Sensitivity Parameters as Coordinates of Unidentified Information

Observed Features and Unidentified Coordinates

A missing-data model can fit the observed data well and still leave important features of the complete-data distribution undetermined. The observed distribution fixes, for example, how often each response pattern occurs and the distribution of the outcomes actually recorded in each pattern. It does not, in general, fix the distribution of outcomes that were not recorded. A useful way to describe this gap is to choose sensitivity parameters: explicit coordinates for the unidentified features, with the observed-data features held fixed.

The word “coordinate” is deliberate. It means that a parameterization labels the possible complete-data laws consistent with the same observed-data law. It does not mean that the coordinate is estimated from the observed data, nor that every numerical value is permissible. Probability constraints, support restrictions, and any stated model assumptions restrict which values define a valid law.

This distinction builds on the earlier material on what the observed likelihood establishes. In population terms, the observed-data law determines the pattern probabilities and the distributions of \(Y_{\mathrm{obs}}\) within those patterns. The unobserved conditional distributions are additional ingredients required to specify a complete-data law. Any sensitivity analysis must state how those ingredients are parameterized and what values are considered plausible.

A Factorization That Separates What Is Fixed

For each pattern \(r\), let \(P_{R,Y_{\mathrm{obs}}}\) denote the observed-data probability measure for the pattern and its observed outcomes. Let \(g_r(dy_{\mathrm{mis}}\mid y_{\mathrm{obs}})\) be a conditional probability distribution for the missing outcomes given the observed outcomes in pattern \(r\). The full joint law is assembled by combining these pieces. This is a measure identity, so it applies whether observed outcomes are discrete, continuous, or a mixture of both.

Here \(A\) and \(B\) are measurable sets of possible observed and missing outcomes. The conditional distribution \(g_r\) is defined almost everywhere under the observed-data law for pattern \(r\); its values at observed-outcome values having probability zero do not affect the joint law. When densities exist, the same factorization can be written as a joint density \(f_{R,Y_{\mathrm{obs}}}(r,y_{\mathrm{obs}})\) multiplied by the conditional density of \(Y_{\mathrm{mis}}\) given \((R=r,Y_{\mathrm{obs}}=y_{\mathrm{obs}})\).

The observed-data law supplies the measure in this expression. A sensitivity model supplies \(g_r\), perhaps through one or more parameters. To be valid, the resulting conditional distributions must be normalized probability distributions and must respect the outcome’s support. If a pattern has probability zero, its conditional distribution is not learned from the data and its contribution to this observed-data factorization is zero; claims about that pattern require additional assumptions.

Worked Example: A Risk-Difference Coordinate

This example also shows why a sensitivity parameter must come with a scale and interpretation. A risk difference is measured in probability points. Other coordinates may be more natural for different scientific questions, but they can have different constraints and different behavior near the boundaries of the parameter space.

Worked Example: An Odds-Ratio Coordinate

The risk-difference and odds-ratio coordinates describe the same kind of unidentified contrast, but they are not interchangeable numerically. In the preceding example, \(\gamma=2\) corresponds to a risk difference \(0.5714-0.40=0.1714\). Choosing a coordinate therefore affects how experts express and vary assumptions; it does not make the underlying missing outcomes observable.

Worked Example: Coordinates Across Multiple Patterns

Derivation and Practical Checks

The factorization gives a direct justification for this way of working. Fix an observed-data law \(P_{R,Y_{\mathrm{obs}}}\). For each positive-probability pattern, choose a valid conditional distribution \(g_r\). Integrating \(g_r\) over all possible missing-outcome values gives one, so integrating the factorization over \(B\) equal to the whole missing-outcome space returns \(P(R=r,Y_{\mathrm{obs}}\in A)\). Thus the constructed law reproduces the fixed observed-data distribution. Conversely, any complete-data law has conditional distributions of the missing components given the pattern and observed components (up to almost-everywhere qualifications), and hence yields such a factorization.

This argument establishes a representation, not a claim that all conditional distributions are scientifically reasonable. A proposed sensitivity model can fail as a probability model or impose unintended implications. Useful checks include:

• Support and normalization: each conditional distribution must assign total probability one and respect the outcome’s possible values. For binary outcomes, every specified conditional risk must be in \([0,1]\).
• Observed-data preservation: changing sensitivity coordinates should not alter the fixed response-pattern probabilities or the distributions of recorded outcomes within patterns.
• Boundary behavior: odds, log odds, and other transformed coordinates may be undefined or infinite at probability boundaries. Inspect those cases instead of relying on an interior formula.
• Dependence structure: for several missing outcomes, marginal sensitivity parameters alone may not specify their joint distribution. The joint conditional law may require additional coordinates.
• Scientific interpretation: state what contrast a coordinate represents, its scale, and the values examined. A numerically convenient parameter is not automatically an interpretable or plausible one.

Comparisons and Limitations

A pattern-mixture specification uses the conditional distributions of outcomes within response patterns as its building blocks. A selection-model specification instead describes response probabilities conditional on outcomes. The earlier material on equivalent selection and pattern-mixture representations shows that these can be alternative factorizations of a joint law; neither factorization, by itself, makes an unidentified feature observable. Sensitivity coordinates can be used in either representation, but their meanings and convenient scales may differ.

A sensitivity parameter is also distinct from a parameter that the observed likelihood identifies. It is possible to estimate the observed-data features while the sensitivity coordinate remains unsupported by the observed likelihood. Assigning a prior distribution to such a coordinate, as in Bayesian analyses under nonidentification, can produce a posterior distribution, but that posterior’s information about the unidentified feature depends on the prior and model assumptions as well as the observed data.

The main limitation is fundamental: observed-data fit cannot determine whether a particular missing-outcome conditional distribution is correct when alternatives produce the same observed-data law. Sensitivity analysis makes the dependence of conclusions on such choices visible; it does not empirically verify the choices. Results should therefore report both the fixed observed features and the assumed sensitivity coordinates, with a scientifically justified range or set of scenarios. If a chosen parameterization rules out plausible distributions or fails to capture important dependence, conclusions drawn from it can be misleading.

Common Mistake

Wrong reasoning: “The observed records give us the respondent outcome distribution, so the missing group must have that distribution too.” That conclusion silently sets the missing-outcome conditional distribution equal to the respondent distribution. It is an assumption, not a consequence of the observed data.

Correction: Write down the observed quantities that are fixed, then identify which conditional distributions remain unspecified. If equality across response patterns is scientifically defensible, state it as a restriction and recognize that it is not tested by the observed-data distribution alone. Otherwise, define sensitivity coordinates and examine how conclusions change across valid values.