Positivity Diagnostics for Exposure Conditions

Why Exposure Support Needs Its Own Diagnostic

Under interference, a person’s treatment assignment alone does not describe the condition relevant to a causal contrast. The exposure mapping also summarizes relevant assignments among peers, such as the number or proportion of treated neighbors. A contrast may therefore require observations in several combined conditions—for example, treated with no treated neighbors and untreated with no treated neighbors. A randomized design can assign treatment to everyone while still making one of these exposure conditions impossible or rare.

This is a design-support issue, not merely a question of whether the dataset contains many individuals. Network structure, the assignment mechanism, and the chosen exposure mapping jointly determine how likely each unit is to receive each exposure condition. Diagnostics should be tied to the particular contrast and population being studied.

Definition: Let $Z$ denote the vector of treatment assignments generated by a specified assignment mechanism, and let $E_i(Z)$ be the exposure condition assigned to unit $i$ by a chosen exposure mapping. The assignment probability for condition $e$ is $$ \pi_i(e)=P\{E_i(Z)=e\}. $$ For a target set of units, exposure positivity for condition $e$ requires $\pi_i(e)>0$ for every unit whose causal outcome under $e$ is needed. A contrast between conditions $e$ and $e'$ requires positive probability for both conditions for the units in the target of that contrast.

Here, $Y_i(e)$ denotes unit $i$’s potential outcome under exposure condition $e$, as defined by the exposure mapping. Interpreting it as the relevant potential outcome requires the mapping to capture the assignment features relevant to the causal question. Positivity does not establish that assumption; it asks whether the design can produce the conditions the contrast calls for.

In a randomized design, the assignment mechanism supplies the probabilities $\pi_i(e)$. They may differ across units: a central network member with many neighbors can have a different probability of a particular peer-exposure condition from a member with only one neighbor. For observational assignment, the analogous diagnostic would involve probabilities conditional on measured covariates, but the examples here concern known randomized mechanisms.

Structural Zeros and Observed Sparsity

A condition has a structural zero for a unit when its assignment probability is zero. No assignment allowed by the design can give that unit the condition. More observations from the same design do not solve this problem for that unit; the target contrast or the design must change.

An observed count of zero is different. If $\pi_i(e)>0$, the condition is possible for that unit, but a particular realized assignment may still contain no units in it. That absence is a finite-sample support problem, not proof that the condition is structurally impossible. Conversely, a nonzero count does not establish adequate support: a few observations, especially those with small assignment probabilities, may contribute little stable information.

Worked Example: Assignment Probability and Expected Exposure Count

Hypothetical design. Consider 20 disjoint four-leaf star networks, with one center in each star. Every person is assigned treatment independently with probability $0.5$. The exposure of interest for a center is: the center is treated and exactly two of its four leaves are treated. Find the center’s exposure probability, the expected number of centers receiving this condition, and the probability that none receive it.

For a given center, its own treatment is independent of the leaf assignments. The probability that exactly two of four leaves are treated is $$ \binom{4}{2}(0.5)^2(0.5)^2 =6(0.0625) =0.375. $$ The center must also be treated, which has probability $0.5$. Thus $$ \pi_i(e)=0.5(0.375)=0.1875. $$

Let $N_e$ be the number of the 20 centers in this exposure condition. By adding the 20 indicator expectations, $$ E(N_e)=20(0.1875)=3.75 \text{ centers}. $$ Because the stars are disjoint and all assignments are independent, the 20 center exposure indicators are independent. Therefore, the probability that no center receives the condition is $$ P(N_e=0)=(1-0.1875)^{20}=0.8125^{20}\approx 0.0157. $$

Suppose the realized assignment contains two centers in the condition. Two is below the design expectation of 3.75, but it is a possible realization. The design probability is positive, so this is not a structural zero. The realized count still signals that estimates relying on this condition may be imprecise, particularly if the comparison condition is also sparse.

