Choosing Box‑Cox vs Manual Log‑Plus‑Constant for Right‑Skewed Responses with Zeros in Minitab Regression

Answer to the Core Question

When zeros are in a right‑skewed response, Box‑Cox in Minitab cannot be used because the logarithm is undefined for zero. The manual log(x + c) approach remains available, but Minitab does not publish a rule for picking the constant c. The analyst must decide on a constant that makes all observations strictly positive while keeping the transformed distribution as close to normal as possible.

Likely Explanation (Based on Common Practice)

• Choose the smallest positive value in the data set as c, or a small fixed value such as 1.
• Test a handful of candidate constants, compute the skewness of the transformed variable, and pick the one that yields the lowest skewness.
• Once c is fixed, the transformation is deterministic; diagnostics (residual plots, normal probability plots, leverage) will be consistent across Minitab releases.
• Box‑Cox remains unavailable until zeros are removed or shifted; if you add a constant and then apply Box‑Cox, the λ estimate is stable across releases.

Confirmed Facts (Documented by Minitab)

• Box‑Cox requires strictly positive data; zeros prevent its use.
• Minitab does not provide a built‑in rule for selecting the constant in the log‑plus‑constant method.
• After a constant is chosen, the log‑plus‑constant transformation preserves diagnostic consistency across software versions.

Practical Steps for Your Case

1. Identify the smallest positive observation minPos in the response variable.
2. Generate a set of candidate constants, e.g., c = 0.1, 0.5, 1, minPos (or c = minPos if that is > 0).
3. For each c, compute y_c = log(x + c) in Minitab (use the Transform → Logarithm option and supply c in the “Add constant” field).
4. Plot the histogram and Q‑Q plot of each y_c and calculate the skewness statistic (Minitab’s Descriptive Statistics → Skewness).
5. Choose the c that gives the smallest absolute skewness and visually appears most symmetric.
6. Run the regression on the selected y_c and inspect residual diagnostics. Repeat the same steps in the next Minitab release to confirm consistency.

When to Consider Alternatives

• If zeros constitute a large portion of the data (e.g., > 25 %), a small constant may still leave residual skewness; consider a square‑root or Yeo‑Johnson transformation.
• For very small constants, numerical stability can be an issue; verify that the transformed values are not too close to the machine epsilon.

Missing Diagnostic Detail

Could you let me know the percentage of zero observations in your data set? If zeros are a substantial fraction, that might influence whether a simple log‑plus‑constant is adequate or whether a different transformation is preferable.