Transformations for proportions and percents (more advanced)
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Data that are proportions (between 0 and 1) or percents (between 0 and
100) often benefit from special transformations. The most common is the
^logit^ (or logistic) transformation, which is

            logit p = log (p / (1 - p)) for proportions

         OR logit p = log (p / (100 - p)) for percents

where p is a proportion or percent.

This transformation treats very small and very large values
symmetrically, pulling out the tails and pulling in the middle around
0.5 or 50%. The plot of p against logit p is thus a flattened S-shape.
Strictly logit p cannot be determined for the extreme values of 0 and 1
(100%): if they occur in data, there needs to be some adjustment.

One justification for this logit transformation might be sketched in
terms of a diffusion process such as the spread of literacy. The push
from zero to a few percent might take a fair time; once literacy starts
spreading its increase becomes more rapid and then in turn slows; and
finally the last few percent may be very slow in converting to literacy,
as we are left with the isolated and the awkward, who are the slowest to
pick up any new thing. The resulting curve is thus a flattened S-shape
against time, which in turn is made more nearly linear by taking logits
of literacy. More formally, the same idea might be justified by
imagining that adoption (infection, whatever) is proportional to the
number of contacts between those who do and those who do not, which will
rise and then fall quadratically. More generally, there are many
relationships in which predicted values cannot logically be less than 0
or more than 1 (100%). Using logits is one way of ensuring this:
otherwise models may produce absurd predictions.

The logit (looking only at the case of proportions)

            logit p = log (p / (1 - p))

can be rewritten

            logit p = log p  - log (1 - p)

and in this form can be seen as a member of a set of ^folded^
^transformations^

     transform of p = something done to p -
                          something done to (1 - p).

This way of writing it brings out the symmetrical way in which very high
and very low values are treated. (If p is small, 1 - p is large, and
vice versa.) The logit is occasionally called the ^folded log^. The
simplest other such transformation is the ^folded root^ (that means
square root)

   folded root of p = root of p  - root of (1 - p)

As with square roots and logarithms generally, the folded root has the
advantage that it can be applied without adjustment to data values of 0
and 1 (100%). The folded root is a weaker transformation than the logit.
In practice it is used far less frequently.

Two other transformations for proportions and percents met in the older
literature (and still used occasionally) are the ^angular^ and the
^probit^. The angular is

            arcsin(root of p):

that is, the angle whose sine is the square root of p. In practice, it
behaves very like

             0.41          0.41
            p     - (1 - p)    ,

which in turn is close to

             0.5          0.5
            p    - (1 - p)   ,

which is another way of writing the folded root. The probit is a
transformation with a mathematical connection to the normal (Gaussian)
distribution, which is not only very similar in behaviour to the logit,
but also more awkward to work with. As a result, it is now rarely seen
in any but more advanced applications, where it retains some advantages.

Also see:
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Reasons for using transformations                         help @trreason@

Review of most common transformations                     help @trreview@

Psychological comments -- for the puzzled                 help @trpsych@

How to do transformations in Stata                        help @trstata@


