Why Efficacy and Safety Must Be Considered Together
A dose can improve the probability of an effective response while also increasing the probability of a harmful event. Comparing efficacy alone or safety alone therefore does not necessarily identify the dose with the best overall profile. A benefit-risk decision needs outcomes defined on the same patients, under the same regimen and follow-up period, and a criterion that makes clear how those outcomes are combined.
There is more than one defensible criterion. A decision maker may require both an effective response and no specified safety event; may score efficacy and safety outcomes using explicit weights; or may use a net clinical benefit that assigns benefits and harms a common decision scale. These criteria answer different questions, so their rankings need not agree. The earlier material on Utility-Based Dose Optimization addresses the general idea of optimizing a valued outcome; here the focus is on representing and combining efficacy and safety outcomes jointly.
The event definitions are part of the decision, not a detail to choose after seeing the dose ranking. Specify the response threshold, the safety event or composite, the follow-up times, and the population to which the predictions apply. If response and safety are measured over different periods, the joint event may not represent a coherent clinical decision. Likewise, a safety event can be serious even when it is uncommon, while a mild event may be tolerable at a higher frequency; a probability alone does not express that distinction.
Joint Probabilities and Dependence
For binary efficacy and safety outcomes, the joint distribution has four cells: response with event, response without event, no response with event, and neither outcome. The probability of joint success is one of those cells. Equivalently, it is the efficacy probability minus the probability of both response and the safety event:
Knowing only the two marginal probabilities \(P(E\mid d)\) and \(P(S\mid d)\) does not generally determine the probability of joint success. Multiplying \(P(E\mid d)\) by \(1-P(S\mid d)\) is valid only when the efficacy and safety events are independent at that dose. This is a substantive assumption: patients with a response may have a different safety-event risk from patients without a response.
The bounds follow from the restrictions that an intersection cannot exceed either event probability and that probabilities cannot be negative. In particular, \(P(E\cap S)\) is at most \(\min(p_E,p_S)\), and at least \(\max(0,p_E+p_S-1)\). Subtracting those limits on \(P(E\cap S)\) from \(p_E\) gives the stated bounds for \(P(E\cap S^c)\). These bounds may be wide, but they show precisely what the marginal predictions do and do not establish.
Worked Example: Ranking Doses by Joint Success
Suppose a hypothetical model predicts binary outcomes by a common assessment time for three candidate doses. In each cell below, the probability is for the paired efficacy and safety outcomes in the same patient. The safety event is considered undesirable.
| Dose | Response and event | Response, no event | No response and event | Neither |
|---|---|---|---|---|
| Low | 0.04 | 0.56 | 0.06 | 0.34 |
| Medium | 0.12 | 0.63 | 0.06 | 0.19 |
| High | 0.30 | 0.54 | 0.05 | 0.11 |
Each row sums to 1. For the low dose, the efficacy probability is \(0.04+0.56=0.60\), and the joint success probability is the response-without-event cell, \(0.56\). For the medium dose, efficacy is \(0.12+0.63=0.75\), while joint success is \(0.63\). For the high dose, efficacy is \(0.30+0.54=0.84\), while joint success is \(0.54\). Thus, among these three candidates, the medium dose maximizes the specified joint success probability, even though the high dose has the greatest efficacy probability.
The dependence matters. At the medium dose, the safety-event probability is \(0.12+0.06=0.18\). If independence were assumed, the calculated joint success would be \(0.75(1-0.18)=0.75(0.82)=0.615\), rather than the joint-model value \(0.63\). The difference occurs because the model assigns \(0.12\) probability to response and event together, whereas independence would imply \(0.75(0.18)=0.135\). The joint probability should be taken from an appropriate joint model or data source, not reconstructed from marginals unless independence is justified.
Net Clinical Benefit and Weighted Outcomes
A joint-success criterion treats every patient who meets both conditions as a success and every other patient as not meeting that criterion. It may be suitable when the decision requirement is explicitly “effective without this event,” but it does not distinguish the different ways a patient can fail that requirement. A weighted outcome can retain more of that information by assigning a value to each outcome state.
For example, let \(B_E\) be the decision value of meeting the efficacy criterion and \(C_S\) the cost assigned to experiencing the safety event. Under an additive scoring rule, with no interaction between the two outcomes, the expected net clinical benefit at dose \(d\) is:
This expression follows by assigning each patient the score \(B_E I(E)-C_S I(S)\), where \(I(\cdot)\) is 1 when the event occurs and 0 otherwise, and taking its expectation. If a patient can experience both outcomes, the additive rule gives that patient both the efficacy value and the safety cost. If that is not an acceptable representation—for example, if the safety consequence changes the value of an efficacy response—score the four joint states directly instead. The weights must be clinically and decisionally justified; they are not estimated from the efficacy and safety probabilities alone.
Worked Example: Ranking Doses by Net Clinical Benefit
Use the hypothetical marginal probabilities from the preceding table. Suppose, for illustration, that meeting the response criterion is assigned \(+10\) decision units and the safety event is assigned \(-25\) units. These stipulated weights make the score an additive expected value; they are not clinical estimates.
