Quantile-Based Delta Adjustments

Why Specify Shifts at Particular Quantiles?

A sensitivity assumption about a missing outcome need not affect every part of its distribution equally. For example, a hypothetical clinical judgment might be that the lower end of the missing-outcome distribution is close to a reference distribution, while its upper end is substantially worse. A single mean shift cannot express that pattern. A quantile-based delta adjustment specifies how selected points in the conditional distribution are shifted, then asks whether those shifts define a coherent distribution.

Let $Y$ be an outcome, $X=x$ the covariates for an individual, and $R=1$ or $R=0$ indicate whether the outcome is observed or missing. Let $Q_{\mathrm{ref}}(u\mid x)$ be the quantile function of a chosen reference distribution for $Y$ conditional on $X=x$. This is a reference assumption for missing outcomes, not a distribution identified from the missing outcomes themselves. For example, it may come from extrapolating a model fitted to observed outcomes under a reference missingness assumption.

Definition: Under an additive quantile-based delta adjustment, the conditional quantile function assumed for missing outcomes is $$ Q_{\mathrm{mis}}(u\mid x) = Q_{\mathrm{ref}}(u\mid x)+\delta(u,x), \qquad 0<u<1. $$ Here $u$ is a probability level, and $\delta(u,x)$ is the assumed shift at that quantile for covariate value $x$. The shift has the same units as $Y$. The resulting curve must be a valid quantile function, not merely a collection of plausible shifted values.

A shift at $u=0.5$ concerns the conditional median; shifts near $u=0.1$ or $u=0.9$ concern the lower or upper part of the distribution. The shift is an assumption about the unobserved conditional distribution. The observed data do not, by themselves, identify its value for outcomes that are missing.

Conditions for a Valid Adjusted Quantile Curve

Under the inverse-CDF convention, a quantile function is $Q(u)=\inf\{y:F(y)\geq u\}$, for $0<u<1$, where $F$ is a cumulative distribution function. Such a quantile function is nondecreasing and left-continuous on the interior of its probability range. Thus a proposed adjustment must preserve both properties. Nondecreasingness is a necessary check against quantile crossing, but by itself it is not sufficient for being an inverse-CDF quantile function.

In practice, the relevant object to check is the sum $Q_{\mathrm{ref}}(u\mid x)+\delta(u,x)$, not the delta curve alone. The delta may decrease with $u$, provided that the reference quantile curve increases enough for the adjusted curve to remain nondecreasing. Conversely, even a seemingly reasonable set of shifts can make the adjusted values decrease as $u$ increases. If shifts are specified only at selected probability levels, an interpolation rule is also needed to define the curve between them.

For an interior quantile curve, being nondecreasing and left-continuous gives the standard inverse-CDF form, with endpoint or tail behavior chosen to match the intended support. One way to see why these shape conditions matter is to construct a random outcome from the curve.

Justification: Suppose $Q_{\mathrm{mis}}(u\mid x)$ is nondecreasing and left-continuous for $0<u<1$, with appropriate endpoint or tail behavior. Let $U$ be uniform on $(0,1)$ and define $Y_{\delta}=Q_{\mathrm{mis}}(U\mid x)$. This construction gives a distribution whose inverse-CDF quantile function is $Q_{\mathrm{mis}}$. Indeed, for a fixed $u$, if $y<Q_{\mathrm{mis}}(u\mid x)$, left-continuity ensures that values just to the left of $u$ remain above $y$; monotonicity then makes the probability of $Y_{\delta}\leq y$ less than $u$. If $y\geq Q_{\mathrm{mis}}(u\mid x)$, monotonicity gives $Q_{\mathrm{mis}}(v\mid x)\leq y$ for every $v\leq u$, so that probability is at least $u$. Therefore the smallest $y$ with cumulative probability at least $u$ is $Q_{\mathrm{mis}}(u\mid x)$.

Worked Example: A Constant Shift Across Quantiles

Suppose a hypothetical reference distribution for a continuous outcome has $Q_{\mathrm{ref}}(u)=10+20u$ points for $0<u<1$. Consider the assumption $\delta(u)=2$ points at every quantile. The reference quartiles are $Q_{\mathrm{ref}}(0.25)=15$, $Q_{\mathrm{ref}}(0.50)=20$, and $Q_{\mathrm{ref}}(0.75)=25$ points.