Diagnosing a Contrast Across Conditions

For a contrast between $e$ and $e'$, inspect support for both conditions rather than reporting the count for only one. A compact diagnostic table can list, for each condition and relevant unit or network stratum, the assignment probability, the expected count, and the observed count. Expected counts are sums of unit-specific probabilities: $$ E(N_e)=\sum_{i\in\mathcal T}\pi_i(e), $$ where $\mathcal T$ is the target set. This formula does not require units’ exposure indicators to be independent. The distribution of $N_e$, including the chance of zero observed units, can depend strongly on dependence induced by the assignment design.

A positive expected count is not a guarantee of realized support. Nor does a reasonable total expected count guarantee support in every subgroup or network stratum that the estimand requires. For example, a condition might be common among low-degree units but absent among high-degree units. If the target contrast includes both, an overall count can conceal the lack of support in the latter group.

Worked Example: A Structural Zero in a Sparse Network

Hypothetical design. Three units form a path: unit 1 is connected to unit 2, and unit 2 is connected to unit 3. Exactly one of the three units is selected for treatment, with each of the three assignments equally likely. Define exposure as the unit’s own treatment and the fraction of its neighbors treated. Examine the center unit, unit 2.

The three possible assignment vectors, written in unit order, are $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$. Under $(1,0,0)$, unit 2 is untreated and one of its two neighbors is treated, so its exposure is $(0,1/2)$. Under $(0,1,0)$, its exposure is $(1,0)$. Under $(0,0,1)$, its exposure is again $(0,1/2)$. Hence $$ \pi_2\{(0,1/2)\}=\frac{2}{3}, \qquad \pi_2\{(1,0)\}=\frac{1}{3}, \qquad \pi_2\{(0,0)\}=0. $$

A contrast between own treatment and no own treatment while the center has no treated neighbors would require the conditions $(1,0)$ and $(0,0)$. The first occurs with probability $1/3$, but the second is structurally impossible under this fixed-one-treated design. Thus the specified direct contrast at peer exposure zero is not supported for the center by this design. Observing additional repetitions of the same assignment mechanism would not create the missing condition; the design or target contrast would need to change.

Effective Sample Size Under Unequal Assignment Probabilities

Even when a condition is observed, unequal assignment probabilities can make the usable information highly uneven. In inverse-probability calculations, a unit observed in a rare condition receives a larger weight, $1/\pi_i(e)$, than a unit observed in a common condition. A few large weights can make an estimate sensitive to a small number of units.

For a realized exposure condition $e$, define $I_i(e)=1$ if unit $i$ is observed in that condition and $I_i(e)=0$ otherwise. The inverse-probability weight is $w_i(e)=I_i(e)/\pi_i(e)$, for units with positive $\pi_i(e)$. A commonly used diagnostic is the Kish effective sample size: $$ n_{\mathrm{eff}}(e)= \frac{\left(\sum_i w_i(e)\right)^2} {\sum_i w_i(e)^2}, $$ provided at least one weight is nonzero. It summarizes how concentrated the weights are: equal positive weights give an effective sample size equal to the number of such weights, while unequal weights give a smaller value. It is a diagnostic, not a literal count of independent observations or a guarantee of precision.

Worked Example: Four Observations but a Smaller Effective Sample Size

Hypothetical diagnostic data. Five focal units have assignment probabilities for condition $e$ of $0.50,\ 0.50,\ 0.25,\ 0.10,$ and $0.10$. In the realized assignment, units 1, 2, 3, and 5 receive $e$; unit 4 does not. Calculate the effective sample size for $e$.