At the low dose, \(NCB=10(0.60)-25(0.10)=6.00-2.50=3.50\) units per patient. At the medium dose, \(NCB=10(0.75)-25(0.18)=7.50-4.50=3.00\) units per patient. At the high dose, \(NCB=10(0.84)-25(0.35)=8.40-8.75=-0.350\) units per patient. Under these weights, the low dose ranks highest, unlike the joint-success criterion, which selected the medium dose.
The difference is not a contradiction. The joint-success criterion rewards the probability of a particular combined state, whereas this net-benefit score assigns a specified cost to each safety event and a specified value to each response. A decision should report the criterion and weights so that a reader can understand why a dose ranks first. If the weights are uncertain or contested, show how the ranking changes under plausible alternatives rather than presenting one score as uniquely determined by the data.
When Dependence Is Not Available
If only marginal predictions are available, the independence calculation can be shown as a clearly labelled scenario, not as an established joint probability. The bounds provide a second, assumption-light summary. If a dose must exceed a joint-success threshold, the bounds can reveal whether the available information is sufficient: a lower bound above the threshold guarantees it under any dependence structure consistent with the marginals, while an upper bound below the threshold rules it out. If the threshold lies between the bounds, more information about dependence is needed to decide.
Worked Example: What Marginal Probabilities Can Establish
For a hypothetical dose, suppose \(P(E)=0.80\) and \(P(S)=0.20\). With no dependence information, the joint-success probability must lie between:
Independence would give \(0.80(0.80)=0.64\), but this is only one value in the allowable range. To see why, one joint distribution attaining the lower bound has probabilities \(0.20\) for response with event, \(0.60\) for response without event, \(0.00\) for no response with event, and \(0.20\) for neither. The response probability is \(0.80\), the event probability is \(0.20\), and joint success is \(0.60\). A distribution attaining the upper bound has corresponding probabilities \(0.00\), \(0.80\), \(0.20\), and \(0.00\), giving the same marginals but joint success \(0.80\). Thus, if the decision threshold were \(0.70\), the marginal predictions alone would not establish whether the dose met it.
Assumptions, Diagnostics, and Limitations
A dose comparison is interpretable only when the predictions refer to a common target population, regimen, efficacy definition, safety definition, and time horizon. The model or data must support the joint distribution used in the calculation. Check that all four cell probabilities are between 0 and 1 and sum to 1, and that the marginal probabilities obtained by adding cells agree with any separately reported marginal predictions. A mismatch can indicate a calculation error or inconsistent model outputs.
Also examine whether the joint prediction is credible, not merely arithmetically valid. Compare predicted and observed paired outcome frequencies where suitable data are available, and check whether the relationship between response and safety is represented plausibly. A ranking that changes materially when dependence assumptions or outcome weights change is fragile and should be reported as such. Predictions outside the range of doses or populations supported by the data require particular caution.
Common limitations include poorly measured or rare safety events, uncertain clinical thresholds, and composite endpoints that combine events of different importance. A single joint-success probability can conceal whether failures are mostly lack of efficacy or safety events. An additive net-benefit score can conceal differences among patients and can be misleading if its weights or additivity assumption are inappropriate. Neither criterion, by itself, demonstrates that changing the dose causes an outcome difference; that interpretation depends on the design and assumptions supporting the underlying predictions.
Choosing and Reporting a Criterion
| Approach | What it answers | Main requirement or limitation |
|---|---|---|
| Joint success probability | How often the efficacy criterion is met without the specified safety event | Requires a joint prediction or a justified dependence assumption; other outcome states are grouped together |
| Weighted expected outcome | Which dose has the largest expected score under stated outcome values | Requires defensible weights and a stated rule for patients with both outcomes |
| Net clinical benefit | Whether valued efficacy gains outweigh assigned safety costs on a common scale | Ranking depends on the scale, weights, and any additivity assumptions |
A useful report states the estimand and time horizon, the joint event or scoring rule, the source of the dependence information, and the candidate-dose results. When feasible, show both marginal efficacy and safety predictions alongside the combined criterion. This makes clear whether a selected dose has a balanced profile or achieves its ranking through one particularly favorable component.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A dose has \(P(E)=0.70\) and \(P(S)=0.10\). What joint-success probability would independence imply?
- For \(P(E)=0.65\) and \(P(S)=0.25\), give the bounds on \(P(E\cap S^c)\) without a dependence assumption.
- Suppose efficacy is assigned \(+8\) units and a safety event \(-20\) units. At a dose with \(P(E)=0.60\) and \(P(S)=0.15\), calculate the additive net clinical benefit.
- Why can two dose criteria select different doses without either calculation being wrong?
Solutions. (1) Under independence, \(P(E\cap S^c)=0.70(1-0.10)=0.63\). This is conditional on the independence assumption. (2) The lower bound is \(\max(0,0.65-0.25)=0.40\), and the upper bound is \(\min(0.65,1-0.25)=0.65\); the marginals do not identify a single value. (3) \(NCB=8(0.60)-20(0.15)=4.80-3.00=1.80\) units per patient under the stipulated additive weights. (4) A joint-success rule values the probability of one specified outcome state, whereas a weighted or net-benefit rule assigns values to outcomes. Different explicit criteria can therefore produce different rankings.