13-MeasurementErrorInX

13-MeasurementErrorInX - EXST7034 - Regression Techniques...

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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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EXST7034 - Regression Techniques 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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13-MeasurementErrorInX - EXST7034 - Regression Techniques...

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