Log-Binomial Regression And Risk Ratios

Why Use a Log Link for a Binary Outcome?

For a binary outcome, the conditional mean is the probability of the event. A regression model for that mean can therefore describe how risk changes with covariates. Logistic regression models the log odds; a log-binomial model instead models the log risk. This makes exponentiated coefficients directly interpretable as relative changes in risk, conditional on the other covariates.

Let $Y_i$ indicate whether the event occurs for observation $i$, with $Y_i=1$ for an event and $Y_i=0$ otherwise. Write $\mathbf{x}_i=(1,x_{i1},\ldots,x_{ip})^\mathsf{T}$ for the covariate vector, including an intercept, and let $\boldsymbol{\beta}=(\beta_0,\beta_1,\ldots,\beta_p)^\mathsf{T}$. The risk at covariate profile $\mathbf{x}_i$ is $p_i=P(Y_i=1\mid\mathbf{x}_i)$.

Definition: The binary log-link model, also called the log-binomial model, specifies $Y_i\mid\mathbf{x}_i\sim\operatorname{Bernoulli}(p_i)$ and $$ \log(p_i)=\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta}, \qquad p_i=\exp(\mathbf{x}_i^\mathsf{T}\boldsymbol{\beta}). $$ The model is defined at covariate profiles for which $0<p_i<1$. Thus its linear predictor must give a valid probability at each profile where the model is used.

The name “log-binomial” comes from the binomial response distribution with a log link for its mean. Here the response is Bernoulli, the binomial distribution with one trial. The link function is $g(p)=\log(p)$; its inverse is $g^{-1}(\eta)=\exp(\eta)$. Unlike a link whose inverse maps every real number into the probability range, the exponential function can exceed one. The probability-range requirement is therefore part of specifying a valid model, not an optional check.

Why Exponentiated Coefficients Are Risk Ratios

A conditional risk ratio compares the event probabilities at two covariate profiles while holding specified covariates fixed. For a one-unit increase in covariate $x_j$, let $\mathbf{e}_j$ denote the vector that increases its coordinate by one and leaves all other coordinates unchanged. The coefficient $\beta_j$ describes that change on the log-risk scale.

Theorem: Suppose the log-binomial model holds at both $\mathbf{x}$ and $\mathbf{x}+\mathbf{e}_j$, with positive risks at both profiles. Then the conditional risk ratio for the one-unit increase in $x_j$, holding the other covariates fixed, is $\exp(\beta_j)$.

Proof. By the model, the risk at the first profile is $\exp(\mathbf{x}^\mathsf{T}\boldsymbol{\beta})$. At the second profile it is \[ \exp\big((\mathbf{x}+\mathbf{e}_j)^\mathsf{T}\boldsymbol{\beta}\big) =\exp(\mathbf{x}^\mathsf{T}\boldsymbol{\beta}+\beta_j). \] Dividing the second risk by the first gives \[ \frac{\exp(\mathbf{x}^\mathsf{T}\boldsymbol{\beta}+\beta_j)} {\exp(\mathbf{x}^\mathsf{T}\boldsymbol{\beta})} =\exp(\beta_j). \] The ratio does not depend on the fixed values of the other covariates, although the risks being compared do. $\square$

More generally, increasing $x_j$ by $\Delta$ units, with the remaining covariates unchanged, gives a risk ratio of $\exp(\Delta\beta_j)$. For an indicator variable coded zero or one, $\exp(\beta_j)$ compares the risk when the indicator is one with the risk when it is zero, at the same values of the other covariates. For a continuous variable, it is the ratio for the particular one-unit change used by its measurement scale.

The result also expresses a multiplicative structure: at fixed covariates, the model multiplies the risk by the same factor for each one-unit increase in $x_j$. This follows directly for any change vector $\mathbf{d}$ whose two profiles are within the model’s valid range.

Derived result: For two covariate profiles $\mathbf{x}$ and $\mathbf{x}+\mathbf{d}$, the log-binomial model implies $$ \frac{p(\mathbf{x}+\mathbf{d})}{p(\mathbf{x})} =\exp(\mathbf{d}^\mathsf{T}\boldsymbol{\beta}). $$ Derivation: Substitute the model risks into the ratio: $$ \frac{\exp((\mathbf{x}+\mathbf{d})^\mathsf{T}\boldsymbol{\beta})} {\exp(\mathbf{x}^\mathsf{T}\boldsymbol{\beta})} =\exp\big((\mathbf{x}+\mathbf{d})^\mathsf{T}\boldsymbol{\beta} -\mathbf{x}^\mathsf{T}\boldsymbol{\beta}\big) =\exp(\mathbf{d}^\mathsf{T}\boldsymbol{\beta}). $$ This applies only when both risks are defined and positive under the model.

Worked Coefficient Interpretations

Worked Example: A Binary Treatment Indicator

Suppose treatment is coded $x=0$ for control and $x=1$ for treatment, and the fitted model is $\log(p)=\beta_0+\beta_1x$, with $\beta_0=\log(0.08)$ and $\beta_1=\log(1.5)$. Find the two fitted risks and interpret the treatment coefficient.

