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August 31, 2006
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JWBK119-09
Surrogate PCIs for Nonnormal Data 111
9.2.5 Box--Cox power transformation
15
Box and Cox proposed a useful family of power transformations on the necessarily
positive response variable X given by
⎧ λ
⎨ X − 1
X (λ) = λ , for λ = 0, (9.3)
1n X, for λ = 0.
⎩
This continuous family depends on a single parameter λ which can be estimated by
the method of maximum likelihood as follows.
First, a value of λ from a selected range is chosen. For the chosen λ we evaluate

2
1
L max =− ln ˆσ + ln J (λ, X)
2
n

2
1
=− ln ˆσ + (λ − 1) ln X i ,
2
i=1
where
n n
∂W i λ−1

J (λ, X) = = X , for all λ,
i
∂ X i
i=1 i=1
n 2 2
so that ln J (λ, X) = (λ − 1) ln X i . The estimate of ˆσ for fixed λ is ˆσ = S (λ) /n,

i=1
where S(λ) is the residual sum of squares in the analysis of variance of X(λ). After
calculating L max (λ) for several values of λ within the range, L max (λ) can be plotted
against λ. The maximum likelihood estimator of λ is obtained from the value of λ that
maximizes L max (λ). With the optimal λ * value, each of the X data specification limits
is transformed into a normal variate using equation (9.3). The corresponding PCIs
are calculated from the mean and standard deviation of the transformed data using
equations (9.1) and (9.2).
9.2.6 Johnson transformation
Johnson 16 developed a system of distributions based on the method of moments,
similar to the Pearson system. The general form of the transformation is given
by
z = γ + ητ (x; ε, λ) , η > 0, −∞ <γ < ∞,λ > 0, −∞ <ε < ∞, (9.4)

where z is a standard normal variate and x is the variable to be fitted by a Johnson
distribution. The four parameters, γ , η, ε, and λ are to be estimated, and τ is an
arbitrary function which may take one of the following three forms.

9.2.6.1 The lognormal system (S L )
x − ε

τ 1 (x; ε, λ) = log , x ≥ ε (9.5)
λ