Skip to contents

Choosing a transformation: Box-Cox in practice

Source: vignettes/box-cox.Rmd
box-cox.Rmd

Box & Cox (1964) introduced a one-parameter family of power transformations,

y(λ)={(xλ−1)/λλ≠0log⁡(x)λ=0, y(\lambda) = \begin{cases} (x^\lambda - 1)/\lambda & \lambda \neq 0 \\ \log(x) & \lambda = 0, \end{cases}

and a procedure for choosing λ\lambda by maximum likelihood. The goal is to find a scale on which the residuals are approximately normal and homoscedastic — the assumptions that classical inferential tools, including Shewhart charts, presuppose.

shewhart_box_cox() returns the profile log-likelihood, the maximiser λ̂\hat \lambda, and a 95% confidence interval based on the chi-square approximation to twice the log-likelihood drop.

A textbook example 

set.seed(2025)
y <- rlnorm(200, meanlog = 0, sdlog = 0.5)   # log-normal -> lambda = 0
bc <- shewhart_box_cox(y)
bc
#> 
#> ── Box-Cox profile likelihood ──────────────────────────────────────────────────
#> • n = 200
#> • lambda_hat = 0
#> • 95% CI: [-0.25, 0.2]

The optimal lambda is near zero (log transformation), and the 95% CI should cover zero. Let’s plot the profile:

If the CI for λ\lambda contains 1, no transformation is needed (the data are approximately normal as is). If it contains 0, take logs. If it contains 0.5, take square roots — and so on.

Interaction with shewhart_regression(model = "auto")

The "auto" model in shewhart_regression() calls shewhart_box_cox() internally on the response (with a +1 shift to keep zeros valid) and selects among linear, log, loglog according to the value of λ̂\hat \lambda:

  • |λ̂−1|≤0.1|\hat \lambda - 1| \le 0.1 → linear
  • |λ̂−0|≤0.1|\hat \lambda - 0| \le 0.1 → log
  • |λ̂−0.5|≤0.1|\hat \lambda - 0.5| \le 0.1 → loglog
  • otherwise default to linear with a warning

This is a guidance step, not a guarantee. Always inspect the residual diagnostics afterwards via shewhart_diagnostics().

When not to transform

If your data are counts, proportions, times-to-event, or other quantities with a known parametric family, model that family explicitly. Box was clear about this: if you can model the right distribution, do so. Transforms exist for the case where the right distribution isn’t tractable and a normal approximation on a suitably-chosen scale is the best available compromise.

The c, u, p, and np charts in this package implement that advice: they support limits = "poisson" (or "binomial") for exact distribution-aware limits, instead of relying on a transformation to coerce counts into approximate normality.

References

  • Box, G. E. P., & Cox, D. R. (1964). An Analysis of Transformations. Journal of the Royal Statistical Society B, 26(2), 211-252.
  • Atkinson, A. C. (1985). Plots, Transformations and Regression. Oxford.
  • Box, G. E. P., Hunter, W. G., & Hunter, J. S. (2005). Statistics for Experimenters (2nd ed.). Wiley.
  • Sakia, R. M. (1992). The Box-Cox Transformation Technique: A Review. The Statistician, 41(2), 169-178.