Estimating equations: an introduction via glm
Estimating Equations: refers to equations of the form
which are solved for to get estimates . Examples:
1.
The normal equations in linear regression:
2.
The likelihood equations:
where
is the log-likelihood.
3
Heteroscedastic Errors
If plots and/or tests show that the error variances
depend on i there are
several standard approaches to fixing the problem, depending on the nature of the
dependence.
If the variances are known except for a constant factor then we
Instructions: This is an open book test. You
may use notes, text, other books and a
calculator. Your presentations of statistical
analysis will be marked for clarity of
explanation. I expect you to explain what
assumptions you are making and to comment
if
Multivariate Analysis of Covariance
This example concerns a one way layout with 45 patients in 4 groups (defined by
body type and obesity status). There are 4 biochemical measurements made on the
urine of each subject: three are treated as response variab
Example: transformation vs glm
At each of 5 doses of some drug 40 animals were tested and the number dying were
recorded. The log doses are -3.204, -2.903, -2.602, -2.301, and -2.000 and the
numbers surviving are 7, 18, 32, 35, 38. A plot of the data is
A
INCLUDING CATEGORICAL COVARIATES
options pagesize=60 linesize=80;
data scenic;
infile 'scenic.dat' firstobs=2;
input Stay Age Risk Culture Chest Beds School
Region Census Nurses Facil;
Nratio = Nurses / Census ;
R1 = -(Region-4)*(Region-3)*(Region-2)/6;
R
Non-linear Regression
In a regression model of the form
we call the model linear if
for some covariates
and a parameter
however, we have non-linear models such as
general
. Often,
or in
for some function f which need not be linear in
. If the errors
are h
Power and Sample Size Calculations: Examples
SAMPLE SIZE NEEDED using t test: SAND and FIBRE example.
Now for the same assumed values of the parameters how many replicates of the basic
design (using 9 combinations of sand and fibre contents) would I need
Power and Sample Size Calculations
Definition: The power function of a test
procedure in a model with parameters is
Definition: The non-central distribution t with
non-centrality parameter and degrees of
freedom is the distribution of
where Z is
, U is an
Theory of Generalized Linear Models
If Y has a Poisson distribution with parameter
then
for k a non-negative integer. We can use the method of maximum likelihood to
estimate k if we have a sample
with mean
is
. If we observe
This function of
of independen