Asymmetric Loss for Underdosing and Overdosing

Why the Direction of a Dose Error Matters

Two dose recommendations can be equally far from a target yet have different clinical consequences. Recommending too little may leave disease inadequately controlled; recommending too much may increase toxicity. Which error is more harmful depends on the drug, indication, patient, and clinical context. A symmetric distance measure treats deviations of the same size alike, so it cannot express this directional difference.

The introductory material on Quantifying Clinical Utility of Dose Recommendations motivates evaluating recommendations by their consequences, not only by their numerical accuracy. Here the focus is a specific scoring choice: an asymmetric loss function that assigns different penalties to underdosing and overdosing. The weights make the clinical priorities explicit; they are not automatically determined by the observed dose errors.

Definition: Let $d_i^*$ be the specified target dose for patient $i$, and let $d_i$ be the recommended dose. The underdose amount is $(d_i^*-d_i)_+=\max(d_i^*-d_i,0)$, and the overdose amount is $(d_i-d_i^*)_+=\max(d_i-d_i^*,0)$. With nonnegative harm weights $c_U$ and $c_O$, the asymmetric absolute loss is $$ L_i=c_U(d_i^*-d_i)_+ + c_O(d_i-d_i^*)_+. $$ Here $c_U$ and $c_O$ express loss per dose unit below and above target, respectively. If dose is measured in mg and loss in illustrative harm units, the weights have units of harm units per mg.

If $d_i=d_i^*$, both positive-part terms are zero and the loss is zero. If $d_i<d_i^*$, only the underdose term contributes; if $d_i>d_i^*$, only the overdose term contributes. When $c_U=c_O$, this reduces to a constant multiple of absolute error. When the weights differ, direction matters. The weights should be chosen and justified for the clinical decision at hand, rather than tuned merely to make a preferred policy appear better.

Assumptions and Interpretation

This score requires a target dose that can be treated as a meaningful reference for each patient, a recommendation expressed on the same dose scale, and harm weights that represent the relative consequences of deviations. In the examples, the target is treated as known and fixed. In practice, an estimated target may be uncertain, and the score then evaluates recommendations relative to that chosen reference—not necessarily relative to the patient's unknowable true optimum.

The linear form assumes that each additional mg of deviation has a constant marginal penalty on a given side of the target. It also assumes that penalties add across patients when a group mean is calculated. These are modelling choices, not clinical facts. The loss is a decision-relevant summary, not a direct measurement of toxicity, treatment failure, or health outcome unless it has been explicitly linked and validated against such outcomes.

For $n$ patients given equal weight, the mean asymmetric loss is

\bar{L}=\frac{1}{n}\sum_{i=1}^{n}L_i.

It is reported in the units used for the loss weights. Equal patient weighting describes the average loss per evaluated patient in that sample; it does not mean that all patients have equal clinical importance. If patient weights are used, their definition and normalization must be stated. A comparison also requires the same target definition, weights, patient set, and aggregation rule for every policy.

Why the Score Can Change a Comparison

Write $e_i=d_i-d_i^*$ for the signed dose error. Absolute error is $|e_i|$, and it is unchanged if an error switches from below target to above target. In contrast, asymmetric loss is $c_U|e_i|$ when $e_i<0$ and $c_O|e_i|$ when $e_i>0$. Thus two policies can have the same total absolute error but different losses if their errors point in different directions.

This follows directly from the definition: for an underdose of size $a>0$, $L=c_Ua$; for an overdose of the same size, $L=c_Oa$. Their difference is

L_{\text{under}}-L_{\text{over}}=a(c_U-c_O).

So if $c_U>c_O$, the underdose incurs more loss than an equally large overdose. This conclusion concerns the specified loss model and weights; it does not establish that underdosing is universally more harmful.

