Regression Analysis of Count Data and Development of Statistical Models

1
Regression Analysis of Count Data and Development
of Statistical Models

• Lecture 7
• Part II

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Statistical Models For Crash Data
Recap Poisson Model
The PDF of the Poisson regression for yi given
xi is
The mean and variance are given by
The mean function is given by
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Statistical Models For Crash Data
Recap Poisson-gamma Model (NB)
The PDF of the Poisson-gamma regression for yi
given xi is
The mean and variance are given by
The mean function is given by
Note
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Statistical Models For Crash Data
Statistical fit (Goodness of fit)
There are various methods for estimating the
statistical fit of models. The most common one is
to compute the deviance between the prediction
and the estimated values. This is similar to the
sum of square errors described before for the
multivariate linear regression with a normal
error structure. The methods include Deviance
Statistics The Scaled Deviance Akaikes
Information Criterion Pseudo R-Squared
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Statistical Models For Crash Data
Statistical fit (Goodness of fit)
The deviance statistic is defined as twice the
difference between the maximum log-likelihood
achievable and the log-likelihood of the fitted
model
When competitive models are compared, the model
with the lowest deviance offers the best
statistical fit. A note of caution this is only
valid when the dispersion parameter F is the same
for each competitive model.
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Statistical Models For Crash Data
Statistical fit (Goodness of fit)
The deviance statistic for the Poisson model is
the following
The deviance statistic for the Poisson-gamma
model is the following
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Statistical Models For Crash Data
Statistical fit (example)
Regression Analysis Response
variate Y Distribution Negative binomial with
parameter k 2.4683 Link function Log Fitted
terms Constant L_F1 L_F2 Residual d.f.
252, deviance 285.4 Summary of analysis

mean deviance d.f.
deviance deviance ratio Regression
2 88.2 44.085 38.93 Residual
252 285.4 1.133 Total
254 373.6 1.471
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Statistical Models For Crash Data
Statistical fit (example)
Estimates of parameters
estimate s.e. t(252)
Constant -9.81 1.56
-6.29 L_F1 0.816
0.150 5.45 L_F2
0.3732 0.0629 5.93
MESSAGE s.e.s are based on the residual
deviance Aggregation parameter
k se 2.4683 0.3645
-2 x log-likelihood The values above
indicate this ß0 exp(-9.81) 0.00005501 ß1
0.816 ß2 0.3732
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Statistical Models For Crash Data
Confidence Intervals
Confidence intervals for GLMs can be computed
using various approaches. They are generally a
little more complicated than the method used for
linear models. Some of these approaches are
approximation. One common approach is to use the
delta method.
The standard error of the estimate of the mean
response at x0 is
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Statistical Models For Crash Data
Confidence Intervals (contd)
Where,
This is similar to the variance-covariance matrix
of linear models
Again, this is in essence the values at x0.
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Statistical Models For Crash Data
Confidence Intervals (contd)
Confidence interval on the mean response is
Confidence interval for predicted values is
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Statistical Models For Crash Data
Confidence Intervals
A recent paper written by Wood (2004) has
provided a direct way for estimating the
confidence intervals for Poisson and
Poisson-gamma models specifically for crash data.
Poisson Model
95 confidence interval on the mean response µ is
given by
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Statistical Models For Crash Data
Confidence Intervals
Poisson Model
95 confidence interval on the predicted response
y is given by
This matrix can be provided by computer programs
Where,
the largest integer less or equal to x
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Statistical Models For Crash Data
Confidence Intervals
Poisson-gamma Model
95 confidence interval on the mean response µ is
given by
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Statistical Models For Crash Data
Confidence Intervals
Poisson-gamma Model
95 confidence interval on the mean response m
(the mean of the gamma distribution) is given by
Remember
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Statistical Models For Crash Data
Confidence Intervals
Poisson-gamma Model
95 confidence interval on the predicted response
y is given by
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Statistical Models For Crash Data
Confidence Intervals
Example
This example is taken from Wood (2004). The
following model predicts the number of rear-end
collisions at signalized intersections
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Statistical Models For Crash Data
Confidence Intervals
In this example, the matrix (DD)-1 is equal to
Compute for F10,000
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Statistical Models For Crash Data
Confidence Intervals
Compute the confidence interval for µ.
For the confidence interval is
Compute the confidence interval for m.
For the confidence
interval is
Compute the confidence interval for y.
For the confidence
interval is
See figure on next page
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Statistical Models For Crash Data
Confidence Intervals
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Statistical Models For Crash Data

• Crash data have often the characteristics that
  the mean µ can be very low (below 0.5)
• Create problems with goodness-of-fit and
  prediction
• Read paper by Wood, G.R. (2004) Generalised
  Linear Models and Goodness of Fit Testing.
  Accident Analysis Prevention, Vol. 34, pp.
  417-427.

