Why a Delta Grid Is a Set of Assumptions
A delta sensitivity analysis asks how conclusions change when the distribution assigned to missing outcomes departs from a reference model. The delta grid is the set of values at which that departure will be examined. Constructing the grid is therefore not a search for values that the observed data estimate. It is a way to express, systematically and transparently, a range of clinically meaningful assumptions about outcomes that were not observed.
A grid is useful only when each value has a clear interpretation. Its range should span plausible departures, including the reference assumption when appropriate. Its spacing should let a reader see how results vary across that range without suggesting more precision than clinical knowledge supports. These choices depend on the outcome, its units, the delta parameterization, and the clinical question.
The grid must be designed for the parameterization actually used. A shift of 2 points on a symptom scale is not interchangeable with a log-odds shift of 2 for a binary outcome. Before choosing values, state what delta changes, for whom, on what scale, and in which direction. For example, if a larger outcome value represents better health, a negative additive delta means worse outcomes among the missing cases than under the reference model.
Building the Range and Spacing
Start by identifying the clinical question and the reference assumption. Then use information that can justify assumptions about missing outcomes: clinical thresholds, expert elicitation, relevant external evidence, or a clearly reasoned combination. The introductory material on delta calibration and clinical elicitation supplies possible sources of such information; here the task is to turn those inputs into a usable set of scenarios.
A clinical threshold is especially useful for interpreting increments. If a 5-point change is considered clinically important on a particular scale, a grid might include values separated by 2 or 2.5 points so readers can see how assumptions approaching that threshold affect results. The threshold does not automatically determine either the endpoints or the spacing: those choices also depend on how far missing outcomes could plausibly depart from the reference distribution and how much resolution is useful.
Specify the model, target missing outcomes, outcome scale, units, and sign convention. A grid without these details cannot be interpreted reliably.
Choose a lower and upper value using thresholds, expert judgments, or external evidence. Record why each bound is plausible; do not choose bounds merely because they produce a desired conclusion.
Relate the increment to clinical units or a recognized threshold. Use enough values to show relevant changes, but do not imply that unsupported decimal precision represents clinical knowledge.
State what each value assumes about missing outcomes relative to the reference model. If multiple deltas are used, describe the combinations as clinical scenarios rather than presenting an unexplained table of numbers.
For a regular grid from \(\delta_{\min}\) to \(\delta_{\max}\) in increments of \(h\), with the interval width divisible by \(h\), the grid contains \(K+1\) values, where \(K=(\delta_{\max}-\delta_{\min})/h\). If the width is not divisible by the proposed increment, either choose a compatible increment or include a shorter final interval and disclose it. Avoid silently dropping a clinically important endpoint.
Spacing controls resolution, not certainty. For an equally spaced grid with interval \(h\), any value between adjacent points is at most \(h/2\) from a grid point. That arithmetic fact can help describe how finely the range is sampled; it does not establish that the true delta is near a grid value, or that the grid covers the true value.
Worked Examples
Worked Example: A Continuous Symptom Score
Suppose a hypothetical trial uses a 0-to-100 symptom-improvement score, where larger values are better. The reference model predicts a mean improvement of 18 points for the missing outcomes in a specified group. Clinical advisers consider missing outcomes between 0 and 6 points worse than that reference plausible, and a 5-point difference clinically important. Construct and interpret a regular grid using 2-point increments.
Here the additive convention is \(Y_{\text{mis}}(\delta)=Y_{\text{mis}}(0)+\delta\), so negative delta means less improvement. The advised range is \(-6\) to 0 points. Its width is \(0-(-6)=6\) points; an increment of 2 points divides that width into \(6/2=3\) intervals. The grid is therefore \(\{-6,-4,-2,0\}\) points.
The second expression shows the shifted reference mean at each grid value: 18 minus 6 is 12, 18 minus 4 is 14, and 18 minus 2 is 16 points, while the reference case remains 18 points. The 2-point spacing is smaller than the 5-point clinical threshold, so the grid includes a scenario close to, but not equal to, a threshold-sized departure. The range reflects the stipulated clinical judgment; neither the mean 18 nor the observed trial outcomes identify which delta is correct.
Worked Example: A Binary Outcome on the Log-Odds Scale
In a hypothetical study, the binary outcome is recovery by 12 weeks. Suppose the reference model assigns a recovery probability of 0.60 to a particular missing-outcome profile. For this example define delta as an additive shift to the log odds of recovery. Clinical advisers consider the odds among missing participants plausibly between half and 1.5 times the reference odds. Construct a grid that includes the reference case and the two stated bounds.
The reference odds are \(0.60/(1-0.60)=1.5\). The corresponding log-odds delta values are the logarithms of the odds ratios: \(\log(0.5)\approx-0.6931\), \(\log(1)=0\), and \(\log(1.5)\approx0.4055\). These endpoints are not symmetric around zero, which is appropriate: the advisers specified an odds-ratio range, not an equal-distance log-odds range.
