Positivity and Empty-Pattern Identification Failures

Why Positivity and Empty Patterns Matter

An outcome distribution can be unidentified for a local reason: in a particular covariate or history stratum, nobody has an observed value of that outcome. This is different from having only a small number of observations. A small number may make estimates unstable; a true zero response probability leaves the observed-data law with no direct outcome information from that stratum. In longitudinal data, the same issue arises when an observation pattern that would reveal an outcome is absent in a stratum.

The key is to specify the target before diagnosing the problem. Is the goal the outcome distribution among people with a particular covariate value, or the outcome distribution in a particular response pattern? Is the relevant outcome observed under any pattern that occurs in the stratum? These distinctions prevent a zero cell from being treated as a generic modelling problem.

Definition: Let \(X\) denote covariates or a recorded history, \(Y\) an outcome (or outcome vector), and \(R\) the observation pattern. For a stratum \(X=x\), response positivity for pattern \(r\) means \(P(R=r\mid X=x)>0\). An empty pattern in a stratum means that the pattern has zero probability there in the population, or that it has zero observed cases in the sample; these are distinct claims. For pattern \(r\), let \(O(r)\) be the set of outcome components observed under that pattern.

The population statement \(P(R=r\mid X=x)=0\) is a structural zero. In contrast, an empty sample cell does not establish a structural zero: a pattern with positive population probability can fail to appear in a finite sample. A further distinction is essential. Positive probability of a pattern does not mean all components of \(Y\) are observed under it. The pattern reveals only the components in \(O(r)\).

What the Observed Data Can Support

In a pattern-mixture representation, the full outcome distribution conditional on \(X=x\) is decomposed across response patterns. Write \(\pi_r(x)=P(R=r\mid X=x)\). For any outcome event \(A\), the decomposition is

P(Y\in A\mid X=x) = \sum_{r:\,\pi_r(x)>0} \pi_r(x)P(Y\in A\mid R=r,X=x).

This is the law of total probability, not an identifying assumption. The observed data identify the pattern probabilities \(\pi_r(x)\), subject to the usual support and sampling requirements. Within a pattern, the data reveal the distribution of components in \(O(r)\). They do not, without further assumptions, reveal the conditional distribution of components missing under that pattern.

If \(\pi_r(x)=0\), the population contains no members of that pattern in that stratum. Its conditional distribution \(P(Y\mid R=r,X=x)\) is not defined by ordinary conditioning on a positive-probability event; importantly, its mixture contribution is also zero. The identification failure for a target such as \(P(Y\mid X=x)\) occurs when the patterns that do occur leave some outcome components unobserved and the observed data provide no restriction on their values. A particularly stark case is a stratum with no observed outcomes at all.

Key distinction: Positivity is a support condition, not an imputation rule. If the only patterns with observed values of a target outcome have zero probability in a stratum, the observed data cannot directly identify that outcome distribution there. Even when an outcome is observed for some people in the stratum, positivity by itself does not identify its distribution among people whose value is missing.

Some methods require stronger, method-specific conditions. For example, inverse-probability weighting requires positive response probability for every target stratum being represented by observed outcomes; otherwise the corresponding inverse probability is undefined. A conditional-independence assumption such as missing at random may provide a route to identification when its assumptions and positivity conditions hold, but it cannot create observed data in a stratum where the relevant response probability is zero. Any claim that a model fills such a gap must therefore rest on restrictions beyond the observed data.

A Mixture Decomposition and Its Implication

For a binary outcome \(Y\), define \(q_r(x)=P(Y=1\mid R=r,X=x)\) whenever \(\pi_r(x)>0\). The outcome prevalence in the stratum is

P(Y=1\mid X=x)=\sum_{r:\,\pi_r(x)>0}\pi_r(x)q_r(x).

If \(Y\) is observed under pattern \(r\), its pattern-specific prevalence \(q_r(x)\) is estimable from that pattern, subject to adequate data. If \(Y\) is not observed under \(r\), then \(q_r(x)\) is not obtained from the observed outcomes under that pattern. Since a binary prevalence must lie between 0 and 1, leaving one or more such terms unrestricted produces bounds rather than a point value.

List the target stratum and patterns.
For each pattern, record its probability in the stratum and which outcome components it observes.

Mark observed and unobserved contributions.
Use the observed-data distribution for components actually seen. Do not assign an observed value to a component missing under a pattern.

Check support, then state the conclusion.
If a needed pattern has zero probability, direct estimation from that pattern is unavailable. Report a bound or state the additional extrapolation assumption required for a point estimate.

