Why Build a Scenario Grid?
A dose-regimen comparison can look simple when only two regimens are involved. But its results may differ across covariate subgroups, and each simulated outcome may also depend on uncertain model parameters. A scenario grid makes those dimensions visible: every combination of regimen, subgroup, and parameter draw has a defined place in the analysis. The resulting table supports comparisons that are traceable to the simulations rather than assembled from mismatched summaries.
The goal is not merely to produce a large table. It is to define the comparisons before simulation, preserve the pairing between scenarios where appropriate, and summarize exposure and threshold attainment at the level of the question. A pooled result can be misleading if it hides a subgroup with a different response, while a subgroup result can be misleading if its uncertainty source is unclear.
Define the Grid and the Quantity to Compare
Before running simulations, state the regimens, subgroup definitions, parameter uncertainty representation, exposure metric, threshold, and target population weights. For example, a grid with two regimens, three subgroups, and 1,000 parameter draws has \(2 \times 3 \times 1{,}000 = 6{,}000\) regimen-subgroup-draw combinations. If each draw also contains simulated individuals, the individual index is an additional dimension, not a substitute for the parameter-draw index.
Exposure must have a precise definition and unit. In the hypothetical examples below, the outcome is steady-state AUC over a 24-hour interval, in mg·h/L. The threshold is a specified target AUC, and attainment for an outcome \(AUC\) means \(AUC \geq T\), where \(T\) is that target. In other applications, use the exposure metric and threshold appropriate to the scientific question.
For a cell, the mean exposure across \(M\) equally weighted parameter draws is the arithmetic mean of the draw-specific cell summaries. The estimated attainment fraction is the number of draw-specific outcomes meeting the threshold divided by the number of outcomes evaluated. If each draw includes individual simulations, define whether attainment is first calculated within each draw and then averaged across draws, or calculated over all equally weighted draw-individual outcomes. These are equivalent only when the draws have equal numbers of equally weighted individuals.
Here \(I(\cdot)\) is an indicator equal to 1 when its condition is true and 0 otherwise. If a draw represents a distribution of individuals, \(AUC_{r,g,m}\) in these expressions should instead be replaced by the draw-specific mean exposure or draw-specific attainment fraction, according to the stated estimand.
Worked Example: Specifying Grid Dimensions
Suppose a hypothetical simulation compares two regimens in three subgroups: lower, middle, and higher baseline clearance. It uses 500 parameter draws. Each parameter draw generates 200 individuals in each subgroup for each regimen. Count the scenario cells and the simulated individual-level outcomes.
There are \(2 \times 3 \times 500 = 3{,}000\) regimen-subgroup-draw cells. Each cell contains 200 individual-level outcomes, so the total number of outcomes is \(3{,}000 \times 200 = 600{,}000\). The grid dimensions and outcome count are different quantities: 3,000 describes the combinations being summarized, while 600,000 describes the individual simulated records. If regimen comparisons are paired using the same parameter and individual random-number draws, those pairings should be retained in the output.
Carry Parameter Uncertainty Through Each Cell
Parameter uncertainty is propagated by simulating or evaluating outcomes under parameter values drawn from the specified uncertainty distribution. Each draw is applied consistently across the relevant cells. When the scientific comparison is between regimens under the same uncertain parameter set, use the same parameter draw for both regimens. This paired construction makes the contrast answer a within-draw question and avoids adding irrelevant variation from independently resampling parameters for each regimen.
For example, under a simple linear pharmacokinetic model at steady state, the AUC over a dosing interval is dose divided by clearance. This relationship assumes linear, time-invariant clearance and the same interval-specific definition of AUC. It is used below only to make the arithmetic transparent; a more complex model would obtain each AUC from its simulated concentration-time profile.
If a parameter draw changes clearance, it changes exposure in every regimen-subgroup cell affected by that clearance parameter. The regimen contrast should therefore be calculated within each draw before summarizing across draws. This preserves the covariance induced by shared parameter draws and gives a direct distribution of paired contrasts.
The same logic applies to threshold attainment: calculate the difference in attainment indicators or draw-specific attainment fractions between regimens within each subgroup and draw, then summarize that contrast. A difference in attainment fractions is expressed in percentage points, not as a percentage change unless a relative contrast is explicitly calculated.
Worked Example: Exposure and Attainment by Regimen and Subgroup
Consider a hypothetical model with two 24-hour regimens: regimen A gives 100 mg every 24 hours, and regimen B gives 150 mg every 24 hours. The lower-clearance subgroup has baseline clearance 8 L/h and the higher-clearance subgroup has baseline clearance 12 L/h. Four illustrative parameter draws multiply each subgroup’s baseline clearance by \(f_m = 0.8,\ 0.9,\ 1.1,\ 1.2\). The same multiplier is used for both regimens within each subgroup and draw. Let the attainment threshold be \(T = 12\) mg·h/L. For this deliberately small illustration, the outcomes are calculated directly from \(AUC_{24}=\text{Dose}/CL\); the four draws are not intended to provide a precise uncertainty interval.
