Sensitivity Analysis for Missing Not at Random

Why Sensitivity Analysis Is Needed

When outcomes are missing not at random (MNAR), the distribution of an unobserved outcome depends on information that remains unobserved, even after conditioning on the measured covariates and observed outcomes. The observed data alone generally cannot determine how far the missing outcomes depart from a missing-at-random (MAR) prediction. A sensitivity analysis makes this uncertainty explicit: specify a reference prediction under MAR, describe plausible departures from it, and examine how the GEE estimates and conclusions change.

The sensitivity parameter is not an extra coefficient that the observed data can reliably estimate. It indexes alternative assumptions about the missing outcomes. The analysis is therefore conditional on the chosen values and on the model used to relate missing outcomes to observed information.

Pattern-Mixture Model and Sensitivity Parameter

Let \(Y_i=(Y_{i1},\ldots,Y_{im_i})^\top\) be the full response vector for cluster \(i\), and let \(R_i\) denote its response pattern. Write \(O_i\) for the observed data in that cluster, including its observed outcomes, covariates, and response pattern. A pattern-mixture representation describes the distribution of outcomes separately within response patterns, \(f(Y_i\mid R_i,X_i)\), together with the distribution of the patterns. Sensitivity modeling specifies how the conditional distribution of missing outcomes within a pattern differs from a reference distribution.

For a continuous outcome, a simple conditional-mean adjustment is \(E_\delta[Y_{ij}\mid O_i,R_i,X_i]=E_{\text{MAR}}[Y_{ij}\mid O_i,X_i]+\delta_j(R_i)\) for a missing response. Here \(\delta_j(R_i)\) is measured in the outcome's units. For a binary outcome, a useful alternative is a conditional log-odds adjustment: \(\operatorname{logit}P_\delta(Y_{ij}=1\mid O_i,R_i,X_i)=\operatorname{logit}p_{\text{MAR},ij}+\delta_j(R_i)\). Then \(\exp\{\delta_j(R_i)\}\) is the assumed odds ratio between the missing-outcome odds and the MAR-predicted odds, conditional on the stated information. These are examples of sensitivity models, not universal choices; the scale and pattern-specific structure should match the scientific question.

A delta adjustment specifies a conditional mean or probability, not necessarily an entire conditional distribution. For GEE scores that are linear in the response vector given covariates and parameters, those conditional means suffice to compute the conditional expected score. Other analyses may require a full predictive distribution, including assumptions about variance or dependence among missing responses.

Using the Assumptions in a GEE

For a full response vector, write the GEE contribution as \(U_i(\beta)=D_i^\top V_i^{-1}\{Y_i-\mu_i(\beta)\}\), where \(\mu_i(\beta)\) is the marginal mean vector, \(D_i=\partial\mu_i(\beta)/\partial\beta^\top\), and \(V_i\) is the working covariance matrix. The sensitivity-adjusted contribution is the conditional expected full-data contribution, \(U_i(\beta;\delta)=E_\delta[U_i(\beta)\mid O_i,R_i,X_i]\). When \(D_i\) and \(V_i\) are functions of the covariates and \(\beta\), rather than of the missing responses themselves, this becomes \(D_i^\top V_i^{-1}\{\tilde Y_i(\delta)-\mu_i(\beta)\}\), where observed entries of \(\tilde Y_i(\delta)\) are the actual responses and missing entries are their assumed conditional expectations.

Proof. By the definition of the adjusted contribution and the law of iterated expectation,

The last expectation is the full-data expectation under the assumed distribution, because integrating the conditional distribution of the missing outcomes over the observed data and response patterns reconstructs that assumed full-data distribution. By the stated hypothesis, this expectation is zero. Hence \(E[U_i(\beta_0;\delta)]=0\). \(\square\)

This result is conditional on the sensitivity model, not a test of whether that model is true. It also does not say that a particular \(\delta\) is identified by the data. With independent clusters and a fixed \(\delta\), one can use the cluster contributions \(U_i(\hat\beta;\delta)\) in sandwich inference, subject to the usual regularity conditions for an estimating-equation estimator. If parameters in the MAR prediction model are estimated, their uncertainty must also be included, for example through stacked estimating equations or a cluster bootstrap that refits the prediction model. Recompute the standard error and interval at each sensitivity value rather than automatically carrying them over from the MAR fit.

Worked Sensitivity Analyses

Choosing and Reporting a Sensitivity Range

The data do not usually supply a unique plausible range for \(\delta\). Choose values using the outcome scale, clinical or scientific knowledge, and the amount of departure that would meaningfully alter interpretation. For example, an additive shift can be calibrated in clinically familiar units, while a log-odds shift can be reported as an odds ratio relative to the MAR prediction. With multiple visits or missingness patterns, a single common shift may be too restrictive; a prespecified set of pattern- or time-specific shifts can represent distinct assumptions.

At each value, refit or solve the sensitivity-adjusted GEE and report the resulting coefficient or contrast, its uncertainty, and the inferential conclusion. A table or plot across the prespecified range makes it clear whether the substantive conclusion is stable or depends on a narrow assumption. If the conclusion changes, report where it changes and what that departure means on the outcome scale. Sensitivity analysis does not select the “correct” scenario; it shows what the data imply under transparent alternatives.

Common Mistakes

• Treating \(\delta\) as estimated from the observed outcomes. The data generally cannot identify departures that concern unobserved responses. State the values as assumptions and justify the range substantively.
• Calling the sensitivity analysis a test of MAR. A result that changes across plausible departures demonstrates sensitivity to assumptions; it does not establish which missingness model is true.
• Using a link-scale shift without interpreting its scale. For a binary conditional model, \(\delta\) is a log-odds shift and \(\exp(\delta)\) is an odds ratio. It is not an additive change in probability.
• Reporting only adjusted point estimates. GEE conclusions also depend on uncertainty. Recompute standard errors and intervals under each scenario, including uncertainty from estimated prediction models.
• Assuming one shift fits every pattern. A common shift is a simplifying choice, not a requirement. If departures could differ by visit or missingness pattern, specify and report that structure.