Why a MAR Reference Is Needed
A delta sensitivity analysis asks how conclusions might change if the distribution of missing values differs from the distribution predicted by a missing-at-random (MAR) imputation model. That question needs a defined starting point. Without a reference, a statement such as “increase the imputed values” is incomplete: it does not say which distribution is being changed, or what the imputed values would have been in the absence of the sensitivity departure.
The reference is the conditional predictive distribution supplied by the chosen MAR imputation model. It is a baseline for comparisons, not proof that MAR is true. Delta indexes alternative distributions relative to that baseline. The key design rule is that the sensitivity family must return exactly to the MAR reference when delta is zero, while observed values remain as recorded at every delta setting.
In practice, a MICE reference distribution comes from the conditional model used for the incomplete variable, together with the other variables included in that imputation step. Predictive draws should reflect the model’s uncertainty as well as individual residual variation; a single fitted conditional mean is not, by itself, a predictive distribution. The reference is conditional on the model and the predictors available to it, so it may differ across individuals.
The MAR interpretation requires that, conditional on the variables represented in the imputation model, missingness does not depend on the unobserved value itself. The imputation model must also be sufficiently well specified for the relevant conditional distributions, and there must be adequate information to estimate them for the covariate patterns where values are missing. These assumptions define the reference analysis. A delta sensitivity analysis then asks what happens under specified departures from that reference; observed data alone generally cannot identify which departure is correct.
What “Anchored” Means in a Distribution
The anchor is a condition on the entire conditional distribution, not merely on one summary such as a mean. At \(\delta=0\), the probabilities or density assigned to possible values must be those of the MAR reference. At nonzero delta, the analyst specifies a different conditional distribution to represent the sensitivity scenario. The family must be a valid distribution for every value of delta considered: its probabilities must be nonnegative and sum to one, or its cumulative distribution function must be nondecreasing, right-continuous, and have the appropriate limits.
A useful way to understand the connection between distributions is through quantiles. For a CDF \(F\), define its quantile function by \(Q_F(u)=\inf\{y:F(y)\ge u\}\), for \(0<u<1\). If \(U\sim\text{Uniform}(0,1)\), then \(Q_F(U)\) is a draw from \(F\). Thus one can generate a reference draw \(Q_{F_{0,i}}(U)\), or a sensitivity draw \(Q_{F_{\delta,i}}(U)\), using the same uniform draw to make the contrast transparent.
The probability identity follows because, for a valid CDF, the event that the quantile draw is no greater than \(y\) corresponds to \(U\le F_{\delta,i}(y)\). Since \(U\) is uniform, that event has probability \(F_{\delta,i}(y)\). At \(\delta=0\), the anchor condition gives \(Q_{F_{\delta,i}}=Q_{F_{0,i}}\), so the sensitivity draw has the reference distribution. This justifies treating the MAR imputation as the zero-delta case rather than as a separate, unrelated analysis.
Worked Example: Checking the Anchor With Conditional Distributions
Suppose, purely hypothetically, a missing outcome can take the values 0, 1, or 2. For an individual with a particular predictor pattern, the MAR reference probabilities are 0.20, 0.50, and 0.30, respectively. A proposed sensitivity scenario has probabilities 0.10, 0.40, and 0.50. Both sets are valid probability distributions because their entries are nonnegative and each set sums to 1.
The reference CDF has \(F_{0}(0)=0.20\), \(F_{0}(1)=0.70\), and \(F_{0}(2)=1\). The proposed sensitivity CDF has \(F_{\delta}(0)=0.10\), \(F_{\delta}(1)=0.50\), and \(F_{\delta}(2)=1\). They differ, so this proposed distribution represents a nonzero-delta scenario; it cannot be the distribution at \(\delta=0\).
To verify the sampling implication, take hypothetical uniform draws \(U=0.15,\ 0.55,\ 0.90\). Under the reference distribution, the corresponding quantile draws are 0, 1, and 2: the first uniform is at most 0.20, the second is above 0.20 but at most 0.70, and the third is above 0.70. Under the proposed sensitivity distribution, the draws are 1, 2, and 2: 0.15 exceeds 0.10, 0.55 exceeds 0.50, and 0.90 is also above 0.50.
If the sensitivity family is evaluated at \(\delta=0\), all three probabilities must instead be 0.20, 0.50, and 0.30. The same uniforms then reproduce reference draws 0, 1, and 2. Comparing the whole probability vector verifies the anchor more fully than checking only the average outcome.
Observed Values and Completed-Data Quantities
The sensitivity distribution describes values that are unobserved. It does not revise the recorded outcome for someone whose outcome was observed. In each completed data set, retain every observed \(Y_i\), and replace only the missing entries with draws from the selected \(F_{\delta,i}\). The resulting completed-data quantity can change with delta because the missing entries change, not because the observed entries have been altered.
Worked Example: Completed-Data Means From a Fixed Observed Sample
Consider a hypothetical sample of six individuals with outcome values, in score points, \(10,\ 12,\ 14,\ 16\), and two missing values. The sum of the four observed values is \(10+12+14+16=52\). Suppose three reference imputations give the missing pairs \((11,15)\), \((12,14)\), and \((10,16)\). Each completed-data mean is calculated over all six individuals.
For the first reference imputation, the completed total is \(52+11+15=78\), so the mean is \(78/6=13.00\) score points. For the second, it is \(52+12+14=78\), giving \(78/6=13.00\). For the third, it is \(52+10+16=78\), also giving \(78/6=13.00\). The average of these three illustrative completed-data means is \((13.00+13.00+13.00)/3=13.00\) score points.
