Lecture 13 Estimation and hypothesis testing for logistic ...
Lecture 13 Estimation and hypothesis testing for
logistic regression
BIOST 515 February 19, 2004
BIOST 515, Lecture 13
Outline
? Review of maximum likelihood estimation ? Maximum likelihood estimation for logistic regression ? Testing in logistic regression
BIOST 515, Lecture 13
1
Maximum likelihood estimation
Let's begin with an illustration from a simple bernoulli case.
In this case, we observe independent binary responses, and we wish to draw inferences about the probability of an event in the population. Sound familiar?
? Suppose in a population from which we are sampling, each individual has the same probability, p, that an event occurs.
? For each individual in our sample of size n, Yi = 1 indicates that an event occurs for the ith subject, otherwise, Yi = 0.
? The observed data is Y1, . . . , Yn.
BIOST 515, Lecture 13
2
The joint probability of the data (the likelihood) is given by
n
L=
pYi(1 - p)1-Yi
i=1
=
Pn
p i=1
Yi(1
-
p)n-Pni=1
Yi.
For estimation, we will work with the log-likelihood
n
n
l = log(L) = Yi log(p) + (n - Yi)log(1 - p).
i=1
i=1
The maximum likelihood estimate (MLE) of p is that value that maximizes l (equivalent to maximizing L).
BIOST 515, Lecture 13
3
The first derivative of l with respect to p is
l n
n
U (p) = = p
Yi/p - (n -
Yi)/(1 - p)
i=1
i=1
and is referred to as the score funcion. To calculate the MLE of p, we set the score function, U (p) equal to 0 and solve for p. In this case, we get an MLE of p that is
n
p^ = Yi/n.
i=1
BIOST 515, Lecture 13
4
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