Box-Cox statistics for help in choosing transformations
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by Richard Goldstein, Qualitas, Brighton, MA  EMAIL goldst@@harvarda.bitnet

^boxcoxg^ lower-search-limit upper-limit step-size varlist [^if^ exp] [^in^ range]

This ado-file is used to search for possible transformations of the dependent
variable in a linear model (regression, anova) using the Box-Cox procedure.

This procedure limits the choice to the "power family", as follows:

                 (y^^lambda-1)/lambda         if lambda not equal to 0
    y^^(lambda) =
                 log(y)                      if lambda equal to 0

This includes many types of transformations, but certainly not all that might
be useful; further, this family is not useful when the dependent variable is a
proportion.  Of course, it is obvious that this family cannot be used when
negative numbers are possible.  Finally, note that this procedure is
specifically aimed at transformations of the dependent variable only.


Note that sometimes the result of ^boxcoxg^ will be a missing value--this means
that this particular transformation is not possible with the data provided.

^boxcoxg^ can be used to search for the best transformation or to examine just
one particular transformation.  In either case, you MUST enter three number
prior to entering the variable list.  The first number is the minimum value of
lambda that you want to search over, the second is the maximum, while the
third number is the step size.  If you just want to see the results for one
particular value of lambda, enter the same number for the minimum and the
maximum and enter any positive value for the step size (DO NOT enter a step
size of 0).  If you want to search over the space 0 (log transform) to 1 (no
transform) with a step size of .1, enter `^boxcoxg 0 1 .1^ <varlist>'.  Negative
numbers are allowed for either the minimum or the maximum.

The data set is automatically cleared when the ado-file is finished so you can
immediately re-enter ^boxcoxg^ with a finer search pattern if you so desire.  For
example, your first search might be from -3 to 3 with a step size of .25; you
might then search from one step smaller than the minimum found to one step
larger with a much smaller step size (e.g., -.5 0 .01).  Do not attempt to use
a step size smaller than .01, as this is not allowed.

The only output is a three-column list: the first column is lambda, the
transformation parameter; the second is the error sum of squares; while the
third is the log-likelihood.  You choose the smalled SSE (largest
log-likelihood) as the best transformation, or something close to this that
makes substantive sense.  The code could be easily modified to allow
likelihood-ratio tests, using the CRC-provided ado file (see STB-1) if the
user cared to do so.  The code could also be easily modified so that the
results, either SSE or log-likelihood, could be graphed if so desired.

Example:

^. us weisberg, replace^
(Weisberg, 1985, p. 149)

^. boxcoxg -2 2 .5 area peri^
lambda          SSE     Log-likelihood
-2.00       90372.63      -142.6462
-1.50       19289.59      -123.3415
-1.00        4708.16      -105.7132
-0.50        1301.88       -89.6446
 0.00         377.30       -74.1632
 0.50         116.26       -59.4482
 1.00         217.96       -67.3039
 1.50        1168.32       -88.2915
 2.00        5493.07      -107.6405