Pattern-Mixture Models for Missing Data

Introduction

Missing data are almost unavoidable in longitudinal clinical trials. Patients may withdraw from treatment, discontinue because of adverse events, become lost to follow-up, miss visits, or remain in the study but fail to provide an outcome measurement.

The statistical problem is not simply that some observations are absent. The more important issue is that the values that would have been observed after dropout are unknown.

For example, suppose a patient discontinues an antidepressant trial because their symptoms worsen. The unobserved future outcome may be systematically different from the outcome that would have been observed had the patient remained in the study.

A primary analysis therefore requires assumptions about the relationship between the observed data, the missing data, and the mechanism generating missingness.

Why Missingness Matters

Consider a two-arm randomized clinical trial with repeated measurements at baseline and Weeks 4, 8, and 12.

Patients 3 and 4 have incomplete outcome trajectories.

The central question is: What should we assume about the outcomes that were never observed?

There is no purely statistical way to determine those values from the observed data alone.

Consequently, missing-data assumptions are part of the inferential model.

Three Common Missingness Mechanisms

Missing-data methodology commonly distinguishes among:

• Missing completely at random (MCAR)
• Missing at random (MAR)
• Missing not at random (MNAR)

MCAR

Under MCAR, the probability of missingness does not depend on either observed or unobserved outcome values, conditional on the relevant study variables.

Conceptually:

where \(R\) denotes the missingness indicator.

MCAR is a strong assumption and is often unrealistic in longitudinal clinical trials.

MAR

Under MAR, missingness may depend on observed information but does not depend on the unobserved outcomes after conditioning on the observed data and covariates included in the model.

Many likelihood-based longitudinal analyses and standard multiple-imputation procedures are valid under MAR, assuming the model is correctly specified.

MNAR

Under MNAR, missingness depends on the unobserved outcome even after conditioning on observed information.

A conceptual representation is:

MNAR assumptions cannot generally be verified from the observed data alone.

Where Pattern-Mixture Models Fit

There are several ways to factor the joint distribution of the outcome and missingness process.

A pattern-mixture model starts with:

where:

• \(Y\) represents the complete outcome vector.
• \(R\) represents the missingness pattern.
• \(f(Y\mid R)\) describes the outcome distribution within a missingness pattern.
• \(P(R)\) describes the probability of each missingness pattern.

The central modeling object is therefore:

for each relevant missingness pattern \(r\).

What Is a Missingness Pattern?

A missingness pattern describes which measurements are observed and which are missing.

For a study with three post-baseline visits, possible patterns might include:

The exact set of patterns depends on the visit schedule and the structure of the missing data.

The Pattern-Mixture Factorization

Suppose there are two missingness patterns:

Then:

More generally:

The observed data inform the outcome distribution for each pattern to different degrees.

The challenge is that some components of \(f(Y\mid R=r)\) involve outcomes that are missing within that pattern.

Why Pattern-Mixture Models Can Be Nonidentified

Suppose patients who discontinue after Week 8 have no Week 12 measurement.

We can estimate the distribution of Week 12 outcomes among completers.

But we cannot directly estimate the distribution of Week 12 outcomes among patients who discontinued at Week 8 because those outcomes were never observed.

The observed data therefore do not uniquely identify the full joint distribution without additional assumptions.

Identification Through an Extrapolation Restriction

A common strategy is to specify an assumption connecting the distribution of missing outcomes to distributions that can be estimated from observed data.

For example, suppose the conditional mean of a missing Week 12 outcome among dropouts is related to the mean among comparable observed patients.

One might specify:

where \(\delta\) is a sensitivity parameter.

If:

the dropout pattern is assumed to have the same conditional mean as the reference pattern.

Other values of \(\delta\) represent departures from that assumption.

Delta Adjustment

The delta-adjustment approach is one of the most practical pattern-mixture sensitivity-analysis methods.

First, estimate the missing outcomes under an MAR model.

Then shift the imputed outcomes by a prespecified sensitivity parameter.

For a continuous endpoint:

For an endpoint in which larger values indicate worse outcomes, the direction of \(\delta\) should be chosen accordingly.

