Basic Analysis and Graphing - SAS

Chapter 2 Performing Univariate Analysis 81
Statistical Details for the Distribution Platform
Generalized Log (Glog)
This distribution is useful for fitting data that are rarely normally distributed and often have non-constant
variance, like biological assay data. The Glog distribution is described with the parameters μ (location), σ
(scale), and λ (shape).
pdf:
φ 1 σ -- x x
------------------------------
+ 2 + λ 2 x + x
log
– μ
2 + λ 2
2 ---------------------------------------------------------
σ( x 2 + λ 2 + x x 2 + λ 2 )
for 0 ≤ λ ; 0 < σ;
– ∞ < μ < ∞
The Glog distribution is a transformation to normality, and comes from the following relationship:
1 x + x
If z = --
2 + λ
log------------------------------
2 – μ ~ N(0,1), then x ~ Glog(μ,σ,λ).
σ 2
When λ = 0, the Glog reduces to the LogNormal (μ,σ).
Note: The parameter confidence intervals are hidden in the default report. Parameter confidence intervals
are not very meaningful for the GLog distribution, because it is a transformation to normality. To show
parameter confidence intervals, right-click in the report and select Columns > Lower 95% and Upper
95%.
All
In the Compare Distributions report, the ShowDistribution list is sorted by AICc in ascending order.
The formula for AICc is as follows:
AICc = -2logL + 2ν + 2ν( ν+
1)
n ------------------------- – ( ν + 1)
where:
– logL is the logLikelihood
– n is the sample size
– ν is the number of parameters
If your data has negative values, the ShowDistribution list does not include those distributions that require
data with positive values. If your data has non-integer values, the list of distributions does not include
discrete distributions. Distributions with threshold parameters, like Beta and Johnson Sb, are not included
in the list of possible distributions.
Statistical Details for Discrete Fit Distributions
This section contains statistical details for the options in the Discrete Fit menu.