Reasons for using transformations
---------------------------------

There are many reasons for transformation. The list here is not
comprehensive.

1. Convenience
2. Reducing skewness
3. Equal spreads
4. Linear relationships
5. Additive relationships

If you are looking at just one variable, 1, 2 and 3 are relevant, while
if you are looking at two or more variables, 4 and 5 are more important.
However, transformations that achieve 4 and 5 very often achieve 2 and
3.

1. ^Convenience^ A transformed scale may be as natural as the original
scale and more convenient for a specific purpose (e.g. percentages
rather than original data, sines rather than degrees).

One important example is ^standardisation^, whereby values are adjusted
for differing level and spread. In general

                       value - level
 standardised value =  -------------.
                          spread

Standardised values have level 0 and spread 1 and have no units: hence
standardisation is useful for comparing variables expressed in different
units. Most commonly a ^standard score^ is calculated as

                       x - mean of x
                  z =  -------------.
                          sd of x

Standardisation makes no difference to the shape of a distribution.

2. ^Reducing skewness^ A transformation may be used to reduce skewness.
A distribution that is symmetric or nearly so is often easier to handle
and interpret than a skewed distribution.

To reduce right skewness, take roots or logarithms or reciprocals (roots
are weakest). This is the commonest problem in practice.

To reduce left skewness, take squares or cubes or higher powers.

3. ^Equal spreads^ A transformation may be used to produce approximately
equal spreads, despite marked variations in level, which again makes
data easier to handle and interpret. Each data set or subset having
about the same spread or variability is a condition called
^homoscedasticity^: its opposite is called ^heteroscedasticity^. (The
spelling ^-sked-^ rather than ^-sced-^ is also used.)

4.  ^Linear relationships^ When looking at relationships between
variables, it is often far easier to think about patterns that are
approximately linear than about patterns that are highly curved. This is
vitally important when using linear regression, which amounts to fitting
such patterns to data. (In Stata, @regress@ is the basic command for
regression.)

For example, a plot of logarithms of a series of values against time has
the property that periods with ^constant rates of change^ (growth or
decline) plot as straight lines.

5. ^Additive relationships^ Relationships are often easier to analyse
when additive rather than (say) multiplicative. So

                  y = a + bx

in which two terms a and bx are added is easier to deal with than

                        b
                  y = ax

                          b
in which two terms a and x  are multiplied. ^Additivity^ is a vital
issue in ^analysis of variance^ (in Stata, @anova@, @oneway@, etc.).

In practice, a transformation often works, serendipitously, to do
several of these at once, particularly to reduce skewness, to produce
nearly equal spreads and to produce a nearly linear or additive
relationship.

Also see:
---------

Review of most common transformations                     help @trreview@

Transformations for proportions and percents              help @trpropor@

Psychological comments -- for the puzzled                 help @trpsych@

How to do transformations in Stata                        help @trstata@