Pattern-Specific Delta Adjustments

The sensitivity parameter does not have to be the same for every pattern.

For example:

where \(\delta_r\) is specific to missingness pattern \(r\).

This allows investigators to distinguish between, for example:

• Patients who discontinued early
• Patients who discontinued late
• Patients who withdrew because of adverse events
• Patients who withdrew for lack of efficacy
• Patients who were administratively censored

A clinically informed sensitivity analysis may therefore be more realistic than applying a single arbitrary shift to all missing observations.

A Simple Longitudinal Example

Suppose a randomized trial measures a continuous symptom score at Weeks 4, 8, and 12.

Lower scores indicate improvement.

The primary estimand is the treatment difference in mean Week 12 score.

Assume:

Suppose 20 active-treatment patients and 30 control patients have missing Week 12 measurements.

A primary MAR analysis produces:

and:

Therefore the estimated treatment difference is:

Because lower scores indicate improvement, the treatment difference of \(-6.0\) favors active treatment.

Applying a Delta Adjustment

Now suppose we want to assess the possibility that patients with missing outcomes would have experienced worse outcomes than predicted under MAR.

Choose:

for missing outcomes in both treatment groups.

The missing outcomes become:

The resulting analysis is then repeated using these adjusted imputations.

The purpose is not to claim that \(+5\) is the true value.

Instead, it asks: Would the clinical conclusion remain the same if missing outcomes were 5 points worse than under the MAR assumption?

Why the Treatment-Group-Specific Delta Matters

In many clinical trials, the missingness mechanism may plausibly differ between treatment groups.

For example, patients receiving active treatment may discontinue because of lack of efficacy less frequently than patients receiving placebo.

Alternatively, adverse-event discontinuation may be more common in the active arm.

A sensitivity analysis might therefore specify:

and:

This can be substantially more informative than assuming the same missing-data behavior in both arms.

Reference-Based Imputation

A related pattern-mixture approach is reference-based imputation.

Instead of specifying an arbitrary numerical shift, the investigator uses information from a reference treatment group to define the post-dropout trajectory.

Common concepts include:

• Jump to reference: after discontinuation, the patient's future outcomes follow the reference group's distribution.
• Copy reference: the patient's future trajectory follows the estimated trajectory from the reference group.
• Copy increments in reference: the patient's future changes follow the changes observed in the reference group.

These assumptions can be attractive when there is a clear clinical rationale for using the control group as a reference after treatment discontinuation.

Pattern-Mixture Models and Multiple Imputation

Pattern-mixture models are often implemented through multiple imputation.

The general strategy is:

Rubin's Rules

Suppose \(M\) multiply imputed datasets produce estimates:

The pooled point estimate is:

The within-imputation variance is:

The between-imputation variance is:

The total variance is:

The resulting pooled estimate and variance form the basis for inference.

What Is a Tipping-Point Analysis?

A tipping-point analysis systematically varies the assumptions about missing outcomes to determine where the statistical conclusion changes.

For example, suppose the primary MAR analysis gives:

with a confidence interval that excludes zero.

We could repeat the analysis under:

The tipping point is the region in which the conclusion changes, for example when the confidence interval first includes zero.

A Two-Arm Tipping-Point Grid

A particularly useful sensitivity analysis allows the treatment and control groups to have different departures from MAR.

For example:

and:

Every combination produces a separate sensitivity analysis.

The exact boundary depends on the data and analysis model.

The value of the grid is that it makes the sensitivity assumptions transparent.

Pattern-Mixture Versus Selection Models

Pattern-mixture and selection models are two different factorizations of the same joint distribution.

Neither factorization is inherently "more correct."

They emphasize different aspects of the unobserved-data problem.

Pattern-Mixture Versus Shared-Parameter Models

A third framework is the shared-parameter model.

A shared-parameter model introduces latent variables that influence both the outcome process and the missingness process.

Conceptually:

This differs from a pattern-mixture model, which conditions directly on the missingness pattern.

Why MAR Is Not Automatically "Pattern-Mixture"

Pattern-mixture modeling and MNAR sensitivity analysis are closely related, but they should not be treated as identical concepts.

A pattern-mixture factorization can represent MAR as a special case when the appropriate restrictions are imposed.

