x, family = binomial(link = "logit"))

Coefficients:
(Intercept)            x  
     -59631         3578  

Degrees of Freedom: 999 Total (i.e. Null);  998 Residual
Null Deviance:      989.6 
Residual Deviance: 0.0004403    AIC: 4

La forma CORRECTA de simular la variable respuesta es mediante el uso de la función rbinom, así:

y <- rbinom(1000, 1, p)
y

   [1] 0 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 1 1 1 0 0 0 0 0 1 1 1 0
  [38] 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 0 1
  [75] 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1
 [112] 1 1 1 1 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 0 0 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0
 [149] 1 0 1 0 1 1 1 0 1 1 1 0 0 1 0 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1 0 1 0 1
 [186] 1 1 1 0 1 0 1 1 1 1 1 1 1 0 1 0 0 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 1
 [223] 1 1 1 1 1 0 1 0 0 1 1 1 0 0 1 1 0 1 1 1 0 0 0 1 0 0 1 0 0 1 1 1 1 1 0 1 1
 [260] 1 1 0 1 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 0 1 0 1 1
 [297] 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1
 [334] 1 1 1 1 1 1 0 1 0 1 1 1 1 0 1 1 1 1 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 1
 [371] 0 1 1 0 0 0 0 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 0 0 1 0 0 0
 [408] 1 1 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 1
 [445] 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0
 [482] 0 1 0 1 1 0 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 1 1
 [519] 1 1 1 1 1 0 1 1 1 1 1 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1
 [556] 1 0 0 1 1 0 1 1 1 1 1 0 1 0 0 0 1 1 0 1 1 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1
 [593] 1 1 1 0 0 1 1 1 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 1 0
 [630] 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 0 1 0
 [667] 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 0 1 1 0 1 0 0 1 1 0 1 1 1 1 1
 [704] 1 0 1 1 1 0 0 1 1 1 1 0 0 0 1 1 0 1 0 0 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 1 1
 [741] 1 1 0 0 1 1 1 1 1 0 1 1 0 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 0 1 1
 [778] 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 0 1 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 0 1 0
 [815] 1 0 0 1 1 1 1 1 0 1 1 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 0 1 1 1 1 1
 [852] 1 1 1 1 0 1 1 1 1 0 1 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 1 1 1
 [889] 0 0 1 1 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 0
 [926] 1 0 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1
 [963] 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 1
[1000] 0

glm(y~x, family=binomial(link="logit"))


Call:  glm(formula = y ~ x, family = binomial(link = "logit"))

Coefficients:
(Intercept)            x  
    -5.2673       0.3198  

Degrees of Freedom: 999 Total (i.e. Null);  998 Residual
Null Deviance:      1208 
Residual Deviance: 983.3    AIC: 987.3

De cierta forma, cuando se simula la variable respuesta de la forma incorrecta, se está perdiendo la aleatoriedad de la variable, puesto que se crea una relación determinística entre las probabilidades de éxito y los valores de la variable. Por esta razón, el algoritmo de estimación de Newton-Raphson no converge, pues la función de verosimilitud no tendrá máximo. Este problema se evade fácilmente cuando, por medio de la función rbinom, se garantiza la aleatoriedad de la variable respuesta.