391:
1284:
1735:
440:
1044:
1593:
1279:{\displaystyle {\begin{aligned}Y_{i}^{1\ast }&={\boldsymbol {\beta }}_{1}\cdot \mathbf {X} _{i}+\varepsilon _{1}\,\\Y_{i}^{2\ast }&={\boldsymbol {\beta }}_{2}\cdot \mathbf {X} _{i}+\varepsilon _{2}\,\\\ldots &\ldots \\Y_{i}^{m\ast }&={\boldsymbol {\beta }}_{m}\cdot \mathbf {X} _{i}+\varepsilon _{m}\,\\\end{aligned}}}
1350:
868:
647:
The observed outcomes might be "has disease A, has disease B, has disease C, has none of the diseases" for a set of rare diseases with similar symptoms, and the explanatory variables might be characteristics of the patients thought to be pertinent (sex, race, age,
1588:{\displaystyle Y_{i}={\begin{cases}1&{\text{if }}Y_{i}^{1\ast }>Y_{i}^{2\ast },\ldots ,Y_{i}^{m\ast }\\2&{\text{if }}Y_{i}^{2\ast }>Y_{i}^{1\ast },Y_{i}^{3\ast },\ldots ,Y_{i}^{m\ast }\\\ldots &\ldots \\m&{\text{otherwise.}}\end{cases}}}
668:
that can be used to predict the likely outcome of an unobserved multi-way trial given the associated explanatory variables. In the process, the model attempts to explain the relative effect of differing explanatory variables on the different outcomes.
1339:
659:
The observed outcomes are the votes of people for a given party or candidate in a multi-way election, and the explanatory variables are the demographic characteristics of each person (e.g. sex, race, age, income,
712:
1674:
1013:
1712:
1049:
1295:
863:{\displaystyle Y_{i}|x_{1,i},\ldots ,x_{k,i}\ \sim \operatorname {Categorical} (p_{i,1},\ldots ,p_{i,m}),{\text{ for }}i=1,\dots ,n}
421:
1684:
508:
450:
331:
480:
1604:
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879:
529:
This article is about modeling a single event with multiple outcomes. For modeling several correlated binary outcomes, see
321:
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487:
1815:
706:
at hand because it is determined by the values of the explanatory variables associated with that observation. That is:
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571:, which is used to model correlated binary outcomes for more than one independent variable.
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53:
1334:{\displaystyle {\boldsymbol {\varepsilon }}\sim {\mathcal {N}}(0,{\boldsymbol {\Sigma }})}
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24:
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1679:
Note that this model allows for arbitrary correlation between the
1786:(Seventh ed.). Boston: Pearson Education. pp. 810–811.
1765:
For details on how the equations are estimated, see the article
1714:
is the identity matrix (such that there is no correlation or
1581:
552:
used when there are several possible categories that the
1669:{\displaystyle Y_{i}=\arg \max _{h=1}^{m}Y_{i}^{h\ast }}
643:, predictor variables, features, etc.). Some examples:
1008:{\displaystyle \Pr=p_{i,h},{\text{ for }}i=1,\dots ,n,}
1697:
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1607:
1353:
1298:
1047:
882:
715:
1707:{\displaystyle \scriptstyle {\boldsymbol {\Sigma }}}
579:
It is assumed that we have a series of observations
556:
can fall into. As such, it is an alternative to the
1706:
1668:
1587:
1333:
1278:
1034:Multinomial probit is often written in terms of a
1007:
862:
656:, presence or absence of various symptoms, etc.).