Similarly, pattern-mixture methods can be used to construct MNAR sensitivity analyses.

A More Formal Longitudinal Formulation

Let the complete outcome vector be:

and let:

where:

The pattern-mixture representation is:

For pattern \(r\), the model becomes:

The observed portion of this distribution can be estimated from the data.

The unobserved portion requires an identifying restriction or sensitivity parameter.

Conditional Mean Structure

For a continuous endpoint, one might specify a pattern-specific multivariate normal model:

where:

• \(\mu_r\) is the mean vector for pattern \(r\)
• \(\Sigma_r\) is the covariance matrix for pattern \(r\)

The model can incorporate baseline covariates and treatment effects.

For example:

Different patterns can therefore have different conditional mean structures.

Pattern-Specific Covariance

The covariance structure can also vary by missingness pattern:

In practice, however, fully pattern-specific covariance matrices can become unstable when some missingness patterns contain relatively few patients.

Analysts may therefore impose common covariance structures or borrow information across patterns.

Why Sparse Patterns Are a Problem

Suppose a study has 10 possible missingness patterns but two patterns contain only three patients each.

Trying to estimate a completely separate multivariate model for every pattern may produce:

• Unstable estimates
• Nonpositive-definite covariance matrices
• Large standard errors
• Convergence problems
• Highly sensitive results

A practical pattern-mixture analysis therefore needs to consider both the statistical model and the amount of information available within each pattern.

Collapsing Missingness Patterns

One possible strategy is to combine clinically similar patterns.

For example:

However, collapsing patterns should have a clinical or statistical rationale.

It should not be done simply because a desired result is obtained.

Monotone Missingness

Many pattern-mixture implementations are easiest when missingness is monotone.

For example:

represent progressively earlier dropout.

This structure allows sequential conditional models such as:

The factorization can then be adapted for the observed and unobserved components of each pattern.

Nonmonotone Missingness

Real clinical-trial data can also contain intermittent missingness.

For example:

means the Week 8 measurement is missing even though later measurements are observed.

Such patterns require careful handling because information after a missing visit can help predict the missing value.

A naive monotone transformation may discard useful information.

Delta Adjustment in a Longitudinal Setting

Suppose the primary MAR model produces an imputed Week 12 outcome \(Y_{12}^{MAR}\).

A delta-adjusted value is:

If lower scores indicate improvement, positive \(\delta\) makes the imputed outcome worse.

If higher scores indicate improvement, the interpretation reverses.

Delta Adjustment Does Not Mean Adding a Constant to Everyone

A common misunderstanding is that delta adjustment simply changes every observed outcome.

That is incorrect.

The adjustment is generally applied to the unobserved/imputed outcomes, not the observed outcomes.

The observed data remain unchanged.

while:

Sequential Delta Adjustment

For longitudinal outcomes, a single final-visit shift may not always be the most clinically realistic assumption.

An analyst might instead specify a trajectory:

so that:

and:

This represents a progressively larger departure from the MAR trajectory.

Jump-to-Reference

Suppose the active-treatment patient discontinues treatment at Week 8.

A jump-to-reference assumption may state that, after discontinuation, their future outcomes follow the control group's conditional distribution.

Conceptually:

This can be useful when the clinical question is: What would have happened had the patient effectively reverted to the reference treatment trajectory after stopping active treatment?

Copy Increments in Reference

Another reference-based assumption is to copy the future change observed in the reference group.

Suppose a patient has an observed Week 8 score \(Y_8\).

If the expected control-group change from Week 8 to Week 12 is:

then the post-discontinuation prediction may use:

This differs from simply setting the patient's Week 12 outcome equal to a control-group mean.

Choosing Sensitivity Parameters

Sensitivity parameters should ideally be motivated by:

• Clinical knowledge
• Observed longitudinal trajectories
• Historical trial data
• External studies
• Reasons for treatment discontinuation
• Expected behavior after discontinuation
• Regulatory expectations for sensitivity analysis

A grid of arbitrary numbers can be useful for exploration, but the primary sensitivity analysis should emphasize clinically interpretable scenarios.

Reason-Based Sensitivity Analyses

The reason for missingness can be especially informative.

