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EXST7034  Regression Techniques
Page 1
Measurement errors in X
We have assumed that all variation is in Y.
Measurement error in this variable
will not effect the results, as long as they are uncorrelated and unbiased,
since they cancel out.
However, we have assumed that X is measured without error, and measurement
error in this variable can cause error.
Since all error is “vertical", we
cannot incorporate this measurement error into our model.
As a result,
this additional error must, in some way, get incorporated into the model
and/or its error.
often it is not true that X is measured without error particularly in meristic
relationships
e
g
.
height of brother
height of sister
body length
scale length
length
weight
Let the measurement error in X be denoted as
X
X
33
3
‡
3
$
œ
where X is the measured value and X is the true value of the variable.
‡
3
3
Then, when fitting the supposed model
Y
X
3!"
3
3
œ
""
%
we are actually fitting
(X
)
‡
3
$
%
and, multiplying out and grouping variability effects,
Y
X
(
)
3
3
"
‡
3
%
$
"
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Page 2
As a result,
a) X is not fixed (measured without error), it is a random variable
‡
3
b) The variance term is not longer independent of X , since
contains X
33
3
$
c) b and b are biased (towards zero) and
!"
lack consistency (ie.
lim P(
)> )=0 where
is some arbitrary,
^
8p_
""
%
%
3
3
positive real number; so
does not tend toward
probabilistically as n
^
3
3
increases infinitely)
d) There are a couple of cases or aspects of the variation in X where variation is
3
not a problem.
a) X may be a random variable, not under the control of the investigator.
3
However, this is not a problem as long as the value of X is measured
3
without measurement error and is known exactly.
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 Fall '08
 Geaghan,J

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