Box-Cox Transformation for Regression

Box-Cox transformation for regression chooses the power lambda for transforming a positive outcome by maximising the profile likelihood of the linear model, with a likelihood-ratio interval.

회귀의 Box-Cox 변환은 선형모형의 프로파일 우도를 최대로 하는 거듭제곱 λ를 골라 양수 결과변수를 변환하고, 우도비로 λ의 신뢰구간을 구하는 방법입니다.

My regression residuals are right-skewed; should I log-transform the outcome or use some other power?

회귀 잔차가 오른쪽으로 치우쳐 있는데, 결과변수를 로그 변환해야 할까요, 다른 변환을 써야 할까요?

The Box-Cox procedure fits the regression for a range of power transformations of the outcome and picks the lambda that maximises the profile log-likelihood. Its confidence interval shows which convenient transformations, such as the logarithm or square root, are consistent with the data.

When to use Box-Cox Transformation for Regression

• A positive continuous outcome has skewed or heteroskedastic residuals.
• A choice between the log, square-root or no transformation needs evidence.
• A variance-stabilising transformation is wanted before a linear model.

When to use something else

• The outcome has zero or negative values; use the Yeo-Johnson transformation or a generalized linear model.
• Effects must be interpreted on the original scale; use a gamma or log-link generalized linear model.

Assumptions and requirements

• The outcome is strictly positive.
• Some power transformation makes the errors approximately normal and homoskedastic.
• The linear predictor is correctly specified.

What the analysis reports

• Maximum-likelihood lambda with likelihood-ratio confidence interval
• Suggested convenient lambda inside the interval
• Likelihood-ratio tests of lambda = 1 and lambda = 0
• Regression refitted on the transformed outcome; residual normality before and after

How to report it

The Box-Cox profile likelihood gave lambda = [value], 95% CI [[value], [value]]; the model was refitted with lambda = [value] (chi2(1) = [value] for no transformation, p = [value]).

Related methods

• Multiple Linear Regression
• Gamma Regression
• Yeo-Johnson Transformation

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