What Is an Individual Forecast?
A forecast is not fully specified by saying “predict this patient’s concentration.” It must identify which future outcome is being predicted, at what time or over what interval, and given what information. The same individual can have different forecasts for different horizons, dosing plans, or information sets.
In an NLME model, observations from individuals are described using population-level parameters and individual-level variation. For a particular individual, a forecast may condition on the specified population model, that individual’s covariates and dosing history, and their observations available at the forecast time. The resulting distribution describes uncertainty about a future outcome under those stated conditions; it is not simply a predicted value.
For clarity, write \(C_i\) for the information being conditioned on. A forecast distribution can then be expressed as \(P(T_i\in A\mid C_i)\), the conditional probability that the future target falls in a set \(A\). For example, \(T_i\) might be a concentration at a particular future time, while \(A\) might be the range below a safety threshold.
It matters whether a quantity is treated as known or remains uncertain. A covariate recorded for the individual may be held fixed; a future dose schedule may be specified as a scenario. Individual random effects, residual variation, or population parameters may instead be treated as uncertain, depending on the forecasting procedure. The conditioning statement and the model must make these choices clear.
Specify the Target Before the Calculation
A target can be a single future measurement, a summary over a time interval, or an event. These are not interchangeable. “Concentration at 24 hours,” “maximum concentration during the next dosing interval,” and “whether concentration ever exceeds a threshold” refer to different random quantities, even if they concern the same individual and regimen.
A useful notation is \(Y_i(t^*)\) for an outcome measured for individual \(i\) at future time \(t^*\). A target could instead be a function of several future outcomes, such as the maximum concentration during an interval. Its distribution depends on the chosen target as well as on the conditioning information.
Observed data and future outcomes also have distinct roles. The observed data are measurements available when the forecast is made; they belong in the conditioning information if the forecast is intended to use them. The future outcome is not yet observed and is the target whose distribution is being described. Confusing the two can make a forecast appear more informative than it is.
Worked Example: A Future Concentration as the Target
Hypothetical setup. At a forecast time, a model and an individual’s available data have been used to specify the conditional distribution of a future concentration \(C\). For this example, the output is stipulated to be \(\ln(C/(1\ \text{mg/L}))\mid C_i \sim N(1.3118,0.25^2)\). Here \(C_i\) denotes the stated conditioning information, including the model, covariates, dosing history, and observed data available at forecast time. The model output is given; the example does not derive how the individual data produced it.
Target and interpretation. The target is one future concentration, in mg/L, at the specified forecast time. The distribution is conditional on \(C_i\), not a description of every possible future measurement in the population.
The median concentration is obtained by exponentiating the mean of the normal log-concentration: \[ \operatorname{median}(C\mid C_i) = \exp(1.3118)\ \text{mg/L} \approx 3.713\ \text{mg/L}. \] The arithmetic mean differs because the target is lognormal. For \(Z\sim N(\mu,\sigma^2)\), \(E[\exp(Z)]=\exp(\mu+\sigma^2/2)\). Therefore, \[ E(C\mid C_i) = \exp(1.3118+0.25^2/2)\ \text{mg/L} = \exp(1.34305)\ \text{mg/L} \approx 3.831\ \text{mg/L}. \] These are two summaries of the same conditional distribution, not two different targets.
The assumptions used here are that the stipulated log-scale distribution is appropriate for this future outcome and that the concentration is positive, as required by the logarithm. Whether the distribution adequately represents the individual and the future time point requires model assessment; a numerical forecast alone does not establish that.
Condition on a Clear Dosing Scenario
A forecast involving treatment is tied to a dosing history or a stated future regimen. Past doses determine the modeled exposure leading up to the forecast time. A planned regimen specifies the scenario under which a future forecast is made; it does not guarantee that the individual will receive exactly those doses. If actual dosing later differs from the plan, the original forecast was conditional on a scenario that did not occur.
A deterministic model trajectory can be useful for checking what the dose history implies under fixed parameter values. But unless stochastic variation is included, that trajectory is not a full forecast distribution. This distinction prevents a precise-looking model calculation from being mistaken for a statement that accounts for all relevant uncertainty.
Worked Example: Two Doses in the Conditioning History
Hypothetical setup. Consider a one-compartment model with instantaneous doses, fixed volume \(V=10\) L, and fixed first-order elimination rate \(k=0.100\ \text{h}^{-1}\). The individual receives \(150\) mg at time 0 and \(150\) mg at time 12 h. The target is the structural concentration at time 24 h. Assume exact dose times and amounts, no additional doses, and no stochastic residual variation for this calculation.
Under these assumptions, the contribution of a dose \(D_j\) given at time \(t_j\) to the concentration at time \(t\) is \((D_j/V)\exp[-k(t-t_j)]\). The total concentration is the sum of the two dose contributions: \[ C(24)=\frac{150}{10}\exp[-0.100(24-0)] +\frac{150}{10}\exp[-0.100(24-12)]\ \text{mg/L}. \]
Substituting the elapsed times gives \[ C(24)=15\exp(-2.4)+15\exp(-1.2) \approx 1.361+4.518 =5.879\ \text{mg/L}. \] The time units cancel in each exponent because \(k\) is in \(\text{h}^{-1}\) and elapsed time is in hours. The result is conditional on this model, these fixed parameter values, and this dose history.
This is a structural trajectory prediction, not a full individual forecast distribution: the example holds the parameters fixed and includes no residual or other stochastic variation. If the target were a future measured concentration, a model for measurement or residual variation would also be needed. If the future dosing plan changed, the target’s distribution would refer to a different scenario.