Applying the formula gives $$ Q_{\mathrm{mis}}(0.25)=15+2=17,\qquad Q_{\mathrm{mis}}(0.50)=20+2=22,\qquad Q_{\mathrm{mis}}(0.75)=25+2=27 $$ points. More generally, $Q_{\mathrm{mis}}(u)=12+20u$, which is increasing and continuous. It defines a valid distribution: under the reference curve, the outcome is uniform from 10 to 30 points; after the shift, it is uniform from 12 to 32 points. The median is 2 points higher under the missing-outcome assumption. This example is a location shift expressed using quantiles, not a quantile-specific departure.

Specifying Different Shifts at Different Quantiles

Quantile-specific adjustments can express assumptions about changes in different parts of the distribution. A specification should state the shifts at chosen probability levels and how they are extended between those levels. A simple interpolation rule is piecewise linear interpolation, but it is an assumption about the whole delta curve, not a consequence of knowing the selected shifts.

Worked Example: Leaving the Lower Quartile Unshifted

For a hypothetical outcome measured in points, suppose the reference quantile function is $Q_{\mathrm{ref}}(u)=10+20u$, so the reference quartiles at $u=0.25,0.50,0.75$ are 15, 20, and 25 points. Assume shifts of 0, 2, and 6 points at those three levels. Define $\delta(u)$ to be 0 up to $u=0.25$, linear from 0 to 2 between $0.25$ and $0.50$, linear from 2 to 6 between $0.50$ and $0.75$, and 6 above $0.75$.

At the specified levels, the arithmetic is $$ Q_{\mathrm{mis}}(0.25)=15+0=15,\qquad Q_{\mathrm{mis}}(0.50)=20+2=22,\qquad Q_{\mathrm{mis}}(0.75)=25+6=31 $$ points. Between the anchors, the delta slopes are $2/0.25=8$ points per probability unit and $4/0.25=16$ points per probability unit. The reference curve has slope 20 points per probability unit, so the adjusted slopes on those intervals are 28 and 36. On the constant-delta portions, the adjusted slope is 20. The resulting curve is continuous and increasing throughout the interior. It is therefore coherent as a quantile curve under the stated interpolation and tail extension.

The assumption says more than “the median shift is 2 points”: it also specifies a 0-point shift at the lower quartile, a 6-point shift at the upper quartile, and a particular pattern between and beyond them. The conclusion concerns the assumed missing-outcome distribution; it does not establish that missing outcomes actually follow it.

Selected quantile shifts should be interpreted conditionally on $x$. If $\delta(u,x)$ varies across covariate values, the specification describes different conditional distributions for different covariate profiles. It does not follow that the same shifts apply to every subgroup. For a simple sensitivity analysis, the covariate profile or subgroup to which each shift applies must be stated clearly.

Crossing, Continuity, and Repairing a Specification

A proposed sequence of adjusted quantiles that reverses their order cannot describe any distribution. For example, if the adjusted 25th percentile is larger than the adjusted median, no cumulative distribution function can have both values as its inverse-CDF quantiles. A second, distinct problem arises when a curve is nondecreasing but not left-continuous: it can also fail to be a quantile function under the stated convention.

Worked Example: Detecting Quantile Crossing and a Continuity Failure

Suppose a hypothetical reference curve is $Q_{\mathrm{ref}}(u)=100+20u$ points. Its values at $u=0.25,0.50,0.75$ are 105, 110, and 115 points. A proposed set of shifts is $+8,0,-8$ points, respectively. The adjusted values are $$ 105+8=113,\qquad 110+0=110,\qquad 115-8=107 $$ points. They decrease as the quantile level rises, so this specification has quantile crossing and is invalid.

One possible revised specification uses shifts $+2,0,-2$ points at the same three levels. The adjusted values become $107,110,113$ points. If the delta is linearly interpolated between those levels and held constant outside them, the reference slope of 20 is reduced by 8 on each interpolated interval, leaving an adjusted slope of 12; outside, the slope remains 20. The full adjusted curve is increasing and continuous. This revision is one coherent alternative, not a data-derived correction or the uniquely correct set of shifts.

