Ordered probit models

1
Ordered probit models
2
Ordered Probit

• Many discrete outcomes are to questions that have
  a natural ordering but no quantitative
  interpretation
• Examples
• Self reported health status
• (excellent, very good, good, fair, poor)
• Do you agree with the following statement
• Strongly agree, agree, disagree, strongly disagree

3

• Can use the same type of model as in the previous
  section to analyze these outcomes
• Another latent variable model
• Key to the model there is a monotonic ordering
  of the qualitative responses

4
Self reported health status

• Excellent, very good, good, fair, poor
• Coded as 1, 2, 3, 4, 5 on National Health
  Interview Survey
• We will code as 5,4,3,2,1 (easier to think of
  this way)
• Asked on every major health survey
• Important predictor of health outcomes, e.g.
  mortality
• Key question what predicts health status?

5

• Important to note the numbers 1-5 mean nothing
  in terms of their value, just an ordering to show
  you the lowest to highest
• The example below is easily adapted to include
  categorical variables with any number of outcomes

6
Model

• yi latent index of reported health
• The latent index measures your own scale of
  health. Once yi crosses a certain value you
  report poor, then good, then very good, then
  excellent health

7

• yi (1,2,3,4,5) for (fair, poor, VG, G, excel)
• Interval decision rule
• yi1 if yi u1
• yi2 if u1 lt yi u2
• yi3 if u2 lt yi u3
• yi4 if u3 lt yi u4
• yi5 if yi gt u4

8

• As with logit and probit models, we will assume
  yi is a function of observed and unobserved
  variables
• yi ß0 x1i ß1 x2i ß2 . xki ßk ei
• yi xi ß ei

9

• The threshold values (u1, u2, u3, u4) are
  unknown. We do not know the value of the index
  necessary to push you from very good to
  excellent.
• In theory, the threshold values are different for
  everyone
• Computer will not only estimate the ßs, but also
  the thresholds average across people

10

• As with probit and logit, the model will be
  determined by the assumed distribution of e
• In practice, most people pick nornal, generating
  an ordered probit (I have no idea why)
• We will generate the math for the probit version

11
Probabilities

• Lets do the outliers, Pr(yi1) and Pr(yi5) first
• Pr(yi1)
• Pr(yi u1)
• Pr(xi ß ei u1 )
• Pr(ei u1 - xi ß)
• Fu1 - xi ß 1- Fxi ß u1

12

• Pr(yi5)
• Pr(yi gt u4)
• Pr(xi ß ei gt u4 )
• Pr(ei gt u4 - xi ß)
• 1 - Fu4 - xi ß Fxi ß u4

13
Sample one for y3

• Pr(yi3) Pr(u2 lt yi u3)
• Pr(yi u3) Pr(yi u2)
• Pr(xi ß ei u3) Pr(xi ß ei u2)
• Pr(ei u3- xi ß) - Pr(ei u2 - xi ß)
• Fu3- xi ß - Fu2 - xi ß
• 1 - Fxi ß - u3 1 Fxi ß - u2
• Fxi ß - u2 - Fxi ß - u3

14
Summary

• Pr(yi1) 1- Fxi ß u1
• Pr(yi2) Fxi ß u1 - Fxi ß u2
• Pr(yi3) Fxi ß u2 - Fxi ß u3
• Pr(yi4) Fxi ß u3 - Fxi ß u4
• Pr(yi5) Fxi ß u4

15
Likelihood function

• There are 5 possible choices for each person
• Only 1 is observed
• L Si lnPr(yik) for k

16
Programming example

• Cancer control supplement to 1994 National Health
  Interview Survey
• Question what observed characteristics predict
  self reported health (1-5 scale)
• 1poor, 5excellent
• Key covariates income, education, age, current
  and former smoking status
• Programs
• sr_health_status.do, .dta, .log

17

• desc
• male byte 9.0g 1 if male
• age byte 9.0g age in years
• educ byte 9.0g years of
  education
• smoke byte 9.0g current smoker
• smoke5 byte 9.0g smoked in past 5
  years
• black float 9.0g 1 if respondent
  is black
• othrace float 9.0g 1 if other race
  (white is ref)
• sr_health float 9.0g 1-5 self reported
  health,
• 
  5excel, 1poor
• famincl float 9.0g log family income

18

• tab sr_health
• 1-5 self
• reported
• health,
• 5excel,
• 1poor Freq. Percent Cum.
• -----------------------------------------------
• 1 342 2.65 2.65
• 2 991 7.68 10.33
• 3 3,068 23.78 34.12
• 4 3,855 29.88 64.00
• 5 4,644 36.00 100.00
• -----------------------------------------------
• Total 12,900 100.00