Uncertainty and Conditioning Are Different Choices
Conditioning says what information is held fixed or supplied when defining a forecast. Uncertainty says which quantities can still vary under the model. For example, conditioning on a recorded body weight does not remove uncertainty in an individual random effect. Conversely, a calculation that fixes a random effect at one value excludes variation in that effect, even if the value was chosen using the individual’s observations.
One way to express the distinction is to let \(U_i\) collect quantities that remain uncertain, such as an individual effect or future residual variation. A conditional forecast for target \(T_i\) can be represented by averaging over the possible values of \(U_i\):
This follows from the law of total probability: partition the possibilities for \(T_i\in A\) by the value of \(U_i\), compute the conditional probability for each value, and average using the distribution of \(U_i\) under the stated information. In a fitted NLME analysis, exactly which quantities are included in \(U_i\), and how their conditional distribution is obtained, depend on the forecasting setup. The equation makes the bookkeeping explicit; it does not prescribe a particular estimation procedure.
Population parameters can also be treated differently in different forecasting analyses. A forecast conditional on fixed fitted parameter values does not include uncertainty in those estimates. A forecast that propagates uncertainty in population parameters does. Neither description is complete unless it states which choice was made. The same principle applies to uncertain future covariates or adherence: either specify them as fixed scenarios or model their uncertainty as part of the forecast.
Worked Example: Forecasting a Threshold Event
Hypothetical setup. Use the conditional lognormal concentration distribution from the first worked example. The target is now the event that the future concentration is below \(2.00\ \text{mg/L}\) at the same specified time, under the same conditioning information.
Standardizing the log concentration gives \[ P(C<2.00\ \text{mg/L}\mid C_i) = P\left( Z< \frac{\ln(2.00)-1.3118}{0.25} \right), \] where \(Z\sim N(0,1)\). The standardized value is \[ \frac{0.6931-1.3118}{0.25}\approx -2.475. \] Thus \(P(C<2.00\ \text{mg/L}\mid C_i)\approx 0.0067\), or about \(0.67\%\), under the stipulated model and conditioning information.
This probability is about a specified future event for an individual under the stated conditions. It is not the probability that the model is correct, nor does it apply automatically to a different time, threshold, dose plan, or information set. The calculation assumes the given lognormal distribution is appropriate and that the threshold and concentration use the same units.
Diagnostics, Alternatives, and Limitations
Before interpreting a forecast, check whether the target and conditioning information match the intended decision. A forecast can be internally calculated correctly and still answer the wrong question if, for example, it uses the wrong forecast time, omits a dose, or treats a future planned regimen as though it were observed fact.
- Check the time ordering. Conditioning observations should be available at the forecast time. Using a measurement collected afterward leaks future information into the forecast.
- Check the regimen. Confirm dose amounts, times, and the status of future doses as planned scenario inputs rather than known outcomes.
- Check the target’s units and definition. A concentration at one time is not an interval maximum or a time-above-threshold target.
- Check what variation is represented. A distribution that omits residual variability or uncertainty in an important input can be too narrow for the intended use.
- Check model compatibility. A model may be unreliable for a covariate range, dose history, or forecast horizon poorly represented by its assumptions or data.
A conditional individual forecast differs from a population-marginal prediction, which averages over the distribution of individuals rather than conditioning on one person’s specified information. It also differs from a deterministic trajectory, which gives a model output at fixed inputs and parameter values but need not represent a distribution of future outcomes. These alternatives can all be useful, but they answer different questions.
A common pitfall is to call any model-generated number an “individual forecast.” For instance, a concentration calculated from fixed parameter values and a dose schedule may be only a structural prediction. The correction is to state whether the result is a point on a deterministic trajectory or a probability distribution, then identify which uncertainties the distribution includes. Another pitfall is to treat observed individual data as merely descriptive when the intended forecast is supposed to condition on them. State explicitly whether those observations informed the forecast.
No conditioning statement can compensate for a misspecified model, unrecorded changes in dosing, or covariates that evolve unpredictably but are treated as fixed. Forecasts are conditional conclusions: they describe outcomes under a model and a specified information set, not guarantees about what will happen.
Check Your Understanding
For each question, identify the target or conditioning choice. Solutions follow each question.
- A model predicts a concentration at 12 h after a stated dose. What additional information is needed to call this a fully specified individual forecast?
Solution: Specify the individual and the conditioning set, including the population model, relevant covariates, dosing history, and observed data available at forecast time. State which quantities remain uncertain and whether the result is a distribution or only a fixed-input trajectory. - Why is a planned dose schedule not the same as an observed dosing history?
Solution: A planned schedule defines the scenario under which a forecast is made. It does not establish that the individual actually received those doses. If actual dosing differs, the forecast’s conditioning scenario differs from what occurred. - A concentration is measured at 8 h, and a forecast is made at 6 h. Can the 8 h measurement be part of the forecast’s conditioning information?
Solution: Not for a forecast genuinely made at 6 h. The 8 h measurement is future information relative to that forecast and including it would introduce information leakage. - What is the difference between a deterministic trajectory and an individual forecast distribution?
Solution: A deterministic trajectory evaluates a model at fixed inputs and parameter values. A forecast distribution describes probabilities for a future target and must state what sources of variation or uncertainty it includes. - What changes when the target is “concentration below a threshold” rather than “concentration at a time”?
Solution: The target changes from a continuous future measurement to an event defined by that measurement and a threshold. The event probability must be calculated under the same stated conditioning information, and it answers a different question from a concentration summary.