Consider three groups:

This can motivate pattern-specific or reason-specific sensitivity parameters.

Pattern-Mixture Models and Estimands

Modern clinical-trial missing-data analyses should be connected to the estimand framework.

Before choosing an imputation strategy, define what treatment effect is being estimated.

For example, the estimand may concern:

• Effect regardless of treatment discontinuation
• Effect while adhering to treatment
• Effect under a hypothetical continued-treatment scenario
• Effect incorporating intercurrent events as part of the outcome

The missing-data strategy should support the chosen estimand rather than silently define it.

Intercurrent Events and Missing Outcomes

Treatment discontinuation and missing outcome assessment are related but are not necessarily the same statistical event.

A patient may discontinue treatment but continue to provide outcome data.

Conversely, a patient may remain on treatment but miss a particular visit.

Therefore:

Pattern-Mixture Models and MMRM

A mixed model for repeated measures, or MMRM, is frequently used as a primary analysis for continuous longitudinal endpoints.

An MMRM can provide valid inference under MAR when its assumptions and model specification are appropriate.

Pattern-mixture sensitivity analyses can then be used to examine departures from the MAR assumption.

This creates a useful analysis structure:

The sensitivity analyses should generally preserve the primary estimand and change only the assumptions about unobserved outcomes.

A Complete Worked Example

Consider a randomized two-arm clinical trial with a continuous endpoint measured at baseline and Weeks 4, 8, and 12.

The endpoint is a symptom score where lower values indicate improvement.

Suppose the primary analysis estimates the Week 12 treatment difference using an MMRM with treatment, visit, treatment-by-visit interaction, baseline score, and other prespecified covariates.

The primary analysis favors active treatment because the confidence interval excludes zero.

Constructing the Sensitivity Analysis

Suppose the investigator believes that patients with missing outcomes may have had worse outcomes than predicted under MAR.

Because larger symptom scores indicate worse outcomes, positive deltas represent worse unobserved outcomes.

Consider:

The delta of zero corresponds to the MAR analysis.

For each delta:

The completed datasets are analyzed using the same prespecified treatment effect model.

Illustrative Sensitivity Results

These numbers are illustrative rather than universal; the actual estimates depend on the data, imputation model, covariance structure, and analysis model.

The important interpretation is that the conclusion remains favorable under moderate departures from MAR but becomes uncertain under sufficiently large departures.

Finding the Tipping Point

In this example, the tipping point occurs between approximately:

More precisely, one could use a finer grid:

and identify the point at which the confidence interval crosses zero.

The interpretation might then be: The treatment conclusion would change only if missing outcomes were approximately 9 points worse than their MAR-imputed values.

Whether that departure is clinically plausible is a separate question.

Statistical Robustness Versus Clinical Plausibility

A sensitivity analysis has two dimensions.

A result that survives an implausibly extreme sensitivity assumption may be statistically reassuring but clinically uninformative.

Conversely, a small tipping point may be concerning if the corresponding departure is clinically realistic.

Pattern-Specific Sensitivity Example

Suppose there are two important dropout patterns:

• Early dropout after Week 4
• Late dropout after Week 8

The investigator might specify:

and:

The assumption is that early discontinuers would have had outcomes further from their MAR predictions than late discontinuers.

This could be justified if early withdrawal is more strongly associated with lack of efficacy.

Pattern-Mixture Sensitivity Analysis Workflow

R Implementation: Basic Delta Adjustment

A simple implementation can be constructed after generating MAR imputations.

delta <- 5

completed_data$Y12_sensitivity <-
  ifelse(
    completed_data$Y12_missing == 1,
    completed_data$Y12_mar + delta,
    completed_data$Y12_mar
  )

In a real multiple-imputation analysis, the adjustment should be applied within each imputed dataset rather than to a single completed dataset.

R Implementation With Multiple Deltas

deltas <- c(0, 2, 4, 6, 8, 10)

results <- lapply(deltas, function(delta) {

  dat <- completed_data

  dat$Y12_sens <- ifelse(
    dat$missing_Y12 == 1,
    dat$Y12_mar + delta,
    dat$Y12_mar
  )

  fit <- lm(
    Y12_sens ~ treatment + baseline,
    data = dat
  )

  coef(summary(fit))["treatmentActive", ]

})

results

For a production analysis, the model, imputation structure, variance calculation, and pooling procedure should match the prespecified statistical analysis plan.

