Principal Strata and Compliance Types

Why Compliance Types Matter

With a binary instrument and a binary treatment, a unit has two potential treatment decisions: the decision it would make if the instrument were set to 0, and the decision it would make if the instrument were set to 1. The instrument reveals only one of these decisions for any one unit. Comparing the two potential decisions nevertheless gives a useful classification of units: principal strata, commonly called compliance types.

This classification describes how treatment responds to the instrument, not how outcomes respond to treatment. It is therefore distinct from an outcome effect. The classification helps clarify what is observed in each instrument group and what assumptions, especially monotonicity, imply about those groups.

Definition: Let $Z\in\{0,1\}$ be a binary instrument and $D(z)\in\{0,1\}$ a unit’s potential treatment decision when the instrument is set to $z$. The unit’s principal stratum is the ordered pair $(D(0),D(1))$. The four possible strata are:

Always-takers: $(1,1)$, so $D(0)=1$ and $D(1)=1$.
Compliers: $(0,1)$, so $D(0)=0$ and $D(1)=1$.
Never-takers: $(0,0)$, so $D(0)=0$ and $D(1)=0$.
Defiers: $(1,0)$, so $D(0)=1$ and $D(1)=0$.

The first coordinate records the decision under $Z=0$; the second records the decision under $Z=1$. This ordering matters. For example, a complier takes treatment when the instrument is 1 but not when it is 0. A defier does the reverse.

From Potential Decisions to Observed Treatment

The observed instrument value determines which potential treatment decision is realized. Under consistency, the observed treatment is $D=D(Z)$. Thus, when $Z=0$, we observe $D(0)$, and when $Z=1$, we observe $D(1)$. The following table applies this rule to every type.

Principal stratum	$D(0)$	$D(1)$	Observed $D$ if $Z=0$	Observed $D$ if $Z=1$
Always-taker	1	1	1	1
Complier	0	1	0	1
Never-taker	0	0	0	0
Defier	1	0	1	0

The table shows why observing a unit’s instrument value and treatment status does not generally reveal its type. For instance, a unit with $Z=0$ and $D=1$ could be an always-taker or a defier. Its unobserved $D(1)$ distinguishes those types. Similarly, $Z=1$ and $D=1$ is compatible with an always-taker or a complier. The full pair of potential decisions is a unit-level, generally unobserved classification.

Worked Example: Classifying Potential Treatment Decisions

Consider four hypothetical units whose potential decisions are known for illustration. Classify each unit and state the treatment that would be observed under each instrument value.

Unit A has $(D(0),D(1))=(1,1)$. It is an always-taker: it receives treatment under either instrument value. If $Z=0$, its observed $D$ is 1; if $Z=1$, its observed $D$ is also 1.

Unit B has $(D(0),D(1))=(0,1)$. It is a complier: its observed $D$ is 0 when $Z=0$, and 1 when $Z=1$.

Unit C has $(D(0),D(1))=(0,0)$. It is a never-taker: its observed $D$ is 0 for either instrument value.

Unit D has $(D(0),D(1))=(1,0)$. It is a defier: its observed $D$ is 1 when $Z=0$, and 0 when $Z=1$.

In an actual study, each unit is observed under only one value of $Z$, so only one entry in its pair is revealed. This illustration classifies units from both potential decisions; it does not suggest that both can ordinarily be measured for the same unit.

How Types Contribute to Instrument-Group Treatment Rates

Let $S$ denote a unit’s principal stratum, and let $\pi_s(z)=P(S=s\mid Z=z)$ be the proportion of type $s$ among units assigned or exposed to instrument value $z$. The observed treatment rate in an instrument group is the sum of the proportions of types that take treatment under that instrument value.

P(D=1\mid Z=0)=\pi_{\text{always}}(0)+\pi_{\text{defier}}(0), \qquad P(D=1\mid Z=1)=\pi_{\text{always}}(1)+\pi_{\text{complier}}(1).

