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Partial Identification and Sensitivity · Tutorial 817 of 952

Covariate-Stratified Attrition Bounds

Learn how to construct, aggregate, and interpret worst-case attrition bounds within covariate strata, including when stratification does and does not improve them.

Advanced 10 min read

What You'll Learn

  • Define stratum-specific response rates, observed outcome means, and valid outcome support limits.
  • Derive lower and upper bounds for each stratum’s full-population mean under unrestricted attrition.
  • Aggregate stratum endpoints using the target population’s covariate distribution.
  • Verify why the bounds are sharp and identify the assumptions they require.
  • Compare stratified bounds with pooled bounds and recognize when covariates tighten the interval.
  • Distinguish identification bounds from statistical confidence intervals.

Why Stratify Attrition Bounds?

When outcomes are missing because some individuals leave a study or are not observed at follow-up, the outcome mean among those observed need not equal the mean for the full target population. Worst-case bounds address this without assuming that missingness is random: they use only the observed outcomes and valid limits on the outcomes that are missing. Covariates can help by defining groups in which those limits are more informative.

The central task is to bound the full-population mean separately within each covariate stratum, then combine those bounds using the stratum proportions in the target population. This is a statement about what the available information identifies, not a claim that missing outcomes take their lower or upper limits in reality.

Definition: Let \(K\) denote a covariate stratum, \(Y\) the outcome of interest, and \(R\) an indicator equal to 1 when \(Y\) is observed and 0 when it is missing. In stratum \(k\), let \(p_k=P(K=k)\), \(q_k=P(R=1\mid K=k)\), and \(m_k=E[Y\mid R=1,K=k]\) when \(q_k>0\). Suppose every outcome in that stratum, observed or not, is known to lie in \([L_k,U_k]\). The target is the full-population mean \(E[Y]\).

The strata must be defined using covariates available for the target population, including for individuals whose outcomes are missing. The weights \(p_k\) describe the target population, not merely the respondents. In a finite population, these are the proportions of all target individuals in each stratum; in a probability formulation, they are the corresponding population probabilities.

Bounds Within One Stratum

Within stratum \(k\), a fraction \(q_k\) of outcomes is observed, with mean \(m_k\). The remaining fraction \(1-q_k\) has an unknown mean, but that mean must lie between \(L_k\) and \(U_k\). Therefore, the full stratum mean is bounded by combining the observed contribution with the smallest or largest possible contribution from missing outcomes.

$$ q_k m_k+(1-q_k)L_k \;\leq\; E[Y\mid K=k] \;\leq\; q_k m_k+(1-q_k)U_k. $$

These expressions also cover complete response: if \(q_k=1\), both endpoints equal \(m_k\). If \(q_k=0\), there is no observed mean \(m_k\) to use; the stratum mean is bounded only by \([L_k,U_k]\). A stratum with no observations therefore contributes its support limits, not an invented observed mean.

Theorem: Under unrestricted attrition within each stratum, the sharp bounds for the full-population mean are $$ \sum_k p_k\{q_km_k+(1-q_k)L_k\} \;\leq\; E[Y] \;\leq\; \sum_k p_k\{q_km_k+(1-q_k)U_k\}. $$ For a stratum with \(q_k=0\), interpret its lower and upper contributions as \(p_kL_k\) and \(p_kU_k\), respectively.

The formula assumes that the stated support limits apply to every outcome in the stratum, that observed outcomes and response indicators are correctly recorded, and that the stratum proportions and response rates refer to the target population. It imposes no restriction on how the missing outcomes are distributed within their support. In particular, it does not assume that observed individuals represent missing individuals.

Worked Example: Two Strata With Different Outcome Supports

Worked Example: Two Strata With Different Outcome Supports

Suppose a target population has 100 people: 60 in stratum A and 40 in stratum B. In A, 48 outcomes are observed, their mean is 20, and every outcome is known to lie between 0 and 40. In B, 20 outcomes are observed, their mean is 70, and every outcome is known to lie between 50 and 100. Find bounds for the full-population mean.

The target weights are \(p_A=60/100=0.6\) and \(p_B=40/100=0.4\). The response rates are \(q_A=48/60=0.8\) and \(q_B=20/40=0.5\).

For A, the stratum-specific lower and upper endpoints are

$$ 0.8(20)+0.2(0)=16, \qquad 0.8(20)+0.2(40)=24. $$

For B, they are

$$ 0.5(70)+0.5(50)=60, \qquad 0.5(70)+0.5(100)=85. $$

Weight these endpoints by the proportions of all target individuals in the strata. The aggregate lower endpoint is

$$ 0.6(16)+0.4(60)=9.6+24=33.6, $$

and the aggregate upper endpoint is

$$ 0.6(24)+0.4(85)=14.4+34=48.4. $$

Thus the full-population mean is bounded by 33.6 and 48.4 outcome units. The limits do not say that the mean has a particular probability of lying in this interval; they describe the range compatible with the stated data and assumptions.

Why the Aggregate Bounds Are Sharp

A bound is sharp when its endpoint can actually be attained by some assignment of the missing outcomes that satisfies the assumptions. Here the lower endpoint is attained by assigning every missing outcome in stratum \(k\) the value \(L_k\). The upper endpoint is attained by assigning every missing outcome the value \(U_k\). These assignments preserve all observed outcomes and respect each stratum’s support.

1
Hold observed outcomes fixed.
For each stratum, the observed fraction contributes \(q_km_k\) to its mean, and this contribution cannot change when assigning values to missing outcomes.
2
Assign missing outcomes to the lower limits.
Setting every missing outcome in stratum \(k\) to \(L_k\) gives the stratum mean \(q_km_k+(1-q_k)L_k\). Weighting and summing gives the stated lower endpoint.
3
Assign missing outcomes to the upper limits.
Setting every missing outcome in stratum \(k\) to \(U_k\) gives \(q_km_k+(1-q_k)U_k\). Weighting and summing gives the stated upper endpoint.

