Covariate-Stratified Identification and Bounds

Why Stratify the Bounds?

When an outcome is missing, the observed outcomes alone generally do not determine the population mean. If the outcome is known to lie within a specified range, that range limits how much the unobserved values can change the mean. Fully observed covariates allow the calculation to be done separately across covariate strata, where outcome ranges or response rates may differ, and then recombined using the population frequency of each stratum.

This is a partial-identification calculation: it describes which population means are compatible with the observed-data distribution and the stated outcome-support restrictions. It does not, by itself, produce a single identified mean or account for sampling uncertainty.

Definition: Let $Y$ be an outcome, $R=1$ indicate that $Y$ is observed, and $C$ be a fully observed covariate taking one of a finite set of values $c=1,\ldots,K$, each with positive population frequency. Write $\pi_c=P(C=c)$ for the population frequency of stratum $c$, and $\rho_c=P(R=1\mid C=c)$ for its response rate. Define the observed-outcome contribution $$ q_c=E\!\left[Y\,1(R=1)\mid C=c\right]. $$ When $\rho_c>0$, $q_c=\rho_c E(Y\mid R=1,C=c)$. When $\rho_c=0$, $q_c=0$; this definition does not require a respondent mean in that stratum.

Assume the outcome in stratum $c$ is known to lie in the closed interval $[L_c,U_c]$, with $L_c\leq U_c$. The interval may be common across strata or may depend on $c$. The observed-data distribution identifies $\pi_c$, $\rho_c$, and $q_c$ in the population-identification sense: these are features of the distribution of the observed covariates, response indicators, and observed outcomes.

Deriving and Combining the Stratum Bounds

For a stratum with missing outcomes, define $\nu_c=E(Y\mid R=0,C=c)$. This conditional mean is used only when $1-\rho_c>0$. Splitting the stratum mean according to response status gives

E(Y\mid C=c) =E\!\left[Y\,1(R=1)\mid C=c\right] + E\!\left[Y\,1(R=0)\mid C=c\right] =q_c+(1-\rho_c)\nu_c.

Since $L_c\leq Y\leq U_c$ for every individual in the stratum, any mean among nonrespondents must also satisfy $L_c\leq \nu_c\leq U_c$. Multiplying by the nonnegative missing fraction $1-\rho_c$ and adding $q_c$ yields the stratum-specific limits. This argument also covers $\rho_c=1$, when the missing contribution is zero.

Theorem: Suppose the finite strata cover the target population, their frequencies $\pi_c$ are known from the observed covariates, and the outcome in every positive-frequency stratum lies in its known support $[L_c,U_c]$. If no additional restrictions are imposed on the missing outcomes, then $$ q_c+(1-\rho_c)L_c \;\leq\; E(Y\mid C=c) \;\leq\; q_c+(1-\rho_c)U_c, $$ and the population mean $\mu=E(Y)$ satisfies $$ \sum_{c=1}^K \pi_c\{q_c+(1-\rho_c)L_c\} \;\leq\; \mu \;\leq\; \sum_{c=1}^K \pi_c\{q_c+(1-\rho_c)U_c\}. $$ These are sharp bounds: every endpoint can be achieved by assigning all missing outcomes in each stratum to the corresponding support endpoint, without changing the observed-data distribution.

The population formula follows from the law of total expectation: $E(Y)=\sum_c P(C=c)E(Y\mid C=c)$. The weights are covariate frequencies, not response rates. Each stratum’s lower limit contributes $\pi_c\{q_c+(1-\rho_c)L_c\}$, and its upper limit contributes $\pi_c\{q_c+(1-\rho_c)U_c\}$.

