The Box-Cox transformation estimates a lambda value, as shown in the following table, which minimizes the standard deviation of a standardized transformed variable. The resulting transformation is Yλ when λ ҂ 0 and ln Y when λ = 0.
The Box-Cox method searches through many types of transformations. The following table shows some common transformations where Y' is the transform of the data Y.
Lambda (λ) value
Algorithm for Johnson transformation
The Johnson transformation optimally selects one of three families of distribution to transform the data to follow a normal distribution.
γ + η ln [(x – ε) / (λ + ε – x)]
η, λ > 0, –∞ < γ < ∞ , –∞ < ε < ∞, ε < x < ε + λ
γ + η ln (x – ε)
η > 0, –∞ < γ < ∞, –∞ < ε < ∞, ε < x
γ + η Sinh–1 [(x – ε) / λ] , where
Sinh–1(x) = ln [x + sqrt (1 + x2)]
η, λ > 0, –∞ < γ < ∞, –∞ < ε < ∞, –∞ < x < ∞
The algorithm uses the following procedure:
Considers almost all potential transformation functions from the Johnson system.
Estimates the parameters in the function using the method described in Chou, et al.1
Transforms the data using the transformation function.
Calculates Anderson-Darling statistics and the corresponding p-value for the transformed data.
Selects the transformation function that has the largest p-value that is greater than the p-value criterion (default is 0.10) that you specify in the Transform dialog box. Otherwise, no transformation is appropriate.
The Johnson family distribution with the variable bounded (B)
The Johnson family distribution with the variable lognormal (L)
The Johnson family distribution with the variable unbounded (U)
For more information on the Johnson transformation, see Chou, et al.1 Minitab replaces the Shapiro-Wilks normality test used in that text with the Anderson-Darling test.