Quantitative Methods

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Quantitative Methods

• Analyzing event counts

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Event Count Analysis

• Event counts involve a non-negative
  interger-valued random variable. Examples are
  the number of bills introduced by a legislator,
  the number of car accidents, etc. Trivia one
  of the earliest recorded uses of the poisson
  distribution was an 1898 analysis of the number
  of Prussian soldiers that were kicked to death by
  horses.
• OLS can generally not be used for event count
  analysis because it will produce biased and
  inconsistent estimates. (The dependent variable
  is not really interval / continuousit is left
  censoredand the data are heteroskedastic.)

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Event Count Analysis

• Poisson models

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Poisson regressionanother example
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Poisson Models

• The poisson distribution function
• (a poisson distribution has a mean and variance
  equal to ?. As ? increases, the distribution is
  approximately normal.

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Poisson Models

• The predicted counts (or incidence rates) can
  be calculated from the results as follows

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Poisson Models

• One can compare incidence rates with the
  incidence rate ratios. The incidence rate
  ratio for a one-unit change in xi with all of the
  variables in the model held constant is e Bi

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Poisson Modelsan example

• --------------------------------------------------
  ----------------
• daysabs b z Pgtz
  eb ebStdX SDofX
• -------------------------------------------------
  ----------------
• gender -0.40935 -8.489 0.000 0.6641
  0.8147 0.5006
• angnce -0.01467 -11.342 0.000 0.9854
  0.7686 17.9392
• --------------------------------------------------
  ----------------

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Poisson Modelsan example

• Being male decreases the of days absent by a
  factor of .66.
• And it decreases the expected of days absent by
  100(.66-1) 33.
• For each point increase in the language score,
  the expected of days absent decreases by a
  factor of .98 (or an expected decrease of
• 100(.98-1) -2))

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Negative Binomial Regression

• Often, there is overdispersion, where the
  variance gt mean. In practice, what this usually
  means of one of two things first, its possible
  that there is some unobserved variable that makes
  some observations have higher counts than others
  (i.e., number of publications of professorsor
  rbi of a sports teamcant assume the mean is
  the same across observations).
• Essentially, this is common with pooled data, and
  unobserved variablesand will look like
  heteroskedasticity. (Example?the school from
  which one graduates).

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Negative Binomial Regression

• The second possibility is that if you have one
  event, it increases or decreases the probability
  that you will have others (i.e., bill sponsorship
  counts)

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Negative Binomial Regression

• A negative binomial regression analysis is
  appropriate in these cases (and if there is no
  overdispersion, a NBR will collapse down to a
  Poisson).
• (Note?there are also alternatives, such as
  zero-inflated (many, many zeros) and
  zero-truncated (no zeros) NBR.)

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Negative Binomial Regression

• Zero inflated models essentially model based on
  the assumption that there is an always zero
  category of cases and a sometimes zero category
  of cases.
• Zero truncated models ? example would be online
  survey of web usage.