Tensions in Practice: An assumption-based point in tension with an assumption-light range¶
Four bounded scores, with two unobserved
Four scores each lie between 0 and 10. Two observed scores are 2 and 4; the other two are missing. If their combined value is assumed to be 6, the full mean is 3. Without that added assumption, their combined value can be anywhere from 0 to 20, so the full mean can range from 1.5 to 6.5. The point and range carry different inferential contracts; the missing values have not been recovered.
Use a specific working estimate
Make a point report under a visible substantive assumption.
Preserve uncertainty about missing values
Report all means compatible with the stated bounds.
Why these aims pull against each other
A point requires a restriction on the missing portion. Keeping all bounded completions avoids that restriction but can leave too wide a range for a decision.
Choose an arrangement to see what changes and what remains difficult.
Observed total and four-person denominator stay fixed. The missing-total restriction changes from one stipulated value to all bounded values, which changes the full-mean claim.
What this choice protects
What it costs
When it fits
Compare the arrangements
State the missing-total assumption
Assume the two missing scores sum to 6, equal to the observed pair’s sum; report (6 + 6)/4 = 3 conditional on it.
| Quantity | Reported | |
|---|---|---|
| Observed | 2 + 4 | 6 |
| Missing | Two values | 6 assumed |
| All four | Mean | 3 if assumed |
- What it protects
- A specific conditional value can be used for an explicitly assumption-based plan.
- What it costs
- The observed scores alone do not establish the missing total, and a decision may depend entirely on this assumption.
- When it fits
- Fits when an external substantive argument supports the restriction and its role is disclosed.
Illustration note: This is a declared finite-population assumption, not a claim that random missingness forces a realized missing pair to equal the observed pair.
Keep the completion range
Retain only the observed scores and the 0-to-10 outcome bounds.
| Quantity | Reported | |
|---|---|---|
| Observed | 2 + 4 | 6 |
| Missing | Two values | 0 to 20 |
| All four | Mean | 1.5 to 6.5 |
- What it protects
- Every compatible full mean remains represented without guessing the missing total.
- What it costs
- The result may be too broad to settle a practical comparison.
- When it fits
- Fits when the reason for missingness is uncertain and the bounds themselves are credible.
Illustration note: Both endpoints are attained by permissible missing pairs: (0,0) and (10,10). This is an identification range, not a confidence interval.
What this illustration does—and does not—establish
The source supplies the structural tension; the invented example makes one relation inspectable. Costs and conditions are part of each arrangement, not exceptions to a universal recommendation.
- No missingness mechanism is inferred from these four records; MCAR, MAR and MNAR are not diagnosed by the table.
- The point has no asserted coverage or frequency guarantee. The range has no assigned probabilities within it.
- Changing the outcome bounds or acquiring actual missing scores would change the comparison.
Source entries
Missing Data Mechanisms (MCAR, MAR, MNAR)
The canonical tension motivates this comparison. The setting, finite values and arrangements are declared editorial illustrations, not measured findings.
Mechanism identifiability vs reliance on assumptions
T1 — Mechanism identifiability vs reliance on assumptions. The MAR-vs-MNAR distinction cannot be tested from observed data alone — any observed data pattern is consistent with both MAR (depending only on observed covariates) and MNAR (depending on unobserved outcomes) mechanisms. This creates a permanent assumption burden: the analyst must either assume MAR (often defensible with rich observed covariates) or explicitly model MNAR (requires substantive assumptions that typically cannot be directly verified). The tension between "pick an assumption and proceed" and "do sensitivity analysis across assumptions" is unavoidable and has shaped modern reporting practice toward requiring both primary analysis and sensitivity checks.
The source operation
(1) Missing data mechanisms classify the process by which observations become missing into three increasingly problematic categories: *MCAR* (missing completely at random) where missingness is independent of all variables observed and unobserved; *MAR* (missing at random) where missingness depends only on observed variables; and *MNAR* (missing not at random) where missingness depends on the unobserved values themselves.