These equations follow directly from the classification table. With $Z=0$, only always-takers and defiers have $D(0)=1$. With $Z=1$, only always-takers and compliers have $D(1)=1$. The remaining types contribute to the untreated shares in their respective groups:

P(D=0\mid Z=0)=\pi_{\text{complier}}(0)+\pi_{\text{never}}(0), \qquad P(D=0\mid Z=1)=\pi_{\text{never}}(1)+\pi_{\text{defier}}(1).

These expressions are valid without assuming that the instrument is independent of potential treatment decisions. But if the types are distributed differently across instrument groups, $\pi_s(0)$ and $\pi_s(1)$ need not be equal. To express both treatment rates using one set of type shares, we need instrument independence: $Z$ is independent of the pair $(D(0),D(1))$. Under that assumption, write $\pi_s=P(S=s)$; then $\pi_s(0)=\pi_s(1)=\pi_s$.

Implication under instrument independence: The difference between the observed treatment rates in the two instrument groups is the share of compliers minus the share of defiers: $$ P(D=1\mid Z=1)-P(D=1\mid Z=0) =\pi_{\text{complier}}-\pi_{\text{defier}}. $$ This is a statement about treatment uptake and type proportions; it does not by itself describe any outcome effect.

To verify the result, substitute the two treatment-rate expressions under independence:

\begin{aligned} P(D=1\mid Z=1)-P(D=1\mid Z=0) &=(\pi_{\text{always}}+\pi_{\text{complier}}) -(\pi_{\text{always}}+\pi_{\text{defier}})\\ &=\pi_{\text{complier}}-\pi_{\text{defier}}. \end{aligned}

The always-taker share cancels because always-takers receive treatment in both groups; the never-taker share does not appear because never-takers receive treatment in neither group. The direction and size of the overall treatment-rate difference therefore depend on the balance between compliers and defiers. A positive difference alone does not prove that there are no defiers.

Worked Example: Treatment Rates With All Four Types

Suppose a hypothetical population has 1,000 units: 100 always-takers, 400 compliers, 400 never-takers, and 100 defiers. Assume the instrument is independent of type and, for simple arithmetic, each instrument group contains the same type proportions. Calculate the treatment rate in each group.

The type shares are $\pi_{\text{always}}=0.10$, $\pi_{\text{complier}}=0.40$, $\pi_{\text{never}}=0.40$, and $\pi_{\text{defier}}=0.10$. In the $Z=0$ group, treatment is received by always-takers and defiers, so $$ P(D=1\mid Z=0)=0.10+0.10=0.20. $$

In the $Z=1$ group, treatment is received by always-takers and compliers, so $$ P(D=1\mid Z=1)=0.10+0.40=0.50. $$

The treatment-rate difference is $0.50-0.20=0.30$. The type-share calculation gives the same result: $0.40-0.10=0.30$, the complier share minus the defier share. The example also shows why the treatment-rate difference is not the complier share when defiers are present: here the difference is 0.30, while the complier share is 0.40.

Monotonicity and What It Changes

A common additional restriction is monotonicity: no unit has a treatment decision that moves opposite to the instrument’s direction. With the coding used here, the restriction is $D(1)\geq D(0)$ for every unit. Since treatment is binary, this rules out the pair $(1,0)$, and therefore rules out defiers. Monotonicity does not rule out always-takers or never-takers; their decisions do not change with the instrument.

D(1)\geq D(0)\ \text{for every unit} \quad\Longleftrightarrow\quad \pi_{\text{defier}}=0.

If instrument independence also holds, the group treatment rates simplify to

P(D=1\mid Z=0)=\pi_{\text{always}}, \qquad P(D=1\mid Z=1)=\pi_{\text{always}}+\pi_{\text{complier}}.