Now suppose a specified nonzero-delta scenario produces the missing pairs \((9,13)\), \((10,14)\), and \((8,12)\) in three corresponding completed data sets. The observed total stays 52 in every calculation. The completed means are \((52+9+13)/6=74/6=12.33\), \((52+10+14)/6=76/6=12.67\), and \((52+8+12)/6=72/6=12.00\) score points. Their average is \((12.33+12.67+12.00)/3=12.33\) score points, using unrounded values for the final calculation.
This arithmetic illustrates how a sensitivity scenario can affect an estimate through the missing values while preserving all observed values. The listed draws are hypothetical; they do not define a particular delta scale or establish which scenario is plausible.
Practical Checks and Failure Modes
Anchoring and implementation should be checked separately. First check that the reference model is a defensible MAR baseline for the intended analysis. Then check the mathematical sensitivity family and finally verify that the software applies it to the right records. A correct anchor cannot rescue a poorly specified reference model, and a good reference model cannot rescue an incorrectly implemented adjustment.
Record the conditional model, predictors, and predictive-draw procedure that define \(F_{0,i}\). Check that predictors are available and relevant for records with missing values, and inspect model fit and chain behavior.
Evaluate the sensitivity specification at \(\delta=0\) and compare it with the reference distribution. For discrete outcomes, compare probabilities at each support value; for continuous outcomes, compare the specified CDFs or their quantiles.
Confirm that observed values are identical across delta settings and that only missing entries receive sensitivity draws. Recompute a simple completed-data summary by hand on a small test set.
Confirm each proposed distribution is valid and assigns draws to allowed values. Review whether the chosen departures are scientifically interpretable; a mathematically valid distribution need not be a credible sensitivity scenario.
Several symptoms suggest that something has gone wrong. If results at \(\delta=0\) differ from the MAR reference, the sensitivity family or implementation is not anchored as intended. If observed entries vary across delta settings, the adjustment has been applied to the wrong records. If probabilities become negative, fail to sum to one, or imply impossible values, the proposed family is invalid for that setting. If imputed distributions are implausible or chains do not adequately explore their predictive distributions, the reference imputation itself needs investigation before interpreting sensitivity results.
How the Reference Analysis Compares With Alternatives
| Approach | What it uses | What it can answer |
|---|---|---|
| MAR reference analysis | The specified conditional imputation distribution \(F_{0,i}\) | What follows under the chosen MAR imputation assumptions |
| Delta sensitivity analysis | A family \(F_{\delta,i}\) anchored to \(F_{0,i}\), evaluated at stated delta settings | How conclusions vary under specified departures from the MAR reference |
| Complete-case analysis | Only records with observed values for the variables needed in the analysis | An analysis of the observed subset; it does not specify a predictive distribution for missing values |
A MAR-only result gives no direct comparison with departures from MAR. A complete-case analysis avoids imputing missing values, but it does not supply a reference distribution for them and can target a different population or be biased, depending on the missingness and analysis assumptions. The delta approach makes its comparison explicit: the MAR analysis is the anchor, and alternative conclusions are conditional on the specified sensitivity distributions.
Delta is not learned from the observed values in the same way as parameters of the reference imputation model. The observed data generally do not reveal the distribution of outcomes among records whose outcomes are missing. Consequently, sensitivity values should be justified and reported as assumptions, not presented as estimates established by the data. The scale and outcome-specific forms of delta require separate choices; here the essential requirement is that every chosen family be clearly tied to its MAR reference.
Common Mistakes
- Using a tautological or vague anchor. Writing that the reference equals itself does not specify the sensitivity condition. State explicitly that \(F_{\delta,i}(y)=F_{0,i}(y)\) when \(\delta=0\), for all relevant \(y\).
- Checking only a summary. Equal means at zero do not guarantee equal distributions. Compare the full conditional distributions or a complete set of probabilities and support points.
- Changing observed values. Delta concerns the missing-value distribution. Keep every observed entry fixed in every completed data set.
- Treating the reference as known truth. MAR is an assumption used to define the baseline, not a conclusion proven by fitting the imputation model.
- Using an inconsistent denominator. A completed-data mean includes every individual, including those whose values were imputed. Count the sample size and verify the total before dividing.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A sensitivity family gives \(F_{\delta,i}(y)=F_{0,i}(y)\) at \(\delta=0\) for one value of \(y\), but not for other values. Is it anchored?
- A hypothetical six-person sample has four observed values totaling 52 and two missing values imputed as 9 and 13. What is the completed-data mean?
- In a hypothetical two-stratum example, the reference probability of outcome 2 is 0.25 in stratum A and 0.75 in stratum B. Each stratum has one missing value, and the two observed outcomes are 2 and 0. What is the completed mean using the stratum-specific reference means?
- At \(\delta=0\), the imputed values match the reference but one observed value differs from the recorded value. What implementation check has failed?
Solutions. (1) No. The equality must hold for the full distribution: \(F_{\delta,i}(y)=F_{0,i}(y)\) at \(\delta=0\) for every relevant \(y\). (2) The completed total is \(52+9+13=74\), so the mean is \(74/6=12.33\) in the outcome’s units. (3) The reference means are \(2(0.25)=0.50\) in A and \(2(0.75)=1.50\) in B. The completed total is \(2+0+0.50+1.50=4.00\), and the completed mean is \(4.00/4=1.00\). (4) The observed-value audit has failed: observed entries must remain unchanged at every delta setting, including zero.