Lecture 1
February 26, 2016
Introduction
The linear model
Y = 0 +
p
X
j X j +
j=1

,
cfw_z
(1)
stochastic error v
cfw_z
:=deterministic part
where y R of interest and X j R is the jth predictor. Linear model is very powerful,
beautiful and exact but t
Lecture 7
September 17, 2016
Last time, we talked about NR local convergence
k 1 2k .
2 CM
(1)
1. Number of correct digits doubles with every iteration and hold assuming 0 is closed to
.
2. Global convergence not guaranteed
3. In practice, a modified NR i
Lecture 8
February 25, 2016
Inference for GLMs
d
Note that in last class we have , and and I1/2
N (0, I) ,where
d
p
0
if a () , . Also I N (0, 1) .
I = X WX
Sanity check (linear model special case)
We have
Y N (X , In )
(1)
1
n
log (2) 2 y X .
2
2
(2)
Lecture 8
September 17, 2016
We have = , where is kdimensional. Want to test H 0 : = 0 .
1. Null does not fully specify distribution.
2. is nuissance parameter.
3. Composite null.
We have
1. Score test
(a) Based on l 0 , 0 properly normalized when 0
Lecture 4
February 26, 2016
Linear models
1. Note that linear models can be written in the following standard form,
y = X + , N 0, 2 In .
(1)
2. Likelihood equation
L (, ) =
n
Y
i=1
1
1
exp 2 yi xi0 2
2
2 2
(2)
and
n
n
1 X
yi xi0 2
l (, ) = log L , 2 =
Lecture 2
September 17, 2016
1
GLM framework
Generally, the GLM framework has 3 components:
1. Randomness  exponential family
y b ( )
Y f (y; , ) = exp
+ c (y, )
a ()
"
#
(1)
2. Systematic: = xt
3. Link function: = () .
Note that we have the following r
BTRY 7180 Generalized Linear Models
March 18, 2016
Chapter 1
Introduction
Recall that we have the linear model
Y =0 +
p
X
j Xj + ,
j=1
where y R of interest and Xj R is the jth predictor. Linear model is very powerful is many case,
it also beautiful and e
Lecture 3
February 26, 2016
We have the following properties for exponential family
1. The likelihood function is
l (, ) = log (y, , ) =
y b ( )
+ c (y, ) .
a ()
(1)
And
" #
y b 0 ( ) 2l
l
l
=
, 2 = Var
.
a ()
(2)
#
Y b 0 ( )
= 0 = E [Y ] = b 0 ( ) .
E
a
Lecture 10
September 17, 2016
Analysis of Deviance
This is just the likelihood ratio test in GLM terminology.
1. Group observations by x1 , (y1 , xi , mi ) for i = 1, . . . , n. We observed predictor xi for mi
times and yi is the avg response at xi . We h
Lecture 5
September 17, 2016
Algorithm for least squares
1. First, we talk about univariate regression. We have y and x in Rn . Then
0
x0y
= xx y .
= (X0X) 1 X0y =
,
y
=
x
kxk 2
kxk 2
2. Next easiest case is that x1 , . . . , xp are orthogonal, i.e., X0