Residuals for Log-Transformed and Bounded Responses

Why the Response Scale Changes a Residual

A PD residual is a discrepancy between an observed response and a model prediction, but its meaning depends on the scale on which the model describes that response. A difference of 5 response units may be informative for one outcome, while a proportional difference is more natural for another. This distinction matters especially when responses are modeled after a logarithmic transformation or are bounded, as with percentages.

For clarity, let \(y_i\) be the observed response for observation \(i\), and let \(\hat y_i\) be its fitted value on the original response scale. Define the raw residual as observed minus predicted. If a model is fitted to \(\log(y_i)\), it also has a residual on the log scale. These are not interchangeable: one is an additive discrepancy in original units, and the other represents a multiplicative discrepancy after exponentiation.

The logarithmic scale is useful when relative rather than absolute differences are central, or when positive responses vary proportionally. A log residual of 0.10 does not mean the response is 0.10 units above prediction. It means the observed-to-predicted ratio on the modeled log scale is \(\exp(0.10)\), about 1.105. Thus, the observed response is about 10.5% above the prediction represented by \(\exp(\hat\eta_i)\).

Log-Scale Residuals and Original-Scale Predictions

Suppose a positive PD response follows a lognormal observation model conditional on the model predictors:

Here \(\eta_i\) is the predicted mean of the log response, and the model assumes errors are centered at zero on the log scale with constant variance \(\sigma^2\), conditional on the predictors. The distributional assumption also implies positive responses and a lognormal response distribution on the original scale. Residual diagnostics should therefore ask whether discrepancies are reasonably described on the log scale, not only whether original-scale differences look small.

Exponentiating \(\eta_i\) gives the conditional median of \(Y_i\), not its conditional arithmetic mean. Because exponentiation is nonlinear, the positive and negative log-scale deviations do not cancel symmetrically on the original scale. Under the stated normal-error model, the arithmetic mean includes a variance correction.

The correction follows directly from the normal moment-generating function. Write \(Y_i=\exp(\eta_i)\exp(\epsilon_i)\). For a normal random variable with mean zero and variance \(\sigma^2\), \(E[\exp(\epsilon_i)]=\exp(\sigma^2/2)\). Therefore:

This result explains a common source of apparent bias. If the scientific target is the arithmetic mean response but predictions are calculated only as \(\exp(\hat\eta_i)\), those predictions are systematically below the model-implied mean when residual variance is positive. Conversely, if the target is the typical response or conditional median, \(\exp(\hat\eta_i)\) is the relevant prediction. The prediction target must be specified before judging bias.

Residuals for Percentages and Other Bounded Responses

A percentage response is bounded. If recorded on a 0-to-100 scale, it can first be expressed as a proportion \(p=y/100\), which lies from 0 to 1. A raw residual in percentage units is easy to communicate: an observed response of 72% and a prediction of 65% differ by 7 percentage points. This does not mean the response is 7% higher; relative to 65%, the difference is about 10.8%.

Raw percentage-point residuals retain an intuitive interpretation but do not by themselves account for the response distribution or its variance. For example, a model may predict a proportion near zero or one, where the response has less room to vary in one direction than near the middle of the range. If a model uses a link function, residuals may also be calculated on the link scale. For the logit link, \(\operatorname{logit}(p)=\log(p/(1-p))\); a link-scale discrepancy is a difference in log odds, not percentage points.

The appropriate probability model depends on how the percentage was generated. If it is a count of successes out of a known number of trials, a binomial model can use both the proportion and its denominator. If it is a continuous proportion, another bounded-response model may be appropriate. A beta model, for example, is commonly defined for values strictly between zero and one; exact zeros and ones require a model that can represent those boundary values or a justified alternative treatment. The residual definition and its expected variability should match the chosen model.

Choosing and Checking the Residual Scale

A residual plot is informative only when its scale is understood. For a log-transformed response, a plot of log residuals against fitted log responses assesses whether the model’s conditional log-scale structure is plausible. Persistent positive residuals at high fitted values, for example, may indicate systematic underprediction on that scale. A plot of raw residuals against fitted original-scale predictions instead emphasizes absolute errors; a similar-looking pattern can have a different interpretation.

For bounded responses, inspect whether residuals are constrained or asymmetric near the boundaries, and whether their spread changes with the predicted proportion. A raw residual near a predicted value of 0.95 cannot extend upward as far as downward, while at 0.50 there is more room in either direction. A funnel or changing spread in raw residuals may therefore reflect the bounded scale, a misspecified variance model, or both. It is not automatically evidence that the mean model is wrong.

A link-scale residual can reduce some scale-related asymmetry, but it does not make every model assumption true. In particular, the difference between observed and fitted logits is not automatically a standardized residual: its expected variance depends on the response model and, for binomial data, on the trial count. For discrete or strongly non-Gaussian responses, simple residual plots may not have the familiar interpretation of residuals from a continuous normal-error model. Diagnostics should be interpreted with the assumed likelihood and sampling structure in view.

Common Mistakes and Limitations

• Calling a log residual an additive response error. A log residual of 0.20 is in log units. It corresponds to a multiplicative ratio of \(\exp(0.20)\), about 1.221, rather than an increase of 0.20 original units.
• Exponentiating a fitted log mean and calling it the arithmetic mean. Under normal log errors, \(\exp(\eta)\) is the median. The arithmetic mean is \(\exp(\eta+\sigma^2/2)\).
• Calling a percentage-point difference a percent difference. A change from 65% to 72% is 7 percentage points, or about 10.8% relative to 65%; these phrases describe different quantities.
• Taking the logit of a boundary value without a model for it. The logit is undefined at 0 and 1. Adding an arbitrary small constant does not resolve the modeling issue and can change residuals substantially.
• Assuming a transformed scale fixes misspecification. A transformation changes residual interpretation and may better represent variance, but does not guarantee a correct mean model, independence structure, or error distribution.

Compared with raw residuals, transformed-scale residuals may better express proportional error or align with the fitted likelihood, but they are less directly interpretable in clinical response units. Back-transformed predictions restore those units, yet their meaning depends on whether the target is a median, mean, or another summary. For bounded responses, percentage-point residuals are clinically transparent, whereas link-scale residuals respect the model’s predictor scale but are less intuitive. Neither scale is universally preferable; report the scale and prediction target explicitly.

These diagnostics also have limits. A few large residuals do not by themselves identify the cause of a problem, and a visually pattern-free plot cannot prove that the model is correct. Log transformation excludes nonpositive observations unless the scientific measurement model provides a defensible alternative. Bounded outcomes require attention to how endpoints arise and to whether the model’s variance and support match the data. Residual interpretation is one component of model assessment, not a substitute for understanding the observation process.