Discrete Probability Changes for Binary Regressors

Why a Binary Regressor Needs a Finite Contrast

In a logit or probit model, a coefficient changes the model’s index, not the probability by a fixed number of percentage points. That distinction matters especially for a binary regressor: its two possible values are zero and one, so the natural comparison is between the model-predicted probabilities at those two values. This is a finite change, not a slope at a point.

The calculation holds the other covariates fixed. It answers a conditional question: according to the fitted model, how does the predicted probability differ between $D=0$ and $D=1$ for the same values of the remaining covariates? It does not, by itself, show that changing $D$ causes the outcome probability to change.

Model and assumptions: Let $Y$ be a binary outcome, $D\in\{0,1\}$ a binary regressor, and $z$ a vector of other covariates. Assume the conditional outcome is Bernoulli with $$ P(Y=1\mid D,z)=F(\beta_0+\beta_D D+z'\gamma), $$ where $F$ is the specified logit or probit CDF and the index is additive in $D$ and $z$. The formulas below are model-based contrasts under this conditional probability specification. Interpreting them as causal effects additionally requires appropriate causal identification assumptions, such as consistency, conditional exchangeability, and positivity.

Definition of the Discrete Probability Change

For a fixed covariate profile $z$, set the index at $D=0$ to $\eta_0(z)=\beta_0+z'\gamma$. At $D=1$, the index is $\eta_0(z)+\beta_D$. The model’s discrete probability change is the predicted probability at one minus the predicted probability at zero.

Definition: The discrete probability change for $D$, at covariate profile $z$, is $$ \Delta_D(z)=P(Y=1\mid D=1,z)-P(Y=1\mid D=0,z) =F\bigl(\eta_0(z)+\beta_D\bigr)-F\bigl(\eta_0(z)\bigr). $$ For fitted coefficients, the estimated change is $$ \widehat{\Delta}_D(z) =F\bigl(\hat{\eta}_0(z)+\hat{\beta}_D\bigr)-F\bigl(\hat{\eta}_0(z)\bigr). $$

For logit, $F=\Lambda$, where $\Lambda(t)=1/(1+\exp(-t))$. For probit, $F=\Phi$, the standard normal CDF. The contrast is signed: a positive value means the model predicts a higher probability at $D=1$, and a negative value means it predicts a lower probability. The difference is expressed in probability units; multiplying it by 100 gives percentage points.

The contrast depends on both $\beta_D$ and the starting index $\eta_0(z)$. Consequently, a coefficient does not generally translate into one common probability-point change for every individual. Even if the coefficient is the same, profiles with different values of $z$ can have different predicted contrasts.

Worked Example: A Logit Contrast at One Covariate Profile

Suppose a fitted logit model predicts whether a hypothetical applicant receives an offer. Let $D=1$ indicate that the applicant completed an optional application component. At one fixed profile of the other covariates, suppose the fitted index is $-1.0$ when $D=0$, and the fitted coefficient is $\hat{\beta}_D=0.8$. Calculate and interpret the discrete probability change.

The index at $D=1$ is $-1.0+0.8=-0.2$. The two fitted probabilities, evaluated before rounding, are $$ \hat{P}(Y=1\mid D=0,z)=\Lambda(-1.0)=0.2689414, \qquad \hat{P}(Y=1\mid D=1,z)=\Lambda(-0.2)=0.4501660. $$ Therefore, $$ \widehat{\Delta}_D(z)=0.4501660-0.2689414=0.1812246\approx 0.1812. $$

At this covariate profile, the fitted model predicts an offer probability about 18.12 percentage points higher for $D=1$ than for $D=0$. This is a conditional model-based difference, not automatically a causal effect. The logit coefficient also implies an odds ratio of $\exp(0.8)$, but that odds ratio is not the probability-point change calculated here.

Why This Is Not a Derivative-Based Marginal Effect

For a continuously varying regressor $x$ in a model with index $\eta$, the derivative of the predicted probability with respect to $x$ is $f(\eta)\beta_x$, where $f=F'$ is the link CDF’s density and $\beta_x$ is the coefficient on $x$. This is a local rate of change for a continuous movement in $x$. A binary regressor instead moves between two distinct values, so its natural effect is the difference between the probabilities at those values.

The distinction can be justified directly. Since $F'=f$, the exact finite change for an index shift $b$ satisfies $$ F(a+b)-F(a)=\int_a^{a+b} f(t)\,dt. $$ The derivative at $a$, multiplied by $b$, is $b f(a)$. It approximates the integral when the shift is sufficiently small and the density changes little over the interval, but it is not generally equal to the integral. A binary-regressor contrast uses the full change from zero to one, with $b=\beta_D$; it does not replace that change with a local slope.

Worked Example: A Probit Contrast and a Derivative Comparison

Consider a hypothetical probit model for whether a patient attends a scheduled appointment. Let $D=1$ indicate receiving a reminder. At a fixed profile of other covariates, suppose the fitted index at $D=0$ is $-0.5$, and $\hat{\beta}_D=0.8$. Calculate the discrete change and compare it with the derivative of a continuous extension at $D=0$.