1802:
1628:
883:
608:possible choices). Along with each observation
596:, of the outcomes of multi-way choices from a
415:
466:introducing citations to additional sources
422:
408:
1683:, so that it doesn't necessarily respect
1271:
1187:
1117:
639:of explanatory variables (also known as
1029:
574:
456:Relevant discussion may be found on the
1685:independence of irrelevant alternatives
1300:
1233:
1149:
1079:
1803:
1778:
693:occurs with an unobserved probability
1729:
702:that is specific to the observation
564:. It is not to be confused with the
433:
13:
1309:
664:The multinomial probit model is a
14:
1827:
1733:
1699:
1324:
1248:
1164:
1094:
449:relies largely or entirely on a
438:
389:
685:data, where each outcome value
337:Least-squares spectral analysis
275:Generalized estimating equation
95:Multinomial logistic regression
70:Vector generalized linear model
1328:
1314:
951:
906:
886:
828:
784:
727:
1:
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1725:
156:Nonlinear mixed-effects model
7:
548:is a generalization of the
358:Mean and predicted response
10:
1832:
1816:Statistical classification
528:
151:Linear mixed-effects model
683:categorically-distributed
562:multiclass classification
531:multivariate probit model
317:Least absolute deviations
598:categorical distribution
560:model as one method of
546:multinomial probit model
65:Generalized linear model
1718:), the model is called
681:are described as being
672:Formally, the outcomes
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864:
396:Mathematics portal
322:Iteratively reweighted
1709:
1671:
1627:
1590:
1336:
1281:
1036:latent variable model
1030:Latent variable model
1010:
865:
641:independent variables
575:General specification
353:Regression validation
332:Bayesian multivariate
49:Polynomial regression
1784:Econometric Analysis
1694:
1605:
1351:
1296:
1045:
880:
713:
477:"Multinomial probit"
462:improve this article
378:Gauss–Markov theorem
373:Studentized residual
363:Errors and residuals
197:Principal components
167:Nonlinear regression
54:General linear model
1811:Regression analysis
1665:
1551:
1524:
1503:
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1450:
1423:
1402:
1223:
1139:
1069:
1022:possible values of
223:Errors-in-variables
90:Logistic regression
80:Binomial regression
25:Regression analysis
19:Part of a series on
1780:Greene, William H.
1745:. You can help by
1720:independent probit
1716:heteroscedasticity
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554:dependent variable
110:Multinomial probit
1793:978-0-273-75356-8
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666:statistical model
558:multinomial logit
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85:Binary regression
44:Simple regression
39:Linear regression
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873:or equivalently
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689:for observation
621:observed values
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301:Ridge regression
136:Multilevel model
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654:body-mass index
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368:Goodness of fit
75:Discrete choice
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650:blood pressure
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460:. Please help
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146:Random effects
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1754:February 2017
1748:
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1741:This section
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479: –
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473:Find sources:
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451:single source
447:This article
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239:Least squares
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172:Nonparametric
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141:Fixed effects
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115:Ordered logit
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1783:
1767:Probit model
1764:
1751:
1747:adding to it
1742:
1719:
1689:
1678:
1597:
1343:
1288:
1033:
1023:
1019:
1018:for each of
1017:
872:
703:
698:
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631:
626:
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617:is a set of
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584:
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569:probit model
567:multivariate
566:
550:probit model
545:
542:econometrics
535:
515:
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491:
484:
472:
448:
296:Non-negative
109:
779:Categorical
604:(there are
306:Regularized
270:Generalized
202:Least angle
100:Mixed logit
1805:Categories
1773:References
1726:Estimation
1575:otherwise.
538:statistics
488:newspapers
345:Background
249:Non-linear
231:Estimation
1700:Σ
1662:∗
1625:
1598:That is,
1562:…
1557:…
1548:∗
1529:…
1521:∗
1500:∗
1479:∗
1447:∗
1428:…
1420:∗
1399:∗
1325:Σ
1305:∼
1301:ε
1263:ε
1244:⋅
1234:β
1220:∗
1200:…
1193:…
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1080:β
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994:…
930:…
852:…
807:…
782:
776:∼
751:…
518:July 2015
458:talk page
212:Segmented
1782:(2012).
1462:if
1382:if
600:of size
327:Bayesian
265:Weighted
260:Ordinary
192:Isotonic
187:Quantile
630:, ...,
502:scholar
286:Partial
125:Poisson
1790:
1289:where
773:
660:etc.).
592:= 1...
588:, for
544:, the
504:
497:
490:
483:
475:
244:Linear
182:Robust
105:Probit
31:Models
1690:When
1344:Then
509:JSTOR
495:books
291:Total
207:Local
1788:ISBN
1484:>
1404:>
540:and
481:news
1749:.
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627:1,i
536:In
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423:e
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