Transcript Limited Dependent Variables: Event Counts

Limited Dependent Variables:
Event Counts
Adapted primarily from John McIver’s
notes, Hoffman’s “Generalized Linear
Models,” and Scott’s “Regression
Models for Categorical and Limited DVs”
Event Counts
• The DV is…
– Event count models are models where the dependent variable is
a count of events: i.e., the number of occurrences in a fixed
domain.
– The domain may be a unit of time (minute, day, year) or units in
fixed time (an individual or geographic unit).
• The DV is not…
– Grouped binary data
• Data which are the number of “successes” (or “failures”) out of some
known number of binary trials (# of failed coups, # successful veto
overrides)
• Political Knowledge measures?
– Ordinal data
• Use ordered logit or ordered probit
Counts as DVs
• Political protests in a nation in a year (Kasler
1996)
• Number of lynchings per county per year in
the South (Tolnay, Deane, and Beck 1996)
• Number of retirements per year on the
Supreme Court (Hagle 1993)
Characteristics of Event Data
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0
• 3) A histogram will indicate
a rapidly decreasing tail,
esp. w/ rare phenomena
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• 2) Counts are integers
(discrete, rather than
continuous variables): 2.7
children??
Density
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• 1) Event counts are nonnegative (lower bound is
zero)
• 4) Distribution is not
normal (in most cases)
– Poisson or negative
binomial
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polpart
Source: 1996 National Black Election Study
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How do we estimate these regression
models?
• Maximum Likelihood Estimation
– Find the parameter of interest (lambda, Beta, p)
given a set of data.
– MLE finds the value of the parameter that makes
the observed data most likely
– Liabilities (or assets…) of MLE:
• Consistency: Sample size increases, bias decreases
• Asymptotic efficiency: Smallest variance among
consistent estimators
• Asymptotic normally distributed: Hypothesis testing
Why not OLS?
• OLS assumes a linear relationship
– This assumption will often produce predicted event counts less
than zero (a logical impossibility).
– This assumption also means that the difference between 0 and
1 event in a given unit is the same as the difference between 10
and 11 events or between 100 and 101 events.
• Heteroskedasticity is likely (and a certainty if events are
distributed as they commonly occur as Poisson
distributed data).
• So OLS is…inaccurate, inconsistent, biased and inefficient.
Yuck.
But not always…
• When OLS is okay…
– As lambda (rate of the event) increases, the DV will increasingly
appear to follow a normal distribution
The Poisson Distribution
• Count variables, especially when measuring a
rare phenomena, often follow a Poisson
distribution.
• Lambda ( ) is known as the rate in the
context of Poisson distribution.
Probability of Number of Events in a
Poisson Distribution
Lambda
2
Number of
Events
0
1
2
3
4
5
P(i)
0.135335
0.270671
0.270671
0.180447
0.090224
0.036089
• If the average number of political acts per
year, based on past data, is 2, then we
expect the probability of one political act in
the next year would be…?
Assumptions of Poisson
1) The mean of the distribution equals its
variance (a.k.a equidispersion)
2) Events that make up the Poisson distribution
are assumed to be independent
– A lack of independence can lead to a violation of
Assumption 1. Known as overdispersion.
• Different distribution is used for these models – the
overdispersed Poisson or the negative binomial.
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Density
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Negative Binomial (overdispersed
data) v. Poisson Distribution
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2
nonepolpart
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• Non-electoral PTP
• Mean = 1.59
• Var = 2.08
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0
2
4
polpart
• Electoral PTP
• Mean = 1.37
• Var = 1.33
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Poisson Regression Model
• Goal
– Estimate the increase in the DV for a unit change in
the IV
– Predict expected counts for various groups
• Intuition
– We use the regression equation to come up with the
expected “log-number” of events and then
exponentiate this quantity to obtain a predicted count
– Interpretation of coefficients is done in a similar way
Poisson Regression: Electoral Participation
• What causes African Americans to participate in more political acts?
• Does education affect the number of political acts?
by educdum: sum polpart
-> educdum = High School or Less
Variable |
Obs
Mean Std. Dev.
Min
Max
-------------+-------------------------------------------------------polpart |
335 1.080597 .9922211
0
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------------------------------------------------------------------------> educdum = More than HS
Variable |
Obs
Mean Std. Dev.
