Why an Overall Treatment Rate Is Not Enough
Suppose a study includes both treated and untreated people. The overall treatment prevalence may be between zero and one, but that fact alone does not show that both treatment options are represented at every relevant covariate profile. For example, one subgroup could contain only treated people and another only untreated people, while the overall sample contains many of each. Conditional positivity describes the more specific requirement: within the covariate profiles that matter for the target population, each treatment must have positive probability.
This distinction matters when using propensity scores, subclassification, or weighting. Those methods rely on treatment probabilities conditional on covariates, not merely on the total number treated. The condition is a population statement; observed counts can provide evidence about it but do not by themselves define it.
“Almost everywhere” means that the condition may fail on a set of covariate values that has probability zero under the target distribution of \(X\), without affecting this definition. If the target population is restricted to a particular set of covariate values, positivity is assessed on that target set, not automatically on every mathematically possible value of \(X\). Some stronger formulations require a fixed \(\epsilon>0\) such that \(\epsilon\le e(x)\le 1-\epsilon\) throughout the target support. That is a uniform bound, stronger than strict positivity almost everywhere.
Conditional positivity is one assumption used in causal analyses, not a substitute for the other assumptions. For example, a causal interpretation also requires conditions concerning consistency and conditional exchangeability, as appropriate to the design and estimand. Positivity says that both treatment options are available in the relevant covariate profiles; it does not establish that measured covariates suffice to control confounding.
Conditional Probabilities Versus Marginal Prevalence
The marginal treatment prevalence is \(P(A=1)\), the probability of treatment without conditioning on covariates. By the law of total probability, it is an average of the conditional probabilities over the covariate distribution:
Here \(F_X\) is the distribution of covariates in the target population. Thus, the marginal quantity compresses information across covariate profiles. It can tell us the overall fraction treated, but not how treatment is allocated within particular profiles.
If conditional positivity holds, then the marginal prevalence is strictly between zero and one. The reverse implication does not hold. To see the forward direction, if \(e(X)>0\) almost surely, then its expectation is positive; likewise, if \(1-e(X)>0\) almost surely, then \(E[1-e(X)]>0\), so \(E[e(X)]<1\). But an average between zero and one can be produced by conditional probabilities equal to zero in some profiles and one in others.
Worked Examples
Worked Example: A Balanced Overall Rate With No Within-Group Treatment Variation
Consider a hypothetical target population with two equally common covariate groups, \(X=0\) and \(X=1\). Suppose every person in group \(X=0\) is untreated and every person in group \(X=1\) is treated. Find the marginal treatment prevalence and check conditional positivity.
The group probabilities are \(P(X=0)=0.5\) and \(P(X=1)=0.5\). The conditional treatment probabilities are \(e(0)=0\) and \(e(1)=1\). Therefore,
Half of the population is treated, so the marginal prevalence appears perfectly interior. But conditional positivity fails in both groups: for \(X=0\), no one has a positive probability of treatment; for \(X=1\), no one has a positive probability of no treatment. The overall rate conceals the complete separation of treatment by covariate group.
Worked Example: Checking Positivity Across Discrete Covariate Profiles
In a hypothetical population, a binary covariate \(X\) has two levels. Suppose \(P(X=0)=0.6\), \(P(X=1)=0.4\), \(e(0)=0.25\), and \(e(1)=0.875\). Calculate the marginal treatment prevalence and assess conditional positivity.
The marginal prevalence is the covariate-proportion-weighted average of the two conditional probabilities:
Within \(X=0\), the probability of treatment is \(0.25\) and the probability of no treatment is \(1-0.25=0.75\). Within \(X=1\), those probabilities are \(0.875\) and \(1-0.875=0.125\). Both probabilities are positive in each group, so conditional positivity holds for these two profiles. The marginal prevalence is also \(0.50\), but it is the profile-specific probabilities—not the marginal value—that establish the condition here.
Worked Example: A Continuous Covariate and an Average Propensity Score
Suppose, hypothetically, that \(X\) is uniformly distributed on \([0,1]\) in the target population and the treatment probability is \(e(x)=0.2+0.6x\). Find the marginal treatment prevalence and determine whether conditional positivity holds.
Because \(X\) has density 1 on \([0,1]\), averaging the propensity score gives
For every \(x\in[0,1]\), \(e(x)\) ranges from \(0.2\) to \(0.8\). Thus \(0<e(x)<1\) throughout the target support, and conditional positivity holds. In this example the probability is also uniformly bounded away from zero and one: \(0.2\le e(x)\le0.8\).
For contrast, keep \(X\) uniform on \([0,1]\) but let \(e(x)=x\). Then \(P(A=1)=\int_0^1x\,dx=0.5\). The condition \(0<e(x)<1\) holds for every \(x\) strictly between 0 and 1; the endpoints have probability zero under this continuous distribution. So conditional positivity holds almost everywhere under the definition above, even though there is no fixed positive lower bound on \(e(x)\) over the whole interval. This illustrates why strict positivity and a uniform bound are distinct conditions.
