Why Save a Baseline MAR Fit?
A sensitivity analysis needs a clearly defined reference point. A baseline analysis under a missing-at-random (MAR) assumption provides that reference: it estimates the target using the observed data and a stated outcome model, without imposing an additional departure from MAR. Later analyses can then be compared with it only if the estimand, outcome scale, analysis population, and model are kept aligned.
The baseline fit is not a test of whether MAR is true. MAR concerns the relationship between missingness and unobserved outcomes after conditioning on observed information; that relationship cannot generally be verified from observed data alone. The practical task here is narrower: specify and fit a defensible MAR analysis, inspect whether the implementation behaved as intended, and save enough information to reproduce and compare the result.
Specify the Reference Analysis Before Fitting
Write down the estimand and the model before running software. For example, in a two-arm longitudinal study, the estimand might be the treatment-group difference in mean symptom score at week 8, in symptom-score points, among the randomized participants. A repeated-measures model could include treatment, visit, and treatment-by-visit terms, with a specified within-subject covariance structure. The week-8 contrast is then calculated from the fitted model parameters. If baseline score is included, state how it enters the model and how the target contrast is defined.
The MAR analysis requires more than choosing an estimation command. Its principal assumptions should be made explicit:
- Outcome-model adequacy. The likelihood or imputation model represents the relevant outcome distributions, treatment and time effects, and within-subject dependence sufficiently well for the target estimand.
- MAR conditional on observed information. Given the observed outcomes and covariates used by the analysis, missingness does not further depend on the missing outcomes. This is an identifying assumption, not a conclusion from a convergence message or a missingness table.
- Ignorability conditions for likelihood inference. When the missingness mechanism is omitted from the likelihood, the outcome and missingness parameters must be distinct, and the mechanism must satisfy the MAR restriction. If these conditions do not hold, the omitted mechanism cannot simply be disregarded.
- Suitable information and support. The observed data must contain enough information to estimate the model, and relevant covariate and treatment combinations must not be unsupported. Sparse patterns or near-separation can make estimates unstable even when the model is formally specified.
- For multiple imputation, proper imputation. Imputed values must reflect uncertainty about missing outcomes and be generated from an imputation model that is compatible enough with the analysis model for the intended inference. The imputation model should include the analysis variables and useful observed predictors of outcomes or missingness.
Likelihood and multiple imputation are alternative implementation routes, not interchangeable labels for the same computation. A likelihood fit uses each subject's observed outcome components and integrates over missing components under the specified joint outcome model. Multiple imputation generates several completed data sets under an imputation model, fits the analysis model to each, and retains the resulting estimates and uncertainty for subsequent combination. The later pooling step should use the saved per-imputation results; a single imputation or an average of completed data sets does not by itself account for imputation uncertainty.
Record the estimand, population, outcome scale, covariates, time points, model, variance structure, missing-data method, and software settings before fitting.
Verify subject identifiers, treatment coding, visit labels, outcome units, duplicate records, and the observed-versus-missing indicators. Reconcile the analysis sample size with the encoded patterns.
Use the prespecified likelihood or imputation approach. Retain warnings and fit status rather than treating a returned estimate as proof of successful estimation.
Check model-specific diagnostics, store the estimate and uncertainty in a stable format, and preserve the specification, data version, software details, and run record.
Worked Example: Check the Analysis Patterns
Worked Example: Check the Analysis Patterns
Suppose a hypothetical two-arm trial has 120 randomized participants, 60 per arm, with symptom scores at baseline, week 4, and week 8. Baseline scores are complete. At week 4, 54 participants in each arm have observed scores; at week 8, 48 in the control arm and 46 in the intervention arm have observed scores. Before fitting a repeated-measures likelihood model, check the pattern counts.
At week 4, the number missing is \(60-54=6\) in each arm, so \(6+6=12\) week-4 scores are missing. At week 8, the numbers missing are \(60-48=12\) and \(60-46=14\), giving \(12+14=26\) missing week-8 scores. There are \(120\times 3=360\) planned outcome records. Since baseline is complete, the observed count is \(120+54+54+48+46=322\), and the missing count is \(12+26=38\). The check \(322+38=360\) reconciles the totals.
The analysis can use the observed components of each subject's outcome vector rather than discarding everyone with a missing follow-up. For this to be an ignorable MAR likelihood analysis, the joint outcome model must be appropriate and the missingness mechanism must satisfy MAR conditional on the observed data, with distinct parameters. The pattern counts confirm data bookkeeping; they do not establish those assumptions.
Save Results That Can Be Compared
A useful output record makes it possible to identify exactly what was fitted and to distinguish a change in assumptions from a change in implementation. Keep one record per reported parameter and, for multiple imputation, one record per imputation as well as the analysis-level metadata. Include the estimand name and units, estimate, standard error (\(SE\)), interval limits and method, sample size, outcome model, missing-data method, convergence or imputation diagnostics, and run identifier. Save the covariance matrix when later contrasts may require more than one parameter.
For likelihood fits, retain the log likelihood, convergence status, iteration count, gradient or other optimization diagnostic when available, and any boundary or singularity warnings. For multiple imputation, retain the number of imputations, random-number seed, imputation settings, and each imputation's estimates and \(SE\) values. Also record the data version or checksum, analysis code version, software and package versions, and the exact model specification. A seed alone is not enough to reproduce a run if the data, code, or software settings have changed.
Use an explicit scenario label such as baseline_MAR, not a vague label such as “main run.” Keep the estimand and parameter names stable across scenarios. A minimal record might contain: scenario label, estimand, parameter, estimate, \(SE\), interval limits, method, sample size, fit status, and run metadata. For MI, the imputation index belongs in the per-imputation result table. Save results at adequate precision; round only for presentation.
