Residual Error Patterns Across Predicted Concentrations

Why Plot Residuals Against Predicted Concentration?

A residual-versus-prediction plot asks whether the discrepancies between observed and predicted concentrations change systematically across the concentration range. This matters because residual error models make different predictions about the spread of those discrepancies. A plot can reveal a pattern that is difficult to see in a single overall residual summary, but it is a diagnostic, not a model-selection rule: an apparent pattern can also reflect sampling variation, dependence, or problems with the predictions themselves.

For observation \(j\) from subject \(i\), let \(C_{ij}\) be the observed concentration and let \(\hat C_{ij}\) be the model-predicted concentration used for diagnosis. The prediction should be identified clearly; in a typical diagnostic it is an individual prediction conditional on the fitted subject effects. The raw residual is the observation minus that prediction. The residual is measured in the same concentration units as \(C_{ij}\).

To interpret the vertical spread, begin with a residual model whose conditional mean is zero and whose variance is specified as a function of predicted concentration. The following descriptions use \(\hat C_{ij}\) as the concentration scale at which the error variance is evaluated. They concern the residual distribution conditional on the model prediction and relevant subject effects, under the assumed residual model. They do not say that every observed residual must be near zero or that a finite plot will display the theoretical pattern perfectly.

The corresponding conditional residual standard deviations are \(\sigma_a\), \(\sigma_p\hat C_{ij}\), and \(\sqrt{\sigma_a^2+\sigma_p^2\hat C_{ij}^{\,2}}\), respectively, for nonnegative predicted concentrations. Thus constant error suggests roughly even vertical spread; proportional error suggests spread increasing roughly in proportion to prediction; and combined error includes a nonzero low-concentration component plus an increasing component. These are patterns to assess, not shapes that each individual point must follow.

Summarizing Spread Within Prediction Bins

A scatterplot can be dense, so it is often useful to divide predictions into ranges, or bins, and summarize the residuals in each range. Bins might be defined by prediction intervals or chosen to have similar numbers of observations. For bin \(b\) containing \(n_b\) residuals, calculate its mean and sample standard deviation:

The bin mean \(\bar e_b\) describes the average signed discrepancy in that range. A mean consistently above or below zero can indicate systematic overprediction or underprediction in that range. The sample standard deviation \(s_b\) describes the within-bin spread around that bin's own mean. It is a descriptive estimate, not an exact value of the model's residual standard deviation. It varies with finite sample size and can be unrepresentative when residuals are dependent or when predictions and residual properties vary substantially within a bin.

A constant shift in all residuals in a bin changes \(\bar e_b\), but does not change \(s_b\): subtracting the bin mean removes that shift. Shifts that vary among observations within the bin can contribute to the spread. For example, if different prediction levels within a wide bin have different conditional residual means, their deviations from the bin mean add a between-level component to the measured spread. This is one reason to avoid overly broad bins.

For independent residuals with conditional means \(\mu_k\) and variances \(v_k\), the expected sample variance within a bin is

This follows by writing each residual as its mean plus a zero-mean deviation. The expected sum of squared deviations from the sample mean separates into the sum of the individual variances, adjusted for estimating the mean, and the squared differences among the conditional means. If every conditional mean is shifted by the same constant, then \(\mu_k-\bar\mu=0\) for all \(k\), so that shift does not inflate the expected sample variance. If the means differ within the bin, the second term is positive. Dependence changes this calculation, so the displayed expression relies on independence.

Reading the Three Main Patterns

Under proportional error, dividing a residual by its prediction gives a scale-free diagnostic when predictions are positive: the standard deviation of \(e_{ij}/\hat C_{ij}\) is approximately \(\sigma_p\). Under combined error, that ratio is not constant; its variance is \(\sigma_a^2/\hat C_{ij}^{\,2}+\sigma_p^2\), so the additive component matters most at low predictions. Ratios should not be used near zero, where division by a small prediction can magnify noise.

Diagnostics, Alternatives, and Limits

Begin by plotting raw residuals against the predictions used to calculate them and include a horizontal zero reference. Look for both a trend in the center and a change in vertical spread. Add binned means to inspect signed bias and binned standard deviations or another clearly defined spread summary to inspect dispersion. State the bin boundaries or rule and the number of observations in each bin; a spread estimate from a tiny bin should not be given the same visual weight as one based on many observations.

A raw residual plot is not the only view. Standardized residuals divide raw residuals by a model-based residual standard deviation; if that variance model is appropriate, their spread should be more nearly stable across prediction. They can help show remaining structure, but a standardized plot can conceal misspecification if the same questionable variance model is used to scale the residuals. Residuals against time can reveal time-localized structure that a prediction plot does not show. These views answer different diagnostic questions, so none replaces the others.

The interpretation of bins relies on the conditional-mean and variance assumptions. A visible nonzero binned mean is not evidence for a particular variance function: variance describes spread, while the mean describes signed bias. Conversely, a widening cloud does not by itself identify proportional error. If observations from the same subject are dependent, treating all residuals as independent can make a pattern look more convincing than it is. Summaries should therefore be read alongside the sampling structure and other diagnostics, not as independent formal tests.

Predicted concentrations are estimated, not fixed known values. Their uncertainty, and the method used to obtain individual predictions, can affect the residual pattern. A wrong structural prediction can also leave systematic residual patterns; a variance model should not be used automatically to absorb every mismatch. Concentrations near an assay's quantification limit, censored observations, or a narrow prediction range can make the visual distinction among error structures especially difficult.

A common mistake is to see a residual cloud that gets wider at higher predictions and conclude that proportional error has been proved. The correction is to ask whether the standard deviation is approximately proportional to prediction, whether a nonzero additive component remains at low predictions, and whether the pattern is stable across sensible bin choices and diagnostic views. These plots help assess compatibility with a residual structure; they do not alone establish it or replace comparing plausible models with the full data and scientific context.