Choosing Point Forecasts: Mean, Median, and Mode

One Forecast Distribution, Several Point Forecasts

After updating an individual’s parameter distribution with observed concentrations, a forecast for a future concentration is still a distribution, not a single number. Numerical integration can represent that posterior predictive distribution while incorporating uncertainty about the individual’s parameters and future observation. A point forecast compresses that distribution to one value. The appropriate value depends on what kinds of errors matter for the decision.

The mean, median, and mode are not competing estimates of one uniquely correct answer. Each summarizes a different feature of the predictive distribution, and each has a decision-theoretic interpretation under particular conditions. For a symmetric distribution they may coincide; for a skewed concentration forecast they can be well separated.

The distribution used for these summaries must be the posterior predictive distribution for the future concentration, rather than merely the parameter posterior or a concentration curve evaluated at one parameter estimate. In particular, future observation variability should be included if the forecast concerns an observed future concentration. The conditioning information and future sampling occasion must be held fixed while comparing candidate point forecasts.

Squared-Error Loss Selects the Predictive Mean

Suppose errors are penalized quadratically, so an error twice as large incurs four times the loss. For a candidate forecast \(a\), define squared-error loss as \(L(Y,a)=(Y-a)^2\). Assume the posterior predictive distribution has a finite second moment, so the expected loss exists. Write its mean as \(m=E[Y\mid\text{data}]\). Expanding around \(m\) gives the justification:

The cross-term is zero because \(E[Y-m\mid\text{data}]=0\). The variance does not depend on \(a\), and \((m-a)^2\) is minimized at \(a=m\). Thus the posterior predictive mean uniquely minimizes expected squared-error loss, provided the variance is finite. This is a property of the full predictive distribution, not a claim that the mean is the most typical or most likely concentration.

Absolute-Error Loss Selects a Predictive Median

If the cost of an error increases in proportion to its magnitude, use absolute-error loss \(L(Y,a)=|Y-a|\). Assume the posterior predictive distribution has a finite first absolute moment. A median \(a\) is a value with at least half the predictive probability at or below it and at least half at or above it. Any such median minimizes expected absolute-error loss.

For a continuous predictive distribution with cumulative distribution function \(F\), the expected absolute loss as a function of \(a\) has derivative \(2F(a)-1\), wherever that derivative exists. When \(F(a)<0.5\), increasing \(a\) decreases expected loss; when \(F(a)>0.5\), increasing \(a\) increases it. A point where \(F(a)=0.5\) is therefore a minimizer. With discrete probability or a flat portion of the distribution, there can be multiple medians; the general condition is that each side of the chosen value has probability at least one half.

Skewed Concentrations Separate the Summaries

Concentrations are positive, and uncertainty in pharmacokinetic parameters can yield a right-skewed predictive distribution. A lognormal distribution is a useful hypothetical illustration of how its summaries differ. If \(\log Y\) is normally distributed with mean \(\mu\) and variance \(\sigma^2\), then \(Y\) has median \(\exp(\mu)\), mean \(\exp(\mu+\sigma^2/2)\), and mode \(\exp(\mu-\sigma^2)\). These are summaries on the original concentration scale.

What “Mode” Means for a Forecast

For a discrete predictive distribution, a mode is a value with the largest predictive probability. For a continuous distribution, the density mode is a value where the predictive density attains its maximum. A density mode describes where the distribution is most concentrated locally; it is not generally the value with the smallest expected squared or absolute error.

There is an important distinction from ordinary zero-one loss. For a continuously distributed \(Y\), the probability of \(Y\) equaling any particular point \(a\) is zero. Consequently, the loss \(L(Y,a)=1\) if \(Y\ne a\) and \(0\) otherwise has expected loss one for every point forecast. It cannot select a mode. A continuous density mode is instead a modal decision under a density-based criterion, or a limiting local criterion: for a small fixed tolerance \(\epsilon>0\), select a forecast maximizing \(P(|Y-a|<\epsilon\mid\text{data})\). For a smooth density, this probability is approximately \(2\epsilon p(a\mid\text{data})\) for small \(\epsilon\), so maximizing it approaches maximizing the density. This interpretation depends on the tolerance being defined on the concentration scale of interest.

Choosing and Checking a Point Forecast

A defensible choice starts by specifying what a forecast error means for the decision. Use the predictive mean when large errors should be penalized disproportionately, a median when errors are judged by their absolute magnitude, and a mode when the most concentrated or most probable local outcome is the target. In a clinical decision, none of these losses may represent the actual consequences of under- and over-prediction; a decision-specific loss may then be more appropriate.

When the predictive distribution is computed numerically, check that each summary is calculated from the same normalized distribution and on the intended scale. Integration or simulation can represent the predictive mean and cumulative probabilities; locating a mode also requires estimating or evaluating the predictive density. A coarse grid can miss a narrow peak, and smoothing choices can shift a numerical mode. If there are multiple peaks, report that the mode is not unique or state the criterion used to select among them.

A mode depends on the variable’s scale: transforming concentration to log concentration changes the density through a Jacobian, so the mode on the log scale need not transform to the mode on the concentration scale. Means and medians also transform differently: the median of a strictly increasing transformation is the transformed median, but the mean generally is not. Always identify the scale on which the forecast and loss are defined.

• Tail sensitivity: a few large predictive values can raise the mean substantially. Check whether the predictive tail is adequately represented and whether squared error is clinically meaningful.
• Multimodality: a single mean may lie in a low-density region between peaks. In that case it remains the squared-error action, but it may be a poor description of a typical outcome.
• Numerical instability: compare integration accuracy or simulation precision for the mean and median, and assess the density grid or smoothing method for the mode.
• Scale mismatch: do not report a mode found on log concentration as though it were the concentration-scale mode.

A common mistake is to call the mode “the most likely future concentration” and therefore choose it for every forecast. For a continuous outcome, an exact point has probability zero, and the density peak only has a direct local-probability interpretation under a specified scale and small tolerance. The correction is to name the predictive distribution, state the loss or modal criterion, and select the summary that matches it.