Biostatistical Methods: Survival Analysis and Logistic Regression
STAT 574

Fall 2013
Example from Table 2.1 data
Yjk number of chronic medical conditions for women that
use similar practitioner services.
j = 1 town; j = 2 country
k = 1, 2, . . . , kj where k1 = 26 and k2 = 23.
For town , Y1 = 1.423 and SD = 1.17.
For country, Y2 = 0.913 a
Biostatistical Methods: Survival Analysis and Logistic Regression
STAT 574

Fall 2013
Bayes theorem. Statistical framework.
is some unknown parameter.
is typically a vector, = (1 , 2 , . . . , r ).
r = 1 is the scalar case or one parameter model.
y = (y1 , y2 , . . . , yn ) is a data vector. Provides information
about .
f (y ) is the sa
Biostatistical Methods: Survival Analysis and Logistic Regression
STAT 574

Fall 2013
Settings for a BinaryBinomial response model
Sample of target population. Each individual is
independent and classied by success, failure.
Response
Predictors for each individual
Y1=0
x11,x12,
,x1p
Y2=0
x21,x22,
,x2p
Y3=1
x31,x32,
,x3p
Individuals with s
Biostatistical Methods: Survival Analysis and Logistic Regression
STAT 574

Fall 2013
Review of concepts for the Normal Linear Model
Yi N (i , 2 ), i = 1, 2, . . . , n.
i = E (Yi ) = xit linear predictor so g (i ) = i . is the
vector of parameters
The usual notation for the model is,
Y = X +
Y is the vector of observations, X design matrix
Biostatistical Methods: Survival Analysis and Logistic Regression
STAT 574

Fall 2013
Nominal Logistic Regression
Random variable Y that corresponds to K categories (blue,
red, green).
1 , 2 , . . ., K probabilities for each category (population).
n independent observations of Y.
We count Yi , i = 1 . . . , K number of times we observe
cat
Biostatistical Methods: Survival Analysis and Logistic Regression
STAT 574

Fall 2013
Notes on Survival analysis
Develop models for survival times Y .
f (y ) is its pdf and F (y ) its distribution function.
Survivor function: probability of survival beyond time y or
S (y ) = P [Y > y ] = 1 F (y ).
Hazard function: probability of death in a
Biostatistical Methods: Survival Analysis and Logistic Regression
STAT 574

Fall 2013
Exponential family of distributions
Probability distribution on Y that has the form,
f (y ) = exp [a(y )b() + c ()d (y )]
a(), d () are functions of y .
b(), c () are functions of .
b() is called the natural parameter.
In fact,
E (a(Y ) =
c ()
b ()c () c
Biostatistical Methods: Survival Analysis and Logistic Regression
STAT 574

Fall 2013
Count data
Refers to number of times and events. Frequency data.
Number of tornados per month, hurricanes per year.
Often modeled with a Poisson distribution of parameter .
f (y ; ) =
e y
; y = 0, 1, 2, . . . ,
y!
is the average number of ocurrances.
Poi
Biostatistical Methods: Survival Analysis and Logistic Regression
STAT 574

Fall 2013
Stat 574: GLMs and Survival Analysis
Gabriel Huerta
Department of Mathematics and Statistics
University of New Mexico
Albuquerque, NM, E.U.A.
Chapter 1, summary notes.
UNM
Poisson regression example
Ni number of virus mutations for patient i = 1, 2, . . .
Biostatistical Methods: Survival Analysis and Logistic Regression
STAT 574

Fall 2013
Some Examples on Gibbs Sampling and
MetropolisHastings methods
S420/620 Introduction to Statistical Theory, Fall 2012
Gibbs Sampler
Sample a multidimensional probability distribution from
conditional densities.
Suppose d = 2, = (1 , 2 ). Set an initial p