Min
Max
-------------+-------------------------------------------------------polpart |
517 1.560928 1.198619
0
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Poisson regression in Stata
• Generic code:
– poisson dv iv (poisson polpart educdum)
Interpretation
• Signs indicate the effect on the expected
number of counts.
• Incident Rate Ratios
– In the Poisson case, the quantity of interest is
known as the incidence rate – that is, λ. The
natural way to compare two observations, then, is
the “incidence rate ratio” (or IRR).
Incidence Rate Ratios
• For a binary covariate XD, we can think of the
IRR as the ratio…
That is, we can tell the relative change in the incidence
rate for a one–unit change in any given variable Xk by
simply exponentiating its coefficient estimate βk.
Interpretation: Expected Counts and Incidence Rate Ratios
Formula for Expected Counts
• In our case, then:
– Expected number of acts among those w/ HS educ or less (x=0):
• exp (0.0775137) = 1.08
– Expected number of acts among those w/ more than HS educ
(x=1):
• exp (0.0775137 + 0.3677671) = 1.56
• This means that the incidence rate for those with more than
a HS education is 1.56 /1.08 = 1.44 times that for those with
a HS education or less
• We can also calculate percent differences between these
groups:
– Percent difference = (1.56 – 1.08) / 1.08 = 44% increase in
political acts
An extended model
Quantities of Interest
In the example, this means that the estimated IRR
for the education variable is equal to
exp(0.10274) = 1.11.
• This means that a one–unit change in the level
of education variable corresponds to an
estimated IRR 1.11.
– i.e., increasing the level of education of a respondent
by one year increases the estimated incidence rate by
a factor of 1.11 or about 11% more political acts,
cetaris parabus.
Stata reports irr’s as well
Percent Change in Expected Count
• For an 8 unit increase in education (min to max),
this means we will see (all else equal):
Calculating Expected Counts
• For a typical case (education =4.08 [some college], contacted =
0, efficacy =0.49, female = 1), the predicted count would be:
E(Y|mean of Xi) = exp[−0.434 + (0.103 × 4.08) + (0.462 × 0)
+ (0.365*0.49) + (-0.051*1)]
= exp(0.11409)
= 1.12
Expected Counts
• You can accordingly calculate the change in expected
counts by calculating the predicted count for different
values of Xi, and taking the difference.
– The expected count for the same person (on the previous
slide), but who was contacted would be = exp(0.57609) =
1.78
– So, being contacted results in (1.78−1.12) ≈ 0.67 increase in
political acts.
– Note that 1.78/1.12 = 1.59, which is the same as the IRR for
a one unit change in contacted.
• Stata way:
– “predict polpart1, n” where ‘n’ provides counts rather than
‘p’ for probability
Expected Political Acts as Education
Increases (other IVs at mean or mode)
XB
1
-0.20352
0.815857365
2
-0.10078
0.904135774
3
0.001964
1.001966193
4
0.104704
1.110382181
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0.207444
1.230529129
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0.310184
1.363676366
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0.412924
1.511230566
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0.515664
1.674750607
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0.618404
1.855964047
Number of Electoral Political Acts
Source: 1996 NBES
Number of Acts
Education
Number of Political
Acts
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1
2
3
4
5
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7
R's Level of Education
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Alternatives to Poisson
• The assumption that the mean equals the
variance is often unrealistic
– Overdispersed data: Variance exceeds the mean
– Problems:
• Poisson is consistent, but inefficient
• SEs are biased downward using Poisson resulting in
larger z-values (incorrect inferences)
• Solutions:
a) Extradispersed Poisson Regression
b) Negative binomial regression model
Extradispersed Poisson Regression
Model
• Accounts for the fact that the variance of the DV differs
from the mean
– Affects only the standard errors of the model
• SEExtradispersed = SEUnadjusted * sqrt(dispersion)
– Point estimates are the same (rates, IRRs, predicted
counts)
• In Stata:
– glm dv ivs, family(poisson) link(log) scale(dev) irls
– predict dv, mu Note that we use ‘mu’ instead of ‘n’
which is the general command asking fro predicted values
when using glm.
