Probit Selection on the Current Outcome

Why Use a Probit Selection Equation?

A dropout model must describe how the chance of leaving depends on information about a participant at the time of the decision. In a current-outcome selection model, the dropout chance at visit $t$ depends on $Y_t$, the outcome at that same visit. If a participant drops out before that outcome is recorded, $Y_t$ is unavailable in the observed data, but it remains part of the model: the selection process is allowed to depend on this unobserved current value.

The probit link is one way to express this dependence. It maps a linear predictor to a probability using the standard normal cumulative distribution function. Its coefficients describe changes on a latent standard-normal scale, not changes in log odds. That distinction matters when interpreting parameters or comparing a probit specification with the logistic current-outcome model.

Definition: Let $R_t=1$ mean that the outcome at visit $t$ is observed, and $R_t=0$ mean that dropout has occurred by that visit. For participants still observed at $t-1$, a current-outcome probit selection equation is $$ P(R_t=0\mid R_{t-1}=1,Y_1,\ldots,Y_t) =\Phi(\alpha_t+\gamma Y_t), $$ where $\Phi$ is the standard normal cumulative distribution function. This particular equation assumes that, among those at risk, dropout depends on the current outcome through the displayed linear predictor and not on the earlier outcomes.

The equation applies only to participants still at risk of dropout at the start of visit $t$. It is a discrete-time conditional dropout probability, not the probability of ever dropping out. The intercept $\alpha_t$ may vary by visit; $\gamma$ may be shared across visits or allowed to vary if the model is specified that way. The equation does not itself specify the distribution of the outcomes. A longitudinal outcome model is needed to describe those outcomes and, together with the selection model, to form a joint model.

Latent-Variable Form and the Probit Scale

The probit equation has a useful latent-variable representation. Imagine a latent dropout score $D_t^*$ for a participant still observed at $t-1$:

D_t^*=\alpha_t+\gamma Y_t+\varepsilon_t, \qquad \varepsilon_t\sim N(0,1), \qquad R_t=0\ \text{if and only if}\ D_t^*>0.

This representation gives the stated probability directly. Conditional on $Y_t=y$, dropout occurs when $\varepsilon_t>-(\alpha_t+\gamma y)$. Since $\varepsilon_t$ is standard normal and symmetric about zero,

\begin{aligned} P(R_t=0\mid R_{t-1}=1,Y_t=y) &=P\{\varepsilon_t>-(\alpha_t+\gamma y)\}\\ &=1-\Phi\{-(\alpha_t+\gamma y)\}\\ &=\Phi(\alpha_t+\gamma y). \end{aligned}

The latent disturbance variance is fixed at 1. This is a scale convention: the magnitude of a latent score is not directly observed, so its scale must be fixed to identify the coefficients. If the disturbance instead had standard deviation $\sigma_\varepsilon$, the probability would be $\Phi\{(\alpha_t+\gamma y)/\sigma_\varepsilon\}$; without an additional scale constraint, only the coefficients relative to $\sigma_\varepsilon$ would determine the probabilities.

This is different from the standard logistic latent-variable convention, whose error distribution has variance $\pi^2/3$, rather than 1. Consequently, a probit coefficient and a logistic coefficient with the same numerical value do not represent the same change in dropout probability or the same latent-scale effect. A rough conversion sometimes used to align the central portions of the links is to multiply a probit slope by about 1.6 to obtain a logistic slope. It is only an approximation: the normal and logistic distribution functions have different shapes, so there is no exact universal coefficient conversion.

Worked Example: Interpreting a Standardized-Outcome Probit Slope

Suppose a hypothetical fitted dropout model uses $Z_t$, the current outcome standardized so that one unit represents one reference standard deviation. The estimated equation is

P(R_t=0\mid R_{t-1}=1,Z_t=z)=\Phi(-1.0+0.8z).

At the reference outcome $z=0$, the dropout probability is

\Phi(-1.0)=0.1587.

