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[NMusers] RE: Failure to arrive at expected parameter estimates

From: HUI, Ka Ho <matthew.hui_at_link.cuhk.edu.hk>
Date: Thu, 19 May 2016 15:26:35 +0000

Dear all,

We just identified that the cause of the problem is model misspecification,=
 which happens for small values of x near zero for a logarithmic function. =
We managed to solve the problem by using a shift of the x-axis by using thi=
s:
C=THETA(1)
B=THETA(2)
                S=THETA(3)
F=C+B*LOG(FACTOR1+S)

Thanks!
Matthew


From: HUI, Ka Ho
Sent: Thursday, May 19, 2016 4:18 PM
To: nmusers_at_globomaxnm.com
Subject: Failure to arrive at expected parameter estimates

Dear all,

I have some data x (input) and y (output), with 'inverse' heteroscedasticit=
y, where variance is greater for smaller x.
The data file is attached (data.txt).
After filtering off all data with FILTER1=1 and FILTER2=1, the binned d=
ata plot looks like this (Question.jpg).
Most data points are at small x (43.3% are between 0-10, 12.9% are between =
10-20, 9% are between 20-30, 34.8% for the rest, data are more sparse at la=
rger x)

Blue points are the mean, red and purple points show the 5th and 95th perce=
ntiles in each bin. Green points are the SD in each bin. Curve estimations =
has been done and the equation for the means are shown as equation (1) and =
that for the SDs are shown at the bottom.

Here is a template for our first control stream, written according to the r=
esults of curve estimation for means:
$INPUT ID DV FILTER1 FILTER2 FACTOR1 MDV
$DATA data.txt IGNORE=_at_ IGNORE=(FILTER1.EQ.1,FILTER2.EQ.1)
$PRED
                C=THETA(1)
                B=THETA(2)
                F=C+B*LOG(FACTOR1) ;Relationship as shown in equation (1)
                Y=F+EPS(1)
                DUMMY=ETA(1)
$THETA
                (-20, -0.5, 20) ;C, curve estimation result is -0.4465
                (-20, 1, 20) ;B, curve estimation result is 1.0266
$OMEGA
                0 FIXED
$SIGMA
                2
$EST METHOD=1 INTERACTION MAXEVAL=9999 PRINT=1
$COV
$TABLE ...

The fitted parameters are illustrated by equation (3), which is obviously b=
iased below for x > 100. The bias was also observed in residual plots.

To explain also for the heteroscedasticity, we tried another control stream=
, written according to the results of curve estimation for SD:
$INPUT ID DV FILTER1 FILTER2 FACTOR1 MDV
$DATA data.txt IGNORE=_at_ IGNORE=(FILTER1.EQ.1,FILTER2.EQ.1)
$PRED
                C=THETA(1)
                B=THETA(2)
                C_SD=THETA(3)
                B_SD=THETA(4)
                W=C_SD*B_SD**FACTOR1 ;Relationship as shown in the equati=
on at the bottom
                F=C+B*LOG(FACTOR1)
                Y=F+(W*EPS(1)) ;Variance depends on FACTOR1
                DUMMY=ETA(1)
$THETA
                (-20, -0.5, 20) ;C, curve estimation result is -0.4465
                (-20, 1, 20) ;B, curve estimation result is 1.0266
                (-20, 0.72, 20) ;C_SD, curve estimation result is 0.7529
                (-20, 1, 20) ;B_DD, curve estimation result is 0.9962
$OMEGA
                0 FIXED
$SIGMA
                1 FIXED
$EST METHOD=1 INTERACTION MAXEVAL=9999 PRINT=1
$COV
$TABLE ...

The fitted parameters are illustrated by equation (2), which is still biase=
d.

Despite the fact that most data points concentrate at small x, which may ha=
ve contributed to the bias at large x, we observed the fitted parameters (e=
quation (2)/equation(3)) and note that these two equations are in fact over=
-estimating the means even at small x, and therefore we have no idea why th=
ese two equations resulted. We tried different initial estimates but in vai=
n.

It would be great if someone can give any advice!

Thanks!
Matthew

Received on Thu May 19 2016 - 11:26:35 EDT

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