Worked Calculations

Worked Example: Binary Outcome in Two Strata

Consider a hypothetical population with two covariate strata. The outcome is binary, so its support is $[0,1]$ in both strata. Suppose stratum A has frequency $\pi_A=0.60$, response rate $\rho_A=0.80$, and respondent mean $E(Y\mid R=1,A)=0.25$. Stratum B has frequency $\pi_B=0.40$, response rate $\rho_B=0.50$, and respondent mean $E(Y\mid R=1,B)=0.60$. The strata frequencies sum to $1$, and both response rates are positive, so the respondent means are defined.

In A, $q_A=0.80(0.25)=0.20$. Thus the conditional mean is bounded by $0.20+(1-0.80)(0)=0.20$ and $0.20+(1-0.80)(1)=0.40$. In B, $q_B=0.50(0.60)=0.30$, giving limits $0.30+(1-0.50)(0)=0.30$ and $0.30+(1-0.50)(1)=0.80$.

Weighting by stratum frequency gives the population lower limit $0.60(0.20)+0.40(0.30)=0.24$, and upper limit $0.60(0.40)+0.40(0.80)=0.56$. Therefore the compatible population mean lies between 0.24 and 0.56, or between 24% and 56% for this binary outcome.

Here stratifying does not tighten the bounds compared with using the common support $[0,1]$ directly. The overall observed-outcome contribution is $0.60(0.20)+0.40(0.30)=0.24$, and the overall missing fraction is $0.60(0.20)+0.40(0.50)=0.32$. The pooled limits are $0.24+0.32(0)=0.24$ and $0.24+0.32(1)=0.56$, the same as the stratified limits.

Worked Example: Stratum-Specific Outcome Ranges

Suppose a hypothetical bounded score has support $[0,10]$ in stratum A and $[0,20]$ in stratum B. Let $\pi_A=0.70$, $\rho_A=0.75$, and $E(Y\mid R=1,A)=4$. Let $\pi_B=0.30$, $\rho_B=0.50$, and $E(Y\mid R=1,B)=12$. These respondent values lie within their stated stratum supports; the frequencies sum to one, and the positive response rates make both respondent means defined.

For A, $q_A=0.75(4)=3$. The stratum mean lies between $3+0.25(0)=3$ and $3+0.25(10)=5.5$. For B, $q_B=0.50(12)=6$. Its limits are $6+0.50(0)=6$ and $6+0.50(20)=16$.

The population lower limit is $0.70(3)+0.30(6)=3.90$. The upper limit is $0.70(5.5)+0.30(16)=3.85+4.80=8.65$. Thus the population mean score is bounded by 3.90 and 8.65 points.

For comparison, ignoring the narrower support in A and using the coarser common range $[0,20]$ gives the same observed contribution, $3.90$, and an overall missing fraction of $0.70(0.25)+0.30(0.50)=0.325$. The resulting upper limit is $3.90+0.325(20)=10.4$, which is wider. The reduction to 8.65 comes from using the valid, narrower support in A—not from stratification alone.

Worked Example: A Stratum With No Observed Outcomes

In a hypothetical population, stratum A has frequency $0.25$, no respondents, and outcome support $[2,8]$. Stratum B has frequency $0.75$, response rate $0.80$, respondent mean $5$, and the same support $[2,8]$. The frequencies sum to one; the observed mean in B is within support. There is no respondent mean to calculate in A, so use $q_c$, not $\rho_c E(Y\mid R=1,C=c)$, as the observed contribution.

In A, $\rho_A=0$ and $q_A=0$, so its conditional mean lies between $0+(1-0)(2)=2$ and $0+(1-0)(8)=8$. In B, $q_B=0.80(5)=4$; its limits are $4+0.20(2)=4.4$ and $4+0.20(8)=5.6$.

The population lower limit is $0.25(2)+0.75(4.4)=0.50+3.30=3.80$. The upper limit is $0.25(8)+0.75(5.6)=2.00+4.20=6.20$. The compatible population mean is therefore between 3.80 and 6.20 outcome units. No undefined respondent mean appears in either calculation.