No compatible assignment can produce a mean below the first construction or above the second, because every missing value is constrained to its stratum’s support. Since both endpoint assignments are feasible, neither endpoint can be improved without adding assumptions or information. This establishes sharpness for the specified supports and strata.

When Covariates Tighten the Bounds

Stratification can improve the bounds when it supplies valid, more informative limits on missing outcomes. For example, a broad outcome range may be appropriate for the whole population, while substantive knowledge establishes narrower ranges for particular covariate groups. Using those narrower limits prevents missing outcomes in a stratum from being assigned values that are impossible for that group.

However, dividing the data into strata does not automatically tighten worst-case bounds. If the same support \([L,U]\) applies in every stratum, aggregating the stratum-specific endpoints gives exactly the pooled worst-case bounds. To see why, the observed contribution summed across strata is \(\sum_k p_kq_km_k\), while the total missing fraction is \(\sum_kp_k(1-q_k)\). The pooled lower endpoint is that observed contribution plus \(L\) times the missing fraction, exactly the sum of the stratum-specific lower contributions when every \(L_k=L\). The same argument applies to the upper endpoint with \(U_k=U\).

Worked Example: Comparing Stratified and Pooled Bounds

Use the data from the two-stratum example. There are 68 observed outcomes: 48 in A with mean 20, and 20 in B with mean 70. If we ignore the strata and use only the global support \([0,100]\), compare the pooled bounds with the stratified bounds.

The observed outcome total is

$$ 48(20)+20(70)=960+1400=2360. $$

The overall observed fraction is \(68/100=0.68\), and the pooled observed mean is \(2360/68\approx34.7059\). Therefore, using the global support, the pooled lower endpoint is

$$ 0.68(34.7059)+0.32(0)=23.6, $$

and the pooled upper endpoint is

$$ 0.68(34.7059)+0.32(100)=23.6+32=55.6. $$

The stratified interval, 33.6 to 48.4, is narrower than the pooled interval, 23.6 to 55.6. The improvement comes from using the tighter, group-specific supports \([0,40]\) and \([50,100]\), not from stratification alone. If every stratum instead had the common support \([0,100]\), stratified aggregation would return the pooled endpoints exactly.

Using the Correct Target Weights

The stratum endpoints must be weighted by the target population’s stratum proportions. Weighting by the share of respondents in each stratum generally targets a different population, because response rates may differ across strata. If the target population is specified separately from the observed sample, its covariate distribution must be used in the aggregation.

Worked Example: Aggregating for a Specified Target Population

Suppose the target population is 70% in stratum A and 30% in stratum B. In A, half of outcomes are observed, the observed mean is 10, and the support is \([0,20]\). In B, 80% of outcomes are observed, the observed mean is 30, and the support is \([20,40]\). Find bounds for the target-population mean.

In A, the endpoints are

$$ 0.5(10)+0.5(0)=5, \qquad 0.5(10)+0.5(20)=15. $$

In B, they are

$$ 0.8(30)+0.2(20)=28, \qquad 0.8(30)+0.2(40)=32. $$

Use the target weights \(0.7\) and \(0.3\), not weights based on the respondent counts:

$$ \text{Lower}=0.7(5)+0.3(28)=3.5+8.4=11.9, $$
$$ \text{Upper}=0.7(15)+0.3(32)=10.5+9.6=20.1. $$

The target-population mean is therefore bounded by 11.9 and 20.1 outcome units. These calculations use the specified target composition even if the observed respondents have a different distribution across A and B.

Interpretation and Common Pitfalls

These endpoints concern identification under worst-case missing outcomes. They are not confidence limits and do not account for sampling variability in estimated response rates, observed means, or stratum proportions. If those quantities are estimated from a sample, uncertainty about the endpoints is a separate inferential problem.

  • Using observed ranges as guaranteed supports. The smallest and largest observed outcomes need not bound the unobserved outcomes. A support must be justified as applying to all individuals in the stratum, not inferred automatically from the observed sample.
  • Weighting by respondents instead of the target population. Respondent proportions can differ from target stratum proportions when attrition rates vary. The aggregation weights must match the estimand.
  • Claiming that covariates always narrow the interval. With a common support and unrestricted missingness, stratification alone leaves the pooled worst-case endpoints unchanged. Tighter valid supports or additional restrictions are needed for an improvement.
  • Treating an endpoint assignment as a prediction. Assigning all missing values to a support limit constructs a feasible worst case. It does not assert that this extreme pattern is likely or actually occurred.
Key takeaway: Bound the mean within each stratum using its response rate, observed mean, and valid outcome limits; then aggregate with target-population stratum weights. The resulting bounds are sharp under unrestricted attrition, and covariates tighten them only when they provide genuinely more informative restrictions.

Check Your Understanding

Try these questions using the definitions and results above.

  1. In one stratum, 75% of outcomes are observed with mean 12, and all outcomes lie in \([4,20]\). Calculate the lower and upper bounds for the stratum mean.
  2. Why must the aggregate endpoints be weighted by target-population stratum proportions rather than respondent proportions?
  3. If every stratum has the same known support \([0,1]\), does stratifying by covariates alone tighten the unrestricted worst-case bounds for the overall mean? Explain.
  4. What assignment of missing outcomes attains the lower endpoint, and why does that establish sharpness?
  5. Explain why the observed minimum and maximum are not automatically valid support limits for missing outcomes.

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