A Short Justification of the Marginal Implication
The claim that conditional positivity implies an interior marginal prevalence can be justified without assuming a uniform bound. Let \(Z=e(X)\), so \(0\le Z\le1\). If \(Z>0\) almost surely, then the events \(\{Z\ge1/k\}\), for positive integers \(k\), together cover the event \(\{Z>0\}\). Since that event has probability one, at least one of these sets has positive probability. For that \(k\),
Applying the same argument to \(1-Z\), which is positive almost surely under conditional positivity, gives \(E[1-Z]>0\). Since \(E[Z]+E[1-Z]=1\), it follows that \(E[Z]<1\). Therefore \(0<P(A=1)<1\). This establishes only the forward implication; the first worked example shows why the converse fails.
How to State and Assess the Condition
A precise positivity statement identifies the treatment, the covariates, and the target population. For a binary treatment, the condition is that both \(P(A=1\mid X=x)\) and \(P(A=0\mid X=x)\) are positive for almost every \(x\) in the target covariate distribution. In a discrete setting, this means checking every covariate profile with positive target-population probability. In a continuous setting, it concerns the conditional probability function over the target support, with “almost everywhere” interpreted relative to the target distribution.
This is not the same as checking that the data set contains at least one treated and one untreated person. Nor is it enough to check that a fitted model gives an average predicted treatment probability between zero and one. Those checks concern overall counts or averages. The defining condition is conditional. Estimated propensity scores may help describe the fitted conditional probabilities, but those estimates are not themselves proof that the population condition holds.
A further distinction is useful: positivity is defined relative to a target population, while a model for \(e(x)\) is an estimation strategy. A fitted model could be used to estimate conditional probabilities, but the assumption itself does not require a logistic model, or any particular functional form. In discrete data, conditional probabilities can be written directly for each profile. With continuous covariates, a model or other estimation method is generally needed to summarize how the probability varies with \(x\); that modeling choice does not change the definition.
| Quantity or check | What it describes | Does it establish conditional positivity? |
|---|---|---|
| \(P(A=1)\) | Overall treatment prevalence | No; it averages across covariate profiles |
| \(P(A=1\mid X=x)\) for target profiles | Treatment probability within covariate profiles | Yes, when both treatment probabilities are positive almost everywhere |
| Observed total counts by treatment | How many treated and untreated people were observed | No; totals do not show how treatment is distributed across profiles |
| A uniform bound \(\epsilon\le e(x)\le1-\epsilon\) | A stronger lower and upper probability bound | It implies strict positivity, but is stronger than strict positivity almost everywhere |
Common Mistakes and Limits
- Using an interior marginal rate as proof. An overall prevalence of 0.50 is compatible with profiles having treatment probabilities of zero and one. Check probabilities conditional on covariates.
- Confusing positivity with balance. Positivity requires both treatment options to have positive probability within relevant profiles. It does not assert that treated and untreated groups have identical covariate distributions.
- Confusing strict positivity with a uniform bound. Requiring \(0<e(x)<1\) almost everywhere does not necessarily imply that all probabilities stay a fixed distance from zero and one.
- Ignoring the target population. A covariate profile outside the target population's support does not determine positivity for that target. State the population over which the condition is being asserted.
- Treating a sample count as the definition. A lack of observed treatment variation in a small sample is not, by itself, a mathematical definition of a zero population probability. The assumption concerns the population conditional probabilities.
Conditional positivity is a support condition on treatment assignment given covariates. It tells us whether both treatment options are represented in principle for the covariate profiles relevant to the target population. It does not alone guarantee precise estimation, establish exchangeability, or determine the causal estimand. Those are separate questions.
Check Your Understanding
Answer each question using conditional probabilities, not only overall treatment counts.
- A population has two equally common covariate groups. The treatment probabilities are 0.1 and 0.9. Calculate the marginal treatment prevalence and state whether conditional positivity holds.
- Suppose a continuous covariate has a distribution supported on \([0,1]\), and \(e(x)=0.3+0.2x\). What is the range of treatment probabilities on that support, and does strict conditional positivity hold?
- Can \(P(A=1)=0.4\) establish conditional positivity? Explain.
- For \(X\) uniform on \([0,1]\) and \(e(x)=x\), does strict positivity almost everywhere hold? Does a uniform positive lower bound on \(e(x)\) hold?
Solutions. (1) The marginal prevalence is \((0.5)(0.1)+(0.5)(0.9)=0.05+0.45=0.50\). Both conditional probabilities are strictly between zero and one in both groups, so conditional positivity holds. (2) Since \(e(x)=0.3+0.2x\), its range is \([0.3,0.5]\). It is strictly between zero and one throughout the support, so conditional positivity holds; it is also bounded away from both endpoints. (3) No. The value 0.4 is an average over covariate profiles and does not show whether either treatment probability is zero or one within some profiles. (4) Yes, strict positivity holds almost everywhere: \(0<x<1\) except at the two endpoints, which have probability zero under the uniform distribution. No uniform positive lower bound holds on all of \([0,1]\), because \(e(x)=x\) can be arbitrarily close to zero.