Baseline Risk and Absolute Treatment Benefit

Why Baseline Risk Matters

A relative effect can be the same across groups while the number of outcomes prevented differs. If a treatment reduces an adverse outcome by the same proportion in two groups, a group with a higher untreated risk will generally have more events prevented per treated person. This distinction matters when describing who benefits, comparing treatment strategies, or communicating results in clinically meaningful units.

The key comparison is between a group’s risk under the comparator and its risk under treatment, over the same follow-up period. The comparator risk is often called the baseline risk. Here, “baseline” refers to the risk under the comparator condition, not necessarily to a risk measured before treatment or to a person’s immutable underlying susceptibility.

Definition: Let \(A=0\) denote the comparator and \(A=1\) the treatment, and let \(Y=1\) indicate the adverse outcome by a specified time. For a pretreatment subgroup \(S=s\), define \(r_0(s)=P(Y(0)=1\mid S=s)\) and \(r_1(s)=P(Y(1)=1\mid S=s)\). The subgroup baseline risk is \(r_0(s)\). The absolute risk reduction is \(ARR(s)=r_0(s)-r_1(s)\), and the risk ratio is \(RR(s)=r_1(s)/r_0(s)\), when \(r_0(s)>0\). Positive \(ARR(s)\) indicates fewer adverse outcomes under treatment.

These are causal quantities because they compare potential outcomes: the outcome that would occur under the comparator and the outcome that would occur under treatment. In practice, they can be estimated under suitable assumptions. For example, in a randomized trial, random assignment within the relevant subgroups, consistency of the treatment versions, a nonzero chance of each assignment in each subgroup, and no interference between participants support identification of these risks from the corresponding observed group risks. In an observational study, additional conditions are needed to justify the causal comparison; a difference in observed risks alone does not establish a treatment effect.

For the calculations in this tutorial, suppose the outcome is adverse, the time horizon and outcome definition are the same across groups, and treatment has a common risk ratio \(\theta\) across the subgroups being compared. This last condition is a substantive model assumption, not something guaranteed by the arithmetic. If the estimated relative effects vary, or if the common-ratio model is inappropriate, the formulas below should not be presented as established subgroup effects.

From a Common Relative Effect to Absolute Benefit

If the common risk ratio is \(\theta\), then the treatment risk in subgroup \(s\) is \(r_1(s)=\theta r_0(s)\). Substitution into the definition of absolute risk reduction gives a direct conversion from baseline risk to absolute benefit.

\begin{aligned} ARR(s) &= r_0(s)-r_1(s)\\ &= r_0(s)-\theta r_0(s)\\ &= r_0(s)(1-\theta). \end{aligned}

When \(0\leq\theta<1\), treatment lowers the adverse-outcome risk under this model, and the absolute reduction is proportional to baseline risk. If one subgroup has twice the baseline risk of another, its absolute reduction is twice as large, provided the common risk ratio applies and both baseline risks are positive. The relative reduction, \(1-\theta\), is the same in each subgroup; the number of percentage points reduced is not.

Key result: Under a common risk ratio \(\theta\), \(ARR(s)=r_0(s)(1-\theta)\). For two subgroups \(s_1\) and \(s_2\), \(ARR(s_1)-ARR(s_2)=(1-\theta)\{r_0(s_1)-r_0(s_2)\}\). Thus, if treatment is beneficial and one subgroup has a higher baseline risk, that subgroup has a larger absolute risk reduction under the common-ratio model.

This is one reason effect modification depends on the effect scale. Equal relative effects do not imply equal risk differences. A statement such as “the treatment effect is the same in both groups” is incomplete unless it specifies whether “same” refers to a relative effect, an absolute effect, or another scale. The calculation does not establish that a common relative effect is true; it shows what absolute benefits follow if it is true.

Worked Example: Same Relative Reduction, Different Absolute Benefits

Suppose, hypothetically, that a treatment has a risk ratio of \(0.70\) in two groups over the same follow-up period. Assume the adverse-outcome baseline risks are \(20\%\) in Group H and \(5\%\) in Group L, and assume the common risk ratio is appropriate for both groups. Calculate each treatment risk and absolute risk reduction.

Group H. The treatment risk is \(0.70\times0.20=0.14\), or \(14\%\). The absolute risk reduction is \(0.20-0.14=0.06\), or 6 percentage points.

Group L. The treatment risk is \(0.70\times0.05=0.035\), or \(3.5\%\). The absolute risk reduction is \(0.05-0.035=0.015\), or 1.5 percentage points.

Both groups have a relative reduction of \(1-0.70=0.30\), or \(30\%\). But the absolute reduction is four times as large in Group H: \(0.06/0.015=4\), matching the ratio of their baseline risks, \(0.20/0.05=4\). These hypothetical calculations illustrate the model’s consequence; they do not show that real groups share a risk ratio of \(0.70\).

Calculating Benefits From Hypothetical Risk Data

When risks are given as counts, calculate each risk using the number of people with the outcome divided by the number in the relevant treatment-by-subgroup group. Then calculate the risk ratio and risk difference from those risks. The arithmetic describes the causal effects only if the design and assumptions support interpreting those observed risks as estimates of the relevant potential-outcome risks.

Worked Example: Comparing Subgroup Risks in a Hypothetical Trial

Imagine a hypothetical randomized trial with 1,200 participants assigned to treatment and 1,200 assigned to comparator in each of two pretreatment risk groups. In the higher-risk group, 240 comparator participants and 168 treated participants experience the adverse outcome. In the lower-risk group, the corresponding counts are 60 and 42. Assume randomization within each subgroup, consistency, positive assignment probability, no interference, and the same outcome definition and follow-up period. Calculate the observed risks and contrasts.

