From Index Coefficients to Probability Effects
In a binary response model, a coefficient describes how a regressor shifts the model's index, not directly how many probability points the response changes. The probability effect depends both on that coefficient and on the covariate values at which the effect is evaluated. In logit and probit models, the link function makes this dependence explicit: a density evaluated at the index scales each coefficient.
This tutorial derives the local effect of a continuous regressor while holding the other regressors fixed. The result is a pointwise marginal effect, meaning a derivative evaluated at a specified covariate vector. It describes the model's rate of change at that point; it is not automatically a finite-change effect or a causal effect.
Model and Derivative Formulas
Assume the conditional response probability has the single-index form \(p(x)=G(x^\top\beta)\), where \(x\) includes the regressors and an intercept, \(\beta\) is the coefficient vector, and \(G\) is a differentiable cumulative distribution function. Write the index as \(\eta=x^\top\beta\). The formulas below apply to a continuous regressor that enters this index linearly and additively, so that \(\partial\eta/\partial x_j=\beta_j\). For a binary outcome, the model specifies \(Y\mid X=x\sim\text{Bernoulli}(p(x))\).
In logit, \(G=\Lambda\), the logistic cumulative distribution function. In probit, \(G=\Phi\), the standard normal cumulative distribution function, with density \(\phi\). Differentiating the conditional probability gives the following results.
\(p(x)=G(\eta)\), where \(\eta=x^\top\beta\).
\(\partial p(x)/\partial x_j=G'(\eta)\,\partial\eta/\partial x_j\).
Since \(\partial\eta/\partial x_j=\beta_j\), the effect is \(G'(\eta)\beta_j\), evaluated at the covariate values of interest.
For logit, \(\Lambda'(\eta)=\Lambda(\eta)[1-\Lambda(\eta)]\): differentiating \(1/[1+\exp(-\eta)]\) gives \(\exp(-\eta)/[1+\exp(-\eta)]^2\), which equals \(\Lambda(\eta)[1-\Lambda(\eta)]\). For probit, the derivative of the standard normal cumulative distribution function is its density, \(\Phi'(\eta)=\phi(\eta)\). These identities yield the two formulas by the chain rule.
In either model, the derivative separates into a coefficient and an index-dependent scale factor. The coefficient determines the direction of the effect when the scale factor is positive: a positive coefficient gives a positive derivative, and a negative coefficient gives a negative derivative. The scale factor determines how strongly that index shift translates into a probability change at the specified index. In logit it is largest at \(\eta=0\), where it is \(1/4\), and becomes smaller as the index moves away from zero. The probit scale factor is \(\phi(\eta)\), also largest at zero and decreasing as \(|\eta|\) increases.
Worked Example: A Logit Effect at Given Covariates
Suppose a hypothetical model for whether a client completes a program is \[ P(Y=1\mid X=x)=\Lambda(\eta),\qquad \eta=-1+0.8x_1-0.4x_2, \] where \(x_1\) is weekly study hours and \(x_2\) is travel time to the program in tens of minutes. Find the pointwise effects of both continuous regressors at \(x_1=2\) hours and \(x_2=1\) ten-minute unit.
First calculate the index and probability: \[ \eta=-1+0.8(2)-0.4(1)=0.2,\qquad p(x)=\Lambda(0.2)=\frac{1}{1+\exp(-0.2)}\approx0.5498. \] The logit scale factor is \[ \Lambda(0.2)[1-\Lambda(0.2)]=(0.5498)(0.4502)\approx0.2475. \]
The weekly-study-hours coefficient is \(0.8\), so its pointwise effect is \[ \frac{\partial p(x)}{\partial x_1}=0.2475(0.8)\approx0.1980. \] At these covariate values, the modeled completion probability is increasing locally by about 0.198 probability units per additional weekly study hour, or about 19.8 percentage points per hour as a local rate.
The travel-time coefficient is \(-0.4\), so its pointwise effect is \[ \frac{\partial p(x)}{\partial x_2}=0.2475(-0.4)\approx-0.09901. \] Thus the probability is decreasing locally by about 0.09901 probability units per one-unit increase in \(x_2\). Since one unit of \(x_2\) is ten minutes, this is about 9.90 percentage points per ten additional travel minutes as a local rate. These are model-based derivatives, not claims that changing either regressor would cause the stated probability change.
Probit Effects at Specified Values
The probit calculation uses the same chain rule but a different derivative of the link. The standard normal density is \(\phi(\eta)=(1/\sqrt{2\pi})\exp(-\eta^2/2)\). It is evaluated at the index before multiplying by the coefficient. The coefficient alone is therefore not the probability derivative.
Worked Example: Probit Effects at a Zero Index
Consider a hypothetical probit model for whether a patient attends a scheduled follow-up: \[ P(Y=1\mid X=x)=\Phi(\eta),\qquad \eta=-1.2+0.1x_1+0.2x_2, \] where \(x_1\) is years of education and \(x_2\) is years of prior experience. Calculate both pointwise effects at \(x_1=10\) years and \(x_2=1\) year.
