Limited 

Dependent Variables

Ciaran S. Phibbs

November 13, 2013
Limited Dependent Variables

 0-1, small number of options, small 

counts, etc.

 The dependent variable is not continuous, 

or even close to continuous. 
Outline 

 Binary Choice 

 Multinomial Choice

 Counts 

 Most models in general framework of 

probability models

–Prob (event/occurs)
Basic Problems

 Heteroscedastic error terms 

 Predictions not constrained to match 

actual outcomes

– Real problem with predicted values being 

negative when a negative number isn’t 

possible
Yi

 = βo + βX + εi 

Yi=0 if lived, Yi=1 if died

 Prob (Yi=1) = F(X, β) 

 Prob (Yi=0) = 1 – F(X,β) 

OLS, also called a linear probability model 

 εi is heteroscedastic, depends on βX

 Predictions not constrained to (0,1) 
Binary Outcomes 

Common in Health Care

 Mortality

 Other outcome

– Infection

– Patient safety event

– Rehospitalization <30 days

 Decision to seek medical care 
Standard Approaches 

to Binary Choice-1 

 Logistic regression 
Advantages of Logistic 

Regression 

 Designed for relatively rare events 

 Commonly used in health care; most 

readers can interpret an odds ratio 
Standard Approaches 

to Binary Choice-2 

 Probit regression (classic example is 

decision to make a large purchase)

y* = βX + ε 

 y=1 if y* >0

 y=0 if y* ≤0
Binary Choice

 There are other methods, using other 

distributions. 

 In general, logistic and probit give about 

the same answer. 

 It used to be a lot easier to calculate 

marginal effects with probit, not so any 

more
Odds Ratios vs. Relative Risks

 Standard method of interpreting logistic 

regression is odds ratios. 

 Convert to % effect, really relative risk 

 This approximation starts to break down 

at 10% outcome incidence 

Can Convert OR to RR

 Zhang J, Yu KF. What’s the Relative Risk? A 

Method of Correcting the Odds Ratio in 

Cohort Studies of Common Outcomes. JAMA 

1998;280(19):1690-1691.

 RR = OR 

 (1-P0) + (P0 x OR) 

Where P0 is the sample probability of the 

outcome 
Effect of Correction for RR

From Phibbs et al., NEJM 5/24/2007, ≈20% mortality

Odds Ratio Calculated RR

2.72 2.08

2.39 1.91

1.78 1.56

1.51 1.38

1.08 1.06
Extensions 

 Panel data, can now estimate both 

random effects and fixed effects models. 

The Stata manual lists over 30 related 

estimation commands

 All kinds of variations. 

– Panel data

– Grouped data
Extensions 

 Goodness of fit tests. Several tests. 

 Probably the most commonly reported 

statistics are:

– Area under ROC curve, c-statistic in SAS 

output. Range 0.50 to 1.0. 

– Hosmer-Lemeshow test

– NEJM paper, c=0.86, H-L p=0.34
More on Hosmer-Lemeshow Test

 The H-L test breaks the sample up into n (usually 10, 

some programs (Stata) let you vary this) equal groups 

and compares the number of observed and expected 

events in each group.

 If your model predicts well, the events will be 

concentrated in the highest risk groups; most can be 

in the highest risk group.

 Alternate specification, divide the sample so that the 

events are split into equal groups. 
Estimate Note for 

Very Large Samples

 If you have very large samples; millions, it takes a lot 

longer to estimate a maximum likelihood model than 

OLS

 But, same X matrix, so the p-values for OLS are 

approximately the same as a logit model. Can use 

OLS for model development, and only estimate the 

final models with logit or other maximum likelihood 

model.
Multinomial Choice

 What if more than one choice or 

outcome? 

 Options are more limited

– Multivariable Probit (multiple decisions, 

each with two alternatives)

– Two different logit models (single decision, 

multiple alternatives)
Logit Models for Multiple 

Choices 

 Conditional Logit Model (McFadden)

– Unordered choices

 Multinomial Logit Model

– Choices can be ordered.
Examples of Health Care Uses 

for Logit Models for Multiple 

Choices 

 Choice of what hospital to use, among 

those in market area

 Choice of treatment among several 

options 
Conditional Logit Model
Conditional logit model

 Also known as the random utility model

 Is derived from consumer theory

 How consumers choose from a set of options

 Model driven by the characteristics of the 

choices.

