backtransformed

Backtransformed refers to the process of applying an inverse transformation to results that were previously transformed for analysis, in order to report findings on the original data scale. In practice, analysts transform data to stabilize variance, meet model assumptions, or linearize relationships; common transforms include logarithmic, square-root, and Box-Cox transformations. After fitting a model or performing analysis on the transformed data, the results—such as predictions, means, or confidence intervals—are backtransformed to the original scale for interpretation.

A key consideration is that backtransformed point estimates can be biased because the transformation is nonlinear. For example, predicting on a log scale and backtransforming with the exponential does not yield the same expected value as the original scale unless bias correction is applied. Techniques include the use of a bias-corrected backtransformation or a smearing estimator, and resampling methods such as bootstrap to obtain intervals on the original scale.

Constructing confidence or prediction intervals on the original scale also requires care. Methods include the delta method, which approximates the distribution after backtransformation, or bootstrap-based percentile intervals.

In fields such as statistics and signal processing, backtransformation may also refer to applying the inverse transform to recover the original signal or data domain after analysis in an alternative representation.

See also: inverse transformation, Box-Cox transformation, log transformation, Jensen's inequality, bootstrap.