The index at these values is \[ \eta=-1.2+0.1(10)+0.2(1)=-1.2+1.0+0.2=0. \] The predicted probability is \(\Phi(0)=0.5000\), and the probit scale factor is \(\phi(0)=1/\sqrt{2\pi}\approx0.3989\).
For education, the coefficient is \(0.1\), giving \[ \frac{\partial p(x)}{\partial x_1}=\phi(0)(0.1)=(0.3989)(0.1)\approx0.03989. \] At this point, the modeled attendance probability increases locally by about 0.03989 probability units, or 3.989 percentage points, per additional year of education.
For prior experience, the coefficient is \(0.2\), giving \[ \frac{\partial p(x)}{\partial x_2}=\phi(0)(0.2)=(0.3989)(0.2)\approx0.07979. \] The modeled probability increases locally by about 7.979 percentage points per additional year of experience at this point. Both calculations hold the other regressor fixed. The figures are local derivatives; they do not imply that a full one-year change must produce exactly that finite change in probability.
Why the Covariate Values Matter
Even when a coefficient is constant, a pointwise effect usually is not. As the regressors change, the index changes, which changes the density multiplying the coefficient. The next hypothetical calculation illustrates this for both links. It is a comparison of formulas using the same index and coefficient values, not a claim that two separately fitted models would have identical coefficients.
Worked Example: Effects at Low, Central, and High Indices
Suppose the index is \(\eta=-2+0.5x\), where \(x\) is a continuous measure in hours, and consider \(x=0\), \(x=4\), and \(x=8\). The corresponding indices are \(-2\), \(0\), and \(2\). For illustration, calculate the derivative for a link with coefficient \(0.5\).
In logit, the scale factor at \(\eta=\pm2\) is approximately \(0.1050\), while at zero it is \(0.2500\). Thus the derivatives at \(x=0,4,8\), respectively, are \[ 0.1050(0.5)\approx0.05250,\qquad 0.2500(0.5)=0.1250,\qquad 0.1050(0.5)\approx0.05250. \]
For probit, \(\phi(-2)=\phi(2)\approx0.05400\) and \(\phi(0)\approx0.3989\). The corresponding derivatives are \[ 0.05400(0.5)\approx0.02700,\qquad 0.3989(0.5)\approx0.1995,\qquad 0.05400(0.5)\approx0.02700. \]
In both sets of calculations the index coefficient is unchanged, but the probability derivative is greatest near index zero and smaller at the two more distant indices. The numerical sizes differ across links because their density functions differ. The calculations show why a coefficient cannot be read as one universal probability-point effect.
Using Estimated Coefficients and Interpreting Results
In an applied analysis, the coefficients are generally estimated. For a requested covariate vector \(x\), calculate the fitted index \(\hat{\eta}=x^\top\hat{\beta}\), evaluate the appropriate scale factor at that index, and multiply by the estimated coefficient \(\hat{\beta}_j\). This plug-in calculation gives the fitted pointwise marginal effect at that vector. Report the regressor's units and describe the result as a local rate of change in predicted probability.
The derivative is defined with the other regressors held fixed, and the linear-index assumption matters. If a regressor enters through a nonlinear transformation, its derivative with respect to the original variable must also be included by the chain rule. The formulas here use the specified additive, linear-in-regressors index. Nor does a conditional probability model alone establish a causal interpretation: a probability derivative describes the fitted conditional relationship unless additional causal assumptions and design support a causal claim.
Common Mistake
A common error is to report \(\hat{\beta}_j\) itself as the change in probability. That coefficient shifts the index; the probability derivative also includes \(\Lambda(\hat{\eta})[1-\Lambda(\hat{\eta})]\) for logit or \(\phi(\hat{\eta})\) for probit. A second error is to treat a derivative as an exact change for a finite increase. A derivative is a local rate: for a small change \(\Delta x_j\), the probability change is approximately \((\partial p/\partial x_j)\Delta x_j\), with the derivative evaluated at the starting covariate values. For larger changes, the probability should be recalculated at the new covariate values rather than assuming the initial derivative remains constant.
Check Your Understanding
Use the definitions and formulas above to answer these questions.
- In a logit model with index \(\eta=x^\top\beta\), write the pointwise marginal-effect formula for a continuous regressor \(x_j\).
- Why does the same probit coefficient produce different pointwise marginal effects at indices zero and two?
- In a logit model, suppose \(\beta_j=-0.6\) and the index at a specified covariate vector is zero. What is the pointwise marginal effect, and what determines its units?
- Explain why a pointwise derivative is not generally the exact probability change for a one-unit increase in a regressor.
- Does a positive pointwise marginal effect by itself show that increasing the regressor causes the binary outcome probability to rise? Explain.