Why Simulate Error and Missing Observations?
A design simulation needs more than a model for the underlying concentration-time profile. Measured concentrations vary around that profile, and some scheduled samples may not be collected or may become unavailable after a participant leaves the study. Ignoring these processes can make a design look more informative, more feasible, or less biased than it would be under plausible study conditions.
The earlier material on Building a Data-Generating Model and Simulating Covariates and Random Effects supplies the subject-level predictions to which residual error is added. Here, the focus is on the additional variability of observations and on whether those observations are recorded. The distinction matters: residual error changes the value of an observation, whereas missingness determines whether the observation is available at all.
Residual-Error Models
Residual error represents unexplained observation-level variation after accounting for the model prediction and subject-level effects. A basic simulation commonly assumes errors are conditionally independent across samples, have mean zero on the modeled scale, and follow the specified distribution given the predictions and simulated subject characteristics. These assumptions are choices, not universal facts: repeated measurements may have correlated errors, and the spread may change with concentration.
With additive error, the error has a constant standard deviation in the units of the measured quantity. With proportional error, its standard deviation scales with the prediction. A common combined model adds the variances of the two components. The combined form below uses a normal distribution on the original measurement scale; it is not the only possible proportional-error parameterization.
Here \(\sigma_{\mathrm{add}}\) has the same units as \(Y_{ij}\), while \(\sigma_{\mathrm{prop}}\) is dimensionless. For positive predictions, proportional error can equivalently be simulated as \(Y_{ij}=f_{ij}(1+\sigma_{\mathrm{prop}}Z_{ij})\), where \(Z_{ij}\sim N(0,1)\). This normal-on-the-original-scale model can generate negative values, especially when predictions are small or proportional variability is large. If negative concentrations are impossible under the intended measurement process, a lognormal model or another positive-valued error distribution may be more appropriate; its mean and variance interpretation must be specified explicitly.
Worked Example: Generating a Combined-Error Observation
Suppose a hypothetical model predicts \(10.0\) mg/L at a sample time. The assumed additive standard deviation is \(0.50\) mg/L and the proportional standard deviation is \(0.20\). Assume independent normal components and use simulated standard-normal draws \(Z_{\mathrm{add}}=0.60\) and \(Z_{\mathrm{prop}}=0.80\).
The combined residual variance is
Thus the residual standard deviation is \(\sqrt{4.25}=2.0616\) mg/L. Using the independent-component representation gives
The variance calculation and the generated value serve different purposes: the variance describes the assumed distribution across repeated simulations, while 11.90 mg/L is one possible realization. At a prediction of \(0.50\) mg/L, the same parameters give a residual standard deviation of \(\sqrt{0.50^2+(0.20\times0.50)^2}=0.5099\) mg/L, illustrating how the additive component dominates at low predictions.
The combined variance follows directly from the independent-component construction. If \(\epsilon_{\mathrm{add}}\) and \(\epsilon_{\mathrm{prop}}\) are independent, mean-zero errors with variances \(\sigma_{\mathrm{add}}^2\) and \(\sigma_{\mathrm{prop}}^2f_{ij}^2\), then
The covariance term is zero because the components are independent. If they are correlated, this variance expression needs an additional covariance term. Simulating one normal residual with the combined variance produces the same marginal normal distribution as the sum of independent normal components, though it does not preserve separate component draws for diagnostics.
Simulating Missed Samples and Dropout
A missed sample is an unavailable observation at a particular scheduled time; a later scheduled sample may still be collected. Dropout is a subject-level event after which subsequent observations are generally unavailable. These processes should not be conflated: intermittent missed samples can leave a participant with later observations, while monotone dropout produces a run of missing samples after the dropout time.
For intermittent missingness, one can generate \(R_{ij}\) as a Bernoulli variable with an observation probability \(q_{ij}\). For dropout, define a dropout hazard at each follow-up among subjects who have not already dropped out. For instance, a hazard can depend on baseline covariates or previously observed outcomes. The simulation must state which history is available to the mechanism and whether dropout is independent of the current unobserved outcome. A mechanism depending on that unobserved outcome is MNAR and requires generating the outcome and missingness jointly according to an explicit model.
A convenient implementation order is to generate the model prediction, generate the residual realization, and then apply the observation mechanism. For an observation-independent mechanism, applying the missingness indicator does not change the distribution of generated values among scheduled samples; it only masks some of them. If missingness depends on current outcome, however, the order cannot be treated as an innocuous masking step: the dependence itself must be represented.
Worked Example: Expected Counts Under Intermittent Missed Samples
Consider a hypothetical study with 200 participants: 120 in subgroup A and 80 in subgroup B. Each participant has two follow-up samples in addition to a baseline sample, and baseline is assumed observed for everyone. At each follow-up, suppose the probability of observation is \(q_A=1/(1+\exp(-3))\) in subgroup A and \(q_B=1/(1+\exp(-2.2))\) in subgroup B. Assume follow-up attendance is independently generated at each visit conditional on subgroup, so missingness can be intermittent rather than monotone dropout.
The probabilities are \(q_A\approx0.952574\) and \(q_B\approx0.900250\). The expected observed count at one follow-up is
Using the unrounded probabilities, the expected total number of recorded samples across baseline and both follow-ups is
These are expectations under the stated mechanism; a simulated replicate will have integer counts that vary around them. An implementation should draw the observation indicators for each participant and visit rather than assigning fractional observations to a replicate. The mechanism here is independent of outcomes; it would not represent outcome-related dropout.
