Point Identification Versus Partial Identification

Why the Set of Compatible Values Matters

A missing-data analysis may produce a single number, a range, or no useful restriction on the target at all. Those outcomes are not interchangeable. A single estimate from a fitted model does not, by itself, establish that the target is determined by the observed data: the number may depend on an assumption about values that were never observed. To assess what the data and assumptions actually determine, ask which target values are compatible with both.

Identification is a property of the relationship between the observed-data distribution, the assumptions, and the target. It is not a measure of how large a sample is or how precisely a statistic has been estimated. We will use a binary outcome $Y$, a response indicator $R$ that equals 1 when $Y$ is observed and 0 otherwise, and the population prevalence $P(Y=1)$ as a running example.

Definition: Let $O$ denote the observed-data distribution and let $A$ denote a stated collection of assumptions. The compatible full-data distributions are all full-data distributions that, under $A$, could have generated $O$. The identification set for a target $\theta$ is the set of target values attained across those compatible distributions: $$ \Theta_I(O,A)=\{\theta(P): P\text{ is compatible with }O\text{ and }A\}. $$ The target is point identified if this set contains exactly one value. It is partially identified when the set contains multiple values but the data and assumptions still restrict it to an informative proper subset of the target's possible values. It is not point identified whenever the set contains more than one value. “Unidentified” is often used for this broad case; when distinguishing it from partial identification, it usually means that no useful restriction beyond the target's basic parameter space has been obtained.

The terminology around “unidentified” is not entirely uniform. The mathematically precise object is always the identification set. A non-singleton set establishes failure of point identification; whether it is called partial identification or simply nonidentification depends on whether its restrictions are informative for the question at hand. A set can also be empty: that means the observed distribution and assumptions are incompatible, not that the target has been identified.

A Binary Outcome With Missing Values

Suppose the observed-data distribution gives the response rate $P(R=1)=0.60$ and the prevalence among respondents $P(Y=1\mid R=1)=0.40$. The outcome prevalence among nonrespondents, $P(Y=1\mid R=0)$, is not observed. Write this unknown prevalence as $u$. Without further restrictions, $u$ can take any value in $[0,1]$. The population prevalence $p=P(Y=1)$ follows from the law of total probability:

p=P(Y=1\mid R=1)P(R=1)+P(Y=1\mid R=0)P(R=0) =0.40(0.60)+u(0.40)=0.24+0.40u.

Because $u$ ranges from 0 to 1, $p$ ranges from 0.24 to 0.64. Each value in this interval can be generated by a corresponding value of $u$, so the interval is not merely a conservative summary: it is the exact identification set under these assumptions. In this example the only restrictions are the observed distribution and the logical requirement that a probability lies between 0 and 1.

Worked Example: Deriving the Unrestricted Identification Set

Treat the response rate and respondent prevalence above as exact features of a hypothetical population’s observed-data distribution. Find the identification set for the population prevalence when no restriction is imposed on nonrespondents’ outcomes.

The missing fraction is $P(R=0)=1-0.60=0.40$. Let $u=P(Y=1\mid R=0)$. Since $Y$ is binary, $0\leq u\leq1$. Substituting into the total-probability formula gives:

$$ p=0.60(0.40)+0.40u=0.24+0.40u. $$

The expression is increasing in $u$, so its minimum occurs at $u=0$ and its maximum at $u=1$:

\Theta_I=[0.24+0.40(0),\;0.24+0.40(1)]=[0.24,\;0.64].

Thus, the observed response rate and respondent prevalence alone do not point identify the population prevalence. They do rule out values below 0.24 or above 0.64 under this setup. The endpoints correspond to all nonrespondents having $Y=0$ and all nonrespondents having $Y=1$, respectively.

How Assumptions Change the Set

An assumption can shrink an identification set by excluding some otherwise compatible full-data distributions. If the restriction pins down the target to one value, it gives point identification conditional on that restriction. If it narrows the set but leaves multiple values, it gives partial identification. The observed data alone need not reveal whether an untestable restriction about missing outcomes is true.

Worked Example: A Sensitivity Restriction Produces a Narrower Set

Use the same hypothetical observed-data distribution. Suppose an analyst is willing to assume that the nonrespondent prevalence differs from the respondent prevalence by no more than 0.10 below or 0.20 above. This is the explicit restriction $0.30\leq u\leq0.60$. Find the resulting set for $p$.

The total-probability expression remains $p=0.24+0.40u$. It is increasing in $u$, so substituting the two endpoints gives:

\Theta_I=[0.24+0.40(0.30),\;0.24+0.40(0.60)] =[0.36,\;0.48].

The restricted set is narrower than $[0.24,0.64]$, but it is not a singleton. The prevalence is therefore partially identified under this restriction, not point identified. This conclusion is conditional on the stated range for $u$; the data used here do not verify that range. Sensitivity parameters, introduced in the earlier material on sensitivity parameters, are one way to describe such departures explicitly rather than conceal them inside a single fitted value.

Worked Example: Point Identification Under an Explicit Assumption

Again use the same response rate and respondent prevalence. Now assume $Y$ is independent of $R$, so the prevalence among nonrespondents equals the prevalence among respondents. What value does this assumption imply for the population prevalence?

