Missing observations: bound a result before imputing values
An observed-only average can answer a different question from the intended full-sample average when some outcomes are missing. Before filling the blanks with AI estimates, calculate what the unavailable observations could change under explicit limits. A wide bound is useful evidence that the result remains unidentified.
The R complete.cases documentation describes finding rows with no missing values. That is a mechanical filter, not a guarantee that the remaining rows represent the target population. The bounding worksheet below is an independent research method, not a property supplied by that function.
Define the target and a defensible range
Write the population, observation unit, outcome and period before inspecting missing values. Distinguish a missing outcome from an entity outside the target population. Excluding an ineligible entity and losing an eligible entity’s outcome have different consequences for the denominator.
Let N be the eligible count, m the missing count and S the sum of known outcomes. If each unavailable value is known to lie between L and U, the full-sample mean lies between (S + mL)/N and (S + mU)/N. This is an arithmetic identification range, not a sampling confidence interval. It requires genuine outcome limits, not convenient limits invented to make the result narrow.
For unbounded outcomes, a finite interval may be unavailable. Do not impose zero as a lower bound on a financial measure that can be negative. Likewise, a reported percentage can exceed 100 under some definitions; the label percent alone does not establish its allowable range.
Hypothetical ten-company worksheet
Assume a fictional research universe has ten eligible companies. Eight disclosed values sum to 480 points. The comparable metric is explicitly bounded from zero to 100, and two companies’ values are missing.
| Calculation | Hypothetical result |
|---|---|
| Observed-only mean | 480/8 = 60 |
| Full-sample lower bound | (480 + 2 × 0)/10 = 48 |
| Full-sample upper bound | (480 + 2 × 100)/10 = 68 |
| Mean after filling both with 60 | 60, conditional on that imputation |
The available data permit a full-sample mean from 48 to 68. The observed-only 60 is not automatically wrong, but treating it as the full-population answer needs additional assumptions. AI-generated values of 60 do not make those assumptions true.
Now suppose separate evidence supports a scenario in which each missing value is at most 40. Under that scenario, the upper bound becomes (480 + 80)/10 = 56, while the lower bound remains 48. Label it a restricted scenario unless the evidence establishes a hard limit.
Investigate why values are unavailable
Compare missingness across observed size, industry, reporting format and prior-period characteristics. A pattern can reveal selection risks but cannot, by itself, establish the missing outcomes. If weaker performers are less likely to disclose the metric, an observed-only average can be systematically optimistic.
Separate assumptions about missingness from the calculation. Assuming missingness is unrelated to outcomes differs from assuming it can be explained by observed covariates. Both differ from a situation where disclosure still depends on the unavailable outcome after those covariates are considered. A convenience sample cannot resolve the distinction by testing only observed rows.
Keep raw statuses, the bound, any imputed dataset and the imputation rule in separate outputs. Do not publish standard errors that treat guessed values as newly observed independent evidence. Recheck when missing disclosures arrive or a proposed bound loses its factual basis.
For nonnegative fixed weights, replace the equal-weight numerator with the known weighted sum plus each missing item’s weight multiplied by its own lower or upper limit. Keep all eligible weights in the denominator. Unknown weights create another identification problem; a bound on outcomes alone may no longer bound the intended weighted mean in the same way.
A later disclosure should update the corresponding item, not silently change the universe. Keep the original interval as a dated result and show how the new observation narrows it. A revised target population belongs in a separate comparison.
FAQ
Is the bound a 95% confidence interval?
No. It is the range compatible with specified outcome limits and known values. Sampling uncertainty is a separate problem.
Does filling blanks with the observed mean preserve validity?
It preserves one arithmetic mean, but can conceal selection and understate uncertainty. It requires assumptions about the unavailable outcomes.