6.5.2. What to do when data are non-normal

Often it is possible to transform non-normal data into approximately normal data

Non-normality is a way of life, since no characteristic (height, weight, etc.) will have exactly a normal distribution. One strategy to make non-normal data resemble normal data is by using a transformation. There is no dearth of transformations in statistics; the issue is which one to select for the situation at hand. Unfortunately, the choice of the "best" transformation is generally not obvious.

This was recognized in 1964 by G.E.P. Box and D.R. Cox. They wrote a paper in which a useful family of power transformations was suggested. These transformations are defined only for positive data values. This should not pose any problem because a constant can always be added if the set of observations contains one or more negative values.

The Box-Cox power transformations are given by

The Box-Cox Transformation

$$ \begin{eqnarray} x(\lambda) & = & \frac{x^\lambda -1}{\lambda} \,\,\,\,\, & \lambda \ne 0 \\ x(\lambda) & = & \mbox{ln}(x) & \lambda = 0 \, . \end{eqnarray} $$

Given the vector of data observations \( x = x_1, \, x_2, \, \ldots, \, x_n\), one way to select the power \(\lambda\) is to use the \(\lambda\) that maximizes the logarithm of the likelihood function

The logarithm of the likelihood function

$$ f(x,\lambda) = -\frac{n}{2} \mbox{ln}\left[ \sum_{i=1}^{n}{\frac{(x_i(\lambda) - \bar{x}(\lambda))^2}{n}} \right] + (\lambda - 1)\sum_{i=1}^{n}{\ln(x_i)} \, , $$ where $$ \bar{x}(\lambda) = \frac{1}{n}\sum_{i=1}^{n}{x_i(\lambda)} $$

is the arithmetic mean of the transformed data.

Confidence bound for \(\lambda\)

In addition, a confidence bound (based on the likelihood ratio statistic) can be constructed for \(\lambda\) as follows: a set of \(\lambda\) values that represent an approximate \(100(1-\alpha)\) % confidence bound for \(\lambda\) is formed from those \(\lambda\) that satisfy $$ f(x,\lambda) \ge f(x,\hat{\lambda}) - 0.5 \chi^2_{1-\alpha, \, 1} \, ,$$ where \(\hat{\lambda}\) denotes the maximum likelihood estimator for \(\lambda\) and \(\chi_{1-\alpha, \, 1}^2\) is the \(100(1-\alpha)\) percentile of the chi-square distribution with 1 degree of freedom.

Example of the Box-Cox scheme

To illustrate the procedure, we used the data from Johnson and Wichern's textbook (Prentice Hall 1988), Example 4.14. The observations are microwave radiation measurements.

Sample data

0.15	0.09	0.18	0.10	0.05	0.12	0.08
0.05	0.08	0.10	0.07	0.02	0.01	0.10
0.10	0.10	0.02	0.10	0.01	0.40	0.10
0.05	0.03	0.05	0.15	0.10	0.15	0.09
0.08	0.18	0.10	0.20	0.11	0.30	0.02
0.20	0.20	0.30	0.30	0.40	0.30	0.05

Table of log-likelihood values for various values of \(\lambda\)

The values of the log-likelihood function obtained by varying \(\lambda\) from -2.0 to 1.9 are given below.

\(\lambda\)	LLF	\(\lambda\)	LLF	\(\lambda\)	LLF
-2.0	7.1146	-0.6	89.0587	0.7	103.0322
-1.9	14.1877	-0.5	92.7855	0.8	101.3254
-1.8	21.1356	-0.4	96.0974	0.9	99.3403
-1.7	27.9468	-0.3	98.9722	1.0	97.1030
-1.6	34.6082	-0.2	101.3923	1.1	94.6372
-1.5	41.1054	-0.1	103.3457	1.2	91.9643
-1.4	47.4229	0.0	104.8276	1.3	89.1034
-1.3	53.5432	0.1	105.8406	1.4	86.0714
1.2	59.4474	0.2	106.3947	1.5	82.8832
-1.1	65.1147	0.3	106.5069	1.6	79.5521
-0.9	75.6471	0.4	106.1994	1.7	76.0896
-0.8	80.4625	0.5	105.4985	1.8	72.5061
-0.7	84.9421	0.6	104.4330	1.9	68.8106

This table shows that \(\lambda = 0.3\) maximizes the log-likelihood function (LLF). This becomes 0.28 if a second digit of accuracy is calculated.

The Box-Cox transform is also discussed in Chapter 1 under the Box Cox Linearity Plot and the Box Cox Normality Plot. The Box-Cox normality plot discussion provides a graphical method for choosing \(\lambda\) to transform a data set to normality. The criterion used to choose \(\lambda\) for the Box-Cox linearity plot is the value of \(\lambda\) that maximizes the correlation between the transformed \(x\)-values and the \(y\)-values when making a normal probability plot of the (transformed) data.