# Choosing a transformation: Box-Cox in practice

    library(shewhartr)
    library(ggplot2)

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

\[ y(\lambda) = \begin{cases} (x^\lambda - 1)/\lambda & \lambda \neq 0 \\ \log(x) & \lambda = 0, \end{cases} \]

and a procedure for choosing \(\lambda\) 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 \(\hat \lambda\), 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:

    autoplot(bc)

If the CI for \(\lambda\) 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 \(\hat \lambda\):

- \(|\hat \lambda - 1| \le 0.1\) → `linear`
- \(|\hat \lambda - 0| \le 0.1\) → `log`
- \(|\hat \lambda - 0.5| \le 0.1\) → `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.