Conditional Independence Given Covariates

Why Condition on Covariates?

An instrument need not be assigned by a lottery to be useful, but a nonrandomized instrument requires a clear account of why its assignment is plausibly unrelated to the potential outcomes and treatment choices that matter. Measured covariates can help: if instrument assignment is independent of those potential quantities within levels of the covariates, the instrument satisfies a conditional version of independence. This is a substantive assumption, not a consequence of including covariates in a regression.

For example, suppose access to a specialized clinic affects whether a patient receives a treatment. Clinic access is not randomly assigned: patients in urban and rural areas may differ in health, resources, or care options. Conditioning on relevant characteristics measured before access is determined may make comparisons more credible. It does not, by itself, establish that the instrument has no direct effect on the outcome, that it changes treatment receipt, or that all important common causes have been measured.

Definition: Let $Z$ be a binary instrument, $D(z)$ the treatment receipt that would occur under instrument value $z$, and $Y(d)$ the outcome that would occur under treatment value $d$. Let $X$ denote measured covariates determined before the instrument. Conditional instrument independence is $$ Z \perp \{Y(0),Y(1),D(0),D(1)\}\mid X. $$ It means that, at each covariate value $x$ in the target population, the distribution of the full set of potential outcomes and treatment receipts is the same for $Z=1$ and $Z=0$. Equivalently, for every possible set $A$ of values of these potential quantities, $$ P\big(\{Y(0),Y(1),D(0),D(1)\}\in A\mid Z=1,X=x\big) = P\big(\{Y(0),Y(1),D(0),D(1)\}\in A\mid Z=0,X=x\big). $$

The condition is stronger than equality of the observed means of a few baseline covariates. Potential outcomes are not jointly observed for any person, so this assumption cannot be directly tested from the observed data. Covariate balance can reveal some problems, but balance does not prove independence from unmeasured variables or from the unobserved potential outcomes.

What the Conditional Statement Does—and Does Not—Say

Conditional independence allows the instrument to be associated with covariates in the population. For instance, urban patients might be more likely than rural patients to live near a clinic. The requirement is that, after fixing urban or rural status (and other appropriate pre-instrument covariates), instrument assignment is unrelated to the potential treatment and outcome quantities. Thus, independence marginally across the whole population is not required.

Conversely, an apparent association between the instrument and an outcome can arise because the instrument groups contain different covariate distributions. Conditional independence addresses this by making the comparison within covariate values. The identifying claim still requires that the covariates suffice to remove relevant selection: if clinicians use unrecorded severity to direct patients toward the instrument, conditioning on recorded age and location may not be enough.

The covariate set must also be chosen with care. It should consist of variables determined before the instrument. A variable caused by the instrument may be a mediator or a selection variable; conditioning on it can change the comparison being made, and can introduce bias. “Adjust for more variables” is not a general solution to confounding.

Worked Example: Independence Within Two Covariate Strata

Imagine a hypothetical population divided into urban and rural patients. For illustration, group each patient by their complete potential-response profile $G$, which records the joint values of $Y(0),Y(1),D(0),D(1)$. The profiles cannot all be observed for an actual patient; this hypothetical table illustrates what the assumption means in a population.

Covariate stratum	Profile	Share when $Z=0$	Share when $Z=1$
Urban	A	60%	60%
Urban	B	40%	40%
Rural	A	30%	30%
Rural	B	70%	70%

Within the urban stratum, the profile distribution is identical across instrument groups: $0.60$ for A and $0.40$ for B. Within the rural stratum, it is also identical: $0.30$ for A and $0.70$ for B. If A and B exhaust the possible profiles in this illustration, these equalities give conditional independence in each stratum.

The profile distributions differ between urban and rural patients, so the table does not imply that $Z$ is independent of the potential quantities without conditioning on $X$. Nor could an analyst establish this assumption by tabulating $G$ in real data, because the full profile is unobserved. The example separates the meaning of the assumption from the evidence that can actually be assessed.

Additional Conditions in a Nonrandomized Instrument Example

Conditional independence is only one part of an instrumental-variable argument. The earlier material on IV assumptions establishes the roles of relevance, exclusion, and monotonicity. In a covariate-adjusted setting, those conditions need to make sense for the covariate strata being used, alongside consistency and a suitable instrument-support condition.

