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st: RE: Generating correlated binary data

From   "FEIVESON, ALAN H. (AL) (JSC-SK) (NASA)" <>
To   "''" <>
Subject   st: RE: Generating correlated binary data
Date   Tue, 26 Apr 2005 08:41:12 -0500

Bryan - any random error terms you add must be done in the expression for
logit(pij) - NOT added to the binary variable. You may ceratinly add a
within-group "e_ij" random error term to 

logit(pij) = b0 + B1*x1ij + B2*z1ij + u0j 

but all this does is affect pij. The actual generation of the binary
dependent variable comes from an operation such as

gen u=uniform()
gen y = (u < pij)

Al Feiveson

-----Original Message-----
[]On Behalf Of Bryan W.
Sent: Tuesday, April 26, 2005 8:29 AM
Subject: st: Generating correlated binary data

I am trying to generate correlated binary data to model random effects with 
xtlogit. I plan to do several Monte Carlo type studies. My question is 
about the specific form of the data to generate. I have two predictors, x1 
(within cluster predictor) and z1 (between cluster predictor), and the 
outcome will be binary. The model takes this form:

logit(pij) = B0j + B1*x1ij + B2*z1ij

B0j = b0 + u0j


logit(pij) = b0 + B1*x1ij + B2*z1ij + u0j

where there are i observations within cluster j.

The issue I have is whether there should be a within-cluster error term 
added to the model when generating data. That is, should the model to 
generate binary data appear like this (the .3 appear in the equation 
because I copy and pasted from code I used, see below)

generate y1_w_error =.3 + .3*x1 +.3*z1 + u0j + e

or like this

generate y1_wo_error =.3 + .3*x1 + .3*z1 + u0j ?

With linear models I know the "e" should be included, and in fact my code 
below generates data for continuous outcomes that behave well, but I'm 
uncertain about the logistic case.

My code is listed below.

set obs 30
generate float z1= round(uniform())
generate float u0j= invnorm(uniform())*1.69
generate n = 30
range id2 1 30
expand n	
generate float x1= round(uniform())
generate float e= invnorm(uniform())

* Option 1 -- model with level 1 error
generate y1_w_error =.3 + .3*x1 +.3*z1 + u0j + e
generate p_w_error = exp(y1_w_error)/(1+exp(y1_w_error))
generate binary_y_w_error = uniform()<=p_w_error

* Option 2 -- model without level 1 error
generate y1_wo_error =.3 + .3*x1 + .3*z1 + u0j
generate p_wo_error = exp(y1_wo_error)/(1+exp(y1_wo_error))
generate binary_y_wo_error = uniform()<=p_wo_error
Bryan W. Griffin
Curriculum, Foundations, & Reading
P.O. Box 8144
Georgia Southern University
Statesboro, GA 30460-8144

Phone: 912-681-0488

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