Delta-Adjusted Pattern-Mixture Sensitivity Analysis

Why Adjust Imputations?

Multiple imputation under a missing-at-random assumption supplies draws for missing values from a predictive distribution informed by observed data. That assumption is often a useful starting point, but it cannot be verified from the observed data alone. A sensitivity analysis asks a focused question: how would the result change if, within a missingness pattern, missing outcomes tended to differ from their missing-at-random predictions?

A pattern-mixture model describes the outcome distribution separately for each pattern of observed and missing variables. A delta adjustment modifies the conditional distribution used to impute missing values in a specified pattern. The delta is a sensitivity parameter: it describes a departure from the MAR imputation model rather than an effect estimated from the observed data.

The pattern \(R\) can encode which variables are observed, not merely whether \(Y\) is observed. In the simple case of one incomplete outcome, it is often enough to distinguish responders from nonresponders. With multiple incomplete variables, different patterns can receive different adjustments. The adjustment is applied conditional on the observed information used by the imputation model, such as \(X\) and any observed components of \(Y\).

What the Observed Data Can and Cannot Identify

For a pattern in which some outcome values are missing, write its conditional distribution as \(f(y_{\mathrm{obs}},y_{\mathrm{mis}}\mid r,x)\). It can be factored into the observed-data distribution and a conditional distribution for the missing part:

The observed-data likelihood for a case in that pattern depends on \(f(y_{\mathrm{obs}}\mid r,x)\), because integrating the second factor over all possible missing values gives one. It does not identify the conditional distribution of the missing values. A delta adjustment makes a transparent choice about that unidentified part; it does not recover a parameter that the observed data estimate.

To verify the result, integrate the joint model over the missing values:

The conditional density integrates to one for every fixed \(y_{\mathrm{obs}},r,x\). The pattern probability and observed-data density are unchanged by a replacement of that conditional density. This is why a sensitivity analysis reports results across specified delta values rather than presenting a single delta as data-estimated. \(\square\)

Choosing and Applying an Adjustment

For a continuous outcome, an additive shift is easy to interpret and preserves the shape and conditional variance of each MAR predictive distribution. If \(Y_{\mathrm{mis}}^{\mathrm{MAR}}\mid y_{\mathrm{obs}},r,x\) has mean \(\mu_r\) and variance \(v_r\), then the shifted draw has mean \(\mu_r+\delta_r\) and variance \(v_r\). The change is in outcome units: for example, a shift of \(-4\) points means missing scores are assumed to average four points below their MAR predictions, conditional on the model information.

For a binary outcome, adding a constant to a zero-or-one draw is not suitable. A common alternative shifts the predicted log odds. If the MAR predicted probability is \(p\), define the adjusted probability by:

Here \(\delta\) is a log-odds shift; \(\exp(\delta)\) is the multiplicative change in odds relative to the MAR prediction. A value of zero returns the MAR probability. For counts or other restricted outcomes, the adjustment must respect the support of the outcome; the continuous additive rule should not be applied in a way that creates impossible values.

How Pattern-Specific Deltas Change a Population Mean

The effect of a delta on an overall mean depends both on its size and on the proportion of the population in the adjusted pattern. Let \(R\) take values \(1,\ldots,K\), let \(P(R=r)=\pi_r\), and let \(\mu_r^{\mathrm{MAR}}\) be the MAR-implied mean outcome in pattern \(r\). If the conditional mean in each pattern is shifted by \(\delta_r\), then the overall mean is:

This identity is a direct application of the law of total expectation. Conditional on \(R=r\), the adjusted mean is \(\mu_r^{\mathrm{MAR}}+\delta_r\). Averaging those conditional means using the pattern probabilities gives:

Subtracting the MAR mean \(\sum_r\pi_r\mu_r^{\mathrm{MAR}}\) proves that the change is \(\sum_r\pi_r\delta_r\). In a setting where only the nonresponding pattern is shifted, the overall mean change is the nonresponse proportion multiplied by that pattern’s delta. \(\square\)

A Practical Multiple-Imputation Workflow

A delta-adjusted analysis is most useful when it is implemented as a planned set of interpretable scenarios. For each fixed scenario, retain the observed values, generate predictive imputations under the MAR model, adjust only the missing values in the designated patterns, and apply the same analysis to every completed data set. The parameter uncertainty and predictive variation in the imputations should be retained rather than replaced by fitted means.

Interpreting Results and Avoiding Common Errors

Rubin’s combining rules summarize uncertainty from sampling and imputation under a given imputation model and fixed sensitivity scenario. They do not express uncertainty about which delta scenario is correct. A sensitivity analysis therefore presents results across scenarios; it should not treat a range of delta values as though it were automatically incorporated into the usual multiple-imputation standard error. If substantive knowledge supports a probability distribution over delta values, that uncertainty requires an explicit additional model and analysis.

Choose delta values on a scale a reader can interpret. For a continuous outcome, relate shifts to meaningful units or a clinically relevant difference. For a binary outcome, describe the odds multiplier as well as the resulting probabilities at representative MAR risks, because the same odds shift does not produce the same probability change for every starting risk. It is also useful to identify a tipping point: a delta value at which an important conclusion, such as whether an interval includes zero, changes.

• Wrong: “The data estimate the delta.” The missing outcomes are unobserved, so their conditional distribution is not identified by the observed-data likelihood. Correction: justify delta values substantively and report the conclusions conditional on them.
• Wrong: “Shift every value in the data set.” That changes observed outcomes as well as the hypothesized missing outcomes. Correction: apply the adjustment only to missing draws in the selected patterns.
• Wrong: “Use the same numerical delta for every outcome type.” A five-unit additive shift and a five-unit log-odds shift have different meanings and scales. Correction: define the adjustment on an outcome-appropriate scale and state its interpretation.
• Wrong: “A single pooled standard error covers all sensitivity assumptions.” The standard error within a scenario does not account for uncertainty about the delta choice. Correction: show estimates and intervals across the specified scenarios.

The central benefit of delta adjustment is not that it makes the unobserved outcomes known. It exposes the consequence of a clear, interpretable departure from MAR, pattern by pattern, while preserving the observed data and the ordinary uncertainty represented within each multiple-imputation analysis.