Generalized Linear Model

Generalized Linear Model A Unified Theory Various responses Binary Ordinal Count Polytomous

Generalized Linear Model A Unified Theory

Mean Structure Ordinary Linear Model Generalized Linear Model

Link Functions Link function: Log link Logit link Log-log link Probit link

Logistic Regression Model Link functions

Poisson Regression Model Link Functions Example: Auto Insurance

Ordinal Regression Model Link Functions

Polytomous Regression Model Link Functions

Polytomous Regression Properties Therefore,

Deviance Likelihood function Deviance Objective: Measuring discrepancy (like residual sum of squares)

Normal Example Likelihood function Deviance

Poisson Example Likelihood function Deviance

Analysis of Deviance Model d.f. Discrepancy s.s. 1 11 1000 A 8 500 3 A+B 6 200 300 2 A+B+A*B

Pearson Residuals Define where Example: Normal Distribution

Deviance Residual Deviance Define Example: Normal Distribution

Logistic Regression

Binary Responses Example Properties Credit approval, employment Response can only take one of the two possible outcomes Covariates could be anything

Logistic Regression Model Link functions

Case Study Objective: Comparing site preference for lizard Data Source: Fienberg (1970b) Variables Response: Site preference (Sunny/Shady). Discretized perch height and diameter Time of the data (Early, Mid, Late) Species: Grahami and Opalinus

Statistical Model

SAS Program proc genmod data=A0; class site diameter height time species; freq number; model species = diameter height time site /dist=bin link=logit p r type3; run;

Logit Link

Probit Link

Poisson Regression

Count Responses Example Properties Auto accidents, service request Properties Constant arriving rate Independent waiting time Waiting time is memorylessness Then, the number of requests per unit time has to be a Poisson random variable

Poisson Distribution Probability Density Function Mean and Variance

Normalization Transformation Define transformation Limiting distribution (why?)

Variance Stabilization Transformation Define It can be obtained then Therefore

Poisson Regression Model Link Functions Example: Auto Insurance

Case Study Objective: What cause the wave damage to cargo ships Data Source: Lloyd’s Register of Shipping by J. Crilley and L. N. Heminway Variables Ship type: A – E. Year of construction: 60-64, 65-69, 70-74,75-79 Period of operation: 60-74, 75-79 Aggregate months services

Statistical Model Log(expected number of incidents) = log(aggregate month services) + (effect due to ship type) + (effect due to year of construction) + (effect due to service period)

SAS Program proc genmod data=A0; class type year period; model number = type year period logMonth/dist=P link=log p r type3; run;

Ordinal Regression

Ordinal Responses Example Properties Preference data No numerical meaning Order dose matters Consecutive categories can be collapsed into one

Ordinal Regression Model Link Functions

Case Study Objective: Which cheese customers like most? Data Source: Experiment done by Dr. Graeme Newell Variables Cheese type: A – D. Response: 1 – 9 with larger value = better

Statistical Model

SAS Program proc genmod data=A0; class type; freq total; model pref = type/dist=multinomial link=cumlogit p r type3; run;

Progress Report Due: Next week before the class Requirement: Electronic submission by e-mail In one zipped file, no other format will be accepted The zip file should contain: Finalize project proposal in WORD or PDF format Cleaned data set: In SAS format Preliminary/Descriptive Analysis Report Please refer to the sample directory structure on server

New Project A animal study Requirement Four treatment groups with one is control Ordinal responses were measure on 17 consecutive days Question: Is there a treatment effect? Requirement Formal report as before Due: In two weeks (Dec. 13th)