R Multiple-Imputation Framework

Packages such as mice can be used to generate multiple imputations.

library(mice)

imp <- mice(
  analysis_data,
  m = 100,
  method = "pmm",
  seed = 2026
)

The imputed datasets can then be extracted and modified according to the pattern-mixture sensitivity assumption.

datasets <- lapply(
  1:100,
  function(i) complete(imp, i)
)

A delta-adjusted analysis can then be repeated within every imputed dataset.

Pooling the Results in R

For the primary MAR analysis, a typical multiple-imputation workflow may use:

fit <- with(
  imp,
  lm(Y12 ~ treatment + baseline)
)

pooled <- pool(fit)

summary(
  pooled,
  conf.int = TRUE
)

For a sensitivity analysis, the same principle applies after the missing outcomes have been modified according to the pattern-mixture assumption.

SAS Implementation Concept

In SAS, pattern-mixture sensitivity analyses can be implemented using PROC MI, DATA steps, and PROC MIXED or another prespecified analysis model.

proc mi data=analysis_data
        out=mi_data
        nimpute=100
        seed=2026;

   class treatment;

   var baseline week4 week8 week12;

run;

After the MAR imputations are generated, a delta adjustment can be applied to the imputed observations.

data mi_sensitivity;

   set mi_data;

   if missing_flag=1 then
      week12_sens = week12 + 5;
   else
      week12_sens = week12;

run;

The analysis can then be repeated within each imputation and combined using PROC MIANALYZE.

proc mixed data=mi_sensitivity;
   by _Imputation_;

   class treatment;

   model week12_sens =
         treatment
         baseline
         / solution;

   ods output SolutionF=estimates;

run;

proc mianalyze data=estimates;
   modeleffects treatmentActive;
run;

The exact implementation depends on the imputation model, endpoint, covariance structure, and estimand.

Do Not Treat Code as the Method

A sensitivity analysis is defined by its statistical assumptions, not by the software used to implement it.

The same pattern-mixture assumption can be implemented in R, SAS, Stata, Python, or other software.

The critical questions are:

• What outcomes are assumed to be missing?
• What distribution is assumed for those outcomes?
• What identifies that distribution?
• How does the assumption differ from MAR?
• How is uncertainty propagated through the analysis?

Common Mistake: Complete-Case Analysis

A common response to missing data is simply to analyze only patients with complete Week 12 measurements.

This can be inefficient and can introduce bias when completion status is associated with outcomes or treatment effects.

Complete-case analysis may be valid under restrictive assumptions, but it should not automatically be treated as a robust missing-data strategy.

Common Mistake: Last Observation Carried Forward

Another historical approach is last observation carried forward, or LOCF.

This replaces a missing value with the patient's most recent observed value.

For example:

LOCF implicitly imposes assumptions about the future trajectory of patients after dropout.

Those assumptions are often difficult to justify scientifically and do not automatically produce valid uncertainty estimates.

Common Mistake: Assuming MAR Is Proven

MAR cannot generally be proven from observed data.

The missingness mechanism involves unobserved outcomes, so distinguishing MAR from MNAR is fundamentally limited by nonidentification.

The appropriate goal is therefore not to "prove MAR."

Instead, analysts should:

• Use clinically justified primary assumptions.
• Conduct prespecified sensitivity analyses.
• Assess robustness to plausible departures.
• Explain the implications of the missing-data assumptions.

Common Mistake: Arbitrary Delta Values

A sensitivity analysis using delta values of:

is not automatically informative.

The values should correspond to meaningful clinical differences.

For example, a 5-point shift may be substantial for one endpoint and trivial for another.

The sensitivity parameter should therefore be interpreted in the units and scale of the endpoint.

Common Mistake: Ignoring the Direction of the Endpoint

Suppose higher values indicate improvement.

Then:

means better unobserved outcomes.

If higher values indicate worse disease severity, the same numerical shift means worse outcomes.