Why the Bounds Are Sharp

The inequalities establish that no compatible mean can fall outside the stated limits. To see that the limits cannot be improved using only the given information, hold the entire observed-data distribution fixed. In each stratum, assign every missing outcome the value $L_c$. The observed outcomes, response indicators, covariates, and stratum frequencies remain unchanged, and the resulting population mean equals the lower limit. Assigning every missing outcome $U_c$ instead produces the upper limit.

This construction works when $\rho_c=0$: all outcomes in that stratum are unobserved, and assigning them to $L_c$ or $U_c$ still leaves the observed data unchanged. When $\rho_c=1$, there are no missing outcomes in the stratum, so its contribution is already determined by $q_c$, and its lower and upper limits coincide. These arguments establish sharpness relative to the stated support restrictions and observed-data distribution; additional credible restrictions could change the bounds.

Diagnostics, Alternatives, and Limitations

Before interpreting a result, check that the strata form a partition of the target population, that their frequencies are nonnegative and sum to one, and that each outcome range is valid for every individual in its stratum. Check the observed values against those ranges as well: an observed value outside $[L_c,U_c]$ signals an incorrect support specification, data problem, or mismatch between the target and the assumed model. Calculate $q_c$ directly from the observed-data distribution or as $\rho_c$ times the respondent mean only when $\rho_c>0$.

A frequent mistake is to calculate a respondent mean in every stratum and substitute it into a formula, even when a stratum has no respondents. That conditional mean is undefined when $\rho_c=0$. The correction is to use $q_c=E[Y1(R=1)\mid C=c]$, which equals zero in that stratum, and apply the limits $q_c+(1-\rho_c)L_c$ and $q_c+(1-\rho_c)U_c$.

Another common error is to assume that stratification must narrow the population bounds. With the same support $[L,U]$ in every stratum, the weighted lower limit simplifies to $\sum_c\pi_cq_c+L\sum_c\pi_c(1-\rho_c)$; the upper limit has the same form with $U$. Those are exactly the pooled bounds. Stratification can be informative when valid stratum-specific supports differ, or when additional restrictions link outcomes across or within strata. Without such information, merely dividing the data into groups does not reveal the missing outcomes.

These bounds are identification limits for a population distribution, not confidence limits calculated from a finite sample. In practice, $\pi_c$, $\rho_c$, and $q_c$ are estimated, and their sampling variability can matter. Very small strata may produce unstable estimates; overly fine stratification can leave little information for estimating response rates or observed contributions. The support assumptions are also substantive: a narrow but unjustified range can make bounds look precise without making them trustworthy. Model-based extrapolation or restrictions on missing outcomes may yield narrower conclusions, but those require assumptions beyond the unrestricted-support calculation here.

Key takeaway: Within each covariate stratum, the observed-outcome contribution is $q_c$, while the missing fraction contributes between $(1-\rho_c)L_c$ and $(1-\rho_c)U_c$. Weight these limits by the population stratum frequencies. This formulation remains defined even when a stratum has no observed outcomes.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A stratum has $\rho_c=0.60$, respondent mean $4$, and support $[0,10]$. What are its conditional mean bounds?
Why is $q_c$ preferable to $\rho_c E(Y\mid R=1,C=c)$ as the general definition of the observed contribution?
Two strata share support $[0,1]$. Does stratifying alone necessarily tighten the population-mean bounds? Explain.
State a construction that shows the population lower limit is sharp.

Solutions. (1) $q_c=0.60(4)=2.4$; the bounds are $2.4+0.40(0)=2.4$ and $2.4+0.40(10)=6.4$. (2) The respondent conditional mean is undefined if $\rho_c=0$, whereas $q_c=E[Y1(R=1)\mid C=c]$ is defined and equals zero there. (3) No. With common support, the weighted stratum limits simplify to the pooled limits using the overall observed contribution and missing fraction. (4) Keep the observed-data distribution fixed and assign all missing outcomes in each stratum to its lower support endpoint $L_c$; the resulting population mean equals the weighted lower limit.