This decomposition also clarifies why positivity and identification are related but not interchangeable. A positive pattern probability can make some pattern-specific quantities observable, but not necessarily the entire outcome vector. Conversely, a pattern with zero probability contributes nothing to the mixture, although its absence can still reveal that no within-pattern estimate is available for that stratum.

Worked Example: No Responses in a Covariate Stratum

Suppose \(Y\) is binary and \(X\) has two strata, \(A\) and \(B\), with \(P(X=A)=0.80\) and \(P(X=B)=0.20\). In stratum \(A\), assume \(Y\) is observed for everyone and its prevalence is 0.40. In stratum \(B\), no outcomes are observed. What can be said about the overall prevalence \(P(Y=1)\), without assumptions connecting the strata?

Let \(q_B=P(Y=1\mid X=B)\). Because \(Y\) is binary and never observed in \(B\), the observed data impose only \(0\leq q_B\leq1\). The law of total probability gives

P(Y=1)=P(X=A)P(Y=1\mid X=A)+P(X=B)q_B.

Substituting the given values,

$$ P(Y=1)=0.80(0.40)+0.20q_B=0.32+0.20q_B. $$

At \(q_B=0\), the prevalence is 0.32; at \(q_B=1\), it is 0.52. Thus the sharp range based on these data and the binary-outcome restriction is \([0.32,0.52]\). Both endpoints are achievable by assigning every unobserved outcome in \(B\) to 0 or to 1, respectively. A point estimate inside that interval would require an additional assumption or information source.

Worked Example: A Response-Rate Bound Within One Stratum

In a hypothetical covariate stratum, outcomes are observed for 25% of people. Among those with observed outcomes, 40% have \(Y=1\). Assume the remaining outcomes are completely unobserved and make no assumption about their distribution. Find the sharp bounds on the stratum prevalence.

The observed contribution to the prevalence is the response fraction multiplied by the observed prevalence:

P(R=1\mid X=x)P(Y=1\mid R=1,X=x)=0.25(0.40)=0.10.

The missing fraction is \(1-0.25=0.75\), not 0.25. If none of those missing outcomes equals 1, their contribution is 0; if all equal 1, their contribution is 0.75. Therefore,

P(Y=1\mid X=x)\in[0.10,\;0.10+0.75]=[0.10,\;0.85].

These bounds are sharp: the lower endpoint is attained when all missing outcomes are 0, and the upper endpoint when all are 1. The observed response group does not determine which value within the interval is the actual stratum prevalence.

Worked Example: An Empty Longitudinal Pattern

Consider two binary outcomes, \(Y_1\) and \(Y_2\), in a fixed history stratum. Let pattern \(R=11\) mean both outcomes are observed and \(R=10\) mean only \(Y_1\) is observed. Suppose \(P(R=11\mid X=x)=0.60\), \(P(R=10\mid X=x)=0.40\), and patterns \(01\) and \(00\) are absent in this stratum. Among those in pattern \(11\), the observed prevalence of \(Y_2=1\) is 0.30. Find the data-supported range for \(P(Y_2=1\mid X=x)\).

The pattern \(11\) contributes \(0.60(0.30)=0.18\) to the prevalence. In pattern \(10\), \(Y_2\) is unobserved. Let \(q_{10}=P(Y_2=1\mid R=10,X=x)\), which can range from 0 to 1 without further restrictions. The absent patterns have zero mixture weight. Thus,

P(Y_2=1\mid X=x)=0.60(0.30)+0.40q_{10} =0.18+0.40q_{10}\in[0.18,\;0.58].

The empty patterns do not supply observations for \(Y_2\), and the positive pattern \(10\) does not observe \(Y_2\) either. Hence the observed pattern \(11\) alone does not identify the stratum prevalence. The interval follows directly from allowing the unobserved \(Y_2\) values in pattern \(10\) to range from all 0 to all 1.

Structural Zeros, Sample Zeros, and Diagnostics

Worked Example: An Empty Sample Cell Is Not Proof of a Structural Zero

Suppose a study has 10 people in a high-risk stratum, and each independently has response probability \(p=0.20\). The probability that the sample contains no responses in this stratum is

P(\text{zero responses})=(1-p)^{10}=(0.80)^{10}=0.1074.

So a zero response count can occur with probability about 10.7% even though the population response probability is positive. The empty sample cell is evidence of sparse information, not proof that the population response probability equals zero. If, instead, the population response probability is truly zero, no sample size from that same mechanism will produce an observed outcome in that stratum.

This distinction affects analysis. With a zero observed response count, an estimated response probability is zero and its inverse is undefined. With only one response among 20 people, the estimated response probability is \(1/20=0.05\), giving an inverse weight of \(1/0.05=20\). That large weight signals instability; it does not repair the lack of support. The assumed value \(p=0.20\) in the calculation above is a hypothetical population value used to show why a sample zero is possible.