Independence gives $P(Y=1\mid R=0)=P(Y=1\mid R=1)=0.40$, so $u=0.40$. Substitution yields:

$$ p=0.60(0.40)+0.40(0.40)=0.24+0.16=0.40. $$

Under the independence assumption, every compatible distribution has $p=0.40$, so the target is point identified conditional on that assumption. This is not evidence that independence is true: the assumption concerns how the missing outcomes relate to response, and those outcomes are unavailable for direct comparison. Nor does point identification mean that a finite-sample estimate would equal 0.40 exactly; sampling uncertainty remains when the observed-data distribution is estimated from a sample.

Why the Identification Set Is Exact

An interval is an exact identification set only if it includes every compatible target value and excludes every incompatible one. In the unrestricted binary example, this can be verified directly. For any $p\in[0.24,0.64]$, define $u=(p-0.24)/0.40$. The bounds on $p$ imply $0\leq u\leq1$, and that value of $u$ reproduces $p=0.24+0.40u$. Conversely, any compatible $u\in[0,1]$ produces a $p$ between 0.24 and 0.64. Thus the interval contains exactly, and not merely at least, the compatible values.

Key result: To establish point identification, show that all full-data distributions compatible with the observed distribution and assumptions give the same target value. To establish partial identification, characterize the multiple compatible target values and show that they form a proper restriction of the target’s possible range. Reporting a single value from within a non-singleton set does not establish point identification.

The same logic applies beyond a prevalence. For any target, the identification set is obtained by mapping the compatible full-data distributions through the target function. The set may be a single value, an interval, a collection of disconnected values, or—in a poorly specified problem—empty. An interval description is appropriate only when it accurately represents the set; disconnected compatible values should not be silently filled in.

Identification Is Not Sampling Uncertainty

The calculations above treat the observed-data probabilities as known. In an actual sample, the response rate and respondent prevalence are estimated and vary across repeated samples. That sampling uncertainty is separate from identification uncertainty, which arises because multiple full-data distributions agree with the same observed-data distribution and assumptions. An analysis may therefore have both a range of compatible target values and uncertainty about the endpoints of that range.

For example, replacing 0.60 and 0.40 with sample estimates would produce an estimated identification region, not a guarantee that the population region has those exact endpoints. Confidence procedures for partially identified targets must account for sampling variation in the relevant observed quantities. Simply placing a standard confidence interval around one arbitrarily selected value does not account for the unobserved-outcome ambiguity.

Comparison, Diagnostics, and Limitations

Conclusion	Identification-set description	What it supports
Point identified	A singleton	One target value follows from the observed distribution and assumptions.
Partially identified	Multiple values in an informative proper subset of the target space	The target is restricted to a set, but a unique value is not determined.
Not point identified; sometimes called unidentified	Multiple values; possibly the entire basic target space	The stated information does not determine a unique value; it may provide little or no useful restriction.
Incompatible specification	An empty set	No full-data distribution satisfies both the observed distribution and the assumptions.

The main alternatives are not simply “use a model” or “use no model.” One can report a point estimate conditional on a strong identifying assumption, report a set under weaker restrictions, or conduct sensitivity analysis across explicitly varied assumptions. A point model can be concise but may hide how much its answer depends on extrapolation to missing values. A set can be more transparent but may be wide and less decisive. Sensitivity analysis describes how conclusions change across assumptions; it does not make an untestable assumption empirically verified.

Several checks help reveal errors in an identification argument. The set should respect the target’s logical range; here, $p$ must be in $[0,1]$. Substituting each endpoint into the defining relationship should reproduce the claimed endpoint. Every assumption used to narrow the set should be stated, and the observed data should not be treated as if they directly measured outcomes for nonrespondents. If the resulting set is empty, check the data restrictions and assumptions for incompatibility rather than interpreting emptiness as exceptionally strong identification.

A common mistake is to say, “The model returned $p=0.40$, so the prevalence is point identified.” The correction is to ask whether all full-data distributions compatible with the observed data and stated assumptions produce 0.40. In the running example, that is true under the independence assumption, but not under the observed-data information alone: without that assumption, every value from 0.24 to 0.64 is compatible. A point output can therefore reflect a modeling choice, not identification by the observed data.

Identification results are conditional, not assumption-free certificates. A restriction can be mathematically sufficient to produce a singleton while remaining substantively questionable. Conversely, a wide partial-identification region may be an honest description of limited information rather than a defect in calculation. The practical task is to state the target, observed information, assumptions, and resulting set clearly enough that readers can see which conclusions are data-driven and which depend on extrapolation.

Check Your Understanding

For each question, distinguish what follows from the observed information from what follows only after an assumption is added.

In the unrestricted example, what is the identification set if the response rate is 0.50, the respondent prevalence is 0.30, and $u$ can range from 0 to 1?
With response rate 0.60 and respondent prevalence 0.40, suppose $u$ is restricted to $[0.40,0.50]$. Find the prevalence set and classify identification.
What does an empty identification set imply about the observed distribution and assumptions?
Why does a point estimate from a fitted model not, on its own, establish point identification?

Solutions. (1) $p=0.50(0.30)+0.50u=0.15+0.50u$, so the endpoints are $0.15$ and $0.65$, giving $[0.15,0.65]$; the target is not point identified. (2) $p=0.24+0.40u$, so the endpoints are $0.24+0.40(0.40)=0.40$ and $0.24+0.40(0.50)=0.44$. The set $[0.40,0.44]$ is non-singleton and informative, so this is partial identification under the stated restriction. (3) No full-data distribution can satisfy both the observed information and all stated assumptions; at least one restriction or data specification must be reconsidered. (4) Point identification requires every compatible full-data distribution to yield the same target. A fitted model may select one value by imposing assumptions without showing that alternatives are incompatible.