 Individual characteristics “cancel out” but can 

be included indirectly. For example, in 

hospital choice, can interact individual 

characteristic with distance to hospital

 Can express the results as odds ratios. 
Estimation of McFadden’s Model

 Some software packages (e.g. SAS) 

require that the number of choices be 

equal across all observations. 

 LIMDEP, allows a “NCHOICES” 

options that lets you set the number of 

choices for each observation. This is a 

very useful feature. May be able to do 

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

Estimates

 Study I did looking at elderly serviceconnected veterans choice of VA or nonVA hospital

Log distance 0.66 p<0.001

VA 2.80 p<0.001
Multinomial Logit Model
Multinomial Logit Model

 Must identify a reference choice, model 

yields set of parameter estimates for each 

of the other choices, relative to the 

reference choice

 Allows direct estimation of parameters 

for individual characteristics. Model can 

(should) also include parameters for 

choice characteristics 
Independence of Irrelevant 

Alternatives 

 Results should be robust to varying the number 

of alternative choices 

– Can re-estimate model after deleting some of 

the choices.

– McFadden, regression based test. Regression- Based Specification Tests for the Multinomial 

Logit Model. J Econometrics 1987;34(1/2):63- 82. 

 If fail IIA, may need to estimate a nested logit 

model
Independence of Irrelevant 

Alternatives - 2 

 McFadden test can also be used to test for 

omitted variables.

 For many health applications, doesn’t 

matter, the models are very robust (e.g. 

hospital choice models driven by 

distance). 
Count Data (integers)

 Continuation of the same problem; 

dependent variable can only assume 

specific values and can’t be <zero

 Problem diminishes as counts increase 

 Rule of Thumb. Need to use count data 

models for counts under 30 
Count Data

 Some examples of where count data 

models are needed in health care

– Dependent variable is number of outpatient 

visits

– Number of times a prescription of a chronic 

disease medication is refilled in a year

– Number of adverse events in a unit (or 

hospital) over a period of time
Count Data

 Poisson distribution. A distribution for 

counts. 

– Problem: very restrictive assumption that mean 

and variance are equal
Count Data

 In general, negative binomial is a better choice. 

Stata (nbreg), test for what distribution is part of the 

package. Other distributions can also be used.
Interpreting Count Data Models

 lnE(event rate) = Bx 

 Incidence Rate Ratio = eB, like on odds 

ratio, with a similar interpretation

 Example, effect of average RN tenure on 

unit on infection rate 

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

 More common to see OLS used for 

counts than for binary or very limited 

choices. 

 Real problem with OLS when there are 

lots of zeros. Will result in reduced 

statistical significance. Can go in 

opposite direction when counts are larger. 
Example, OLS vs. Count Data Model, Effect 

on Statistical Significance (t-values)

Count Data Model OLS 

RN HPPD 0.18 -1.44 

LPN HPPD -1.78 -3.62 

Aide HPPD 0.65 -1.92 

36 
Notes for Count Data Models-2 

 30 is a rule of thumb, but should still 

consider a count model if most are small 

counts 

 Need to consider distribution and data 

generating process. If mixed process, 

may need to split sample 
Example of Mixed Data 

Generating Processes

 Predicting LOS for newborns 

– Well babies, all with LOS<= 5 days, clearly 

a count

– Sick newborns, can have very long LOS

 Solution, separate samples, use count 

model for well babies and OLS for sick 

babies 
Other Models

 New models are being introduced all of 

the time. More and better ways to 

address the problems of limited 

dependent variables. 

 Includes semi-parametric and nonparameteric methods. 
Reference Texts

 Greene. Econometric Analysis

 Wooldridge. Econometric Analysis of 

Cross Section and Panel Data

 Maddala. Limited-Dependent and 

Qualitative Variables in Econometrics
Journal References

 McFadden D. Specification Tests for the 

Multinomial Logit Model. J 

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

 Zhang J, Yu KF. What’s the Relative 

Risk? A Method of Correctingthe Odds 

Ratio in Cohort Studies of Common 

Outcomes. JAMA 1998;280(19):1690-

1691. 
42

Next lecture

Cost as the Dependent Variable

Paul Barnett

November 20, 2013