Thus, under both independence and monotonicity, the difference in treatment rates equals the complier share. This conclusion concerns the proportion of units whose treatment decision is changed by the instrument. It does not identify which observed units are compliers, because their unobserved potential decision remains unknown.

Monotonicity also changes what can be inferred from an individual’s observed pair $(Z,D)$. Under this restriction, a unit with $Z=0,D=1$ must be an always-taker: a defier is ruled out. Likewise, a unit with $Z=1,D=0$ must be a never-taker. But a unit with $Z=0,D=0$ could be a complier or a never-taker, and a unit with $Z=1,D=1$ could be a complier or an always-taker. Monotonicity narrows the possibilities; it does not reveal every type.

Worked Example: Recovering Type Shares Under Monotonicity

In a hypothetical study, suppose the observed treatment rate is 0.18 among units with $Z=0$, and 0.63 among units with $Z=1$. Assume instrument independence and monotonicity. Find the four principal-stratum shares.

Monotonicity implies $\pi_{\text{defier}}=0$. Under independence, the $Z=0$ treatment rate is the always-taker share, so $\pi_{\text{always}}=0.18$. The $Z=1$ treatment rate is the sum of always-takers and compliers, so $$ \pi_{\text{complier}}=0.63-0.18=0.45. $$

The never-taker share is the fraction untreated when $Z=1$, because under monotonicity a unit untreated at $Z=1$ is a never-taker. Hence $$ \pi_{\text{never}}=1-0.63=0.37. $$

The shares sum to $0.18+0.45+0.37+0=1.00$, as required. These calculations identify population proportions under the stated assumptions, not the stratum membership of each individual unit.

What the Classification Does—and Does Not—Assume

Compliance type is defined entirely by potential treatment decisions. Neither an outcome model nor an exclusion restriction is needed to define the four types or derive the treatment-status table. Exclusion concerns how the instrument may affect outcomes; it does not alter which treatment decision a unit would make under each instrument value. Conversely, instrument independence matters when translating treatment rates across instrument groups into common population type shares. Without independence, the group-specific shares may differ.

Relevance is also a group-level property: it concerns whether the instrument is associated with a change in observed treatment. A positive treatment-rate difference under independence means that the complier share exceeds the defier share. Only when monotonicity additionally rules out defiers does that difference equal the complier share. These statements should not be confused with the stronger and generally unavailable claim that every unit responds to the instrument in the same way.

Common Mistakes

Reversing the ordered pair. The first entry is $D(0)$, and the second is $D(1)$. Thus $(0,1)$ denotes compliers, not defiers.
Claiming that an observed unit is a complier because its treatment matches the instrument. For $Z=1,D=1$, for example, the unit could be a complier or an always-taker. The unobserved potential decision prevents direct classification.
Equating a treatment-rate difference with the complier share without checking assumptions. With instrument independence but possible defiers, the difference is the complier share minus the defier share. Monotonicity is needed to eliminate the defier share.
Assuming the instrument groups have the same type composition automatically. This equality follows from independence of $Z$ and the potential treatment decisions; it is not a consequence of the type definitions alone.

Key takeaway: Principal strata classify units by the pair $(D(0),D(1))$. The classification gives exact mappings from type to observed treatment in each instrument group. Independence allows group treatment rates to be written using common type shares, while monotonicity rules out defiers and simplifies those relationships.

Check Your Understanding

Use the ordered pair convention $(D(0),D(1))$ in each answer.

Classify a unit with potential treatment decisions $(1,0)$, and state its observed treatment under each instrument value.
Which principal strata can produce $D=1$ when $Z=0$? Which can produce $D=1$ when $Z=1$?
Under instrument independence, express the difference in treatment rates between $Z=1$ and $Z=0$ using the complier and defier shares.
What does monotonicity rule out, and which two types can be inferred from $Z=0,D=1$ or $Z=1,D=0$ under that assumption?
Why does observing one unit’s instrument value and treatment status generally not reveal its principal stratum?