At one standard deviation above the reference, $z=1$, the linear predictor is $-1.0+0.8(1)=-0.2$, so the dropout probability is $\Phi(-0.2)=0.4207$. The model therefore gives a difference of $0.4207-0.1587=0.2620$, or 26.20 percentage points, between these two conditional probabilities.

The positive slope means that higher current outcomes are associated with a higher conditional dropout probability in this model. The slope $0.8$ is a change in the probit linear predictor per standard deviation; it is not an 80% increase in dropout probability, a risk difference, or a log odds ratio. The calculated probabilities are conditional on being at risk at the visit.

Outcome Units and Coefficient Interpretation

For an unstandardized outcome, $\gamma$ is measured in probit-index units per outcome unit. Its sign indicates the direction of the modeled association: $\gamma>0$ makes the conditional dropout probability increase with $Y_t$, while $\gamma<0$ makes it decrease. The size of $\gamma$ alone does not give a constant probability change. Because the normal cumulative distribution function is nonlinear, a one-unit increase in $Y_t$ has different probability implications at different starting values.

If $Y_t$ is transformed to $Z_t=(Y_t-c)/s$, where $c$ and $s>0$ are fixed reference location and scale values, the coefficient on $Z_t$ describes a one-$s$-unit change on the original scale. The intercept then refers to the dropout probability at $Y_t=c$, since $Z_t=0$ there. Choosing and reporting the outcome scale is important: a coefficient per one point is not directly comparable to a coefficient per ten points or per standard deviation.

Worked Example: Using an Outcome Measured in Points

Consider a hypothetical model for a clinical score measured in points:

P(R_t=0\mid R_{t-1}=1,Y_t=y)=\Phi(-0.70+0.12y).

At a score of 4 points, the linear predictor is $-0.70+0.12(4)=-0.22$, giving a conditional dropout probability of $\Phi(-0.22)=0.4129$. At 8 points, it is $-0.70+0.12(8)=0.26$, giving $\Phi(0.26)=0.6026$. The model-implied difference is $0.6026-0.4129=0.1897$, or 18.97 percentage points, between scores of 4 and 8 points.

This calculation uses the model’s conditional probabilities for people still observed at the previous visit. It does not show that raising a person’s score would cause dropout risk to change. It also does not imply that every four-point increase produces the same 18.97-percentage-point change; the probit curve’s slope varies over the outcome range.

What Changes Compared With Logistic Selection?

A logistic current-outcome selection model uses the same risk-set and outcome timing but applies the logistic cumulative distribution function to the linear predictor. The logistic model has a linear predictor in log odds; in that model, its current-outcome coefficient is a change in log odds per outcome unit. The probit model instead has a linear predictor on the standard-normal latent scale. It does not have a constant odds ratio for a one-unit outcome increase.

Feature	Probit selection	Logistic selection
Link from linear predictor to probability	Standard normal cumulative distribution function, $\Phi$	Logistic cumulative distribution function
Latent error scale under standard convention	Variance 1	Variance $\pi^2/3$
Current-outcome coefficient	Change in probit index per outcome unit	Change in log odds per outcome unit
Constant odds ratio for a one-unit increase?	No	Yes, under the specified linear logistic model

Worked Example: Why Equal Numerical Slopes Do Not Match

Compare two hypothetical models, both using a standardized current outcome $z$. The probit model has predictor $-0.5+0.6z$; the logistic model has the same numerical predictor but applies the logistic cumulative distribution function instead.

At $z=0$, the probit probability is $\Phi(-0.5)=0.3085$. The logistic probability is $1/\{1+\exp(0.5)\}=0.3775$. At $z=1$, both predictors equal $0.1$: the probit probability is $\Phi(0.1)=0.5398$, while the logistic probability is $1/\{1+\exp(-0.1)\}=0.5250$.

The probabilities differ despite identical numerical intercepts and slopes. The two equations use different links and different latent scales. This comparison does not establish that either model is better; it shows why coefficients should not be compared as if their numerical values had the same interpretation.

