Limited 

Dependent Variabl

Ciaran S Phibbs Ciaran S. Phibbs

May 30, 2012
Limited Dependent Varia

 0-1, small number of options, sma

coun

ts, e

tc. 

 The dependent variable is not con

The dependent variable is not conor even close to continuous. 
Outline 

 Binary Choice  Multinomial Choice  Counts  Most models in general framewor

probability models 

–P b( ro event/occurs) 
Basic Problems 

 Heteroscedastic error terms  Predictions not constrained to mat

actual outcomes, real problem witactual outcomes, real problem wit

predicted values being negative wnegative num

ber 

i ibl sn’t possibl

e
Y = βo + βX + 

Yi 

Yi=0 if lived, Yi=1 if died 

Prob (Yi=1) = F(X, β) 

P b ro (Y0) 1 F(X β) i=0) = 1 – F(X,β) 

OLS, also called a linear probability 

εi is heteroscedastic, depends o

Predi ict ons not constrai d ( 

ned to (
Binary Outcomes Binary Outcomes

Common in Health Car

 Mortality Mortality  Other outcome 

– Infection– Pi f at

ient sa

fety event 

– Rehos

pitalization <30 da

ys 

 Decision to seek medical care 
Standard Approaches Standard Approaches

to Binar

y Choice-1

 Logistic regression Logistic regression 
Advantages of Logistic Advantages of Logistic

Re

gression 

 Designed for relatively rare events  Commonly used in health care; m

readers can interpret an odds ratio readers can interpret an odds ratio
≤0

Standard Approaches Standard Approaches

to Binary Choice-2

 Probit regression (classic example 

Probit regression (classic example 

decision to make a large purchase

y* = βX + ε
 

y=1 if y* >0 y=1 if y* >0y0 = if y* ≤0
Binary Choice 

 There are other methods, using oth

distributions. 

 In 

general

, lo

gistic and 

probit 

give 

the same answer. 

 It used to be a lot easier to calcula

It used to be a lot easier to calcula

marginal effects with probit, not smore 
Odds Ratios vs. Relative R

 Standard method of interpretinglo

regression is odds ratios. 

 Convert to % effect, really relative  This approximation starts to break 

This approximation starts to break 

at 10% outcome incidence 

Can Convert OR to RR

 Zhang J, Yu KF. What’s the Relative 

Method of Correcting the Odds Ratio Method of Correcting the Odds Ratio 

Cohort Studies of Common Outcomes1998;280(19):1690-1691. 

RR = OR 

(1-P

0) + (P0 x OR) 

Where P0 is the sample probability of th

outcome 
Effect of Correction for R

From Phibbs et al., NEJM 5/24/2007, 

≈20% m

Odds Ratio Calculated 

2.72 2.08 

2 39 . 1 91 . 1 78 . 1 56 . 1.51 1.38 1.08 1.06 
Panel data

Extensions 

 Panel data, can now estimate both 

random eff ects and fi xed e ff ects m

The Stata manual lists 34 related estimation commands 

 All kinds of variations. 

– Panel data – Grouped data 
Extensions 

 Goodness of fit tests. Several test Probably the most commonly repo

statistics are: statistics are: 

– Area under ROC curve, c-statistic in 

output. Range 0 50 . to 10 . .

– Hosmer-Lemeshow test – NEJM paper, c=0.86, H-L p=0.34 
so e a s et va t

More on Hosmer-Lemeshow 

 The H-L test breaks the sample up into n (u

so

me 

p og r

r

a

ms (Stata) let you va

r

y 

this) equ

and compares the number of observed and eevents in each 

group. 

 If your model predicts well, the events will 

concentrated in the highest risk groups; mosconcentrated in the highest risk groups; mos

in the highest risk group. 

 Alternate specification divide the sample so 

Alternate specification, divide the sample so 

events are split into equal groups. 
o e to est ate a a u e ood 

-

Estimate Note for Estimate Note for

Ver

y Lar

ge Sam

ples

 If you have very large samples; millions, it 

l

o

nger to estimate 

a m

axim

um lik

elihood m

OLS 

 But same X matrix so the p But, same X matrix, so the p 

-values for OL

values for OL

approximately the same as a logit model. C

OLS for model development and only estimOLS for model development, and only estim

final models with logit or other maximum li

model. 
Multinomial Choice

 What if more than one choice or 

ou

tcome?

 Options are more limited Options are more limited 

– Multivariable Probit (multiple decis

each ih l ) h w

i

t

h two a

lternat

ives) 

– Two different lo

git models 

(sin

gle d

multiple alternatives) 
Logit Models for Multip

Logit Models for Multip

Choices Choices 

 Conditional Logit Model (McFadd

– Unordered choices Unordered choices 

 Multinomial Logit Model 

– Choices can be ordered. 
Examples of Health Care 

Examples of Health Care 

for Logit Models for Multfor Logit Models for Mult

Choices

 Choice of what hos

pital to use

, am

those in market area

 Ch io

ice o

f 

treatment among severa 

o

ptions 
Conditional Logit Mod
d v dua c a acte st cs ca ce out 

Conditional logit mode

 Also 

known as the ran

dom utility mo

d

 Is derived from consumer theory  How consumers choose from a set of o Model driven b

y the characteristics of 

choices. 

