Why the Effect Scale Matters
A clinical estimand specifies the treatment conditions, outcome, population, and summary used to express the treatment contrast. Even after those elements are fixed, the contrast can be expressed on different scales. For a quantitative outcome, a mean difference reports an additive change in the outcome's units. For a binary outcome, a risk difference is additive, while a risk ratio and an odds ratio are relative measures. These summaries answer related but distinct questions.
The scale is part of the effect summary, not a cosmetic choice made after calculation. A difference of 5 percentage points and a 20% relative reduction can describe the same pair of risks, but they emphasize different features. The appropriate choice depends on the clinical question, how the outcome is coded, and which summary is useful for interpretation. No one scale is universally best.
The mean difference and risk difference have additive null value 0. The risk ratio and odds ratio have multiplicative null value 1. With a binary outcome, risks and risk differences are proportions; multiplying a risk difference by 100 expresses it in percentage points. A mean difference retains the outcome's measurement units, such as points or milligrams per deciliter. Ratios have no units.
Interpretation also depends on which event is coded as 1. If the event is an undesirable outcome, a negative risk difference or a ratio below 1 favors condition 1, relative to condition 0. If the event is a desirable outcome, the direction reverses. Always name the event and the treatment-condition ordering rather than calling a result simply “better.”
Conditions for a Meaningful Treatment Contrast
A scale can be calculated from two arm-specific summaries, but its causal interpretation depends on what those summaries represent. The target population, treatment conditions, outcome definition, and time point must be held fixed when comparing scales. The treatment conditions must be sufficiently well specified for the outcomes \(Y^1\) and \(Y^0\) to have clear meanings.
In a randomized trial, random assignment supports comparing the outcome distributions between assigned groups for the population and treatment conditions represented by the trial, subject to the trial's design and implementation. Interpreting observed outcomes as the intended estimand also requires consistency between assigned and received treatment conditions as defined by the estimand, and outcome information or missing-data handling that supports estimation of the specified arm summaries. In nonrandomized data, treatment groups may differ in ways that affect outcomes; the contrast then needs appropriate identification assumptions and methods before it can be interpreted causally.
These conditions concern identification of \(\mu_a\) or \(p_a\). They do not make one scale interchangeable with another. Once the same arm-specific means or risks are defined, the chosen scale determines how their contrast is summarized.
Binary Outcomes: Additive and Relative Summaries
A risk is a probability over a defined population and time horizon. For example, a 12-week risk of hospitalization is not the same quantity as a 1-year risk. The contrast should therefore state both the event and the period. For binary outcomes, the risk difference gives the absolute change in event probability; the risk ratio gives the event risk under condition 1 as a multiple of the risk under condition 0; and the odds ratio compares event odds.
Worked Example: Four Summaries of the Same Clinical Results
Suppose, hypothetically, that a randomized trial estimates a 12-week risk of a defined undesirable event of \(p_1=0.18\) under the new treatment and \(p_0=0.30\) under control. Assume the target population, event definition, and follow-up period are the same for both risks, and that the trial's design and outcome handling support these arm-specific estimates.
Risk difference. The new-treatment risk minus the control risk is
This is a reduction of 0.12, or 12 percentage points, in 12-week event risk under the new treatment relative to control. The unit is percentage points, not percent.
Risk ratio. The ratio of the risks is
The estimated event risk under the new treatment is 0.60 times the control risk. Equivalently, it is 40% lower relative to control, since \(1-0.60=0.40\). This is a relative comparison of risks.
Odds ratio. The event odds are \(0.18/(1-0.18)=0.2195\) under the new treatment and \(0.30/(1-0.30)=0.4286\) under control. Thus,
The event odds under the new treatment are about 0.512 times the control odds. This is not a 48.8% reduction in event risk: \(1-0.512=0.488\) describes the relative reduction in odds, not in probability.
All three binary summaries use the same event and the same two risks. They differ in scale and interpretation; their numerical values are not expected to match.
The relation between the risk ratio and odds ratio makes the distinction explicit. When \(p_0>0\) and both risks are less than 1,
The odds ratio equals the risk ratio only in special cases, such as equal risks. If both risks are small, then \(1-p_0\) and \(1-p_1\) are near 1, so the odds ratio may approximate the risk ratio. That approximation should not be assumed for common events.
Worked Example: Why the Odds Ratio Is Not a Risk Ratio
Consider a hypothetical comparison in which the event risks are 0.40 under treatment and 0.50 under control. Assume the binary event, target population, and time horizon are identical between conditions.
The risk difference is
a 10-percentage-point lower event risk under treatment. The risk ratio is
so the treatment risk is 0.80 times the control risk, or 20% lower relative to control. The odds ratio is
Thus the odds are about 0.667 times as high, a relative reduction in odds of about 33.3%. The event is common in both groups, so the odds ratio is notably farther from 1 than the risk ratio. Reporting the odds ratio as “a 33.3% reduction in risk” would be incorrect.