Assumptions, Diagnostics, and Limitations

The probit selection specification is meaningful under several conditions. Dropout must be represented as a discrete-time event among participants still at risk. The timing convention must make the current outcome $Y_t$ conceptually relevant to the dropout decision at $t$, including when that outcome is not observed after dropout. The linear predictor $\alpha_t+\gamma Y_t$ and the probit link must be adequate descriptions of the conditional dropout probability over the outcome range. Finally, the standard-normal latent scale is a model convention, not a separately observed fact.

Useful checks include whether fitted probabilities stay between zero and one, whether the estimated current-outcome effect is stable under reasonable changes to outcome scaling, and whether the model’s conditional dropout patterns are consistent with the observed dropout rates across visits and among observed outcome ranges. Extreme estimates, very large standard errors, or fitted probabilities close to zero or one for much of the relevant range can indicate weak information, a poorly scaled outcome, or an overly restrictive specification. Such checks cannot directly confirm how dropout depends on outcomes that are missing.

That last limitation is central. When $Y_t$ is missing because dropout occurs at $t$, the data generally do not reveal the dropout probability at each value of that missing outcome without further modeling assumptions. The probit form supplies one such assumption; it does not make the dependence empirically verifiable merely by choosing a different link. Inference therefore depends on the outcome model and selection model together, and on assumptions that should be stated and examined, including through sensitivity analyses.

The main alternatives have different implications. Logistic selection gives a direct conditional odds-ratio interpretation, while probit selection is natural when a standard-normal latent threshold is a useful representation. The complementary log-log link is another link for discrete-time hazards, but it encodes a different probability curve. No link is automatically correct. The choice should reflect the intended interpretation, modeling context, and sensitivity of substantive conclusions—not numerical convenience alone.

Common pitfall: Treating $\gamma$ as a log odds ratio because the model resembles a logistic equation is incorrect. In the probit model, $\gamma$ is a slope on the standard-normal index. To describe probability changes, substitute relevant outcome values into $\Phi(\alpha_t+\gamma Y_t)$ and report the resulting conditional probabilities in context.

From Equation to Interpretation

State the risk set.
Specify that the equation concerns participants with $R_{t-1}=1$, who can still drop out at visit $t$.

Name the outcome scale.
Give the units of $Y_t$, or define the standardized outcome and its reference values.

Interpret the index slope.
Use the sign of $\gamma$ for direction, and identify its units as probit-index units per outcome unit.

Calculate conditional probabilities.
For meaningful outcome values, evaluate $\Phi(\alpha_t+\gamma y)$; do not report the probit slope as an odds ratio.

State the modeling limits.
Explain that current outcomes may be unobserved at dropout and that the selection dependence is an assumption requiring sensitivity assessment.

Check Your Understanding

Answer each question, then compare with the solutions below.

In a probit dropout model, what does a positive current-outcome coefficient imply, and what does it not imply?
Why is the latent disturbance variance conventionally fixed at 1?
For the model $\Phi(-0.4+0.5z)$, calculate the conditional dropout probability at $z=0$ and $z=1$, and give the difference in percentage points.
Why is a probit coefficient not a constant odds ratio?
Why can observed dropout patterns not, by themselves, verify the modeled dependence on outcomes missing at dropout?

Solutions. (1) A positive coefficient means the model assigns higher conditional dropout probabilities to higher current outcomes among participants at risk. It is not a causal effect or a fixed percentage-point change. (2) The latent scale is not observed, so fixing the error variance identifies the scale of the coefficients; the standard probit convention sets it to 1. (3) At $z=0$, $\Phi(-0.4)=0.3446$. At $z=1$, the index is $0.1$, so $\Phi(0.1)=0.5398$. The difference is $0.5398-0.3446=0.1952$, or 19.52 percentage points. (4) The coefficient is a slope in the probit index, and the probit transformation does not make log odds linear in the outcome. (5) For participants dropping out, the current outcome is unobserved; the dependence of dropout on that value is supplied by the selection-model assumption rather than directly observed.