Example 5.3: Normal data, .
. We saw previously that and se. Find the two forms of confidence interval for .
The asymptotic normality approach gives a confidence interval for as follows.
We have already obtained that
Here . Then, and , so
Therefore the confidence interval is
where
is the maximum likelihood estimate of .
Alternatively, we can derive confidence intervals using the profile deviance. The first step is to evaluate the profile log-likelihood for . Recall,
Fixing and maximising with respect to , the maximum occurs at , where
Thus,
and so
Hence,
For a simulated dataset of size with , the profile deviance is plotted in the left panel of Figure Figure LABEL:nproflik (Link), leading to a 95% confidence interval of for .
The connection between profile likelihood and likelihood is shown in
the right panel of the figure. In this figure contours of the 2-dimensional
likelihood surface are plotted, together with the path
of the profile likelihood for .
Example 5.4: Normal data, other s.
. We have already obtained that
If . Then, and , so
If , with known.
Then, and , so,
In each case an approximate confidence interval is obtained as
Example 5.5: Gamma Distribution, ctd.
and , the population mean. Thus, and . Hence,
where
and
Example 5.6: Simple Linear Regression, ctd.
with (known) explanatory variables and also known. We obtained,
Now, suppose we wish to obtain a confidence interval for based on the profile-likelihood function. We have, for fixed and ,
at the maximum, and so
Recall also that
Hence, we obtain
For the data simulated in Figure 3.7, the corresponding profile deviance for is plotted in Figure Figure 5.2 (Link), leading to a 95% confidence interval for equal to .
This latter example illustrates an important use of the likelihood function for model discrimination. Suppose we are interested in assessing whether there is a ‘significant’ linear relationship between the observations and the explanatory variables .
If there were no such relationship then the true value of would be zero.
Hence, a test of the linear regression model, versus the simpler assumption that each of the responses have a common mean, is equivalent to the test that .
Since 0 falls outside of our 95% confidence interval for , this gives reasonably strong evidence that ; we say that the hypothesis is rejected at the 5% significance level. But could we have reached this conclusion without having to go through the entire procedure of producing the profile deviance function?
In fact, yes. All that is necessary is to evaluate the profile likelihood at (equivalent to fixing and maximizing the log-likelihood with respect to the other parameters), and compare this value with the maximum likelihood value under the complete model where is unconstrained. Doubling the difference gives the deviance, whose value can be compared with the distribution to check for significance.