Translating Odds Ratios Into Probability Changes

From a Logit-Scale Effect to a Probability Change

A logistic concentration-effect model describes change on the log-odds scale, but clinical decisions are often framed in terms of event probabilities: how likely is a response at one concentration compared with another, and by how many percentage points does that probability change? Translating between these scales requires a baseline probability. The same odds ratio can correspond to quite different absolute probability changes at different baselines.

The focus here is a comparison between two specified concentrations under a logistic model. The comparison may come from a linear concentration effect or from a nonlinear concentration function. The essential quantity is the difference in the model’s linear predictor between those concentrations. To interpret it as a concentration contrast, all other model inputs must be held fixed.

Definition: For an event with probability \(p\), the odds are \(p/(1-p)\), and the logit is \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). If \(p_0\) and \(p_1\) are the event probabilities at two specified concentrations, the odds ratio for the contrast is \[ \mathrm{OR}_{1:0}=\frac{p_1/(1-p_1)}{p_0/(1-p_0)}. \] The risk difference (probability difference) is \(\mathrm{RD}_{1-0}=p_1-p_0\). A positive risk difference indicates an increase in event probability; a negative one indicates a decrease.

In this tutorial, “risk” means the probability of the event being modelled; it does not imply that the event is necessarily harmful. If the modelled event is a beneficial response, a positive difference means a higher probability of that response.

The Probability Translation

Write the logistic model at concentration \(C\) as \(p(C)=1/\{1+\exp[-\eta(C)]\}\), where \(\eta(C)\) is the linear predictor. For concentrations \(C_0\) and \(C_1\), let \(\Delta\eta=\eta(C_1)-\eta(C_0)\). Because the logit is the log odds, exponentiating this difference gives the odds ratio:

\mathrm{OR}_{1:0} =\exp\{\eta(C_1)-\eta(C_0)\}.

If the probability at \(C_0\), denoted \(p_0\), is known, then the odds at \(C_0\) are \(p_0/(1-p_0)\). Multiplying those odds by \(\mathrm{OR}_{1:0}\) gives the odds at \(C_1\). Converting those new odds back to a probability yields:

p_1 =\frac{\mathrm{OR}_{1:0}p_0} {1-p_0+\mathrm{OR}_{1:0}p_0}.

The absolute probability change is then

\mathrm{RD}_{1-0} =p_1-p_0 =\frac{p_0(1-p_0)(\mathrm{OR}_{1:0}-1)} {1-p_0+\mathrm{OR}_{1:0}p_0}.

These equations require \(0<p_0<1\) and an odds ratio for the same event, population or covariate profile, and specified concentration contrast. They do not require a linear concentration effect: the odds ratio can be calculated from any specified \(\eta(C)\). The probability translation is conditional on the stated baseline and model inputs. A population-average probability may differ if the model contains individual-specific effects or covariates that vary across people.

Worked Translations

Worked Example: Translating a Given Odds Ratio at a Low Baseline Risk

Suppose a hypothetical binary PK/PD model predicts an event probability of \(p_0=0.0800\) at a reference concentration. The odds ratio for a specified higher concentration is \(2.40\), with all other predictors held fixed. Calculate the new event probability and risk difference.

First convert the baseline probability to odds: \(0.0800/(1-0.0800)=0.0800/0.9200=0.08696\). The odds at the higher concentration are \(2.40(0.08696)=0.2087\). Converting those odds to a probability gives

p_1=\frac{2.40(0.0800)}{0.9200+2.40(0.0800)} =\frac{0.1920}{1.1120} =0.1727.

Thus the modelled event probability rises from 8.00% to 17.27%. The risk difference is \(0.1727-0.0800=0.0927\), an increase of 9.27 percentage points. The odds are multiplied by 2.40; the probability is not. In this example, the probability is approximately 2.16 times the baseline probability, not 2.40 times it.

Worked Example: Using a Linear Concentration Coefficient

Consider the hypothetical model \(\operatorname{logit}\{p(C)\}=\alpha+\beta C\), where \(\beta=0.0180\) per ng/mL. Suppose the baseline probability is \(p_0=0.250\) at \(C_0=10\) ng/mL, and the contrast is to \(C_1=40\) ng/mL. Calculate the model-implied odds ratio, new probability, and risk difference.

The concentration change is \(40-10=30\) ng/mL. In this linear model, \(\Delta\eta=\beta(C_1-C_0)=0.0180(30)=0.540\) log-odds units. Therefore,

\mathrm{OR}_{1:0}=\exp(0.540)=1.716.

Using \(p_0=0.250\), the translated probability is

p_1=\frac{1.716(0.250)}{0.750+1.716(0.250)} =\frac{0.4290}{1.1790} =0.3639.

The predicted event probability at 40 ng/mL is approximately 36.39%, compared with 25.00% at 10 ng/mL. The risk difference is \(0.3639-0.250=0.1139\), or an increase of 11.39 percentage points. The coefficient is per ng/mL, so using it for a 30 ng/mL contrast requires multiplying by 30 before exponentiating.

Worked Example: Translating a Nonlinear Concentration Effect

Suppose a hypothetical nonlinear model has \(\eta(C)= -2.20+1.50C/(20+C)\), with concentration \(C\) measured in ng/mL. Compare \(C_0=10\) ng/mL with \(C_1=40\) ng/mL. Calculate the two probabilities, the odds ratio, and the risk difference.