At the lower bound, shifted odds are \(1.5\times0.5=0.75\), giving probability \(0.75/(1+0.75)=0.4286\). At the reference value, the probability remains 0.60. At the upper bound, odds are \(1.5\times1.5=2.25\), giving probability \(2.25/(1+2.25)=0.6923\). Thus the three scenarios correspond approximately to recovery probabilities of 0.429, 0.600, and 0.692 for this profile. They are illustrative model implications of the stipulated odds ratios, not estimates of the missing participants’ recovery probabilities.
Worked Example: Organizing Two Clinical Assumptions
Consider a hypothetical comparison of two treatment groups using an outcome where higher is better. Let \(\delta_T\) and \(\delta_C\) be additive shifts, in points, for missing outcomes in the treatment and control groups respectively. Experts judge a 4-point worse-than-reference shift plausible in either group. They also consider a scenario in which missing treatment outcomes are 4 points worse than reference while missing control outcomes follow the reference model. Show a small grid that separates a shared-shift scenario from this differential scenario.
Use the clinically motivated one-dimensional values \(\{0,-2,-4\}\) points for a shared shift, meaning \(\delta_T=\delta_C\). For the stated differential scenario, use \((\delta_T,\delta_C)=(-4,0)\). A compact scenario set is therefore \((0,0),(-2,-2),(-4,-4),(-4,0)\). The final pair is not another point on the shared-shift path: it encodes an additional assumption that the departures differ by group.
For example, if the reference predicted mean improvements are 20 points in treatment and 17 in control, the \((-4,0)\) scenario gives shifted means \(20-4=16\) and \(17+0=17\) points. The contrast of these shifted means is \(16-17=-1\) point, compared with the reference contrast \(20-17=3\) points. This arithmetic illustrates why the differential scenario has a distinct clinical interpretation. It does not assert that these means or shifts are identified by the data. If every combination of three values for each group were included, there would be \(3\times3=9\) pairs; the four selected scenarios are a deliberate, interpretable subset rather than an exhaustive Cartesian grid.
Checking the Grid and Its Limitations
Before using a grid, check that the reference model is suitable for the intended analysis, the delta scale matches its mathematical role, and every endpoint is clinically defensible. Inspect the implied outcomes at the bounds. A shift that places predictions outside a bounded outcome's possible range, or creates a clinically incoherent pattern, signals that the parameterization or range needs reconsideration. For bounded outcomes, an additive shift may be inappropriate near the boundaries; a transformed-scale delta or another constrained model may be needed.
Also check whether the spacing supports the intended comparisons. A very coarse grid can miss important variation between scenarios. An extremely fine grid can create a false impression of precision, especially when experts can justify only broad ranges. Unequal spacing can be useful when clinical thresholds or plausible scenarios cluster in one region, provided the reason for the denser region is stated. Do not report more decimal places than the scale and clinical inputs warrant.
The main alternatives are not interchangeable. A single delta value is simpler but examines only one assumption. A regular grid makes change across a range easy to compare. A threshold-based or scenario-based set can be more interpretable when clinical knowledge is concentrated at meaningful boundaries, even if the spacing is irregular. A multi-parameter grid can represent distinct assumptions by group or time, but its number of combinations grows quickly and may obscure the scenarios that matter. Choose the simplest set that covers the clinically relevant assumptions.
Observed data can help assess the reference model and reveal inconsistencies in predictions, but they generally cannot determine how unobserved outcomes differ from that reference without additional assumptions or information. In particular, a denser grid does not make delta more data-identified. Nor does a clinically plausible range guarantee that the true value lies within it. The analysis is a sensitivity assessment conditional on the selected assumptions, not evidence that the assumptions themselves have been verified.
A useful report gives the delta definition and units, the rationale for the range and spacing, the reference case, any scenario-specific assumptions, and the limitations of those choices. This makes the sensitivity analysis reproducible and allows a clinical reader to judge whether the scenarios are relevant without confusing plausibility with statistical identification.
Check Your Understanding
Answer each question using the stated delta convention. Solutions are included below the questions.
- A higher-is-better score uses an additive delta. What does \(\delta=-3\) mean, and what units should it have?
- A clinically supported range is \([-8,0]\) points, with a proposed spacing of 3 points. Does this spacing divide the range exactly? Give a grid that includes both endpoints.
- For a binary outcome, the reference probability is 0.40 and delta is a log-odds shift. What probability corresponds to an odds-ratio multiplier of 2?
- Why does adding more points to a delta grid not make delta identified by observed data?
Solutions. (1) The missing outcomes are shifted 3 score points lower than under the reference model; delta is measured in score points. (2) The width is \(0-(-8)=8\) points and \(8/3\) is not an integer, so 3-point spacing does not divide the range exactly. One endpoint-preserving choice is \(\{-8,-5,-2,0\}\), with a final 2-point interval; alternatively, choose a compatible spacing such as 2 points. (3) Reference odds are \(0.40/0.60=0.6667\). Doubling gives odds \(1.3333\), so the probability is \(1.3333/(1+1.3333)=0.5714\). (4) The grid lists assumptions selected for sensitivity analysis. More values provide finer evaluation of those assumptions but do not supply information about outcomes that were not observed or identify the missing-data departure.