Useful diagnostics begin with cross-tabulating response patterns against covariates and relevant recorded histories. Report stratum counts and pattern counts, not just an overall response percentage. Then ask whether zeros are dictated by design or eligibility, or whether they may be finite-sample zeros. In longitudinal studies, inspect which patterns reveal each outcome component; a positive total sample size does not guarantee that any observed pattern reveals the component of interest.

A model that borrows information across strata can return predictions where the data contain no direct outcome observations. Those predictions may be useful, but they are driven by modelling restrictions such as a shared regression form, smoothness, or a conditional-independence assumption. They should not be described as identified by the observed data alone. Likewise, simply collapsing strata may restore apparent positivity but changes the target and can hide important heterogeneity.

How This Failure Differs From Nearby Problems

Situation	What the data establish	What remains unresolved
Positive response probability, with observed outcomes in a stratum	The outcome distribution among those observed there can be estimated, given adequate data.	The distribution among people whose outcomes are missing may still require assumptions.
Structural zero response probability in a target stratum	The response pattern has zero mass in that stratum.	A component observed only under that pattern has no direct observations there; other occurring patterns may reveal other components. Unsupported components may require extrapolation or bounds.
Zero sample count but positive population probability possible	The sample contains no cases in the cell.	The population probability and outcome distribution are not established as zero or absent.
Positive longitudinal pattern that omits a target component	Components observed under that pattern can be described.	The distribution of the omitted component within the pattern is not directly observed.

Compared with weighting, a parametric outcome model may extrapolate across unsupported strata, but it does so by imposing a functional form rather than by recovering missing support. A pattern-mixture analysis makes the unobserved pattern-specific distributions explicit; absent data leave those components unrestricted unless assumptions are added. Bounds state what follows without selecting one such assumption. These approaches answer different questions, and none makes a structural zero disappear.

Common Mistakes and Limitations

Using the response fraction as the missing fraction. If 25% have observed outcomes, then 75% are missing. In a binary-outcome bound, it is the missing 75% that can contribute anywhere from 0 to 0.75.
Calling every empty sample cell a population zero. A finite sample can miss a pattern with positive probability. Structural zeros require a population or design-based justification.
Assuming a positive pattern probability identifies every outcome. A pattern reveals only its observed components; it says nothing directly about components missing under that pattern.
Treating model predictions as data identification. Predictions in a stratum with no outcome observations depend on assumptions linking that stratum to observed strata.
Confusing no direct information with no possible restrictions. External knowledge or explicit assumptions can narrow bounds, but those restrictions must be stated and justified rather than attributed to the empty cell.

Nonparametric conclusions are limited by support. Bounds can be wide, and a point estimate may be unavailable, even with a large total sample, if the relevant outcome-observation patterns are absent from a target stratum. Conversely, a small sample cell is primarily a precision and stability warning unless there is evidence that the population probability is actually zero.

Key takeaway: For each target stratum and outcome component, verify that a pattern occurring in that stratum actually observes the component. A zero response probability or absent informative pattern blocks direct identification there; any point conclusion beyond the observed support depends on additional assumptions.

Check Your Understanding

Answer each question using only the stated information. Solutions follow the questions.

In a stratum, 30% of outcomes are observed and the observed binary-outcome prevalence is 0.50. With no assumptions about missing outcomes, what are the sharp bounds on the stratum prevalence?
A pattern has population probability zero in a covariate stratum. Does its conditional outcome distribution contribute to the full-data mixture in that stratum?
In a two-outcome study, pattern \(10\) occurs but does not observe \(Y_2\), while pattern \(11\) is absent. What should be checked before claiming that \(Y_2\)'s distribution is identified in the stratum?
Why does a zero response count in a finite sample not, by itself, prove a structural positivity violation?

Solutions. (1) The observed contribution is \(0.30(0.50)=0.15\). The missing fraction is \(0.70\), so the missing contribution ranges from 0 to 0.70; the sharp bounds are \([0.15,0.85]\). (2) No. Its mixture weight is zero, so it contributes zero; a conditional distribution given a zero-probability event is not identified by ordinary conditioning. (3) Check whether any other pattern occurring in that stratum observes \(Y_2\). Pattern \(10\) does not, and the absent pattern \(11\) supplies no cases, so direct information about \(Y_2\) may be absent; identification would require other observed patterns or additional assumptions. (4) If the population response probability is positive, a finite random sample can still contain no responses. The sample zero establishes an empty sample cell, not a population probability of zero.