From:daren.j.austin@gsk.com
Subject:[NMusers] Regression with errors in X
Date:Wed, 29 Jan 2003 15:25:39 +0000

Dear all, 

I am performing a mixed-effects linear regression where I am unwilling
to omit the errors in the X variable. 

My analysis fits 

Y[i,j,k] = a[i] + b[i] X[j,k] + EPS[i] 

where subscript i denotes a study index, j denotes treatment and k time. 
Note that this is a one to many regression (X is used multiple times) and
I am assuming that X is the mean of a normally distributed variable with
measured standard deviation (which I do not wish to ignore
because the CV is 25-40%). 

To incorporate this into my model I have used the following code,
and I have assumed that the residual can vary across studies 

$PRED 
  INCEPT = THETA(1)+ETA(1) 
  SLOPE=THETA(2)+ETA(2) 
  F=INCEPT+SLOPE*XBAR 
  Y=F+SLOPE*THETA(3)*ERR(1)+ERR(2)*EXP(ETA(3)) 
  IPRED=F 
  IRES=IPRED-DLF1 
$THETA 0.1 0.3 0.0441 FIX 
$OMEGA BLOCK(2) 0.1 0.01 0.1 
$OMEGA 0.1 
$SIGMA 1 FIX 0.1 
$EST MAXEVAL=9999 SIGDIGITS=4 METHOD=1 MSFO=msf1 INTER 
$COVARIANCE 
$TABLE ID ETA3 XBAR IPRED IRES NOPRINT ONEHEADER FILE=RUN1.PAR 

THETA(3) is the variance in the X variable and is fixed to 0.0441 (from
measurement). ERR(1) is fixed to unity to ensure that it is N(0,1) for
every observation. Thus the error structure has two terms: 

SLOPE*THETA(3)*ERR(1)
which reflects how far the X variable has been pulled around in the regression and 
ERR(2)*EXP(ETA(3))
which is the residual variancefor a particular study. 

What I would like to do is extract the numerical values of the
first error term, allowing me to plot 'predicted X' vs 'predicted Y'. 
Is there an easy way of obtaining the values of ERR(1) and ERR(2) 
for each observation? 

The objective functions are as follows: 
BASE model: -2167 
Variable ETA: -2213 
ERROR IN X and variable ETA: -2223 

A covariance step is completed, and all parameters uncorrelated. Incorporating
variability in X is clearly warranted. 

Kind regards, 

Daren 

Dr. Daren J. Austin
Principal Pharmacometrician
Clinical Pharmacology Discovery Medicine
GlaxoSmithKline R&D
Tel: 7-711 2073 or +44 (0) 20 8966 2073
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