Computing Fitted Risks for Covariate Profiles

From a Covariate Profile to a Fitted Risk

A logistic-regression table reports estimated coefficients, but a coefficient is not itself an event probability. To answer a question such as “What event probability does the fitted model assign to a 60-year-old patient receiving treatment?”, first translate that profile into the model’s covariates, calculate its linear predictor, and then apply the inverse-logit function. This tutorial focuses on that calculation for specified profiles; it does not quantify uncertainty in the resulting predictions.

For a binary response \(Y\), let \(p(x)=P(Y=1\mid x)\), where \(x\) denotes the covariate information for a profile. In logistic regression, the log odds of the event are modeled as a linear predictor. When the fitted coefficient vector is \(\hat\beta\), the fitted event probability at profile \(x\) is obtained by evaluating that predictor at the profile and transforming it back to the probability scale.

The design vector is the bridge between a description in context and the coefficient estimates. If a model has an intercept, a binary predictor \(A\), and a continuous predictor \(z\), with no interaction, then \(x=(1,A,z)^\mathsf{T}\). The value \(A=0\) or \(A=1\) must follow the model’s coding, and \(z\) must be expressed in the units used to fit the model. A profile is not fully specified until every required predictor value is known.

Why the Inverse-Logit Calculation Works

The logistic model writes the event probability’s log odds as the linear predictor. The fitted probability formula is not a separate modeling assumption: it follows by solving that equation for the probability.

To verify the result, start with the defining logit equation and exponentiate both sides. Then collect the terms involving \(p(x)\).

The solution is valid because probabilities in the logistic model satisfy \(0<p(x)<1\), so the log odds are defined, and \(1+\exp\{\eta(x)\}\) is strictly positive. Substituting \(\hat\eta(x)=x^\mathsf{T}\hat\beta\) gives the fitted value. \(\square\)

This transformation has useful checks. Every finite linear predictor produces a probability strictly between zero and one, and a linear predictor of zero produces a fitted probability of one half. A positive predictor gives a probability above one half; a negative predictor gives one below one half. These checks can reveal sign or arithmetic errors before a prediction is reported.

For completeness, the range follows because \(\exp(\eta)>0\) and the denominator exceeds the numerator by \(1\), so \(0<g(\eta)<1\). At zero, \(g(0)=1/(1+1)=1/2\). Differentiating gives

Thus \(g\) is strictly increasing. Consequently, comparing two fitted risks from the same fitted model can be checked by first comparing their linear predictors: the larger linear predictor always corresponds to the larger fitted risk. This is a computational property, not a statement about causation. \(\square\)

A Reliable Calculation Sequence

A careful prediction is mostly a matter of constructing the correct row of the model matrix. Before doing arithmetic, identify which values correspond to which coefficients. For a categorical predictor, use its reference-category coding; for a continuous predictor, use the recorded unit or transformation; for an interaction, calculate the product of the relevant profile values. Then sum the coefficient contributions before applying the inverse logit.

Continuous Predictors and Units

A continuous predictor enters the linear predictor at the profile’s numerical value, not at an informal description such as “older” or “high.” If age is recorded in years, an age of 50 contributes 50 times its coefficient unless the model instead uses a centered or rescaled age variable. Always write down the actual modeled variable before substituting numbers. Centering or rescaling changes the values used in the calculation, even though the procedure is otherwise the same.

Profiles When the Model Includes an Interaction

An interaction adds a product term to the design vector. Its contribution depends on both profile values, so it must be calculated separately from the corresponding main effects. For a binary treatment \(A\) and a continuous variable \(z\), a model with interaction can be written \(\hat\eta(A,z)=\hat\beta_0+\hat\beta_A A+\hat\beta_z z+\hat\beta_{Az}Az\). At \(A=0\), the product \(Az\) is zero. At \(A=1\), the interaction contribution is \(\hat\beta_{Az}z\). A single coefficient cannot be substituted for the whole profile-specific calculation.

Checks, Interpretation, and a Common Pitfall

A fitted risk is the model’s estimated probability of \(Y=1\) for the specified covariate profile. It is not the observed outcome for a particular person, and it is not automatically a causal probability. Its value depends on the fitted model, the coefficient estimates, the coding, and the profile supplied. When reporting one, name the event and describe the profile so that the number has a clear referent.

Before finalizing a calculation, check that the intercept has been included once; all profile values use the correct coding and units; each interaction product has been computed; and the final result lies between zero and one. Rounding the linear predictor too early can also change the last reported digits, so retain a few digits until the inverse-logit calculation is complete.

• Common mistake: treating the linear predictor as a risk. If \(\hat\eta=1.5\), reporting a fitted risk of 1.5 is impossible because a probability must lie between zero and one. The correction is to transform it: \(\exp(1.5)/\{1+\exp(1.5)\}\approx0.818\).
• Common mistake: omitting an interaction contribution. Substituting only the intercept and main effects does not evaluate the stated interaction model. The correction is to calculate every product term at the profile before summing.
• Common mistake: using the wrong continuous scale. If a model uses centered age in decades, substituting age in years gives the wrong design vector. The correction is to express the profile using the exact variable defined in the fitted model.