Worked Example: Equal Absolute Error, Different Directional Loss

Consider two hypothetical patients, each with a target dose of 100 mg. Policy A recommends 90 mg for the first patient and 110 mg for the second. Policy B recommends 115 mg for the first patient and 95 mg for the second. Suppose, purely for illustration, that $c_U=0.15$ harm units/mg and $c_O=0.05$ harm units/mg.

Symmetric error. Policy A has absolute errors $10$ mg and $10$ mg, so its total absolute error is $10+10=20$ mg and its mean absolute error is $20/2=10$ mg. Policy B has errors $15$ mg and $5$ mg, so its total is also $15+5=20$ mg and its mean is $20/2=10$ mg. By this symmetric measure, the policies tie.

Asymmetric loss for Policy A. The 90 mg recommendation is 10 mg below target, giving $0.15(10)=1.5$ harm units. The 110 mg recommendation is 10 mg above target, giving $0.05(10)=0.5$ harm units. Total loss is $1.5+0.5=2.0$ harm units, or $2.0/2=1.0$ harm unit per patient.

Asymmetric loss for Policy B. The 115 mg recommendation is 15 mg above target, giving $0.05(15)=0.75$ harm units. The 95 mg recommendation is 5 mg below target, giving $0.15(5)=0.75$ harm units. Total loss is $0.75+0.75=1.5$ harm units, or $1.5/2=0.75$ harm unit per patient.

The asymmetric summary favors Policy B under these stipulated weights, even though mean absolute error ties. The reason is that Policy B has less underdose error, which receives the larger penalty. The conclusion is conditional on the weights and target values; it is not a general clinical ranking.

Comparing Policies Across Patients

For a cohort, report enough detail to show how the average arose. The total underdose and overdose amounts are especially useful because they separate the two directions before weights are applied. A single mean loss can conceal whether a result is driven by a few large deviations, many small ones, or a particular direction.

Worked Example: A Ranking Reversal Between Mean Absolute Error and Asymmetric Loss

Suppose three hypothetical patients each have a target dose of 100 mg. Policy A recommends 95, 120, and 105 mg. Policy B recommends 90, 105, and 90 mg. Use $c_U=0.15$ harm units/mg and $c_O=0.05$ harm units/mg, as illustrative values.

Policy A. The errors are an underdose of 5 mg, an overdose of 20 mg, and an overdose of 5 mg. Its total absolute error is $5+20+5=30$ mg, so mean absolute error is $30/3=10$ mg. Its loss is $0.15(5)+0.05(20)+0.05(5)=0.75+1.00+0.25=2.00$ harm units. Mean loss is $2.00/3=0.667$ harm units per patient.

Policy B. Its errors are underdoses of 10 mg and 10 mg, and an overdose of 5 mg. Total absolute error is $10+5+10=25$ mg, and mean absolute error is $25/3=8.33$ mg. Its loss is $0.15(10)+0.05(5)+0.15(10)=1.50+0.25+1.50=3.25$ harm units. Mean loss is $3.25/3=1.08$ harm units per patient.

Mean absolute error favors Policy B because $8.33<10$ mg. The asymmetric loss favors Policy A because $0.667<1.08$ harm units per patient. Policy B's smaller absolute error is concentrated more heavily in underdosing, which is penalized more under these assumptions. Both results are arithmetically correct; they answer different evaluation questions.

Check Sensitivity to the Harm Weights

An asymmetric evaluation is only as defensible as its harm weights. Where the weights are uncertain or contested, report how the comparison changes over plausible values rather than presenting one chosen pair as uniquely correct. This is a sensitivity analysis of the evaluation score, not a claim that the clinical harms themselves have been precisely measured.

Worked Example: Testing Whether the Policy Ranking Depends on the Weights

Use the same three-patient recommendations as in the preceding example. Policy A has a total of 5 mg underdose and 25 mg overdose; Policy B has 20 mg underdose and 5 mg overdose. Therefore their total losses as functions of the weights are

L_A=5c_U+25c_O,\qquad L_B=20c_U+5c_O.