Quantitative Outcomes: Mean Differences
For a quantitative clinical outcome, the mean difference is a direct additive contrast in the outcome's units. Its interpretation requires a meaningful common scale and measurement time across the two conditions. For instance, a difference in mean symptom scores is expressed in score points, not as a percentage change unless a separate relative summary has been defined.
Worked Example: Interpreting a Mean Difference
Suppose a hypothetical trial estimates mean pain scores at week 8 of 4.2 points under a new treatment and 5.0 points under control, on the same 0-to-10 scale. Assume the estimand concerns the same target population and week-8 score under both treatment conditions, and that the arm means are supported by the design and analysis.
The estimated mean difference, new treatment minus control, is
The estimated average score is 0.8 points lower under the new treatment than under control at week 8. Because lower pain scores are favorable, this direction favors the new treatment on the stated outcome. The result is a difference in population means; it does not say that every participant's score decreased by 0.8 points.
Choosing and Reporting a Scale
A useful choice starts with the clinical question, not with whichever number is most striking. An additive scale answers how much the average outcome or event probability differs. A relative scale answers how large one mean, risk, or set of odds is compared with another, depending on the measure. For binary events, reporting both risk difference and a relative measure can make absolute and relative changes visible, provided the event, time horizon, and reference condition are clear.
| Measure | Outcome | Interpretation | Null value |
|---|---|---|---|
| Mean difference | Quantitative | Additive change in outcome units | 0 |
| Risk difference | Binary | Additive change in event probability; often stated in percentage points | 0 |
| Risk ratio | Binary | Event risk in condition 1 divided by event risk in condition 0 | 1 |
| Odds ratio | Binary | Event odds in condition 1 divided by event odds in condition 0 | 1 |
A risk ratio requires \(p_0>0\). An odds ratio as written requires both risks to lie strictly between 0 and 1; observed zero cells can make a direct sample calculation undefined, even when a population parameter is meaningful. Such cases need an appropriate estimation approach rather than a casual interpretation of an undefined ratio. For outcomes with no events in either group, a relative comparison may provide little information; the event definition, sample size, and uncertainty should be considered.
When estimating from trial data, the sample risk in each arm is the number with the event divided by the relevant arm denominator, and the sample mean is the average outcome in that arm. Substituting those estimates into the chosen contrast produces a point estimate. The denominator and handling of missing outcomes must align with the specified estimand; changing them can change the arm summaries rather than merely changing the effect scale.
Common Pitfalls and Diagnostics
- Calling a percentage-point change a percent change. A risk change from 30% to 18% is a decrease of 12 percentage points and a relative decrease of 40%. State which one is meant.
- Interpreting an odds ratio as a risk ratio. Odds are \(p/(1-p)\), not \(p\). This distinction is especially important when events are common.
- Omitting the event or comparison order. “The ratio is 0.8” is incomplete. State the event, the treatment condition in the numerator, the reference condition, and the time period.
- Ignoring outcome coding. A ratio below 1 favors treatment only if the event coded as 1 is undesirable. For a favorable event, the same direction may indicate harm.
- Comparing summaries with different populations or time horizons. A treatment-control contrast at 12 weeks cannot be interpreted as the same estimand as a contrast at 1 year merely because both are labeled risk differences.
- Confusing a scale with an assumption. Choosing an odds ratio does not resolve missing outcomes, confounding, or an unclear treatment condition. The arm-specific quantities must still be identified and estimated appropriately.
A practical diagnostic is to calculate or inspect the underlying arm means or risks before interpreting a contrast. For a binary outcome, check that each risk is between 0 and 1, verify the denominators and event coding, and confirm that the reported ratio matches the stated numerator and reference. If the risks are not small, do not use a rare-event approximation to treat an odds ratio as a risk ratio.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A hypothetical trial reports event risks of 0.12 under treatment and 0.20 under control. What are the risk difference and risk ratio, with contextual interpretations?
- For the same risks, calculate the odds ratio. Is it correct to call it a 40% reduction in risk? Explain.
- A symptom score has means of 3.1 points under treatment and 3.8 points under control. State the mean difference in the treatment-minus-control order and explain its units.
- Name two details that should accompany a reported risk ratio to make its clinical interpretation clear.
Solutions. (1) The risk difference is \(0.12-0.20=-0.08\), an 8-percentage-point lower event risk under treatment. The risk ratio is \(0.12/0.20=0.60\), so treatment risk is 0.60 times control risk, or 40% lower relative to control. (2) The odds ratio is \((0.12/0.88)/(0.20/0.80)=0.1364/0.25\approx0.545\). No: this is a relative comparison of odds, not risk. (3) The difference is \(3.1-3.8=-0.7\) points; the treatment-group mean is 0.7 score points lower. (4) State the event and follow-up period, and specify which treatment condition is in the numerator and which is the reference; also make clear the target population.