Dear Kit,
I realized now that your solution appears based on I having the individual test results. I was looking for an "immediate command" approach as I only have the aggregate data ( i.e. I have only the 2 means, 2 sigmas and number of individuals in each group).
Thanks,
Ben
>>> Kit Baum <[email protected]> 12/9/2007 7:38 AM >>>
Ben said
I am working with normally distributed test results in patients with
and without a disease condition related to alpha (accuracy parameter)
and beta (threshold parameter) such that:
alpha = (2*pi/sqrt(3))*[ (mu_nd - mu_d)/(sigma_nd + sigma_d)]
beta = (sigma_nd - sigma_d)/(sigma_nd + sigma_d)
where mu_nd and sigma_nd are the mean and standard deviation of test
results in N_nd non-diseased patients
and mu_d and sigma_d are the mean and standard deviation of test
results in N_d diseased patients.
How may I compute standard errors and/or confidence intervals for
alpha and beta?
As Ben has specified that the data are normally distributed, all we
have to do is get the two means and sd's in a single coefficient
vector via maximum likelihood:
----- cut here -----
program drop _all
program ben
version 10.0
args lnf mu1 mu2 sigma1 sigma2
qui replace `lnf' = ln(normalden($ML_y1, `mu1', `sigma1')) if
$disease == 0
qui replace `lnf' = ln(normalden($ML_y1, `mu2', `sigma2')) if
$disease == 1
end
sysuse auto, clear
global disease foreign
gen iota = 1
ml model lf ben (mu1: price=iota) (mu2: price=iota) /sigma1 /sigma2
// ml check
ml maximize, nolog
// you can check the results by doing
// ivreg2 price if ~foreign
// ivreg2 price if foreign
// Ben's beta measure
nlcom ([sigma1]_b[_cons] - [sigma2]_b[_cons])/([sigma1]_b[_cons] +
[sigma2]_b[_cons])
// Ben's alpha measure
nlcom 2*_pi*sqrt(3) * (([mu1]_b[_cons] - [mu2]_b[_cons])/([sigma1]_b
[_cons] + [sigma2]_b[_cons]))
---- cut here ----
In this case the 'model' is nothing more than regression on a
constant, allowing mu and sigma to differ across the two classes. You
will get the same mu and sigma if you run that regression (use ivreg2
to get z-stats rather than t-stats, with the sigma divided by n, and
it agrees with ml).
-nlcom- then cranks out the desired point and interval estimates via
the delta method.
Kit Baum, Boston College Economics and DIW Berlin
http://ideas.repec.org/e/pba1.html
An Introduction to Modern Econometrics Using Stata:
http://www.stata-press.com/books/imeus.html
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