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Re: st: RE: simulated data for logistic regression... remedial algebra help?
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From |
Steven Joel Hirsch Samuels <[email protected]> |
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To |
[email protected] |
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Subject |
Re: st: RE: simulated data for logistic regression... remedial algebra help? |
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Date |
Mon, 19 Nov 2007 09:49:40 -0500 |
It's a quadratic.
Let w = proportion of positive tests
p = overall mortality
OR = odds ratio 1 over 2
Solve: p1/(1-p1)=OR(p2/(1-p2) Let Z2 = p2/(1-p2),
p1 = (OR x Z2)/(1+ OR x Z2) =(OR x p2)/(1-p2 + (OR x p2)) = A
Also p = w p1 + (1-w) p2,
p1 = (p -(1 -w)p2)/w =B (you left out the 1-w term in your k2
coefficient)
Equate A and B. Since the denominator in A multiplies the numerator
in B, you have a quadratic equation in p2.
On Nov 18, 2007, at 11:56 PM, Daniel Waxman wrote:
Not sure if the lack of response reflects a lack of clarity in
stating the
problem, or just that the solution isn't obvious to anyone.
In case it is the former, I'll restate the problem:
The goal is to create a simulated data set. To do so, I would like to
determine the probabilities of an outcome (death) given a positive or
negative test result, when the overall mortality rate, the odds
ratio for
mortality as a function of that test and proportion of positive
test results
in the population are known.
The problem reduces (I believe) to solving:
p1/(1-p1)=2(p2/(1-p2)
p2= k1-p1k2
where:
p1 = mortality rate for a positive test
p2 = mortality rate for a negative test
k1 = constant = (overall mortality rate)/(proportion of population
with a
positive test)
k2 = constant = (proportion negative test)/(proportion positive test)
Does this problem look familiar to anyone?
If my poor math skills are not failing me, I believe that I end up
with p1^3
term. Does this sound right? Would it mean that there is no exact
solution?
Any other suggestions for creating simulated data with these
properties?
Dan
-----Original Message-----
From: [email protected]
[mailto:[email protected]] On Behalf Of Daniel
Waxman
Sent: Saturday, November 17, 2007 6:44 PM
To: [email protected]
Subject: st: simulated data for logistic regression... remedial
algebra
help?
I am trying to create a series of simulated data sets for use in
logistic
regression with the following properties:
Mortality (outcome) remains constant. There is a single dichotomous
independent variable whose odds ratio (coefficient) and proportion of
positives can vary between the sets. It all comes down to solving
for the
intercept (`b0'), given the following relationships:
probability_negative=invlogit(`b0’)
probability_positive=invlogit(log(`odds’)+`b0’)
`proportion_positive’*probability_positive+(1-`proportion_positive’)
*probabi
lity_negative=`mortality’
Sad to admit, but I am bumping up against the limitations of my
algebra
skills.
I'd imagine this is trivial for many of you...
i.e.:
**************
clear
set obs 1000
local odds=2
local proportion_positive= .10
local mortality = .05
gen test=uniform()<`proportion_positive’
/*
************solve for `b0' here************
*/
gen probability_negative=invlogit(`b0’)
gen probability_positive=invlogit(log(`odds’)+`b0’)
gen died=uniform() < cond
(test==0,probability_negative,probability_positive)
logistic died test
************************
Thanks.
Dan
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Steven Samuels
[email protected]
18 Cantine's Island
Saugerties, NY 12477
Phone: 845-246-0774
EFax: 208-498-7441
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