This seems obvious, but it is an important source of mistakes when sensitivity analyses are automated across multiple endpoints.

Common Mistake: Changing the Primary Analysis Too

A sensitivity analysis should generally isolate the missing-data assumption.

For example, if the primary analysis uses an MMRM, changing simultaneously to a completely different endpoint definition, covariance model, population, and imputation method makes it difficult to determine what caused the change in the result.

A cleaner sensitivity analysis changes the missing-data assumption while keeping other components of the analysis aligned with the estimand and SAP.

Common Mistake: Using Too Few Imputations

Multiple imputation requires enough imputations to adequately represent between-imputation uncertainty.

When missing information is substantial, using a very small number of imputations can add unnecessary Monte Carlo error.

Modern analyses often use substantially more imputations than the historical default of five or ten, especially when the fraction of missing information is large.

Common Mistake: Treating Sensitivity Analysis as a Fishing Exercise

Investigators should not search over dozens of missing-data assumptions until finding one that produces a preferred conclusion.

Sensitivity analyses should be:

• Prespecified where possible
• Clinically motivated
• Transparent
• Reproducible
• Interpretable

Exploratory analyses can still be valuable, but they should be identified as such.

What Makes a Good Pattern-Mixture Sensitivity Analysis?

Pattern-Mixture Models in Regulatory Submissions

For confirmatory clinical trials, missing-data sensitivity analyses should be documented clearly in the statistical analysis plan and clinical study report.

The documentation should explain:

• The primary estimand
• The primary missing-data assumption
• The primary analysis model
• The missingness patterns
• The sensitivity-analysis assumptions
• The sensitivity parameter values
• The rationale for those values
• The imputation method
• The analysis performed after imputation
• The pooling method
• The effect estimates and confidence intervals
• The tipping point, if applicable

What to Put in the Statistical Analysis Plan

A strong SAP section might specify:

Primary Analysis Versus Sensitivity Analyses

Interpreting a Robust Result

Suppose the primary analysis favors active treatment.

If the conclusion remains favorable under a broad range of clinically plausible pattern-mixture assumptions, the result can be described as robust to those missing-data assumptions.

This does not prove that the treatment effect is true.

It means that the inference is not highly sensitive to the particular unobserved-outcome assumptions examined.

Interpreting a Non-Robust Result

Suppose a very modest departure from MAR causes the confidence interval to include the null value.

That indicates that the treatment conclusion is sensitive to assumptions about unobserved outcomes.

This does not necessarily invalidate the study.

Instead, it should prompt careful discussion of:

• Why outcomes are missing
• Whether the sensitivity assumption is clinically plausible
• How much information is missing
• Whether dropout differs between treatment groups
• Whether the estimand remains appropriate

Pattern-Mixture Models and Dropout by Treatment Arm

A useful diagnostic is to examine missingness rates separately by treatment group.

Different dropout rates do not by themselves prove MNAR.

They do, however, provide useful information for understanding the observed missingness process and motivating sensitivity analyses.

Reasons for Dropout Should Be Examined

A missingness-pattern table should ideally be accompanied by a reason-for- discontinuation summary.

These observed patterns can help determine which sensitivity assumptions are most clinically relevant.

Pattern-Mixture Models for Binary Outcomes

Pattern-mixture concepts are not limited to continuous outcomes.

For a binary endpoint:

the pattern-specific probability can be modeled as:

Sensitivity parameters can then modify the assumed probability of response among patients with a particular missingness pattern.

Because probabilities are bounded between zero and one, additive delta adjustments may be inappropriate.

Instead, sensitivity may be specified on a logit scale:

This guarantees that the resulting probability remains within the valid range.

Pattern-Mixture Models for Time-to-Event Outcomes

Missing-data assumptions for survival endpoints require additional care because censoring and missing event information have a different structure from missing continuous measurements.

A standard survival analysis generally treats censoring according to a conditional independent-censoring assumption.

If informative dropout is suspected, sensitivity analyses may instead alter assumptions about the event distribution after censoring.

The pattern-mixture framework can be extended conceptually, but the specific implementation should reflect the estimand and survival-data structure.

