Why a Delta Shift Can Leave the Valid Range
An additive delta adjustment changes a missing-value draw by a specified amount. For a continuous outcome \(Y\), a simple version is \(Z=Y^{MAR}+\delta\), where \(Y^{MAR}\) is a draw from the imputation distribution under the missing-at-random reference and \(\delta\) is a sensitivity parameter. When the outcome has a genuine lower or upper bound, however, \(Z\) can fall outside the values the outcome can take. For example, a delta shift to a percentage cannot make a valid percentage greater than 100.
Constraining the draw is not merely a computational repair. Truncating a distribution, shifting after a transformation, and adjusting by a fraction of the distance to a bound generally produce different distributions. The choice should reflect the intended sensitivity scenario, not just the desire to eliminate invalid numbers.
These methods assume that the bounds are substantively valid for the quantity being imputed, not merely the smallest and largest values observed in the sample. The bounds should apply to the same measurement scale and units as the outcome. If the observed sample contains no values near a possible endpoint, that alone does not establish a tighter bound. As with other delta adjustments, the resulting values encode sensitivity assumptions; they do not identify the unobserved outcomes from the observed data.
Truncating the Shifted Distribution
One option is to shift the reference draw and then condition its distribution on being within the valid range. For a fixed set of predictors, let \(Y^{MAR}\) have cumulative distribution function \(F_0\), and define the shifted draw \(Z=Y^{MAR}+\delta\). Retaining only draws for which \(L\leq Z\leq U\) gives a truncated distribution.
The denominator is the probability that the untruncated shifted draw is valid: \(P(L\leq Z\leq U)=F_0(U-\delta)-F_0(L-\delta)\). Dividing the shifted density by this probability makes the density integrate to one over the permitted interval. In practice, one can draw from the shifted distribution repeatedly until a valid value is obtained, provided the probability of acceptance is not too small.
Worked Example: Truncating a Shifted Normal Draw
Suppose, for a hypothetical patient with fixed predictors, the MAR imputation distribution is \(Y^{MAR}\sim N(8,3^2)\), measured in units that must lie between 0 and 10. Consider an additive shift of \(\delta=2\). The untruncated shifted variable is \(Z=Y^{MAR}+2\sim N(10,3^2)\), so it can exceed the upper bound.
The probability that a shifted draw is valid is \(P(0\leq Z\leq10)=P(-10/3\leq (Z-10)/3\leq0)\). Using the standard normal cumulative distribution function \(\Phi\),
Thus about half of the shifted draws are retained. For a normal distribution with mean \(\mu\), standard deviation \(\sigma\), and truncation limits \(L,U\), the conditional mean is \(\mu+\sigma\{\phi(a)-\phi(b)\}/\{\Phi(b)-\Phi(a)\}\), where \(a=(L-\mu)/\sigma\), \(b=(U-\mu)/\sigma\), and \(\phi\) is the standard normal density. Here, \(\mu=10\), \(\sigma=3\), \(a=-3.333\), and \(b=0\). Therefore,
Although the untruncated shift raises the mean from 8 to 10, truncation gives a conditional mean of about 7.614 units. This large change arises because the shifted distribution is centered at the upper bound: conditioning on values no greater than 10 retains mostly its lower side. The example illustrates why truncation is a substantive choice rather than a harmless correction.
A related but distinct operation is clipping: replace every value below \(L\) by \(L\), and every value above \(U\) by \(U\). In this example, clipping assigns every draw above 10 exactly the value 10, an event with probability \(0.5\). Truncation instead samples from the distribution conditional on being valid; for a continuous distribution it creates no point mass at 10. Clipping can therefore create artificial piles at the boundaries.
Shifting on a Transformed Scale
If both bounds are finite, a transformation can map the interior of the valid range to the real line. For \(L<y<U\), define the logit transformation of the rescaled outcome by \(g(y)=\log\{(y-L)/(U-y)\}\). A delta shift on this scale adds to the log odds of the fraction of the range used. Applying the inverse transformation ensures that the resulting value remains strictly between the bounds.
Worked Example: A Logit-Scale Shift for a Bounded Outcome
Consider a hypothetical percentage score of 80 on a scale from 0 to 100. Its rescaled value is \(u=(80-0)/(100-0)=0.80\), and its logit is \(g(80)=\log(0.80/0.20)=\log(4)\approx1.3863\). Apply a transformed-scale shift of \(\delta=0.5\).
The shifted value is approximately 86.83 percentage points, which is within the valid range. The parameter 0.5 is not a shift of 0.5 percentage points: it multiplies the odds of the rescaled fraction by \(\exp(0.5)\approx1.649\). More generally, a transformed-scale shift changes the outcome by an amount that depends on its starting value.
