Conditional IV Assumptions Within Covariate Strata

Why State IV Assumptions Within Strata?

An instrument can behave differently across groups defined by characteristics measured before the instrument is assigned. Assignment may be as-if random within clinic but not across clinics, or an instrument may change treatment use substantially for one baseline-risk group and only slightly for another. Stating assumptions within covariate strata makes clear where the IV argument is intended to apply. It also prevents a crucial error: a strong first stage within a group does not, by itself, establish that the instrument is valid there.

Let \(Z\) be a binary instrument, \(D\) a binary treatment, and \(Y\) an outcome. Let \(X\) denote a vector of covariates measured before \(Z\) is assigned. Write \(D(z)\) for the treatment that would occur under instrument value \(z\), and \(Y(d)\) for the outcome under treatment value \(d\). Assume consistency and no interference: the observed treatment is \(D=D(Z)\), and the observed outcome is \(Y=Y(D)\) when exclusion holds. The statements below concern covariate values \(x\) in the target population's support.

Four Assumptions Within a Stratum

Conditional relevance means that the instrument changes treatment probability among individuals with the same covariate value:

This difference is the first-stage contrast within stratum \(x\). For an analysis comparing both instrument levels within that stratum, there must also be instrument overlap: \(0<P(Z=1\mid X=x)<1\). Overlap makes both conditional treatment probabilities estimable in principle; relevance says that their difference is nonzero. They are distinct requirements.

Conditional independence says that, given \(X\), assignment of the instrument is independent of the potential treatment and outcome values:

In words, within each stratum, the people receiving \(Z=1\) and \(Z=0\) must not differ systematically in these potential quantities. This is often motivated by a random or as-if-random assignment mechanism within strata. It is a substantive design or causal assumption, not something established merely by observing similar baseline averages.

Exclusion says that the instrument affects the outcome only through the treatment. In potential-outcome notation, the outcome under instrument value \(z\) and treatment \(d\), written \(Y(z,d)\), must satisfy:

This restriction can be relevant within every \(X=x\) stratum, but conditioning on \(X\) does not make it true. For example, if an encouragement changes access to supportive counseling that itself affects \(Y\), exclusion fails unless that pathway is part of the treatment as defined.

Conditional monotonicity rules out individuals whose treatment moves in the direction opposite to the instrument:

With a binary instrument and treatment, this rules out defiers, for whom \(D(1)=0\) and \(D(0)=1\). It is an individual-level restriction, not a claim that the average first stage is positive. A positive first stage can occur even if some individuals are defiers.

Worked Example: Checking Relevance and Overlap by Stratum

Which Variables Can Define Strata?

A variable used to define these strata must be measured before the instrument is assigned and must represent a characteristic that is fixed for the assignment comparison. Examples can include age at enrollment, baseline disease severity, or clinic when assignment procedures differ by clinic. The timing requirement matters because conditioning on a variable affected by \(Z\) can select people based on a consequence of the instrument rather than compare like baseline groups.

Timing alone is not enough to justify stratification. The choice of \(X\) should be tied to the assignment mechanism and the causal structure. A pre-instrument variable can be a useful adjustment variable, but merely dividing the data by every available baseline measure does not guarantee conditional independence. The relevant claim is that the chosen \(X\) makes instrument assignment independent of the potential outcomes and treatments within strata.

Variables measured after assignment are generally not eligible to define the baseline strata for these assumptions. Examples include treatment receipt, follow-up attendance, a post-assignment biomarker, or whether a participant accepted the encouragement. Such variables may be affected by \(Z\), by \(D\), or by causes of \(Y\). Conditioning on them can change the comparison and may introduce selection bias. They are not converted into legitimate baseline covariates by being recorded in the data.

A useful distinction is between a variable's measurement time and its role in the causal process. A variable recorded before assignment is temporally eligible, but its use must still be justified by the assignment mechanism. Conversely, a post-assignment variable cannot serve as a baseline stratum for conditional independence just because it predicts the outcome well.

Worked Example: A Positive First Stage Does Not Prove Monotonicity

Why the Conditional Assumptions Identify a Within-Stratum Effect

The assumptions have a precise payoff: when they hold in stratum \(x\), the conditional Wald ratio identifies the average treatment effect among compliers in that stratum. This is a local effect for people whose treatment status is changed by the instrument, not automatically an average treatment effect for everyone in the stratum.

To see why, conditional independence allows the observed contrasts between \(Z=1\) and \(Z=0\) within \(x\) to be represented as contrasts between the corresponding potential values. Under exclusion and consistency, the outcome contrast is:

For always-takers and never-takers, \(D(1)=D(0)\), so their contribution to this expression is zero. Under monotonicity, everyone with \(D(1)\ne D(0)\) is a complier, for whom \(D(1)=1\) and \(D(0)=0\). Hence the numerator equals:

The same conditional independence argument expresses the first-stage contrast as:

where monotonicity ensures that only compliers contribute a value of 1 and no defiers contribute \(-1\). Conditional relevance makes this denominator nonzero. Dividing the outcome contrast by the first-stage contrast yields the stated complier average effect. The derivation depends on all of the assumptions: relevance alone cannot replace independence, exclusion, or monotonicity.

Common Mistakes

• Calling a large first stage proof of a valid instrument. A large within-stratum treatment contrast establishes relevance, not conditional independence or exclusion.
• Using a positive first stage to claim there are no defiers. Positive average changes can coexist with some individuals whose treatment moves in the opposite direction.
• Conditioning on a post-instrument variable. Treatment receipt, follow-up attendance, or another consequence of assignment cannot be treated as a baseline stratum just because it is measured precisely.
• Assuming that any baseline covariate guarantees conditional independence. The covariate set must be justified by the assignment mechanism; baseline timing alone is not a proof.
• Treating conditional exclusion as automatic. Stratification does not remove an instrument-to-outcome pathway that bypasses treatment.