?????? Categorical Data Analysis ??? (T.Y. Wang) ??????? (Illinois State University) [e-mail]

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??????Categorical Data Analysis???(T.Y.
Wang)???????(Illinois State University)
tywang_at_ilstu.edu
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• ?????????????(linear regression models)????????

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• ???????
• ?????????,????????
• ?
• ?

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• Heteroscedasticity (???????????,??????????? vs.
  homoscedasticity)

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E(x)p Var(x)p(1-p)
0.1 0.09
0.2 0.16
0.3 0.21
0.4 0.24
0.5 0.25
0.6 0.24
0.7 0.21
0.8 0.16
0.9 0.09
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• ? y ??????? ,??????(variance) ?????? x ????

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• ??????? (?????) (Functional form)
• ????????
• (Nonsensical predictions )
• prvalue, x(age35 k54 wc0 hc0)

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• Example Labor force participation of women (file
  name binlfp2)

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• ?????????,??????????????,??????????

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• Models to be Considered
• Binary Logit (??????) and Probit(?????? )
• Ordinal Ordered Logit (??????)and
  Probit(??????)
• Nominal Multinomial Logit(??????)
• Count Poisson (?? ) and Negative Binomial(???)

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• Maximum Likelihood (??????)Estimation
• ???????????????????(The ML estimate is the value
  that maximizes the likelihood of observing the
  sample data)

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• ????????????????(the method of least
  squares)???????
• ??????????????????????????

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• ??????????????,????????????????????????,??????????
  ?(realization)?

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• ?Latent Variable Model(??????)??

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?0
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 y
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• Structural Model

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• ??????????????????????,???????????????????,???????
  ?????????????????????????

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• ??????????????,????????????

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• ?????????????????(numerical methods)??????????????
  ??

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• ??????????????(????? (start value)),????????????
  ?????,???????????
• ?????????????,?????????(iteration)

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• ??????????????????????????,????????????(convergenc
  e),???????????

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??????(Models for Binary Outcomes)
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• ?Latent Variable Model(??????)??

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?0
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 y
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• Structural Model

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• Distribution of Errors ( )
• binary logit model(????????) if logistic
  distribution
• binary probit model(????????) if normal
  distribution

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• ??????????(Generalized Linear Model) ??
• ????(random component)
• ???? (system component)
• ????(link component)

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• ???? ????????? y,?????????? ??????????????(Bernou
  lli)??

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• ???? ????????
• ???????

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• ????????????? ?????????????, ??????

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?????(logit)??????????????
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• ??(???,odds)
• ????????????????????????
• ??
• ??

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• ??4,??????0.8,???????0.2?????????????4?
• ?????,???

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• ????????
• ??0????,???
• ???????????,???????????????,????????,????

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• ????????
• ??????1 ,???????,???????1???
• ?????1?,?????????????????

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0.01 0.2 0.5 0.7 0.9 0.99
0.99 0.8 0.5 0.3 0.1 0.01
?? 0.01 0.25 1.00 2.33 9.00 99.0
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• odds gt1,?????????????
• 0lt odds lt1,?????????????
• odds 1,?????????????

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• ????????

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• ???(odds ratio) ??????????????

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?? ?? ??(Odds)
?????? 600 5000 0.12
???? 400 60000 0.0067
???(Odds Ratio) 18
?????????,???????????????? ?????????,???????????????? ?????????,???????????????? ?????????,????????????????
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• ??????(logit, the natural log of odds)
• ??????????,????????0???
• ????0.5??,????
• ??????????????????????

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0.1 0.3 0.5 0.7 0.9
0.9 0.7 0.5 0.3 0.1
odds 0.11 0.43 1.00 2.33 9.00
logit -2.20 -0.85 0.00 0.85 2.20
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• ????????????,???????????????(linear in logit)

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• the same change in x have constant effects
• ??????????,????????????????(nonlinear in
  probability)

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• ?????????(exponent)

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• probabilities depends on the start level of xi
  and the values of other variables
• ???????????????? (A Nonlinear Probability Model)

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• ?????????????????????,?????
• ????? ?
• ????,????????,???????(identity)??????????????,????
  ??

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• ??????,?
• ???????????
• ????????????,???0?1????????????????????????

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• ????????????
• ???(Bernoulli)??
• ?????
• binary logit model(????????)
• binary probit model(????????)

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• Example Labor force participation of women (file
  name binlfp2)
• Difference and Similarity between Logit and Probit

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• Hypothesis Testing
• ????,???,p ????
• test (testing effects of single e.g., test k5,
  or multiple coefficients being equal, e.g., test
  hcwc)
• lrtestComparing competitive (nested) models
  using LR test

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• The LR test assesses a hypothesis by comparing
  the log likelihood from the full model (MU) and a
  restricted model (MC) (H0 imposed constraints
  are true). If the constraints significantly
  reduce the log likelihood, the H0 is rejected and
  the full model is preferred

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• Consider the following models
• M1
• M2
• M3

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• When conducting LR test
• the two models must be nested (i.e., a nested
  model is created by imposing constraints on the
  coefficients in the prior model),
• the two models must be fitted on exactly the
  same sample

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• mark and markout to exclude cases with missing
  data
• mark nomiss
• markout nomiss lfpk5 k618 age wc hc lwg inc
• logit lfpk5 inc if nomiss1

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• ??????
• predict????????
• prvalue ? prtab????????(profile)
• prchange?????????????????
• listcoef???????(odds ratio)

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• predict?????????????
• predict prlogit

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• prvalue ? prtab????????(profile)
• prvalue, x(age50 k50 k6180 wc1 hc1)
  rest(mean)
• prtab k5 wc, rest(mean)

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• prchange?????????????????
• marginal change

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• ?????????????, ???????????????? -0.0084
• ?????????????????,????????????????????,?????????,?
  ??

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• discrete change
• ??,

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• discrete change
• ?????????????, ??????????,?????????? 0.34
• ???????????, ?????????????, ??????? 0.19

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• min-gtmax
• ??????????????,????????????????,??????? 0.4372

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• listcoef???????(odds ratio)

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• ?????????,????????????????????

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• ? ????????????,??????

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• ??
• ???????????,??? ???????,??????????? ?

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• ? exp( ) gt1,?,?????????????exp( )?
• ?? exp( ) lt1 ?,?????????????exp( ) ?

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• ?????????????,???????,?????????(???)????0.23?
• ?????????????,???????,?????????(???)????76.8

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• Residuals (???) and Influence (?????)using
  predict

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• ????
• drop if index752
• index345index142
• (data/create or change data/
• keep or drop observations)

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• Scalar Measures of Fit (???)using fitstat
• ??????R2?(Pseudo-R2)
• Information Measures can be used to compare
  models across different samples or to compare
  non-nested models

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• AIC the smaller AIC, the better the fit
• BIC the more negative the BIC, the better the
  fit

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• ??????????
• ????????????????????
• Stata???Wang_PR_BRM.dta