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Statistical Models For Crash Data
Low Mean Issue
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Statistical Models For Crash Data
Time Trend Effects
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Statistical Models For Crash Data
Time Trend Effects
Goal capture changes that vary from year to year
directly into the model.
The model structure is given by the following
Time Trend captured with the intercept (i.e., one
intercept for each year)
Characteristic each year is defined as a
different observation.
Issues Since each site is observed at a
different point in time, a temporal serial
correlation exits and affects the statistical
inferences of statistical models. Therefore, you
need to account for this correlation into the
model.
Modeling approach Generalized Estimating
Equations (GEE) Random-Effects models, etc.
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Bayes Methods

• Originally presented
• Highway Safety Manual
• "After the Crashes Are Counted" Workshop
• Sunday, January 11th, 2004

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Introduction

• The Bayes method approaches the analysis of data
  differently than the classical method
  (frequentist)
• Subjective judgment more easily incorporated with
  the observed data and models
• Treat unknown coefficients of regression models
  as random variables
• Data analysis less limited by the number of
  observations (can be supplemented with subjective
  judgment)
• Computationally intensive (no longer an issue)

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Important Characteristics

• The Bayes method makes inferences from data using
  probability models for quantities that are
  observed and for quantities one is interested to
  learn about
• Bayesian data analysis can be divided into three
  steps
• Setting up a full probability model provide a
  joint probability distribution for all observable
  and unobservable quantities
• Conditioning on observed data calculating and
  interpreting the appropriate posterior
  distribution (conditional probability
  distribution)
• Evaluating the fit of the model and implication
  of the posterior distribution
• Emphasis placed on interval estimation
  (confidence interval) rather than hypothesis
  testing

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Basic Principles
Venn Diagram
A
E1
E2
E3
E4
E5
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Basic Principles
A
E1
E2
E3
E4
E5
Total Probability Theorem
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Basic Principles
Bayes Theorem
If event A occurred, what is the probability that
event Ei also occurred?
A
E1
E2
E3
E4
E5
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Basic Principles
Bayes Theorem
If event A occurred, what is the probability that
event Ei also occurred?
Given the multiplication rule, it can be shown
that
Therefore, we obtain
Using the Total Probability Theorem for P(A), we
get
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Bayes Model
In modeling and data analysis, the Bayes method
can be translated by the following equation
Where, y the observed data (can be defined as
a vector) ? unobserved quantity
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Bayes Model
Terminology for
Posterior probability conditional on y (this
is a joint probability distribution for ? and y)
Prior distribution (can be informative,
non-informative, etc.)
Likelihood function when it is regarded as a
function of ? for a fixed y
Prior predictive distribution (also called the
marginal distribution of y)
e.g., Poisson distribution p(y ?) Po(?)
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Bayes Model

• Hierarchical Models (aka multilevel models)
• They are used when information is available on
  different levels of observational units
• Hierarchical models allow the modeler to
  structure some dependence between the parameters
  under study in a logical manner
• Observable outcomes modeled conditionally on
  certain parameters are known as hyperparameters
• Such hierarchical thinking helps understanding
  multiparameter problems and plays an important
  role in developing computational strategies

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Bayes Model
Ex Hierarchical Model for crash data p(a, ß,
?y)
Assume a and ß (known or unknown)
Hyperparameters (a, ß)
Mean (?) Gamma(a,ß)
Parameter (?)
Poisson distribution with Mean (?)
Observable quantity (y)
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Bayes Model

• Estimation of posterior distribution, p(? y),
  can be accomplished by integrating the full Bayes
  equation (the likelihood and prior probability
  functions)
• The estimation can also be performed by
  simulating the posterior distribution
• Markov Chain Monte Carlo (MCMC) simulation
  techniques are now frequently used for estimating
  the posterior distribution

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Empirical Bayes Model

• The empirical Bayes (EB) method is usually
  employed to simplify the computational difficulty
  associated with the full Bayes method
• The name empirical Bayes arises from the fact
  that the prior distribution is estimated from
  actual data
• The data is used for estimating the
  hyperparameters through MLE or MM
• For the EB, the data is actually used twice
• Once to estimate the hyperparameters
• Once to estimate the posterior distribution

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Empirical Bayes Model

• For the EB method, a different weight is assigned
  to the prior distribution and standard estimate
  respectively
• In safety analyses, the weights are estimated
  with the assumption that the mean (?) for each
  site follows a Gamma distribution
• The EB estimates has been found to outperform
  other estimates, such as the MLE
• The EB framework is presented on next overhead

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Empirical Bayes Model
Expected number of crashes
Where,
EB estimate of expected number of crashes
maximum likelihood estimate
observed number of crashes
Weight factor
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Empirical Bayes Model
Alternative formulation
where
Mean of a Poisson-gamma regression
Dispersion parameter of NB regression
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Empirical Bayes Model
Using the same example shown earlier
F1 24,164 F2 3,392 y10
The values are estimated as follows
Crashes per year
Crashes per year
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Empirical Bayes Model
Observed value 10
Crashes per Year
EB estimate 7.63
MLE estimate 3.9
t
1
2
Year