The index at $D=1$ is $-0.5+0.8=0.3$. Using the standard normal CDF, $$ \widehat{\Delta}_D(z)=\Phi(0.3)-\Phi(-0.5) =0.6179114-0.3085375 =0.3093739\approx 0.3094. $$ Thus the fitted probability of attendance is about 30.94 percentage points higher at $D=1$ than at $D=0$, for this covariate profile.

If one extends the model mathematically to allow $D$ to vary continuously, the derivative at $D=0$ is $\hat{\beta}_D\phi(-0.5)$, where $\phi$ is the standard normal density. Here, $$ 0.8\phi(-0.5)=0.8(0.3520653)=0.2816522. $$ This derivative is a local slope in probability units per unit of $D$ at zero. It is not the binary contrast: the exact change from zero to one is $0.3094$. Since $D$ is observed only at zero and one, the finite contrast is the directly relevant comparison.

Contrasts Across Covariate Profiles

When the goal is to summarize a fitted model across several observations, calculate the binary contrast separately for each observation’s covariates, then average those contrasts if an average is desired. This is an average discrete change. It is not generally the contrast evaluated at an average covariate profile or an average index, because $F$ is nonlinear.

For $n$ profiles $z_i$, the sample average discrete change is $$ \overline{\widehat{\Delta}}_D =\frac{1}{n}\sum_{i=1}^{n} \left[ F\bigl(\hat{\eta}_{0i}+\hat{\beta}_D\bigr) -F\bigl(\hat{\eta}_{0i}\bigr) \right], \qquad \hat{\eta}_{0i}=\hat{\beta}_0+z_i'\hat{\gamma}. $$ Each term compares the same profile with $D$ set to one and to zero. This calculation is distinct from derivative-based average marginal effects and from effects at the means.

Worked Example: Averaging Logit Contrasts Across Three Profiles

For three hypothetical applicants, suppose the fitted logit indices at $D=0$ are $-1.0$, $0$, and $1.0$, and the same fitted binary-regressor coefficient is $\hat{\beta}_D=0.7$. Calculate each contrast and their unweighted average.

For the first profile, the index changes from $-1.0$ to $-0.3$: $$ \widehat{\Delta}_1=\Lambda(-0.3)-\Lambda(-1.0) =0.4255575-0.2689414 =0.1566161. $$

For the second profile, it changes from $0$ to $0.7$: $$ \widehat{\Delta}_2=\Lambda(0.7)-\Lambda(0) =0.6681878-0.5000000 =0.1681878. $$

For the third profile, it changes from $1.0$ to $1.7$: $$ \widehat{\Delta}_3=\Lambda(1.7)-\Lambda(1.0) =0.8455347-0.7310586 =0.1144761. $$

The unweighted average is $$ \overline{\widehat{\Delta}}_D =\frac{0.1566161+0.1681878+0.1144761}{3} =0.1464267\approx 0.1464. $$ Across these three profiles, the average fitted probability is about 14.64 percentage points higher when $D$ is set to one rather than zero. This average is computed from the three individual contrasts; evaluating a single contrast at an average index would be a different calculation.

Interpretation and Common Pitfalls

A discrete probability change is useful when the regressor represents a category, status, or exposure with exactly two values. It reports the model-implied difference on the probability scale, which is often easier to communicate than an index coefficient or an odds ratio. State the outcome, the values being compared, the covariates held fixed, and the units of the difference.

Confusing a coefficient with a probability change. A logit or probit coefficient is an index shift. Calculate $F(\hat{\eta}_0+\hat{\beta}_D)-F(\hat{\eta}_0)$ to obtain the probability contrast.
Using a derivative as the binary effect. The derivative is a local slope for a continuous change. For a zero-to-one switch, subtract the two predicted probabilities.
Ignoring the covariate profile. Nonlinearity means that the same coefficient can yield different probability changes at different starting indices. Give the profile or explain how contrasts are averaged.
Calling a model contrast causal without justification. Setting $D$ to zero or one in a prediction is a model comparison. A causal interpretation requires assumptions that identify the relevant counterfactual probabilities.

Key takeaway: For a binary regressor, compare the fitted probability at $D=1$ with the fitted probability at $D=0$, holding other covariates fixed. This finite difference is not a derivative-based marginal effect, and its probability-point size depends on the covariate profile.

Check Your Understanding

Use the model and definitions above to answer the following questions.

In a logit model, the index at $D=0$ is $-0.4$ and the coefficient on $D$ is $0.6$. Write the expression for the discrete probability change, including the two indices being compared.
Why is the derivative with respect to a continuous extension of $D$ not generally the exact probability change from zero to one?
If the discrete change is $-0.08$, interpret its sign and size in probability points for the stated covariate profile.
Why might two profiles have different probit probability changes even when they share the same coefficient on $D$?
What additional kind of assumptions are needed before interpreting a model-based contrast as a causal effect?