For control, $x=0$, so \[ p(0)=\exp(\log(0.08))=0.08. \] For treatment, $x=1$, so \[ p(1)=\exp(\log(0.08)+\log(1.5)) =0.08(1.5)=0.12. \] The ratio is $p(1)/p(0)=0.12/0.08=1.5=\exp(\beta_1)$. The fitted treatment risk is 1.5 times the control risk, or 50% higher relative to the control risk. The absolute difference is $0.12-0.08=0.04$, which is 4 percentage points.

Worked Example: A Continuous Covariate

Suppose a dose $x$, measured in units of 5 milligrams, has model $\log(p)=\log(0.10)+\log(1.2)x$. Calculate the risks at $x=0$, $x=1$, and $x=2$, and interpret the coefficient.

At $x=0$, the fitted risk is $0.10$. At $x=1$, it is \[ p(1)=0.10(1.2)=0.12. \] At $x=2$, it is \[ p(2)=0.10(1.2)^2=0.144. \] The ratio for one unit, corresponding to a 5-milligram increase, is $0.12/0.10=1.2$ from $x=0$ to $x=1$, and $0.144/0.12=1.2$ from $x=1$ to $x=2$. Thus each 5-milligram increase multiplies the modeled risk by 1.2, when the other covariates, if any, are held fixed. A 10-milligram increase multiplies risk by $1.2^2=1.44$.

Worked Example: An Adjusted Conditional Risk Ratio

Consider a model with treatment $T$, smoking status $S$, and centered age $A$: \[ \log(p)=\log(0.12)+\log(1.5)T+\log(1.2)S+0.01A. \] Compare the fitted risks for $T=0$ and $T=1$ among smokers ($S=1$) whose age is five years above the centering value ($A=5$).

For $T=0$, the linear predictor is $\log(0.12)+\log(1.2)+0.05$, so \[ p(0,1,5)=0.12(1.2)\exp(0.05) \approx 0.12(1.2)(1.051271)=0.151383. \] For $T=1$, the treatment term adds $\log(1.5)$, giving \[ p(1,1,5)=0.12(1.5)(1.2)\exp(0.05) \approx 0.227075. \] Their ratio is approximately $0.227075/0.151383=1.5$. The treatment coefficient therefore gives a conditional risk ratio of 1.5 at these fixed values of smoking and age. The risks vary with those covariates, but the treatment risk ratio does not in this model. Both calculated risks are between zero and one.

What the Interpretation Does—and Does Not—Say

The risk ratio is conditional: it compares risks at specified values of the other covariates. It is not automatically a ratio of risks averaged over a population. Nor does a conditional association become a causal effect merely because it is expressed as a risk ratio; causal interpretation requires additional design and assumptions.

It is also important not to confuse a risk ratio with an odds ratio. In the first worked example, the risks are $0.08$ and $0.12$, so the risk ratio is $1.5$. The corresponding odds ratio is \[ \frac{0.12/(1-0.12)}{0.08/(1-0.08)} =\frac{0.12/0.88}{0.08/0.92} \approx 1.568. \] They are different quantities. The coefficient is exponentiated to obtain a risk ratio here because the model’s link is the logarithm of risk—not because exponentiating any regression coefficient always gives a risk ratio.

A further practical condition is that all modeled risks must be valid probabilities. For example, the one-covariate model in the continuous-covariate example would give risk $0.10(1.2)^k$ at $x=k$. At $k=10$, this is about $0.619$, still below one; at $k=15$, it is about $1.54$, which is impossible as a probability. A fitted model must be used only on a range of covariate profiles where its implied risks are valid. This qualification matters especially when interpreting extrapolations.

Common Mistakes

Calling $\exp(\beta_j)$ an odds ratio. In a log-binomial model it is a risk ratio. Odds ratios arise from a logit link.
Reading a relative change as a percentage-point change. A risk ratio of 1.5 means 50% higher relative risk; it does not mean an increase of 50 percentage points. The percentage-point difference depends on the starting risk.
Ignoring the covariate scale. A coefficient for age measured in years gives a one-year ratio; if age is instead measured in decades, the coefficient has a different one-unit interpretation.
Interpreting the ratio without checking the profiles. The coefficient interpretation requires both compared profiles to have valid risks under the model.
Assuming a conditional ratio is automatically causal or marginal. It is a model-based comparison at fixed covariates; broader interpretations require additional justification.

Key takeaway: In a valid binary log-link model, changing one covariate by one unit while holding the others fixed multiplies the conditional event risk by $\exp(\beta_j)$. The coefficient is therefore a log risk ratio, and its interpretation depends on the covariate’s coding, scale, and the validity of the compared probabilities.

Check Your Understanding

Use the log-binomial model and the coefficient interpretations above to answer these questions.

If a one-unit covariate change has coefficient $\log(0.8)$, what is the conditional risk ratio, and how would you describe it relative to the original risk?
A binary exposure has coefficient $\log(2)$, and the unexposed risk at fixed covariates is $0.15$. Calculate the exposed risk and the risk difference.
Why can the same coefficient give a constant conditional risk ratio but different risk differences at different covariate profiles?
In a model with a continuous covariate measured in years, what change does $\exp(\beta)$ describe? What ratio describes a five-year increase?
What condition must the model’s predicted risks satisfy at the profiles being compared?