How Error and Missingness Change Design Performance
Residual variability generally makes measurements less informative about model parameters than noiseless predictions. Missing samples reduce the available information, and their effect depends on when samples are missed: losing a sample near a time that is especially informative about a parameter can matter more than losing the same number of less informative samples. Bias can also arise if the missingness mechanism changes which outcomes remain represented and the analysis does not account for that mechanism.
The following calculation isolates a simple precision effect. It assumes the observed values at one visit are independent with common variance and that missingness is independent of those values. Those assumptions make the comparison transparent, but it is not a general formula for precision of a nonlinear pharmacometric parameter estimate.
Worked Example: Precision at a Visit With Monotone Dropout
Suppose 100 hypothetical participants are scheduled for a late visit. The concentration at that visit has residual standard deviation \(2.0\) mg/L, and dropout before the visit is independent of the visit concentration, with probability \(0.28\). For a simple estimate of the population mean at that visit, compare the nominal sample size with the expected observed sample size.
With all 100 observations, the standard error under the independent equal-variance assumption is \(2.0/\sqrt{100}=0.2000\) mg/L. The expected number observed after dropout is \(100(1-0.28)=72\). Using 72 as a planning approximation, the standard error is
This approximation is about \(0.2357/0.2000-1=0.1785\), or 17.85%, larger than the full-sample standard error. The realized observed count in a replicate is random, so its actual standard error depends on that count. In a pharmacometric design, the effect on a target parameter also depends on the sampling times, model sensitivities, residual-error structure, and fitting method; this mean-estimation calculation is an illustration, not a substitute for simulating the specified analysis.
A useful comparison is between alternatives that isolate different sources of uncertainty. Additive error gives constant absolute spread; proportional error gives spread that grows with the prediction; combined error allows both. Similarly, an intermittent-missingness model allows return after a missed visit, whereas a dropout model removes later samples after dropout. Independent observation probabilities are simpler to simulate and interpret, but can be inadequate when attendance depends on observed history. More realistic, history-dependent or MNAR models may be necessary when clinically plausible, at the cost of requiring additional assumptions and inputs.
Diagnostics, Failure Modes, and Limitations
Before interpreting design comparisons, check the generated data against the mechanisms specified. Useful diagnostics include residuals plotted against predictions and time, empirical residual means and variances by prediction range, the fraction of simulated values outside plausible measurement limits, counts missing by visit and subgroup, and the distribution of dropout times. Compare these summaries with the intended parameters and probabilities, allowing for ordinary simulation variability.
- Wrong scale or units. An additive standard deviation must use the outcome's units, while a proportional coefficient is dimensionless. Combining a value in different units from the modeled outcome produces a meaningless variance.
- Negative simulated concentrations. A normal model on the original scale can generate negative values. Do not silently truncate them unless truncation is part of the stated data-generating model; truncation changes the distribution and generally its mean.
- Accidental monotone dropout. If the intended process is intermittent missed samples, do not permanently set all later observations missing after the first missed visit.
- Incorrect dropout risk set. A dropout hazard at a visit applies only to participants still at risk of dropout. Applying the same hazard to all participants at every visit does not implement monotone dropout.
- Outcome-dependent missingness treated as independent. If high or low concentrations affect attendance, independent masking will not reproduce the resulting observed-data distribution and may understate bias.
- Confusing expected and realized counts. Expected counts can be non-integer, but each simulated replicate has integer observation counts. Round only when reporting a count approximation, not when constructing replicate data.
No single residual or missingness model is universally adequate. The normal residual models described here are useful when their scale and tail behavior are plausible, but they do not represent assay censoring, correlated residuals, measurement rounding, or all forms of heteroscedasticity without further specification. Missingness probabilities also need justification: a convenient assumed mechanism is not evidence that real participants will follow it. When uncertain, simulate a range of plausible mechanisms and interpret performance as conditional on those assumptions.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A prediction is \(5.0\) mg/L, with \(\sigma_{\mathrm{add}}=0.30\) mg/L and \(\sigma_{\mathrm{prop}}=0.10\). What is the combined residual variance and standard deviation?
- In one replicate, 80 participants are scheduled for a visit and 62 are observed. Is 62 an expected count or a realized count? What might the expected count be if each participant has observation probability 0.75?
- A participant misses the second visit but has a recorded third visit. Is that pattern compatible with monotone dropout? What mechanism could produce it?
- Suppose dropout probability depends on a previously observed adverse-event score but not on the current unobserved concentration after conditioning on the recorded history. Which missingness category describes this mechanism?
Solutions. 1. The variance is \(0.30^2+(0.10\times5.0)^2=0.09+0.25=0.34\;(\mathrm{mg/L})^2\), and the standard deviation is \(\sqrt{0.34}=0.5831\) mg/L. 2. The count 62 is a realized integer count in that replicate. The expected count is \(80(0.75)=60\) observations. 3. No. Monotone dropout prevents later samples after dropout; an intermittent missed-sample mechanism can produce a missed second visit followed by an observed third visit. 4. This is MAR, because missingness depends on observed history and not on the unobserved concentration conditional on that history.