 In

d

i

v

idual 

c

h

a

racterist

ics “ca

ncel out” 

be included indirectly. For example, ihos

pital choice, can interact individua

characteristic with distance to hospital 

 Can ex

press the results as odds ratios. 
Estimation of McFadden’s M

 Some software packages (e.g. SA

require that the number of choices require that the number of choices 

equal across all observations.

 LIMDEP, allows a “NCHOICES” 

o

ptions that lets 

you set the numbe

choices for each observation. Thi

very useful feature very useful feature. May be able t

May be able t

this in Stata (clogit) with “group” 
Example of Conditional L

Example of Conditional L

Estimates

 Stud

y I did lookin gat elderl yservi

connected veterans choice of VA 

VA hospit

al

Log distance 0.66 pVA 2.80 

p 
Multinomial Logit Mod

Multinomial Logit Mod
Multinomial Logit Mod

 Must identify a reference choice, m

yields set of parameter estimates fyields set of parameter estimates f

of the other choices, relative to the re

ference c

h

o

ice 

 Allows direct estimation of param

Allows direct estimation of param

for individual characteristics. Mo(should) also include parameters f

choice characteristics choice characteristics 
Independence of Irreleva

Independence of Irreleva

Alternatives

 Results should be robust to varying th

of alternative choices of alternative choices 

– Can re-estimate model after deleting s

the choices the choices. 

– McFadden, regression based test. Reg

Based Specification Tests for the MulBased Specification Tests for the Mul

Logit Model. J Econometrics 1987;34

82. 

 If fail IIA, may need to estimate a nest

model 
Independence of Irreleva

Independence of Irreleva

Alternatives - 2

 McFadden test can also be used to 

omitted variables. 



For many 

health tiapplications, 

doe

matter

, the models are ver

y robust 

hospital choice models driven by 

distance). 
Count Data (integers) 

 Continuation of the same 

problem

dependent variable can only assum

specific values and can specific values and can t ’be <zero be <zero 

 Problem diminishes as counts incr Rule of Thumb. Need to use coun

modl fe

ls 

for counts un

der 30 
Count Data 

 Some examples of where count da

mod l d d i h lth dels are needed in health care 

– Dependent variable is number of outp

visits 

– Number of times a prescription of a ch

Number of times a prescription of a ch

disease medication is refilled in a year 

– Number ofa dverse event it ( ts in a unit (or

hospital) over a period of time 
Count Data

 Poisson distribution. A distributio

counts. 

– Problem: very restrictive assumption that 

and variance are equal 
Stata b e test o w at d st but o s 

Count Data

 In general, negative binomial is a better cho

Stata 

(

n

b

reg),test for 

w

hat 

d

istribut

i

on is pa

package. Other distributions can also be us
Interpreting Count Data M

 lnE

(event rate

) = Bx 

 Incidence Rate Ratio = e

B, like on 

rati ith i il i t t ti tio, with a s

i

milar 

i

n

terpre

t

ation 

 Example effect of average RN ten

Example, effect of average RN ten

unit on infection rate 

– B= -0.262 IRR = 0.770 
Notes for Count Data Mo

 More common to see OLS used fo

counts than for binary or very limchoices choices 

 Real 

problem with OLS when ther

lots of zeros. Will result in reduce

statistical significance statistical significance 
Notes for Count Data Mod

 30 is a rule of thumb, but should s

consider a count model if most are counts 

 Need to consider distribution and 

generating process. If mixed procmay need to split sample may need to split sample 
Example of Mixed Dat

Example of Mixed Dat

Generatin

g Processes

 Predictin

g LOS for newborns 

– Well babies, all with LOS<= 5 days

a count a count 

– Sick newborns, can have very long 

 Solution, separate samples, use co

mo

d

el for well b

abies an

d OLS fo

babies 
Other Models

 New models are bein

g introduced 

the time. More and better ways to 

address the problems of limited address the problems of limited

dependent variables. 

 Includes semi-parametric and non

parameteric methods parameteric methods. 
Reference Texts 

 Greene. Econometric Analysis  Wooldridge Wooldridge. Econometric Analysi

Econometric Analysi

Cross Section and Panel Data 

 Maddala. Limited-De

pendent and 

Qualitative Variables in Econome
J l R Journal 

R

e

ferences

 McFadden D. Specification Tests 

Multinomial Logit Model. J Multinomial Logit Model.

J

Econometrics 1987;34(1/2):63-82

 Zhang J, Yu KF. What’s the Rela

Ri k? A M th d f C ti th Ri

sk? A 

M

eth

o

d of 

Correctingthe 

Ratio in Cohort Studies of CommoOutcomes. JAMA 1998;280(19):

1691. 
Next lecture

Right

-hand Side Variables hand Side Variables

Ciaran Phibbs Ciaran Phibbs

June 6, 2012