Conditional independence: after conditioning on pre-instrument $X$, instrument assignment is independent of $Y(0),Y(1),D(0),D(1)$. The substantive case for this must address both measured and plausible unmeasured determinants of assignment.
Exclusion: the instrument affects the outcome only through treatment receipt. In potential-outcome notation, the outcome under treatment $d$ does not depend on the instrument value; it is written $Y(d)$, not $Y(d,z)$. This is a substantive claim about the setting, not something achieved by covariate adjustment.
Conditional relevance: the instrument changes treatment receipt within the covariate strata that matter. A common diagnostic contrast is $$ P(D=1\mid Z=1,X=x)-P(D=1\mid Z=0,X=x). $$ If this contrast is zero in a stratum, the instrument provides no treatment-receipt variation there.
Monotonicity: for each individual, $D(1)\geq D(0)$, under the chosen direction of $Z$. This rules out individuals whose treatment receipt would move opposite to the instrument. It is an individual-level restriction, not a claim that every observed subgroup rate must increase without sampling variation.
Instrument positivity: for covariate values in the target population, both instrument values must be possible: $$ 0<P(Z=1\mid X=x)<1. $$ Without support for both values, the within-stratum comparison cannot be formed from the data.
Consistency and no interference: observed treatment and outcome agree with the potential quantities under the instrument and treatment actually received, and one person’s instrument or treatment does not alter another person’s outcome. The instrument and treatment must also be defined clearly enough that the potential outcomes have a consistent interpretation.

Worked Example: A Nonrandomized Clinic-Access Instrument

Consider a hypothetical evaluation of a treatment program. Let $Z=1$ mean living near a specialized clinic and $Z=0$ mean living farther away; let $D=1$ indicate receiving the program. Suppose proximity is not randomized. The analyst measures urban or rural residence, age group, baseline disease severity, and insurance status before proximity is assessed. A hypothetical sample gives the following treatment-receipt counts.

Stratum	$Z$	Number of patients	Received program	Receipt proportion
Urban	0	100	20	0.200
Urban	1	100	45	0.450
Rural	0	80	16	0.200
Rural	1	120	48	0.400

Within the urban stratum, the observed treatment-receipt difference is $$ \frac{45}{100}-\frac{20}{100}=0.450-0.200=0.250. $$ Within the rural stratum, it is $$ \frac{48}{120}-\frac{16}{80}=0.400-0.200=0.200. $$ These positive differences are evidence that proximity is associated with higher program receipt in each observed stratum, which is consistent with conditional relevance. The instrument has observed support in both strata: $P(Z=1\mid X=\text{urban})=100/200=0.500$, and $P(Z=1\mid X=\text{rural})=120/200=0.600$. Neither value is zero or one.

These calculations do not verify conditional independence. The analyst would need a credible argument that, within the measured strata, proximity is not additionally determined by potential outcomes or treatment propensities—for example, by unrecorded health severity or local treatment availability. Exclusion also needs scrutiny: living near the clinic might improve health through other services even for someone who does not receive the program. If so, proximity could affect the outcome through a route other than $D$, violating exclusion. Monotonicity requires a separate argument that proximity does not deter some individuals from receiving the program. The observed receipt proportions alone cannot establish this individual-level condition.

A Short Justification for Within-Stratum Comparisons

Why does conditioning on $X$ matter mathematically? Let $V$ denote the joint vector $\{Y(0),Y(1),D(0),D(1)\}$. Conditional independence says the conditional distribution of $V$ is the same in the two instrument groups at a fixed $x$. Therefore, for any function $g(V)$ with a finite expectation, the distributional equality implies equal conditional expectations:

P(V\in A\mid Z=1,X=x)=P(V\in A\mid Z=0,X=x) \quad\Longrightarrow\quad E[g(V)\mid Z=1,X=x]=E[g(V)\mid Z=0,X=x].

For a direct justification, the expectation of $g(V)$ is an average of its values over the distribution of $V$. If that distribution is identical for $Z=1$ and $Z=0$ within $x$, the averages are identical as well. This is the exchangeability property that makes a within-stratum instrument comparison meaningful under the assumption. It does not establish exclusion or relevance; those concern different links in the IV argument.

Worked Example: A Positivity Failure Hidden by Overall Variation

Suppose a hypothetical program is introduced only in rural clinics. The data include 100 urban patients, all with $Z=0$, and 100 rural patients, split evenly between $Z=0$ and $Z=1$. Overall, both instrument values occur: 150 patients have $Z=0$ and 50 have $Z=1$. But within the urban stratum, $$ P(Z=1\mid X=\text{urban})=\frac{0}{100}=0. $$

Thus, instrument positivity fails for urban patients. There is no observed urban comparison between $Z=1$ and $Z=0$, so the data cannot support a within-urban-stratum instrument contrast. Overall presence of both instrument values does not fix this failure: the instrument values must overlap within the covariate strata relevant to the intended analysis. The analyst could define a narrower target population with support, if substantively appropriate, but should not pretend the missing comparison has been observed.