Pattern-Mixture Models for Count Outcomes

For count outcomes, the conditional distribution might use a Poisson or negative-binomial model.

For example:

Sensitivity assumptions can then modify the conditional mean or other distributional parameters.

Again, the sensitivity parameter should respect the natural scale of the endpoint.

Pattern-Mixture Models and Bayesian Analysis

Pattern-mixture models can also be implemented within a Bayesian framework.

The sensitivity parameter can be assigned a prior distribution:

The posterior distribution then incorporates uncertainty about the sensitivity parameter itself.

This provides a principled way to represent uncertainty about unobserved outcomes, although the choice of prior remains an important scientific assumption.

Frequentist Sensitivity Analysis vs. Bayesian Sensitivity Analysis

Pattern-Mixture Analysis Is a Sensitivity Framework

The most important practical point is that pattern-mixture models are often used not because the analyst believes one particular MNAR assumption is certainly true, but because they provide a structured way to examine uncertainty.

A strong analysis therefore asks:

Reporting the Sensitivity Analysis

A concise clinical-trial report might state that:

• The primary analysis assumed MAR conditional on the prespecified covariates.
• A pattern-mixture delta adjustment was used for sensitivity analysis.
• Positive delta values represented worse outcomes among patients with missing measurements.
• The analysis was repeated across a prespecified range of delta values.
• Results were pooled across multiple imputed datasets.
• The tipping point was identified from the resulting confidence intervals.

The actual numerical results should then be presented in a table or figure.

Recommended Sensitivity-Analysis Table

How to Think About the Tipping Point

A tipping point is not a magical threshold.

It is simply a convenient summary of the sensitivity surface.

For example, if the treatment conclusion changes only when:

then the key question becomes whether a 9-point departure from MAR is clinically plausible.

That question requires subject-matter knowledge, not another statistical test.

Practical Checklist

• Define the estimand before selecting the missing-data method.
• Describe the observed missingness patterns.
• Examine reasons for treatment discontinuation and missing assessments.
• Specify the primary missing-data assumption.
• Use an appropriate primary analysis model.
• Construct clinically meaningful pattern-mixture sensitivity analyses.
• State the direction of every sensitivity parameter.
• Consider treatment-specific or pattern-specific assumptions when justified.
• Use multiple imputation when appropriate to propagate uncertainty.
• Apply Rubin's rules correctly.
• Evaluate a range of plausible sensitivity parameters.
• Identify the tipping point when useful.
• Distinguish statistical robustness from clinical plausibility.
• Prespecify important sensitivity analyses whenever possible.
• Document all assumptions in the SAP and CSR.

Pattern-Mixture Model Decision Framework

The Most Important Concept

Pattern-mixture models provide a way to think directly about the distribution of outcomes within different missingness patterns.

The central problem is that some parts of those distributions involve outcomes that were never observed.

Consequently, additional assumptions are required.

Delta adjustment provides a particularly transparent way to express those assumptions:

The value of the sensitivity analysis is not that it tells us what the unobserved outcomes actually were.

Instead, it tells us how the treatment conclusion behaves if the unobserved outcomes differ from the primary MAR predictions in clinically meaningful ways.

References

Little, R.J.A. (1993). Pattern-mixture models for multivariate incomplete data. Journal of the American Statistical Association, 88, 125–134.
Little, R.J.A. & Rubin, D.B. (2019). Statistical Analysis with Missing Data. 3rd ed. Wiley.
Molenberghs, G. & Kenward, M.G. (2007). Missing Data in Clinical Studies. Wiley.
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Carpenter, J.R., Roger, J.H., Cro, S. & Kenward, M.G. (2014). Analysis of longitudinal trials with missing data: a framework for relevant, accessible assumptions, and inference via multiple imputation. Journal of the Royal Statistical Society: Series C.
Ratitch, B., O'Kelly, M., Tosiello, R., et al. (2013). Exploring the estimand framework in clinical trials and the role of missing data assumptions. Pharmaceutical Statistics.
National Research Council (2010). The Prevention and Treatment of Missing Data in Clinical Trials. National Academies Press.
ICH E9(R1) (2019). Addendum on Estimands and Sensitivity Analysis in Clinical Trials. International Council for Harmonisation.