Higher-risk group. The comparator risk is \(240/1200=0.20\), and the treatment risk is \(168/1200=0.14\). Therefore, \(RR=0.14/0.20=0.70\) and \(ARR=0.20-0.14=0.06\), or 6 percentage points.

Lower-risk group. The comparator risk is \(60/1200=0.05\), and the treatment risk is \(42/1200=0.035\). Therefore, \(RR=0.035/0.05=0.70\) and \(ARR=0.05-0.035=0.015\), or 1.5 percentage points.

The calculated point estimates have the same risk ratio but different absolute reductions, in line with \(ARR=r_0(1-RR)\). The trial’s randomization assumptions support a causal interpretation of subgroup contrasts in expectation, but these sample calculations alone do not establish that the population risk ratios are exactly equal. Sampling variability could make estimated ratios differ from their underlying values, and uncertainty should be considered before drawing strong conclusions.

The absolute benefit can also be expressed as a number needed to treat (NNT) for a specified time horizon: \(NNT(s)=1/ARR(s)\) when \(ARR(s)>0\). It represents the reciprocal of the absolute risk reduction, not a separate effect scale. When the risk reduction is expressed as a proportion, this reciprocal is in people treated per adverse outcome prevented, on average over the specified horizon. In the previous example, the point-estimate NNT is \(1/0.06=16.7\) in the higher-risk group and \(1/0.015=66.7\) in the lower-risk group. Using the usual whole-person convention of rounding up, these are 17 and 67, respectively. They are summaries of the hypothetical point estimates, not guarantees about what will happen to a particular set of patients.

Using the Relationship to Assess a Benefit Threshold

The same equation can be rearranged to ask what baseline risk would be needed to reach a specified absolute benefit, if a relative effect is assumed. This can clarify how a treatment’s absolute benefit changes with baseline risk. It does not, by itself, establish who should receive treatment: decisions may also depend on harms, costs, preferences, and uncertainty in both the baseline risk and relative effect.

Worked Example: Baseline Risk Needed for a Target Reduction

Suppose, hypothetically, that a treatment is assumed to reduce an adverse outcome by \(20\%\) on the relative scale, so \(\theta=0.80\). What baseline risk is required for an absolute risk reduction of at least 3 percentage points? Assume the risk ratio applies at the specified outcome horizon and baseline risk is positive.

The model gives \(ARR=r_0(1-\theta)=0.20r_0\). Requiring \(ARR\geq0.03\) gives \(0.20r_0\geq0.03\). Dividing both sides by \(0.20\), which is positive, gives \(r_0\geq0.15\), or a baseline risk of at least \(15\%\).

For example, if the baseline risk is \(10\%\), the predicted treatment risk is \(0.80\times0.10=0.08\), so the absolute reduction is \(0.10-0.08=0.02\), or 2 percentage points. If the baseline risk is \(20\%\), the predicted treatment risk is \(0.80\times0.20=0.16\), so the reduction is \(0.20-0.16=0.04\), or 4 percentage points. Both calculations use the same assumed relative effect; the difference in absolute benefit comes from baseline risk.

Common Pitfall: Calling the Benefits “The Same”

A common mistake is to see the same risk ratio in two groups and conclude that treatment gives them the same benefit. That reasoning silently treats a relative reduction as if it were an absolute reduction. For example, a \(30\%\) relative reduction applied to a \(20\%\) baseline risk gives a 6-percentage-point reduction, while the same relative reduction applied to a \(5\%\) baseline risk gives a 1.5-percentage-point reduction. The correct conclusion is that the relative reductions are equal under the assumed model, while the absolute benefits differ.

The reverse mistake is also possible: observing different absolute reductions does not, by itself, show that relative effects differ. Baseline risks may differ even when the relative effect is common. Always report the scale and the risks used to calculate it. In particular, identify the outcome, comparator, subgroup, and time horizon; an absolute reduction without these details is difficult to interpret.

Finally, baseline risk is not a shortcut to causal subgroup analysis. It must refer to the risk under a specified comparator in the target subgroup, and the subgroup should be defined before treatment. A risk measured in a different population or at a different follow-up time may not transport to the group of interest. And a constant risk ratio should be treated as an assumption to examine, not as a universal biological rule.

Key takeaway: If the adverse-outcome risk ratio is \(\theta<1\) in every subgroup, absolute risk reduction in subgroup \(s\) is \(r_0(s)(1-\theta)\). Higher comparator risk therefore implies larger absolute benefit under this model, even though the relative reduction is unchanged.

Check Your Understanding

Use the specified outcome and time horizon when interpreting each risk.

A common risk ratio of \(0.75\) is assumed in two groups with baseline risks of \(16\%\) and \(4\%\). What are the treatment risks and absolute risk reductions in each group?
Under a common beneficial risk ratio, why can two groups have different absolute risk reductions even if their relative reductions are equal?
A treatment risk is \(9\%\) and the comparator risk is \(12\%\) in a subgroup. Calculate the risk ratio and absolute risk reduction, and state the units of the latter.
What assumptions are needed before observed treatment and comparator risks in a randomized subgroup can be interpreted as estimates of causal risks?
Explain why the phrase “the treatment benefit is the same in both groups” is incomplete without specifying an effect scale.