1. Generalized Linear Model

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

3. Generalized Linear Model A Unified Theory

4. Mean Structure
Ordinary Linear Model
Generalized Linear Model

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

6. Logistic Regression
Model
Link functions

7. Poisson Regression
Model
Link Functions
Example: Auto Insurance

8. Ordinal Regression
Model
Link Functions

9. Polytomous Regression
Model
Link Functions

10. Polytomous Regression
Properties
Therefore,

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

12. Normal Example
Likelihood function
Deviance

13. Poisson Example
Likelihood function
Deviance

14. 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

15. Pearson Residuals
Define
where
Example: Normal Distribution

16. Deviance Residual
Deviance
Define
Example: Normal Distribution

17. Logistic Regression

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

19. Logistic Regression
Model
Link functions

20. 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

22. Statistical Model

23. 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;

24. Logit Link

25. Probit Link

26. Poisson Regression

27. 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

28. Poisson Distribution
Probability Density Function
Mean and Variance

29. Normalization Transformation
Define transformation
Limiting distribution (why?)

30. Variance Stabilization Transformation
Define
It can be obtained then
Therefore

31. Poisson Regression
Model
Link Functions
Example: Auto Insurance

32. 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

34. 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)

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

43. Ordinal Regression

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

45. Ordinal Regression
Model
Link Functions

46. 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

48. Statistical Model

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

52. Progress Report Due: Next week before the class Requirement:
Electronic submission by
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

53. 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)