Now consider a different proposed adjusted curve equal to 0 for $u<0.5$ and 1 for $u\geq0.5$. It is nondecreasing but not left-continuous at $u=0.5$. Its values of 0 for every $u<0.5$ imply that a distribution with this quantile curve would have $F(0)\geq0.5$. The inverse-CDF quantile at $u=0.5$ would therefore be 0, not 1. This curve fails the inverse-CDF convention even though it has no decreasing segment.

What to Check, and How This Compares With Other Adjustments

Before interpreting a quantile-based sensitivity specification, check that its reference curve is defined for the covariate values of interest, that each delta has the stated outcome units, and that the interpolation and tails are explicit. Then examine the adjusted curve across the full probability range, not just at the selected quantiles. A sparse grid can conceal crossing between anchor points. A curve that is nondecreasing but has a jump in the wrong direction of continuity can also fail the inverse-CDF definition.

The main alternatives make different assumptions:

Approach	What is specified	Important distinction
Mean shift	A change in the conditional mean	It does not, on its own, specify how spread or tail quantiles change.
Location-scale adjustment	A shift in location and a change in residual scale	It imposes a structured distributional change rather than freely selecting quantile shifts.
Quantile-based adjustment	Shifts at selected quantiles and a rule for constructing the full curve	It can express different changes across the distribution, but requires coherence checks and interpolation assumptions.

The earlier material on mean shifts versus full distribution shifts and location-scale delta models provides context for those alternatives; here the distinguishing choice is to anchor sensitivity assumptions at conditional quantiles. Quantile-based adjustments can be useful when clinical concern is specifically about a median or tail, but they do not remove uncertainty about the unobserved outcomes. Results remain conditional on the chosen reference model and delta specification. A few selected shifts alone do not define the rest of the distribution.

Common Mistakes and Limitations

Treating a selected quantile shift as a mean shift. A change at the median says nothing by itself about the mean. The rest of the distribution and its tails also matter.
Checking only the anchor values. Values can be correctly ordered at selected quantiles while an interpolation rule creates crossing between them. Check the full adjusted curve.
Assuming nondecreasingness is sufficient. Under the inverse-CDF convention, the curve must also be left-continuous on the interior, with suitable endpoint or tail behavior.
Leaving interpolation implicit. Shifts at the 25th, 50th, and 75th percentiles do not determine shifts at all other levels. State how the curve is completed.
Interpreting the adjustment as learned from missing outcomes. The missing-outcome shifts are sensitivity assumptions. The observed outcomes can inform the reference model, but do not identify the missing outcomes’ quantile shifts.

A valid quantile curve is a mathematical coherence requirement, not proof that the assumed missing-outcome distribution is clinically plausible. Also, a shift in the outcome’s units may need reconsideration if the outcome’s support or measurement scale makes the shifted values impossible. Validity and plausibility are separate checks.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A reference conditional median is 40 mg/L. The assumed missing-outcome median shift is $-3$ mg/L. What is the adjusted median, and what does the shift mean?
At two probability levels, a proposed adjusted curve has values 18 at $u=0.25$ and 16 at $u=0.50$. What is wrong with this specification?
Why does a list of shifts at the 10th, 50th, and 90th percentiles not, on its own, specify a full missing-outcome distribution?
A proposed curve is nondecreasing but has a right-hand jump at $u=0.5$, with its value at the jump equal to the higher value. What additional condition is violated under the inverse-CDF convention?

Solutions. (1) The adjusted median is $40+(-3)=37$ mg/L. The assumption places the conditional median of the missing outcomes 3 mg/L below the reference conditional median; it is not an estimate learned from the missing outcomes. (2) The adjusted quantile decreases from 18 to 16 as $u$ rises, so quantile crossing makes the proposed curve invalid. (3) The shifts specify only three points. An interpolation rule and tail behavior are needed to define the curve elsewhere, and those choices affect the assumed distribution. (4) The curve is not left-continuous at the jump. Nondecreasingness alone does not make it an inverse-CDF quantile function.