Worked Probit Probability And Effect Calculations

From a Probit Predictor to a Probability

A probit coefficient describes a change on the model’s predictor scale, not a fixed change in probability. To interpret fitted risks and effects, first calculate the linear predictor at the covariate values of interest, then apply the standard normal cumulative distribution function. The same coefficient can correspond to different probability changes at different baseline predictors.

For a binary response with conditional mean \(p(x)=P(Y=1\mid X=x)\), a probit model specifies that the inverse link is the standard normal CDF. Write the linear predictor as \(\eta(x)=x^\mathsf{T}\beta\), where \(x\) is the covariate vector and \(\beta\) is the coefficient vector. Let \(\Phi\) and \(\phi\) denote the CDF and density of a standard normal random variable.

Definition: In a probit model, the fitted probability at covariate profile \(x\) is \(p(x)=\Phi(\eta(x))=\Phi(x^\mathsf{T}\beta)\). For a change from profile \(x_0\) to \(x_1\), the finite probability change is \(p(x_1)-p(x_0)\). For a continuous covariate \(x_j\), its local marginal effect is the derivative \(\partial p(x)/\partial x_j\), with the other covariates held fixed.

The distinction between the two effects matters. A finite change compares two specified profiles; a local effect is a rate of change at one profile. For binary covariates, the finite probability change is generally the relevant probability-scale contrast. A derivative with respect to a binary indicator does not replace that contrast.

Worked Calculations for Binary Covariates

Worked Example: A Binary Treatment at a Lower Baseline Predictor

Suppose the fitted model is \(\eta=-1+0.5A\), where \(A=0\) denotes the reference group and \(A=1\) the treated group. Calculate both fitted probabilities and their difference.

For \(A=0\), the predictor is \(\eta_0=-1+0.5(0)=-1\), so \(p_0=\Phi(-1)=0.1586552539\), approximately \(0.158655\).

For \(A=1\), the predictor is \(\eta_1=-1+0.5(1)=-0.5\), so \(p_1=\Phi(-0.5)=0.3085375387\), approximately \(0.308538\).

The fitted probability difference, calculated from the unrounded CDF values, is \(p_1-p_0=0.3085375387-0.1586552539=0.1498822848\), approximately \(0.149882\). Thus the fitted risk is about 15.0 percentage points higher for \(A=1\) than for \(A=0\) at this predictor profile. The coefficient \(0.5\) is not itself a 0.5 probability increase; it is the change in the probit linear predictor.

For an indicator variable, the model’s coefficient adds a fixed amount to the predictor when the indicator changes from zero to one. But the corresponding probability change depends on the starting predictor: the normal CDF does not increase by a constant amount for equal increments across its entire range.

Local Marginal Effects and Finite Changes

Theorem: Suppose \(p(x)=\Phi(\eta(x))\), where \(\eta\) is differentiable in a continuous covariate \(x_j\). Then, wherever that derivative exists, \(\partial p(x)/\partial x_j=\phi(\eta(x))\,\partial\eta(x)/\partial x_j\). In a linear probit model, \(\eta(x)=x^\mathsf{T}\beta\), this becomes \(\partial p(x)/\partial x_j=\beta_j\phi(x^\mathsf{T}\beta)\).

Proof. The standard normal CDF is \(\Phi(t)=\int_{-\infty}^{t}\phi(u)\,du\). Since \(\phi\) is continuous, the fundamental theorem of calculus gives \(\Phi'(t)=\phi(t)\). Applying the chain rule to \(p(x)=\Phi(\eta(x))\) gives \(\partial p(x)/\partial x_j=\Phi'(\eta(x))\partial\eta(x)/\partial x_j=\phi(\eta(x))\partial\eta(x)/\partial x_j\). For a linear predictor, \(\partial\eta(x)/\partial x_j=\beta_j\), proving the stated expression. \(\square\)

The density factor \(\phi(\eta(x))\) determines how much of the predictor-scale slope appears on the probability scale at a particular profile. It is largest at \(\eta=0\) and decreases as \(|\eta|\) increases. Consequently, a positive coefficient gives a positive local effect, but its size is not constant across profiles.

Worked Example: A Continuous Covariate Over a Half-Unit Increase

Consider \(\eta(x)=-1.2+0.8x\), with all other covariates fixed. Calculate the fitted probability change when \(x\) increases from \(0\) to \(0.5\), the average probability change per unit over that interval, and the local effects at its endpoints.

At \(x=0\), \(\eta(0)=-1.2\), giving \(p(0)=\Phi(-1.2)=0.1150696702\), approximately \(0.115070\). At \(x=0.5\), \(\eta(0.5)=-1.2+0.8(0.5)=-0.8\), giving \(p(0.5)=\Phi(-0.8)=0.2118553986\), approximately \(0.211855\).

Using the unrounded CDF values, the finite probability change is \(p(0.5)-p(0)=0.2118553986-0.1150696702=0.0967857284\), approximately \(0.096786\). This is the fitted change over the specified half-unit interval. Dividing by the interval width, the average change per unit is \(0.0967857284/0.5=0.1935714568\), approximately \(0.193571\).

