Residual Error On Original And Log Concentration Scales

Why the Error Scale Matters

A residual error model describes how an observed concentration varies around a model-predicted concentration. The scale on which the error is defined changes the implied distribution of observations, not just the numerical size of a residual. Additive error treats a given absolute deviation in concentration the same way at low and high predictions. Log-scale error treats a given proportional deviation the same way, so it naturally represents constant relative variability.

Let \(f\) denote the positive concentration predicted for a particular observation, conditional on the structural model and its individual parameter values. Let \(C\) denote the corresponding observed concentration. The residual models below describe the conditional distribution of \(C\) given \(f\); they do not by themselves describe variation in \(f\) caused by differences between individuals.

Additive Error on the Original Concentration Scale

The usual additive model assumes that, conditional on \(f\), the residuals are independent and normally distributed with mean zero and a common variance:

Here \(\sigma_{\mathrm{add}}\) has concentration units, such as mg/L. The zero-mean assumption makes \(f\) the conditional mean of \(C\), and the constant variance assumption gives the same conditional standard deviation at every predicted concentration:

Thus, additive error represents constant absolute variability. Its conditional coefficient of variation, the conditional standard deviation divided by the conditional mean, is \(\sigma_{\mathrm{add}}/f\). It is not constant: with a fixed \(\sigma_{\mathrm{add}}\), relative variability increases as \(f\) decreases.

A normal distribution also assigns positive probability to negative concentrations, even though physical concentrations are nonnegative. This may be negligible when \(f\) is many residual standard deviations above zero, but it can matter when \(f\) is small relative to \(\sigma_{\mathrm{add}}\). The probability is

where \(\Phi\) is the standard normal cumulative distribution function. A substantial value is a warning that an untruncated additive normal model may not represent the observations well near zero.

Normal Error on the Log Concentration Scale

The log-scale model assumes that the log ratio of observed to predicted concentration is normally distributed with mean zero and a common variance:

The ratio \(C/f\) is dimensionless, and \(\sigma_{\log}\) is dimensionless. Equivalently, \(C=f\exp(\epsilon_{\log})\). Because the exponential of any real number is positive, this model assigns probability zero to nonpositive concentrations. It makes multiplicative, rather than additive, deviations symmetric on the modeled scale: residuals of \(+a\) and \(-a\) correspond to concentration ratios \(\exp(a)\) and \(\exp(-a)\).

If \(f\) is expressed in a fixed concentration unit, the distribution of \(C\) is lognormal. To derive its density, set \(u=\log(c/f)\), so that \(c=f\exp(u)\) and \(du/dc=1/c\). Transforming the normal density of \(u\) gives

The factor \(1/c\) is the change-of-variable Jacobian. This density is generally asymmetric on the original concentration scale. In particular, \(f\) is the conditional median, not the conditional arithmetic mean. The moments follow from the normal moment-generating function: if \(U\sim N(0,\sigma_{\log}^2)\), then \(E[\exp(tU)]=\exp(t^2\sigma_{\log}^2/2)\). Taking \(t=1\) and \(t=2\) yields

Consequently, the conditional coefficient of variation is

It does not depend on \(f\). This is the precise sense in which a zero-mean normal error on the log scale gives constant relative variability on the concentration scale. The conditional arithmetic mean exceeds \(f\) because exponentiation is nonlinear; this is sometimes called retransformation bias.

What “Constant Variability” Means

The two models imply different patterns as the prediction changes. Additive error holds the absolute standard deviation fixed, whereas log-scale error holds the coefficient of variation fixed. These are different scientific assumptions, not competing ways to label the same residual.

Choosing and Checking the Scale

The model assumptions should be assessed against the pattern of observations and predictions, not selected solely because one scale is familiar. Under additive error, residuals \(C-f\) should have roughly zero mean, a stable spread in concentration units, and a shape reasonably consistent with a normal distribution. Under log-scale error, residuals \(\log(C/f)\) should have roughly zero mean, a stable spread on the log scale, and a shape reasonably consistent with normality. These are conditional checks: predictions should account for the systematic concentration-time pattern before residual spread is interpreted.

A useful diagnostic is to examine residuals against predicted concentration. A fan-shaped increase in the spread of original-scale residuals can contradict constant additive variance. If the spread of log-ratio residuals is more stable across predictions, that pattern supports considering log-scale error. Conversely, roughly constant absolute residual spread may support an additive model, while log residuals may show changing spread or strong asymmetry. Neither visual pattern proves a model true; sparse observations, measurement limits, or misspecified predictions can obscure the pattern.

The log-scale model requires positive observed concentrations because \(\log(C/f)\) is undefined for \(C\leq0\). A recorded zero or a value below an assay quantification limit cannot simply be logged as though it were an ordinary positive observation. How to model such data requires an explicit treatment appropriate to the observation process. The additive normal model avoids that logarithm but has the negative-tail issue described above. Neither model should be used without considering its support and assumptions.

A combined additive and proportional error model is another common alternative when neither constant absolute nor constant relative variability is adequate. A power variance function provides a different way to let residual variance change with prediction. These alternatives do not remove the need to specify what scale is modeled and to check the resulting variance pattern. In particular, a model with a changing variance is not automatically evidence for a log-scale residual distribution; distributional shape and the mean–median distinction still matter.

These residual models are also limited by their conditional assumptions. They do not establish that the structural predictions are correct, and an apparent residual pattern can reflect structural misspecification as well as an unsuitable error scale. They assume the specified conditional distribution and variance behavior apply across the observations being modeled. They do not, by themselves, represent serial dependence, individual-specific residual variability, or distinct sources of error.

Common Mistakes

• Calling log-scale error “constant standard deviation in concentration.” Its concentration-scale standard deviation increases with \(f\); the coefficient of variation is constant.
• Treating \(f\) as the arithmetic mean under zero-mean log error. In \(C=f\exp(\epsilon_{\log})\), \(f\) is the median, and the mean is \(f\exp(\sigma_{\log}^2/2)\).
• Ignoring units. \(\sigma_{\mathrm{add}}\) is measured in concentration units, while \(\sigma_{\log}\) describes variation in a log ratio and is dimensionless.
• Assuming a normal additive model cannot produce impossible predictions. It assigns positive probability below zero; calculate or assess that probability when predictions are low.
• Taking logarithms of zero or negative observations. The log-scale model requires positive concentrations and needs a deliberate approach to nonpositive or censored data.
• Choosing a scale from a single residual or the largest likelihood alone. Check the assumptions and residual patterns over the relevant prediction range, and compare models on an appropriate basis.