Retransforming Predictions from Log-Outcome Models

Why Retransformation Needs Care

A regression model for a log-transformed outcome can be useful when the outcome is positive and its relationship with predictors is more naturally described on a multiplicative scale. The fitted model produces a prediction for the mean of the log outcome. But the quantity often wanted is the arithmetic mean of the outcome on its original scale. Those are different targets: exponentiating a fitted mean on the log scale does not, in general, recover the arithmetic conditional mean.

The distinction matters in practical reporting. For example, a fitted value of \(1.6\) for log completion time can be exponentiated to obtain a prediction in minutes. That value has a precise interpretation, but it is not automatically the average completion time for all cases with those predictors. To predict that arithmetic average, we need information about the variation in the log-scale errors as well as their mean.

The Log-Outcome Model and Its Retransformation

Let \(Y\) be a positive outcome and let \(x\) denote a specified predictor profile. Suppose the fitted regression model is for the natural logarithm of the outcome:

Here, \(\eta(x)\) is the model's mean log outcome at profile \(x\), and \(\varepsilon\) is the log-scale error. The key condition for interpreting \(\eta(x)\) as the conditional mean of \(\log(Y)\) is \(E(\varepsilon\mid x)=0\). In an estimated model, write the fitted value as \(\hat{\eta}(x)\). The outcome must be positive for its logarithm to be defined.

The conditional geometric mean is defined as \(\exp\{E[\log(Y)\mid x]\}\). By contrast, the arithmetic conditional mean is \(E(Y\mid x)\). To connect them, exponentiate the model equation and take the conditional expectation:

Thus the retransformation factor is \(E\{\exp(\varepsilon)\mid x\}\). Knowing only that \(E(\varepsilon\mid x)=0\) does not determine this factor. It is the distribution of the log-scale error, not just its mean, that determines how to get from the conditional geometric mean to the arithmetic conditional mean.

When Log-Scale Errors Are Normal

A familiar correction is available under a stronger assumption: at each predictor profile, the log-scale error is normally distributed with mean zero and a common variance \(\sigma^2\). Equivalently, the conditional distribution of \(Y\) is lognormal under this model. For a normal random variable with mean zero and variance \(\sigma^2\), the mean of its exponential is \(\exp(\sigma^2/2)\). Therefore:

In practice, the fitted version replaces \(\eta(x)\) and \(\sigma^2\) with their estimates: \(\exp\{\hat{\eta}(x)+\hat{\sigma}^2/2\}\). The normal-error assumption is important: the expression \(\exp(\hat{\sigma}^2/2)\) is not a universal correction for every log-outcome regression. The common-variance assumption is also important; if the log-error variance changes with \(x\), the normal-theory factor changes across profiles.

Why Exponentiating the Fitted Value Falls Short

The difference has a general explanation. If the conditional error mean is zero, then \(E\{\exp(\varepsilon)\mid x\}\) is at least 1. This follows from Jensen's inequality: the exponential function is convex, so the expectation of the exponential is at least the exponential of the expectation.

It follows that, with mean-zero log-scale errors, the arithmetic conditional mean is at least as large as the conditional geometric mean. If the log-scale error varies at that profile, the inequality is strict. The calculation explains why simply exponentiating the fitted log mean will usually underestimate the arithmetic conditional mean, even though the regression has no systematic error on the log scale.

For normally distributed errors, the factor is determined by the variance. Without normality, a different factor can be estimated from exponentiated residuals. This approach is often called a smearing correction. Its central idea is to estimate the average multiplicative adjustment needed to return from the log scale to the original scale.

Choosing and Interpreting the Correction

A residual-based factor is most directly justified when the conditional distribution of the log-scale errors is reasonably similar across predictor profiles. In that setting, an average of exponentiated residuals can estimate a common factor. If the error distribution changes with predictors, the factor \(E\{\exp(\varepsilon)\mid x\}\) may also change. Applying one overall factor to every profile can then give inaccurate arithmetic-mean predictions. A model for the changing error distribution, or a profile-specific correction, may be needed.

The correction changes the predicted target: the uncorrected exponentiation estimates a conditional geometric mean, while the corrected value targets an arithmetic conditional mean. It does not change the original fitted log-scale equation. Nor does a correction guarantee that a model is appropriate: the log-scale relationship and error assumptions still need to be considered. The correction is a way to account for retransformation, not a substitute for checking whether the model describes the data adequately.

Common Mistakes

• Treating \(\exp\{\hat{\eta}(x)\}\) as the arithmetic mean automatically. It is the exponentiated fitted log mean. To estimate the arithmetic mean, account for \(E\{\exp(\varepsilon)\mid x\}\).
• Using the normal correction without a normal-error assumption. The factor \(\exp(\sigma^2/2)\) follows from normal log-scale errors. With other error distributions, estimate or model the appropriate factor instead.
• Assuming mean-zero errors imply no retransformation adjustment. Although \(E(\varepsilon\mid x)=0\), generally \(E\{\exp(\varepsilon)\mid x\}>1\) when the error varies.
• Calling the exponentiated fitted value a median without checking. It represents the conditional median only if the conditional median log-scale error is zero. A zero conditional error mean alone does not guarantee that.