Probability Differences Versus Odds Ratios

One Logistic Effect, Two Scales

A logistic regression coefficient has a direct interpretation on the log-odds scale: for a one-unit increase in a predictor, the odds are multiplied by the exponential of that coefficient, holding the other covariates fixed. But odds are not probabilities. The same multiplication of odds can correspond to a small or a substantial change in fitted probability, depending on the starting probability. This distinction matters whenever an effect is communicated in terms of how many events might occur, rather than only how the odds compare.

Here we compare two quantities for a pair of covariate profiles: their odds ratio and their risk difference, where “risk” means the model’s fitted probability of the binary outcome. The comparison is conditional on the other covariates being held fixed. It is a description of fitted probabilities; by itself, it does not establish a causal effect.

In a logistic regression model, suppose the comparison changes a predictor by one unit and leaves all other covariates unchanged. If its coefficient is \(\beta\), the corresponding odds ratio is \(\theta=\exp(\beta)\). The fitted probability for the reference profile is its baseline risk, denoted here by \(p_0\). Knowing \(\theta\) alone does not give the risk difference: the baseline risk is also needed.

Converting an Odds Ratio to a Risk Difference

The conversion follows directly from the definition of odds. If the reference odds are \(p_0/(1-p_0)\), multiplying them by \(\theta\) gives the comparison-profile odds. Solving those odds for the comparison probability gives an exact expression.

To verify the risk formula, write the comparison odds as \(\theta p_0/(1-p_0)\). A probability \(p_1\) with those odds satisfies \(p_1/(1-p_1)=\theta p_0/(1-p_0)\). Cross-multiplying and collecting the terms in \(p_1\) yields \(p_1(1-p_0+\theta p_0)=\theta p_0\), which gives the stated expression. Subtracting \(p_0\) then gives the difference:

The denominator \(1+(\theta-1)p_0=1-p_0+\theta p_0\) is positive because \(0<p_0<1\) and \(\theta>0\). Thus the sign of the risk difference is the sign of \(\theta-1\): an odds ratio greater than one gives a higher fitted probability, and an odds ratio less than one gives a lower fitted probability. The size of the difference, however, also depends on \(p_0\).

Why the Absolute Change Depends on Baseline Risk

The same odds multiplier acts on different starting odds at different baseline risks. When the baseline risk is near zero, even multiplying small odds may leave the resulting probability relatively low. When the baseline risk is near one, there is little room for a further increase in probability. Between those extremes, the same odds ratio can produce a larger absolute change.

For an odds ratio greater than one, this pattern can be made precise: the risk difference is zero at the limiting baseline risks zero and one, and it has a unique maximum between them. This maximum concerns the model’s risk difference for a fixed odds ratio; it is not an estimate of uncertainty or a claim that the predictor’s effect is causal.

To prove the result, write \(D(p_0)=(\theta-1)p_0(1-p_0)/\{1+(\theta-1)p_0\}\) for \(0<p_0<1\). The denominator is positive, and differentiation gives

The quadratic factor in braces has exactly one positive root. Solving \((\theta-1)p_0^2+2p_0-1=0\) gives the positive root \((\sqrt{\theta}-1)/(\theta-1)=1/(1+\sqrt{\theta})\); the other root is negative. The quadratic factor is positive at \(p_0=0\), changes sign at that positive root, and is negative thereafter. Since \(\theta-1>0\), \(D\) increases before the root and decreases after it. It is therefore the unique maximum on \(0<p_0<1\). At the maximizing baseline, put \(s=\sqrt{\theta}\). Substitution into the risk formula gives \(p_1=s/(1+s)\), while \(p_0=1/(1+s)\), so \(D(p_0)=(s-1)/(s+1)\), as claimed. \(\square\)

Comparing Covariate Profiles

The conversion is useful when translating a fitted logistic model into quantities that directly describe differences in event probabilities. A profile’s baseline risk is obtained from its linear predictor using the inverse-logit function. For a one-unit increase in a predictor with coefficient \(\beta\), keeping the other covariates fixed, the odds ratio is \(\exp(\beta)\). Insert that profile’s fitted baseline risk and this odds ratio into the conversion formula to obtain its comparison risk and risk difference.

Common Mistake and Practical Interpretation

A frequent mistake is to read an odds ratio of two as “the probability doubles.” That interpretation confuses odds with probability. For example, starting at \(p_0=0.50\), an odds ratio of two gives \(p_1=2/3\), not \(1.00\): the probability increases by \(1/6\), or about 16.7 percentage points. At \(p_0=0.20\), the same odds ratio gives \(p_1=1/3\), also not \(0.40\). The correct procedure is to start with the baseline risk, convert it to comparison risk using the odds-ratio formula, and then subtract the baseline risk if an absolute change is wanted.

Another interpretive caution is that risk differences for different profiles answer different conditional questions. They need not summarize one overall effect for a population, since profiles have different covariate values and baseline risks. Here the aim is narrower: to show how the same logistic coefficient translates into different fitted probability changes as the baseline risk varies. The odds ratio describes a multiplicative change in odds; the risk difference describes an additive change in probability. Neither scale can be substituted for the other without specifying the baseline risk.