Transformations May be Necessary Due to:

• Theoretical considerations.
• Dependent variable may have a probability distribution in which the mean is related to the variance.
• Empirical evidence from examination of the residuals.

Variables to be Transformed

• The dependent variable can be transformed. Note: This effects the relationship of the dependent variable with all of the predictor variables in the model.
• Individual predictor variables can be transformed.
• Both dependent and independent variables can be transformed

Major Drawbacks

• Interpretation of the regression involves transformed variables and not the original variables themselves.
• Relationship of the transformed variables to the original variables may be difficult or confusing.
• Transformation may not be able to rectify all of the problems in the original data; the regression analysis may still be suspect.

Log Transformation

1. To linearize regression model with consistently increasing slope.

2. Stabilize variance when variance of residuals increases markedly with increasing Y.

3. To normalize Y when distribution of residuals is positively skewed.

Stata Example

use http://www.gseis.ucla.edu/courses/data/lntrans, clear

scatter y x

generate z = log(y)

scatter z x

regress z x

  Source |       SS       df       MS                  Number of obs =      50
---------+------------------------------               F(  1,    48) = 2916.35
   Model |  365.874096     1  365.874096               Prob > F      =  0.0000
Residual |  6.02190025    48  .125456255               R-squared     =  0.9838
---------+------------------------------               Adj R-squared =  0.9835
   Total |  371.895996    49  7.58971421               Root MSE      =   .3542

------------------------------------------------------------------------------
       z |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
       x |   .9417895   .0174395     54.003   0.000        .906725     .976854
   _cons |    .906511   .1082093      8.377   0.000       .6889417     1.12408
------------------------------------------------------------------------------


predict p

scatter z p x, msym(O i) con(. l) sort

generate p2 = exp(p)

graph y p2 x, msym(O i) con(. l) sort

/* now transform x instead of y */

generate xt = exp(x)

scatter y xt

regress y xt

  Source |       SS       df       MS                  Number of obs =      50
---------+------------------------------               F(  1,    48) =  650.09
   Model |  4.3685e+09     1  4.3685e+09               Prob > F      =  0.0000
Residual |   322552812    48  6719850.24               R-squared     =  0.9312
---------+------------------------------               Adj R-squared =  0.9298
   Total |  4.6911e+09    49  95736235.2               Root MSE      =  2592.3

------------------------------------------------------------------------------
       y |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
      xt |   1.409637   .0552866     25.497   0.000       1.298476    1.520799
   _cons |   493.3881    414.134      1.191   0.239       -339.284     1326.06
------------------------------------------------------------------------------

rvfplot, yline(0) xlabel ylabel

Used to stabilize variance when proportional to the mean of Y; especially when Y approximates a Poisson distribution.

Reciprocal Transformation

To stabilize variance when proportional to the 4th power of mean of Y, i.e., huge increase in variance above some threshold of Y. Purpose is to mimnimize effect of large values of Y. Transformed large Ys will be close to zero, thus large increases in Y will result in only trivial decreases in Y'.

Square Transformation

1. Linearize when X vs Y is curvilinear downward, i.e., slope decreases as X increases..

2. Stabilize variance when it decreases with the mean of Y.

3. Normalize Y when distribution of residuals is negatively skewed.

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