Why Vary the Missing-Data Assumption?
Multiple imputation can account for uncertainty when values are missing, but its usual imputation models rely on a key assumption: after conditioning on the information used in the imputation model, missing values behave like observed values. This is the missing at random (MAR) assumption. It cannot generally be verified using the observed data alone. If missingness still depends on the value that is missing, even after conditioning on observed information, the mechanism is missing not at random (MNAR).
A simple MNAR sensitivity analysis asks a focused question: Would the regression conclusion change if the missing values were systematically different from their MAR-based imputations by a specified amount? A delta adjustment makes that difference explicit. It does not discover the true MNAR mechanism or estimate the right adjustment from the observed data. Instead, it shows how results respond to a range of plausible assumptions.
The shift is applied only to the values that were missing, not to observed values. If multiple imputation was used, the adjustment is applied separately to the corresponding missing values in each imputed data set. The regression is then refitted and its results pooled at each specified delta. Retaining the variation in the imputed values matters: delta shifts their location; it is not a reason to replace all imputed values with one fixed prediction.
Choosing and Interpreting a Delta
First decide which missing variable is under consideration, what direction of departure is plausible, and what unit gives delta a meaningful interpretation. For example, if a missing symptom score is imputed as 12 points under MAR, \(\delta=-3\) means that the sensitivity scenario treats its value as 3 points lower, on average, than the MAR imputation. It does not mean every missing person has precisely the same unobserved score; it is a simple location shift used to test a stated scenario.
A useful analysis evaluates a set, or grid, of values rather than selecting just one. The grid should cover departures that are substantively plausible, including the MAR reference value \(\delta=0\). Knowledge of the measurement scale, subject-matter information, and the likely direction of selection can help motivate the values. There is usually no data-only rule that supplies a uniquely correct MNAR delta.
For an outcome adjustment, ask whether the missing outcomes might be higher or lower than their MAR predictions. For a covariate adjustment, ask the same question about the missing predictor values. A shift must respect the scale: blindly adding a constant to a bounded or categorical variable may create impossible values or meaningless categories. In such cases, the sensitivity scenarios need to be defined on an appropriate scale or through a suitable model for that variable.
Worked Example: Shifting Missing Outcomes and Refitting a Regression
Consider a small, hypothetical data set with predictor \(x\) and outcome \(y\). The outcome for the observation with \(x=2\) is missing. Suppose the MAR imputation predicts \(y=5\), and the other recorded outcomes are as shown below. For illustration, compare the MAR completion with a scenario in which the missing outcome is 1 unit higher.
| \(x\) | Observed or MAR-imputed \(y\) | Outcome after \(\delta=+1\) |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 3 | 3 |
| 2 | 5 (imputed) | 6 |
| 3 | 7 | 7 |
In the MAR completion, the simple linear regression slope is \(2\): the \(y\) values increase by 2 for each one-unit increase in \(x\). Under the shifted scenario, compute the slope from the completed data using \(b_1=S_{xy}/S_{xx}\), where \(S_{xy}=\sum_i(x_i-\bar{x})(y_i-\bar{y})\) and \(S_{xx}=\sum_i(x_i-\bar{x})^2\).
Thus the refitted slope is \(b_1=10.5/5=2.1\). The shift changes the fitted slope in this example; it does not merely change the intercept. The data are deliberately small and hypothetical, so this calculation illustrates refitting rather than supporting a population-level conclusion.
Why a Shift Can Change a Regression Coefficient
For a simple regression with fixed predictor values, let \(M_i=1\) when observation \(i\)'s outcome is missing and \(M_i=0\) otherwise. Suppose the MAR-completed outcome is \(y_i(0)\), and the sensitivity scenario changes it to \(y_i(\delta)=y_i(0)+\delta M_i\). The ordinary least squares slope at each completion is \(S_{xy}/S_{xx}\). Since adding \(\delta M_i\) changes the mean outcome by \(\delta\bar{M}\), its effect on the centered cross-product can be calculated directly:
The last equality holds because \(\sum_i(x_i-\bar{x})=0\). Since the predictor values, and therefore \(S_{xx}\), are unchanged, the slope becomes
This result explains why the pattern of missingness across \(x\) matters. If missing outcomes are concentrated at larger predictor values, changing them can move the slope differently than if they are concentrated at smaller values. If every outcome is missing, then \(M_i=1\) for every observation and \(\sum_i(x_i-\bar{x})M_i=0\): a uniform shift changes the intercept but not the slope. In a multiple regression, covariate adjustment, or multiple imputation, refitting is generally the clearest way to determine the effect; do not assume the coefficient changes by a fixed amount without checking.
Refitting and Comparing Pooled Results
At each delta, use the adjusted completed data to fit the same regression model and pool the estimates and their uncertainty across the imputations. In particular, do not adjust the pooled coefficient after the fact while leaving its standard error unchanged. A changed completion can affect both the estimate and its uncertainty. The model, target coefficient, and other analysis choices should remain the same across the grid so that differences can be attributed to the sensitivity scenario.
