Basic Analysis and Graphing - SAS

82 Performing Univariate Analysis Chapter 2
Statistical Details for the Distribution Platform
Poisson
The Poisson distribution has a single scale parameter λ >0.
pmf:
e λ λ
------------- x
x!
for 0 ≤ λ < ∞; x = 0,1,2,...
E(x) = λ
Var(x) = λ
Since the Poisson distribution is a discrete distribution, the overlaid curve is a step function, with jumps
occurring at every integer.
Gamma Poisson
This distribution is useful when the data is a combination of several Poisson(μ) distributions, each with a
different μ. One example is the overall number of accidents combined from multiple intersections, when
the mean number of accidents (μ) varies between the intersections.
The Gamma Poisson distribution results from assuming that x|μ follows a Poisson distribution and μ
follows a Gamma(α,τ). The Gamma Poisson has parameters λ =ατ and σ =τ+1. The σ is a dispersion
parameter. If σ > 1, there is over dispersion, meaning there is more variation in x than explained by the
Poisson alone. If σ = 1, x reduces to Poisson(λ).
Γx
+ λ
σ -----------
– 1
pmf: ------------------------------------------ σ -----------
– 1 x λ
–-----------
σ – 1
σ for 0 < λ ; 1 ≤ σ ; x = 0,1,2,...
λ
Γ( x + 1)Γ-----------
σ
σ – 1
E(x) = λ
Var(x) = λσ
where
Γ ( . )
is the Gamma function.
Remember that x|μ ~ Poisson(μ), while μ~ Gamma(α,τ). The platform estimates λ =ατ and σ =τ+1. To
obtain estimates for α and τ, use the following formulas:
τ = σ – 1
αˆ
λˆ
= --
τˆ
If the estimate of σ is 1, the formulas do not work. In that case, the Gamma Poisson has reduced to the
Poisson(λ), and λˆ is the estimate of λ.
If the estimate for α is an integer, the Gamma Poisson is equivalent to a Negative Binomial with the
following pmf:
y + r – 1
py ( ) = p for
y
r ( 1 – p) y 0 ≤ y
with r = α and (1-p)/p = τ.