The four nonzero inverse-probability weights are $$ w_1=1/0.50=2,\quad w_2=1/0.50=2,\quad w_3=1/0.25=4,\quad w_5=1/0.10=10. $$ Unit 4 has weight zero for this condition because it was not observed in it. The sum of weights is $2+2+4+10=18$, and the sum of squared weights is $2^2+2^2+4^2+10^2=124$. Therefore, $$ n_{\mathrm{eff}}(e)=\frac{18^2}{124} =\frac{324}{124} \approx 2.61. $$

There are four observed units, but the weight concentration corresponds to an effective sample size of about 2.61 under this diagnostic. In particular, the unit with probability $0.10$ contributes a weight of 10. The value warns that weighted results can be sensitive to a small number of units; it does not mean that exactly 2.61 people were observed or that the estimate has the precision of a simple random sample of that size.

A Practical Diagnostic Sequence

A useful assessment proceeds from the target contrast to the assignment probabilities and then to the realized data. The sequence below keeps structural lack of support distinct from finite-sample sparsity.

Specify the contrast and target.
Name the exposure conditions being compared and the units, subgroup, or network stratum to which the contrast applies. Positivity must be checked for this target, not for an unrelated overall population.

Calculate unit-specific probabilities.
Under the actual assignment mechanism and exposure mapping, calculate $\pi_i(e)$ for each required condition and relevant unit. A zero identifies a structural gap; a small positive value identifies a potentially weakly supported condition.

Summarize expected and observed support.
For each condition, compare $\sum_i\pi_i(e)$ with the realized count $\sum_i I_i(e)$. Break summaries down by target-relevant strata when probabilities or network structures vary.

Inspect weight concentration.
Where inverse-probability weights are used, report their range or distribution and calculate an effective sample size for each condition. A small effective sample size can reveal instability that raw counts hide.

Limit claims to supported contrasts.
If a required condition has zero probability, the design does not identify that contrast for the affected target units. If support is merely sparse, describe the resulting limitations in precision and stability rather than treating the contrast as structurally impossible.

What the Diagnostics Do—and Do Not—Establish

Positivity diagnostics matter because exposure-specific causal estimators need information from the exposure conditions they compare. A condition that is impossible for part of the target population cannot be recovered by a more elaborate calculation based only on the same assignment design. A rare condition may technically satisfy positivity while still producing highly variable estimates, especially when its realized observations receive large weights.

The effective sample size is only one warning signal. It does not account for every source of dependence among network members, and it does not establish that an estimator is unbiased or that its uncertainty has been correctly calculated. Similarly, a large observed count does not prove that all target subgroups are represented. The assignment mechanism, network structure, realized support, and target contrast all remain relevant.

When diagnostics reveal a gap, the response depends on its type. For a structural zero, consider a different estimand or a design that assigns positive probability to the needed conditions. For rare or uneven support, report the limitation, examine the units or strata driving the weights, and avoid presenting a numerically calculable estimate as automatically informative. These diagnostics address support; they do not validate the exposure mapping or establish other requirements for causal interpretation.

Common Mistake

Wrong reasoning: “The condition occurred at least once, so positivity is satisfied and the contrast is adequately supported.”
Correction: Positivity is a statement about assignment probabilities for the target units: each required condition must have positive probability. One realized observation cannot show that this holds for every target unit, and it does not demonstrate adequate finite-sample information. Check unit-specific probabilities, realized counts in both conditions, and weight concentration.

Key takeaway: Diagnose exposure positivity for the specific causal contrast and target population. Separate conditions that the assignment mechanism cannot produce from conditions that happened not to appear, and use effective sample size to flag instability hidden by raw counts.

Check Your Understanding

Use the definitions and diagnostics above to answer these questions.

What does it mean for an exposure condition to have a structural zero for a unit?
A condition has positive assignment probability for every target unit, but its realized count is zero. Is this necessarily a violation of design positivity? Explain.
Why can a network’s overall exposure count conceal a lack of support in a target-relevant degree stratum?
If four units have equal positive weights, what is their Kish effective sample size? How does that compare with four very unequal positive weights?
In the three-unit path example, why can observing more assignments under the same fixed-one-treated mechanism not supply the center’s exposure $(0,0)$?