Diagnostics, Alternatives, and Limitations

Diagnostics can identify warning signs, though none can confirm independence of the full potential-outcome vector. Start by checking whether every covariate used for conditioning is measured before the instrument. Tabulate instrument probabilities by important covariate strata and inspect whether some strata contain few or no observations in one instrument group. Assess treatment-receipt differences within strata and their uncertainty; a positive point estimate based on sparse data may provide little practical information. Examine observed baseline covariate balance within strata as a useful diagnostic, while recognizing that it tests only observed variables.

A regression adjustment can model the relationship between $Z$, $X$, and the observed variables, but a fitted model cannot convert an implausible instrument into a valid one. In particular, a regression that includes $X$ does not establish conditional independence, repair a direct effect of $Z$, or create treatment-receipt variation where none exists. Matching or weighting on measured covariates can help construct comparisons with similar observed $X$, but cannot balance unmeasured factors by construction. These are ways to implement or assess adjustment, not substitutes for the causal assumptions.

Unconditional instrument independence is a stronger design condition: it requires independence without conditioning on $X$, as can be plausible under genuine random assignment. Conditional independence is appropriate when instrument assignment is nonrandom but becomes plausibly as-if independent within well-defined, pre-instrument strata. The latter relies on the adequacy of the adjustment set and sufficient overlap. It can fail when important causes of instrument assignment and potential outcomes remain unmeasured, when covariates are measured poorly, or when stratification creates sparse cells.

A final limitation is that conditioning does not automatically specify how comparisons from different strata should be combined into a single target effect. That requires additional estimand and aggregation choices. Here, the essential point is narrower: before any such comparison can be credible, the instrument must be plausibly independent of the relevant potential quantities within the covariate values being used, and the remaining IV conditions must also be defensible.

Common Mistakes

“The instrument is associated with covariates, so conditional independence fails.” Not necessarily. Conditional independence permits marginal association if the potential quantities are independent of $Z$ within $X$.
“The measured covariates are balanced, so the assumption is proven.” Balance is diagnostic evidence about recorded variables, not proof of independence from unmeasured causes or potential outcomes.
“Including $X$ in a regression makes the instrument valid.” Adjustment does not establish conditional independence or exclusion, and it cannot repair absent overlap or conditional relevance.
“Both instrument values occur in the full sample, so positivity holds.” Positivity must hold within the covariate strata used for the comparison.
“A positive first-stage difference proves monotonicity.” The difference is a group-level contrast; monotonicity rules out individual defiers and is not verified by that contrast.

Key takeaway: Conditional instrument independence means equality of the full potential-outcome and potential-treatment distribution across instrument groups at fixed pre-instrument covariates. In a nonrandomized setting, that assumption must be justified alongside exclusion, conditional relevance, monotonicity, consistency, no interference, and within-stratum instrument support.

Check Your Understanding

Answer each question, then compare with the solutions below.

State conditional instrument independence using $Z$, $X$, and the potential quantities.
Why does observed balance on age and disease severity not prove conditional independence?
In a covariate stratum, the treatment-receipt proportions are 0.35 for $Z=1$ and 0.20 for $Z=0$. What is the conditional first-stage difference, and what does it show?
In one stratum, every person has $Z=0$. Which condition fails, and why can overall instrument variation not repair it?
Give one reason a post-instrument variable should not be included automatically in $X$.

Solutions. (1) $Z\perp\{Y(0),Y(1),D(0),D(1)\}\mid X$: within each relevant $x$, the joint potential-quantity distribution is the same across instrument values. (2) The check covers only observed covariates; unmeasured determinants and potential outcomes remain unobserved. (3) The difference is $0.35-0.20=0.15$, a 15-percentage-point conditional treatment-receipt contrast, consistent with relevance in that stratum but not proof of the other IV assumptions. (4) Instrument positivity fails because $P(Z=1\mid X=x)=0$; variation in other strata does not supply the missing within-stratum comparison. (5) It may be caused by the instrument, so conditioning on it can alter the comparison or introduce bias rather than control a pre-existing confounder.