A confidence interval can help show when a statistical conclusion changes. The following example uses a large-degrees-of-freedom normal approximation for a 95% interval, \(\hat{\beta}\pm1.96SE\), solely to make the arithmetic transparent. In an actual pooled analysis, use the appropriate pooled degrees of freedom and critical value when required. The listed estimates and standard errors are hypothetical results from refitting and pooling at each delta; they are not calculated from the small data set above.
Worked Example: Finding Where an Interval Changes Its Conclusion
Suppose a hypothetical study estimates the regression coefficient for a treatment indicator, measured as the adjusted difference in outcome points between treatment and comparison groups. Across a range of outcome shifts applied to missing outcomes, the pooled results are:
| Outcome shift \(\delta\) (points) | Estimate (points) | SE (points) |
|---|---|---|
| \(-4\) | 0.100 | 0.195 |
| \(-2\) | 0.275 | 0.185 |
| 0 | 0.450 | 0.182 |
| \(+2\) | 0.625 | 0.185 |
At the MAR reference value \(\delta=0\), the estimate is 0.450 points. Using the stated approximation, its interval is calculated as follows:
This interval is entirely above zero. At \(\delta=-2\), the corresponding calculation is \(0.275\pm1.96(0.185)=0.275\pm0.3626\), giving \([-0.0876,\ 0.6376]\), which includes zero. Thus, on this grid, the 95% interval excludes zero at \(\delta=0\) but includes zero at \(\delta=-2\). The sensitivity analysis shows that the interval-based conclusion depends on the assumed shift; it does not show which shift is true. It also does not assign a probability to the shift or include uncertainty about delta in those intervals.
Applying the Same Idea to a Missing Covariate
A covariate sensitivity analysis shifts the missing predictor values, then refits the outcome regression. The interpretation is different from shifting outcomes: a one-unit change in \(\delta_x\) is measured in the predictor's units, and it can alter several regression coefficients through the changed predictor values. If the covariate is bounded, a simple additive shift may need a scale-aware alternative so that the adjusted values remain possible.
Worked Example: Adjusting an Imputed Predictor
Suppose the hypothetical outcome values are \(y=(2,3,6,8)\), while the predictor values are \(x=(1,2,4,5)\). The value \(x=4\) is an MAR imputation; the other predictor values are observed. Compare that completion with the scenario \(\delta_x=+1\), so the missing predictor becomes 5. For the MAR completion, \(\bar{x}=3\), \(\bar{y}=4.75\), \(S_{xx}=10\), and \(S_{xy}=15\). The fitted slope is \(15/10=1.5\).
After the shift, \(x=(1,2,5,5)\), while \(y\) remains unchanged. The new means are \(\bar{x}=3.25\) and \(\bar{y}=4.75\). The centered sums are
The refitted slope is \(16.25/12.75\approx1.275\), rather than 1.5. This calculation shows how a covariate shift can change the fitted relationship. With multiple imputation, perform this adjustment and refit for each imputed data set at every selected \(\delta_x\), then pool the results.
Common Mistakes and Practical Limits
- Treating delta as estimated truth. The observed data generally cannot identify the MNAR shift by themselves. State it as an assumption in a scenario, and examine a defensible range.
- Shifting observed and missing values alike. A delta adjustment targets values that were missing. Changing observed outcomes as well would describe a different analysis.
- Ignoring the units or sign. A positive delta means higher values than the MAR imputations under the definition used here. Always interpret the sign in the units of the adjusted variable.
- Assuming a coefficient must move in a particular direction. The effect depends on which observations are missing, their predictor values, and the model. Refit the model rather than guessing.
- Reporting only the scenario that supports a preferred conclusion. A sensitivity analysis is useful when it shows the range of assumptions examined and where results become materially different.
A delta analysis is a focused probe of one specified departure from MAR, not a complete description of every possible MNAR process. Its conclusions depend on the baseline imputation model, the chosen adjustment scale, and the range of shifts examined. When results change over plausible values, the analysis makes that dependence visible; when they do not, it shows stability only across the scenarios considered.
Check Your Understanding
Answer each question using the definitions and calculations in this tutorial.
- If an outcome is measured in points and \(\delta=-2\), how should the shift be interpreted for a missing value imputed under MAR?
- Why can a constant shift to every outcome leave the regression slope unchanged in a data set where every outcome is missing?
- In a multiple-imputation sensitivity analysis, what should be done to the completed data and regression at each value of delta?
- Why does a confidence interval that changes from excluding zero to including zero across the delta grid not establish which delta is correct?
- A predictor is bounded between 0 and 10. What issue should be checked before applying an additive shift to its missing imputed values?