19
In STATA

• oprobit sr_health male age educ famincl black
  othrace smoke smoke5

20

• Ordered probit estimates
  Number of obs 12900
• 
  LR chi2(8) 2379.61
• 
  Prob gt chi2 0.0000
• Log likelihood -16401.987
  Pseudo R2 0.0676
• --------------------------------------------------
  ----------------------------
• sr_health Coef. Std. Err. z
  Pgtz 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• male .1281241 .0195747 6.55
  0.000 .0897583 .1664899
• age -.0202308 .0008499 -23.80
  0.000 -.0218966 -.018565
• educ .0827086 .0038547 21.46
  0.000 .0751535 .0902637
• famincl .2398957 .0112206 21.38
  0.000 .2179037 .2618878
• black -.221508 .029528 -7.50
  0.000 -.2793818 -.1636341
• othrace -.2425083 .0480047 -5.05
  0.000 -.3365958 -.1484208
• smoke -.2086096 .0219779 -9.49
  0.000 -.2516855 -.1655337
• smoke5 -.1529619 .0357995 -4.27
  0.000 -.2231277 -.0827961
• -------------------------------------------------
  ----------------------------
• _cut1 .4858634 .113179
  (Ancillary parameters)
• _cut2 1.269036 .11282

21
Interpret coefficients

• Marginal effects/changes in probabilities are now
  a function of 2 things
• Point of expansion (xs)
• Frame of reference for outcome (y)
• STATA
• Picks mean values for xs
• You pick the value of y

22
Continuous xs

• Consider y5
• d Pr(yi5)/dxi
• d Fxi ß u4/dxi ßfxi ß u4
• Consider y3
• d Pr(yi3)/dxi ßfxi ß u3 - ßfxi ß u4

23
Discrete Xs

• xi ß ß0 x1i ß1 x2i ß2 . xki ßk
• X2i is yes or no (1 or 0)
• ?Pr(yi5)
• Fß0 x1i ß1 ß2 x3i ß3 .. xki ßk
• - Fß0 x1i ß1 x3i ß3 . xki ßk
• Change in the probabilities when x2i1 and x2i0

24
Ask for marginal effects

• mfx compute, predict(outcome(5))

25

• mfx compute, predict(outcome(5))
• Marginal effects after oprobit
• y Pr(sr_health5) (predict, outcome(5))
• .34103717
• --------------------------------------------------
  ----------------------------
• variable dy/dx Std. Err. z Pgtz
  95 C.I. X
• -------------------------------------------------
  ----------------------------
• male .0471251 .00722 6.53 0.000
  .03298 .06127 .438062
• age -.0074214 .00031 -23.77 0.000
  -.008033 -.00681 39.8412
• educ .0303405 .00142 21.42 0.000
  .027565 .033116 13.2402
• famincl .0880025 .00412 21.37 0.000
  .07993 .096075 10.2131
• black -.0781411 .00996 -7.84 0.000
  -.097665 -.058617 .124264
• othrace -.0843227 .01567 -5.38 0.000
  -.115043 -.053602 .04124
• smoke -.0749785 .00773 -9.71 0.000
  -.09012 -.059837 .289147
• smoke5 -.0545062 .01235 -4.41 0.000
  -.078719 -.030294 .081395
• --------------------------------------------------
  ----------------------------
• () dy/dx is for discrete change of dummy
  variable from 0 to 1

26
Interpret the results

• Males are 4.7 percentage points more likely to
  report excellent
• Each year of age decreases chance of reporting
  excellent by 0.7 percentage points
• Current smokers are 7.5 percentage points less
  likely to report excellent health

27
Minor notes about estimation

• Wald tests/-2 log likelihood tests are done the
  exact same was as in PROBIT and LOGIT

28

• Use PRCHANGE to calculate marginal effect for a
  specific person
• prchange, x(age40 black0 othrace0 smoke0
  smoke50 educ16)
• When a variable is NOT specified (famincl), STATA
  takes the sample mean.

29

• PRCHANGE will produce results for all outcomes
• male
• AvgChg 1 2
  3 4
• 0-gt1 .0203868 -.0020257 -.00886671
  -.02677558 -.01329902
• 5
• 0-gt1 .05096698

30

• age
• AvgChg 1 2
  3 4
• Min-gtMax .13358317 .0184785 .06797072
  .17686112 .07064757
• -1/2 .00321942 .00032518 .00141642
  .00424452 .00206241
• -sd/2 .03728014 .00382077 .01648743
  .04910323 .0237889
• MargEfct .00321947 .00032515 .00141639
  .00424462 .00206252