At 10 ng/mL, \(\eta(C_0)=-2.20+1.50(10/30)=-1.70\), so \(p_0=1/\{1+\exp(1.70)\}=0.1545\). At 40 ng/mL, \(\eta(C_1)=-2.20+1.50(40/60)=-1.20\), so \(p_1=1/\{1+\exp(1.20)\}=0.2315\). The predictor contrast is \(-1.20-(-1.70)=0.500\), giving

\mathrm{OR}_{1:0}=\exp(0.500)=1.649,\qquad \mathrm{RD}_{1-0}=0.2315-0.1545=0.0770.

The model predicts an increase from 15.45% to 23.15%, an absolute change of 7.70 percentage points. The nonlinear concentration function determines the contrast in the predictor; once that contrast is calculated, the probability translation is the same as for a linear model.

Why Baseline Risk Matters

An odds ratio describes a ratio of odds, not a constant difference in probabilities. For a fixed odds ratio greater than 1, a higher baseline probability generally produces a different risk difference than a lower baseline probability. For instance, with \(\mathrm{OR}=2.00\) and \(p_0=0.100\), the formula gives \(p_1=0.200/(0.900+0.200)=0.1818\), so the risk difference is 8.18 percentage points. With the same odds ratio and \(p_0=0.500\), it gives \(p_1=1.000/(0.500+1.000)=0.6667\), so the difference is 16.67 percentage points. The odds ratio is unchanged, but the absolute changes are not.

This matters when communicating exposure-response results. A coefficient or odds ratio alone does not say how many additional patients are expected to experience the event. That requires a baseline probability relevant to the concentration contrast and the population or covariate profile being described. Conversely, the same odds ratio may be clinically consequential in one setting and less consequential in another because the baseline risk differs.

Key result: Given a baseline probability and a model-based odds ratio for a specified concentration contrast, calculate \(p_1\) from \(p_1=\mathrm{OR}_{1:0}p_0/(1-p_0+\mathrm{OR}_{1:0}p_0)\), then report \(p_1-p_0\) as the absolute probability change.

Assumptions, Diagnostics, and Alternatives

The algebraic conversion is exact for the specified odds ratio and baseline probability. Whether those inputs are scientifically appropriate depends on the model and comparison. Check that the contrast uses the intended event definition, concentration units, and direction; that both probabilities refer to the same covariate values; and that the odds ratio was obtained from the difference in linear predictors for the two concentrations. With a nonlinear concentration function, do not treat a coefficient as the effect of every concentration increment unless the model actually specifies a linear change in log odds.

A useful diagnostic is to calculate the probability directly from each linear predictor and compare it with the odds-ratio conversion. They should agree up to rounding. If they do not, check the sign of \(\eta(C_1)-\eta(C_0)\), the concentration units, the reference concentration, and the conversion formula. Also check that a larger odds ratio for the event corresponds to a larger predicted event probability when \(p_0\) is fixed. An inconsistency often indicates that the event or contrast has been reversed.

Alternative summaries answer different questions. A risk difference reports an absolute change in probability; a risk ratio reports a ratio of probabilities. Neither is interchangeable with an odds ratio. A model specified directly on the risk-difference or risk-ratio scale can target those quantities, but it imposes a different mean structure and must produce valid probabilities. For interpretation of a logistic model, converting its predicted probabilities at specified concentrations preserves the fitted logistic model while providing an absolute contrast.

The calculations here treat the baseline probability and odds ratio as given. In applications they are estimated, so uncertainty in both quantities affects the translated probability and risk difference. Plugging point estimates into the equations gives point predictions, not an uncertainty interval. In addition, predictions for one covariate profile should not automatically be described as population-average probabilities.

Common Mistakes

Multiplying probability by the odds ratio. The odds ratio multiplies the baseline odds, not the baseline probability. Convert through odds or use the probability formula.
Reporting an odds ratio as a percentage-point change. An odds ratio of 1.50 is not an increase of 50 percentage points. Compute \(p_1\) and subtract \(p_0\).
Omitting the baseline probability. An odds ratio alone does not determine an absolute probability change. State the baseline risk used.
Using a coefficient without the concentration contrast. For a linear logit effect, multiply the coefficient by the concentration difference before exponentiating.
Changing other predictors between the two predictions. If the goal is the effect of concentration alone, hold other model inputs fixed; otherwise the comparison includes their changes too.

Check Your Understanding

For each question, use the event probability and concentration contrast specified. Solutions follow the questions.

A hypothetical concentration contrast has \(\mathrm{OR}=1.80\) and baseline probability \(p_0=0.200\). Find \(p_1\) and the risk difference.
A logistic model has a coefficient of \(-0.025\) per mg/L. What is the odds ratio for an increase from 4 to 12 mg/L?
For an odds ratio of 2.00, compare the risk differences when baseline probabilities are 0.050 and 0.400. What does the comparison show?
Why is it incorrect to describe an odds ratio of 1.50 as a 50-percentage-point increase in event probability?

Solutions. (1) \(p_1=1.80(0.200)/\{0.800+1.80(0.200)\}=0.360/1.160=0.3103\). The risk difference is \(0.3103-0.200=0.1103\), an increase of 11.03 percentage points. (2) The predictor change is \((-0.025)(12-4)=-0.200\); therefore, \(\mathrm{OR}=\exp(-0.200)=0.8187\). The odds of the event decrease for this concentration contrast. (3) At \(p_0=0.050\), \(p_1=0.100/(0.950+0.100)=0.09524\), giving a risk difference of 0.04524. At \(p_0=0.400\), \(p_1=0.800/(0.600+0.800)=0.5714\), giving a risk difference of 0.1714. The same odds ratio produces different absolute changes at different baselines. (4) The odds ratio multiplies odds, not probability. The probability change depends on \(p_0\), so it must be calculated from both the baseline probability and the odds ratio.