At $c_U=0.15$ and $c_O=0.05$, these are $L_A=5(0.15)+25(0.05)=0.75+1.25=2.00$ and $L_B=20(0.15)+5(0.05)=3.00+0.25=3.25$ harm units, as calculated above.

Now consider a different hypothetical weighting, $c_U=0.03$ and $c_O=0.05$. Then $L_A=5(0.03)+25(0.05)=0.15+1.25=1.40$, while $L_B=20(0.03)+5(0.05)=0.60+0.25=0.85$ harm units. Under these weights Policy B has lower loss. Thus the policy ranking is not invariant to the relative penalties: a transparent evaluation should state the weights and show whether reasonable alternatives alter the conclusion.

Diagnostics, Alternatives, and Limitations

Before interpreting a score, check that its construction matches the intended decision:

Verify direction and units. Confirm that a dose below target enters the underdose term, and that $c_U$ and $c_O$ are multiplied by dose deviations in the stated units. Converting doses from mg to another unit without converting the weights changes the numerical score.
Inspect the two error totals. Tabulate underdose and overdose counts and magnitudes, not just the combined mean. A surprising policy ranking can often be traced to an unexpected direction or a few influential patients.
Check the target definition. A target estimated differently across policies can make their losses incomparable. If targets are uncertain, results should be described as conditional on the chosen reference or evaluated across plausible target values.
Vary the weights. If modest, clinically credible changes reverse the ranking, report that dependence prominently. If a weight is nearly zero, the corresponding error direction contributes very little by construction.
Check for unsupported linearity. If harm grows nonlinearly with deviation, a constant penalty per mg is inadequate. A piecewise quadratic or clinically calibrated outcome-based loss may be more appropriate, but requires its own assumptions and validation.

Mean absolute error remains useful when the goal is numerical closeness and both directions are intentionally treated equally. Separate underdose and overdose summaries preserve direction without imposing a single relative weight. Asymmetric loss combines direction and magnitude into one score, which can aid comparison when the relative harm scale is defensible. None of these summaries alone establishes the actual health benefit or safety of a policy.

A further limitation is aggregation: a cohort mean can hide severe loss for a small subgroup or an individual patient. Report the distribution of losses and relevant subgroup summaries when that matters for the decision. Nor does a loss function make a recommendation safe merely because its average score is low; clinically unacceptable individual recommendations may still require separate safeguards.

Key takeaway: Asymmetric loss makes the relative cost of underdosing and overdosing explicit. It can change a policy comparison even when symmetric error measures tie or favor another policy. The result is meaningful only with a clearly defined target, justified weights, consistent units, and checks of patient-level errors and weight sensitivity.

Check Your Understanding

Answer the questions, then compare your reasoning with the solutions below.

A 100 mg target is paired with a 92 mg recommendation. Identify the underdose amount and the overdose amount.
With $c_U=0.10$ and $c_O=0.04$ harm units/mg, calculate the losses for recommendations of 90 mg and 110 mg when the target is 100 mg.
Two policies have equal total absolute error, but one places more error below target. Which policy has greater asymmetric loss when $c_U>c_O$, all else equal?
Why is a mean asymmetric loss not, by itself, proof that one policy produces better patient outcomes?
What should be checked if a small change to $c_U$ and $c_O$ reverses the policy ranking?

Solutions. (1) The underdose amount is $100-92=8$ mg; the overdose amount is 0 mg. (2) For 90 mg, loss is $0.10(10)=1.0$ harm unit. For 110 mg, loss is $0.04(10)=0.4$ harm unit. (3) The policy with more error below target has greater loss, provided the compared error amounts and other contributions are otherwise equal, because each underdose mg receives the larger weight. (4) The score depends on a chosen target, a specified loss form, and weights; without validation against clinical outcomes, it is a model-based evaluation summary rather than observed evidence of improved health. (5) Check whether the weights are clinically justified and whether the ranking is robust across plausible values; report the sensitivity rather than presenting a weight-dependent ranking as universal.