This approach requires values strictly inside the bounds because \(g(L)\) and \(g(U)\) are infinite. If an outcome can equal an endpoint with positive probability, a continuous transformation of interior values alone does not represent that endpoint mass. The imputation model and adjustment must handle such values explicitly; an arbitrary small offset can change the implied scenario. The transformation also assumes that odds of the rescaled fraction are a meaningful scale on which to express the sensitivity shift.
Direct Bounded-Scale Adjustments
A different option is to express an adjustment as a fraction of the distance remaining to a bound. Rescale the outcome to \(u=(y-L)/(U-L)\), so \(0\leq u\leq1\). For an upward adjustment with \(0\leq\lambda\leq1\), move a fraction \(\lambda\) of the remaining distance to the upper bound. For a downward adjustment, move a fraction \(\lambda\) of the current distance above the lower bound.
Both formulas preserve the interval. For the upward rule, \(u_{\text{up}}\geq0\) and \(u_{\text{up}}\leq(1-\lambda)+\lambda=1\). For the downward rule, \(0\leq u_{\text{down}}\leq u\leq1\). These are bounded-scale adjustments, not constant additive shifts: the change depends on the starting value, and \(\lambda\) is dimensionless.
Worked Example: Moving a Fraction of the Remaining Distance
A hypothetical outcome is \(y=80\) on the range from 0 to 100, so \(u=0.80\). Choose an upward bounded-scale adjustment \(\lambda=0.25\), meaning that one quarter of the distance to the upper bound is added.
The adjusted value is 85, a change of 5 percentage points. At a starting value of 20, the same \(\lambda\) would give \(20+0.25(100-20)=40\), a change of 20 points. Thus \(\lambda=0.25\) does not mean “add 25 percentage points”; it means “move one quarter of the remaining distance toward the upper bound.” For a downward adjustment from 80, the corresponding rule gives \(u_{\text{down}}=0.75(0.80)=0.60\), or 60 on the original scale.
Choosing and Checking a Constraint
The methods encode different assumptions. Truncation conditions the shifted distribution on validity. A transformation applies a constant shift on a chosen unbounded scale and maps back into the permitted interval. A direct bounded-scale rule moves a specified fraction toward a bound, with a change that depends on the starting value. None is automatically the correct interpretation of a subject-matter delta.
| Approach | What is held constant | Important implication |
|---|---|---|
| Truncate the shifted distribution | The additive shift before conditioning | Conditioning can substantially alter the distribution, especially near a bound. |
| Transform, shift, and invert | The shift on the transformed scale | The original-scale change varies with the starting value; endpoints need special handling. |
| Bounded-scale adjustment | The fraction of remaining distance moved | The original-scale change varies with the starting value and the selected direction. |
| Clip invalid draws | The untruncated draw unless it violates a bound | Can create artificial probability mass exactly at a boundary. |
For each method, inspect the fraction of proposed draws that violate the bounds, the fraction of draws at or near each boundary, and the imputed distribution across relevant predictor values. In a truncation approach, a very low acceptance probability is a warning: the constraint is discarding most of the proposed distribution, so the conditional distribution may be far from the intended shifted one. For transformed or bounded-scale adjustments, inspect whether the implied original-scale changes are plausible across the outcome range, not only at its center.
A common mistake is to add a constant delta, clip values outside the range, and describe the result as though it still represented the same constant shift. It does not: clipping changes some draws by more than the specified adjustment and puts mass at a bound. Another mistake is to choose a transformation simply because it guarantees valid values. Validity is necessary, but the transformed scale also determines the meaning of delta. State the scale, direction, bounds, and treatment of endpoint values, then compare the resulting distributions and substantive conclusions across plausible choices.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A shifted continuous draw is clipped to \([0,1]\). What feature can clipping introduce that ordinary truncation of a continuous distribution does not?
- For a variable on \([0,100]\), calculate the upward bounded-scale adjustment with \(y=40\) and \(\lambda=0.20\).
- Why can a logit transformation of a bounded outcome not directly handle an observed value exactly equal to either bound?
- A normal shifted distribution has only a 0.02 probability of falling within the valid range. What diagnostic concern does this raise for rejection-based truncation?
Solutions. (1) Clipping can create a point mass at 0 or 1 because all draws beyond a boundary are set exactly to it; truncation conditions on validity and does not create that boundary mass for a continuous distribution. (2) The remaining distance is \(100-40=60\), so the change is \(0.20(60)=12\), giving \(y_{\text{up}}=52\). (3) The logit of the rescaled outcome is infinite at 0 and 1, so endpoint mass needs explicit treatment rather than a direct finite logit shift. (4) Only 2% of proposed shifted draws are accepted, so nearly all proposals are discarded; the conditional distribution may differ sharply from the intended shifted distribution, and the constraint and sensitivity scenario should be reconsidered.