Basic Analysis and Graphing - SAS

Chapter 2 Performing Univariate Analysis 83
Statistical Details for the Distribution Platform
Binomial
The Binomial option accepts data in two formats: a constant sample size, or a column containing sample
sizes.
pmf: n p for ; x = 0,1,2,...,n
x
x ( 1 – p) n – x 0 ≤ p ≤ 1
E(x) = np
Var(x) = np(1-p)
where n is the number of independent trials.
Note: The confidence interval for the binomial parameter is a Score interval. See Agresti (1998).
Beta Binomial
This distribution is useful when the data is a combination of several Binomial(p) distributions, each with a
different p. One example is the overall number of defects combined from multiple manufacturing lines,
when the mean number of defects (p) varies between the lines.
The Beta Binomial distribution results from assuming that x|π follows a Binomial(n,π) distribution and π
follows a Beta(α,β). The Beta Binomial has parameters p = α/(α+β) and δ =1/(α+β+1). The δ is a
dispersion parameter. When δ > 0, there is over dispersion, meaning there is more variation in x than
explained by the Binomial alone. When δ < 0, there is under dispersion. When δ =0, x is distributed as
Binomial(n,p). The Beta Binomial only exists when n ≥ 2 .
pmf:
n
Γ 1
δ -- – 1 1 Γ x + p -- – 1
Γ n – x + ( 1 – p) 1
δ
-- –
δ 1
--------------------------------------------------------------------------------------------------------------------------
x
Γ p 1 -- –
δ 1 1 Γ ( 1 – p)
-- – 1 Γ 1 n + -- – 1
δ
δ
p 1 – p
for 0 ≤ p ≤1
; max( –------------------- ,–--------------------
) ≤ δ ≤ 1 ; x = 0,1,2,...,n
n – p – 1 n – 2 + p
E(x) = np
Var(x) = np(1-p)[1+(n-1)δ]
where
Γ ( . )
is the Gamma function.
Remember that x|π ~ Binomial(n,π), while π ~ Beta(α,β). The parameters p = α/(α+β) and δ =1/(α+β+1)
are estimated by the platform. To obtain estimates of α and β, use the following formulas:
αˆ
1 – δˆ
= pˆ -----------
δˆ
βˆ
( 1 – pˆ
) 1 – δˆ
= -----------
δˆ