Systematically Misclassified Binary Dependant Variables

SYSTEMATICALLY MISCLASSIFIED BINARY DEPENDANT VARIABLES
tail, would assign herself a 1. A different person who believes she has to have an
extremely high value of the characteristic gives herself a 4. Others with intermediate
values move to the closest node. People between 3 and 3.25 move to 3, while those
between 3.25 and 3.5 move to 3.5. In this case, the rounding errors offset; blue against
red, purple against gray, green against orange, and the two halves of the yellow. The
Likert scale provides an unbiased and consistent estimate of the mean value. The
variance, however, is biased downward because the extreme values are not truncated.
We note that if the location on the distribution is covariate dependent, then covariate
analysis is biased because individual deviations persist. As already noted, the variance of
the distribution of the characteristic is biased downward.
Now suppose that some reason (for example, a bias against being associated with
socially unacceptable behavior) keeps people from centering on the average of the Likert
scale. In other words, the true distribution has a mean different from that offered by the
Likert scale nodes. In terms of Figure 1, we can assume, for example, that the proffered
Likert scale had nodes 1, 2, 3, and 4, with a mean value of 2.5, but the underlying
distribution is as shown. Any respondent with a personal valuation below 1.5 assigns
herself 1. Likewise, a respondent with a personal valuation above 3.5 assigns herself 4.
Other intermediate values are likewise assigned to one of the integer nodes. Now,
matching the colors we get a biased estimate of the mean, not just the variance. The blue
and red areas together have an average value of 2.5, a bias of 0.25 above the mean. The
purple and gray areas together have an average of 2, hence a bias of -0.25. Similarly,
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