Worked Example: Save a Likelihood Estimate
Suppose a hypothetical repeated-measures likelihood fit estimates the week-8 intervention-minus-control contrast as \(-2.50\) symptom-score points, with \(SE=0.90\). For a large-sample, nominal 95% Wald interval, the calculation is
The interval is in symptom-score points. Under the fitted model and its large-sample approximation, it is a nominal 95% confidence interval for the week-8 contrast; it is not a 95% probability statement about this fixed parameter. A report record should retain \(-2.50\), \(0.90\), both interval limits, the Wald method, the estimand, units, model specification, fit status, and MAR label. The interval does not account for uncertainty about whether MAR is a suitable assumption.
Worked Example: Preserve Multiple-Imputation Results
Suppose a hypothetical analysis uses five imputations and fits the same week-8 contrast model in each. The saved estimates, in points, are \(-2.1,-1.8,-2.4,-2.0,-1.7\); the corresponding standard errors are \(0.80,0.82,0.85,0.79,0.81\). The per-imputation records should pair each estimate with its own standard error: imputation 1 has \((-2.1,0.80)\), imputation 2 has \((-1.8,0.82)\), and so on through imputation 5.
The estimates range from \(-2.4\) to \(-1.7\) points, so the saved values reveal visible between-imputation variation. This range is a descriptive check, not a pooled estimate or confidence interval. Save the five paired results, the number of imputations, the seed and imputation settings, and the shared model and estimand. Do not replace these records with a single completed-data fit: that would lose the between-imputation information needed for valid pooling.
Diagnostics and Common Failure Modes
A successful software exit is necessary but not sufficient. For likelihood estimation, inspect convergence warnings, gradient size, iteration history, parameter estimates near boundaries, and whether the covariance estimate is finite and plausible. A singular covariance estimate, extreme standard errors, or substantially different results from reasonable starting values can signal weak identification, a model mismatch, or inadequate data support. Examine fitted outcome distributions and residual or predictive checks appropriate to the model; a missingness-pattern table is useful for understanding data coverage, but it cannot diagnose MAR itself.
For multiple imputation, confirm that all imputations completed and that the imputation model used the intended variables and data types. Review trace plots or other available checks of the imputation process, summaries of observed and imputed values, and the per-imputation analysis estimates. Implausible imputed values, lack of variation across imputations, or extreme instability in the estimates deserve investigation. These checks can reveal implementation problems, but they cannot prove that the imputation model correctly represents the distribution of unobserved outcomes.
Common workflow failures include changing the analysis sample between the baseline and sensitivity runs, silently changing the outcome scale, dropping useful observed predictors from the imputation model, and overwriting an earlier result file. A practical safeguard is to save a read-only run record and compare its specification fields before comparing estimates. If a fit fails, do not label the failed output as a baseline estimate; resolve the failure or document that the intended model could not be estimated.
| Approach | What it uses | What to retain | Important limitation |
|---|---|---|---|
| Likelihood under MAR | Observed outcome components and the specified joint outcome model | Estimate, covariance or \(SE\), fit diagnostics, log likelihood, model and run metadata | Depends on model adequacy and the MAR and ignorability conditions |
| Multiple imputation under MAR | Completed data sets generated from an imputation model conditional on observed information | Per-imputation estimates and \(SE\) values, imputation settings, diagnostics, model and run metadata | Depends on a suitable imputation model and subsequent valid pooling |
| Complete-case analysis | Participants with all required analysis outcomes observed | Analysis sample definition and the same estimand and model details | Can lose information and may target a selected group or be biased unless additional conditions hold |
What the Baseline Does—and Does Not—Establish
The baseline fit is useful because it supplies a reproducible MAR reference estimate and documents the observed-data analysis against which later assumptions can be compared. A difference between a later sensitivity estimate and this reference is interpretable only when the estimand, outcome coding, population, and relevant model choices are aligned. If those change too, the difference cannot be attributed solely to a missing-data assumption.
The baseline does not show that missing outcomes follow MAR, and a close fit to observed outcomes cannot identify the distribution of values that were never observed. Likewise, an apparently stable estimate is not proof that all plausible MNAR departures would have little impact. The proper conclusion is conditional: under the stated outcome model and MAR assumptions, the saved analysis gives the reported estimate and uncertainty.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A repeated outcome model converges with a finite standard error. What does this establish, and what does it not establish about MAR?
- Why should a multiple-imputation workflow save an estimate and standard error from each imputation rather than only the result from one completed data set?
- A hypothetical study plans 100 participants at three visits. Baseline is complete, 90 week-4 scores and 82 week-8 scores are observed. How many planned records are there, and how many are missing?
- A baseline MAR fit and a later sensitivity fit use different outcome units. Can their numerical estimates be compared directly as a measure of sensitivity? Explain.
Solutions. (1) Convergence establishes only that the numerical fitting procedure met its specified stopping criterion and returned an estimate; it does not establish that the outcome model is correct or that missingness is MAR. MAR remains an assumption about dependence on unobserved outcomes. (2) Each completed data set gives one estimate with its own sampling uncertainty; retaining all pairs preserves the information needed to account for variation between imputations as well as within them. One completed-data result omits that between-imputation variation. (3) There are \(100\times3=300\) planned records. The observed count is \(100+90+82=272\), so the number missing is \(300-272=28\). (4) No. The estimates are on different scales, so a numerical difference may reflect unit changes rather than sensitivity to missing-data assumptions. Put both analyses on the same outcome scale and align the estimand before comparing them.