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Density
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Negative Binomial (overdispersed
data) v. Poisson Distribution
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nonepolpart
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• Non-electoral PTP
• Mean = 1.59
• Var = 2.08
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0
2
4
polpart
• Electoral PTP
• Mean = 1.37
• Var = 1.33
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Non-Electoral Participation via Poisson
Non-Electoral Participation via Extradispersed Poisson
Negative Binomial
• Assumes that the variance is larger than the
mean
– More appropriate than Poisson in the common
situation where the events of interest are not
independent
– Follows a different probability mass function
• Stata
– nbreg dv ivs
– nbreg dv ivs, irr
– predict dv1, n
Non-electoral PTP by Negative Binomial
Testing for Overdispersion
• In addition to examining whether or not we can reject
the null that alpha = 0, we can also test for
overdispersion using the log likelihoods from both the
Poisson and the NBRM models:
G2 = 2(ln LNBRM – ln LPRM)
tests the null hypothesis that alpha = 0.
• Distributed as X2 and the two values in the parentheses
are log likelihoods from the NBRM and Poisson
regressions
Which regression model to use?
• No generally accepted rule of thumb regarding how much
extradispersion is allowable before switching from Poisson to
Negative Binomial (Hoffman 2004; Cameron and Tivedi 1998)
– Estimate both Poisson and negative binomial
– Compare results
– If alpha is greater than zero and results differ, use negative
binomial.
– If variance is smaller than the mean (rare), negative binomial is
not appropriate. Extradispersed Poisson will probably be the best
route.
• Differences tend to affect SEs rather than coefficients
(significance of variables rather than estimated coefficients).
Diagnostic Tests for Poisson
Residual analysis
• Compute deviance residuals and predicted counts
– Plot against one another looking for poor fit and influential
observations
• Stata
– predict count, mu
– predict dev1, deviance
– Plot deviance residuals against each IV (if IVs are
continuous random variables)
• Different functional form
– Plot deviance residuals in a normal probability (Q-Q) plot to
examine distribution
• Residuals should fall along diagonal
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deviance residual
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-2
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predicted mean polpart
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Residuals Plotted against Predicted
Counts of Political Acts
•twoway(scatter dev1 count)
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Inverse Normal
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QQ Plot of Residuals Against Normal
Probability
qnorm dev1
•Graph 1 indicates that there may be some observations at the top of
the plot that may be influential or indicate that the model is
misspecified.
•Graph 2 indicates that the residuals generally follow a normal
distribution, indicating our estimator choice is likely appropriate
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Extensions
• Zero-inflated or zero-modified count models
– Number of 0s in a sample exceeds number
predicted under Poisson or negative binomial
• Truncated count model
– Count variables observed only after the first count
occur (“hurdle” models)
• Number of alcoholic beverages in a day (Hoffman 2004)
Empirical Examples of Event Counts
(Poisson Regression)
• D. Cannon (1993) “Sacrificial Lambs or Strategic Politicians?
Political Amateurs in US House Elections.” AJPS 37: 1119-1141.
• J. Robertson (1983) “Inflation, Unemployment and Government
Collapse.” Comparative Political Studies 15: 425-444.
• T. Shields & C. Huang (1995) “Presidential Vetoes: An Event Count
Model.” PRQ 48: 559-572
• J. Spriggs II & P. Wahlbeck (1995) “Calling It Quits: Strategic
Retirement on the Federal Courts of Appeals, 1893-1991.” PRQ 48:
573-597.
• T. Volgy & L. Imwalle (1995) “Hegemonic and Bipolar Perspectives
on the New World Order.” AJPS 39: 819-834.
• M. Koch & S. Cranmer (2009) “Testing the “Dick Cheney”
Hypothesis: Do Governments of the Left Attract more Terrorism than
Governments of the Right?”
References
• Long, J. Scott. 1997. Regression Models for Categorical and
Limited Dependent Variables. Thousand Oaks, CA: Sage
Publications.
• Gujarati, Damodar N. 2003. Basic Econometrics. Singapore:
McGraw-Hill, 4th Edition.
• Hoffman, John P. 2003. Generalized Linear Models. Boston:
Pearson Education Inc.
• Gary King (1988)“Statistical Models for Political Science Event
Counts: Bias in Conventional Procedures and Evidence for the
Exponential Poisson Regression Model.” American Journal of
Political Science 32: 838-863.
• Gary King (1989) “Variance Specification in Event Count Models:
From Restrictive Assumptions to a Generalized Estimator.”
American Journal of Political Science 33: 762-784.
• Gary King (1989) “Event Count Models for International
Relations: Generalizations and Applications.” International
Studies Quarterly, Vol. 33: 123-147.