The local effect at \(x=0\) is \(0.8\phi(-1.2)=0.8(0.1941860547)=0.1553488438\), approximately \(0.155349\) per unit. At \(x=0.5\), it is \(0.8\phi(-0.8)=0.8(0.2896915528)=0.2317532422\), approximately \(0.231753\) per unit. The local effects differ because the predictor, and hence the density factor, differs at the two profiles. Neither endpoint derivative is the finite change itself.

There is also an exact relationship between a finite change and local effects along a continuous path. It explains why averaging local effects over an interval is conceptually different from reporting the derivative at just one endpoint.

Result: For a linear probit predictor \(\eta(x)=a+\beta_jx_j\), holding the other covariates fixed, and any finite endpoints \(u<v\), \(p(v)-p(u)=\int_u^v \beta_j\phi(a+\beta_jt)\,dt\). If \(v>u\), the average probability change per unit on the interval equals the average of the local marginal effect over that interval.

Derivation. The function \(g(t)=\Phi(a+\beta_jt)\) is continuously differentiable, and by the chain rule \(g'(t)=\beta_j\phi(a+\beta_jt)\). The fundamental theorem of calculus therefore gives \(g(v)-g(u)=\int_u^v g'(t)\,dt=\int_u^v\beta_j\phi(a+\beta_jt)\,dt\). Dividing both sides by \(v-u\) gives the average change per unit as \((v-u)^{-1}\int_u^v\beta_j\phi(a+\beta_jt)\,dt\). \(\square\)

Contrasting Baseline Risks

Worked Example: The Same Binary-Covariate Coefficient at Two Baselines

Suppose a binary covariate \(A\) has coefficient \(0.5\). Compare its fitted probability difference for two profiles whose predictors at \(A=0\) are \(-1\) and \(1\), respectively. In each profile, changing \(A\) from zero to one adds \(0.5\) to the predictor.

For the first profile, the predictors are \(-1\) and \(-0.5\). As calculated above, the probabilities are \(0.1586552539\) and \(0.3085375387\), so the difference is approximately \(0.149882\).

For the second profile, the predictors are \(1\) and \(1.5\). The probabilities are \(\Phi(1)=0.8413447461\) and \(\Phi(1.5)=0.9331927987\). Thus the difference is \(0.9331927987-0.8413447461=0.0918480526\), approximately \(0.091848\).

The predictor increment is \(0.5\) in both comparisons, but the probability differences are not equal. The lower-baseline profile has a fitted change of about 0.150, while the higher-baseline profile has a change of about 0.092. These are model-based fitted contrasts; interpreting them causally requires additional design and identification assumptions not supplied by the probit link.

The local-effect formula helps explain the pattern. For a continuous covariate with coefficient \(0.5\), the local effects at predictors \(-1\) and \(1\) are both \(0.5\phi(1)\), since the standard normal density is symmetric: \(\phi(-1)=\phi(1)\). Numerically, each is \(0.5(0.2419707245)=0.1209853623\), approximately \(0.120985\). Equal local effects at these two points do not imply equal finite changes over intervals beginning there: those intervals run from \(-1\) to \(-0.5\) and from \(1\) to \(1.5\), respectively, and the density changes differently across them.

Common Mistakes and a Practical Calculation Order

Reading a coefficient as a probability difference. A probit coefficient changes the linear predictor. Apply \(\Phi\) at both profiles and subtract the resulting probabilities to obtain a finite contrast.
Confusing a derivative with a finite change. The local effect is the derivative at one profile; a finite contrast compares two profiles. For a continuous covariate, the contrast is the integral of local effects along the interval, not simply an endpoint derivative multiplied by the interval width.
Assuming one probability effect applies everywhere. The density factor \(\phi(\eta)\) varies with the predictor, so probability-scale effects vary across profiles even when the coefficient is fixed.
Rounding too early. Subtracting displayed probabilities rounded to six places can produce a difference that does not round correctly to six places. Keep unrounded values for arithmetic, then round the final result.

A reliable calculation follows three steps: compute the predictor for each profile, convert each predictor to a probability with \(\Phi\), and take the contrast required by the question. For a local effect, instead evaluate \(\beta_j\phi(\eta)\) at the stated profile. Name the scale and the profile in the interpretation so that a predictor coefficient, a finite risk difference, and a local rate are not mistaken for one another.

Check Your Understanding

Work each question by identifying whether it asks for a fitted probability, a finite contrast, or a local effect.

For \(\eta=-0.4+0.7A\), where \(A\) is binary, write the expressions for the fitted probabilities at \(A=0\) and \(A=1\), and for their probability difference.
In a probit model with predictor \(\eta=-0.8+0.6x\), state the formula for the local effect of \(x\) at \(x=1\).
Why is a coefficient of \(0.6\) not generally a 0.6-unit increase in fitted probability?
For a continuous covariate, explain how the finite probability change from \(u\) to \(v\) relates to the local effects between those endpoints.
Two profiles have predictors \(-1\) and \(1\). What property of the standard normal density makes the probit local effects equal there when their coefficients are the same?