For regimen A in the lower-clearance subgroup at the first draw, clearance is \(8 \times 0.8 = 6.4\) L/h, so \(AUC_{24}=100/6.4=15.625\) mg·h/L. Repeating the calculation for all cells gives:
| Clearance multiplier | A, lower clearance | B, lower clearance | A, higher clearance | B, higher clearance |
|---|---|---|---|---|
| 0.8 | 15.625 | 23.438 | 10.417 | 15.625 |
| 0.9 | 13.889 | 20.833 | 9.259 | 13.889 |
| 1.1 | 11.364 | 17.045 | 7.576 | 11.364 |
| 1.2 | 10.417 | 15.625 | 6.944 | 10.417 |
All table entries are in mg·h/L. For regimen A in the lower-clearance subgroup, the mean exposure is \((15.625+13.889+11.364+10.417)/4=12.824\) mg·h/L. For regimen B in that subgroup, it is \((23.438+20.833+17.045+15.625)/4=19.235\) mg·h/L. Thus the mean paired regimen contrast is \(19.235-12.824=6.411\) mg·h/L, subject to rounding.
Against the 12 mg·h/L threshold, regimen A attains the target on two of four draws in the lower-clearance subgroup, for \(2/4=0.50\). Regimen B attains it on all four draws, for \(4/4=1.00\). The attainment contrast is therefore \(1.00-0.50=0.50\), or 50 percentage points. In the higher-clearance subgroup, regimen A attains on zero of four draws and regimen B on two of four, so their attainment fractions are 0 and 0.50, respectively. These fractions summarize the specified four illustrative parameter draws; they are not evidence that the true attainment probability is known exactly.
Aggregate Only After Subgroup Results Are Visible
A pooled result requires a target population definition. If subgroup weights represent the intended population, use those weights explicitly. Do not assume that equal weights are appropriate merely because the simulation used the same number of records in each subgroup. For an exposure summary, the population mean is the weighted average of subgroup means. For attainment, it is the weighted average of subgroup attainment probabilities, provided those probabilities refer to the same target population and estimand.
Worked Example: Weighting Subgroup Contrasts
Use the subgroup results from the previous example and suppose the target population is 70% lower-clearance and 30% higher-clearance. The lower-clearance mean exposures are 12.824 mg·h/L for A and 19.235 mg·h/L for B; the higher-clearance means are 8.549 mg·h/L for A and 12.824 mg·h/L for B.
The weighted mean for regimen A is \(0.70(12.824)+0.30(8.549)=11.542\) mg·h/L. For regimen B it is \(0.70(19.235)+0.30(12.824)=17.312\) mg·h/L. The pooled exposure contrast is \(17.312-11.542=5.770\) mg·h/L. Small differences in the last decimal can occur if rounded subgroup means are used.
The weighted attainment fraction for A is \(0.70(0.50)+0.30(0)=0.35\), while for B it is \(0.70(1.00)+0.30(0.50)=0.85\). The weighted difference is \(0.85-0.35=0.50\), or 50 percentage points. This pooled contrast does not replace the subgroup results: the same overall difference can arise from different patterns of subgroup benefit.
Checks, Alternatives, and Limitations
A useful grid is both computationally valid and interpretable. Before interpreting summaries, check that every intended regimen-subgroup-draw combination is present; units agree; exposure and threshold definitions match; and the number of successful, failed, or nonfinite simulations is reported. Verify that parameter draws are shared across regimens when the contrast is intended to be paired. The earlier material on handling failed and nonfinite simulation replicates and validating simulated outputs with invariants addresses those checks; a scenario summary should not silently hide missing or invalid outcomes.
The earlier material on paired scenario comparisons with common random numbers explains why shared draws can improve precision for a contrast. An alternative is to simulate each regimen independently. That can still estimate regimen-specific summaries, but it generally does not preserve the same paired contrast and may add Monte Carlo variation. Another alternative is to report only a pooled summary. This is compact, but it can conceal meaningful subgroup differences and depends on an explicit population weighting scheme.
Parameter uncertainty is not the same as individual variability or uncertainty in subgroup composition. A grid that varies parameters but uses one representative clearance per subgroup does not quantify within-subgroup variability. Likewise, a simulated attainment fraction conditional on a fixed subgroup mixture does not account for uncertainty in the mixture weights unless that uncertainty is modeled. State which sources are included, and avoid describing a small set of parameter draws as a definitive probability distribution or a calibrated confidence interval.
Name the exposure metric, threshold, subgroup definitions, regimen schedules, and target population weights.
Index outcomes by regimen, subgroup, parameter draw, and any individual-simulation dimension; verify expected cell counts.
Calculate exposure and attainment within each cell, then compare regimens within matched subgroup-draw combinations.
Show subgroup summaries before any weighted overall summary, and state what uncertainty and simulation failures are represented.
Check Your Understanding
Answer each question, then compare with the solutions.
- A grid has three regimens, two subgroups, and 250 parameter draws. How many regimen-subgroup-draw cells does it contain?
- In one subgroup and regimen, 72 of 120 equally weighted simulated outcomes meet the threshold. What is the estimated attainment fraction?
- Why should two regimens use matched parameter draws when the goal is a paired regimen contrast?
- Subgroup attainment is 0.80 in a subgroup making up 25% of the target population and 0.40 in the remaining 75%. What is the weighted overall attainment?
Solutions. (1) The grid contains \(3 \times 2 \times 250=1{,}500\) cells. (2) The fraction is \(72/120=0.60\), or 60%; its interpretation is tied to the simulated outcomes and their weighting. (3) Matched draws hold the uncertain parameter realization constant across regimens, so the within-draw contrast is not additionally mixed with independent parameter-sampling variation. (4) The weighted attainment is \(0.25(0.80)+0.75(0.40)=0.20+0.30=0.50\), or 50%.