Logit Versus Probit Dropout Links

Why Compare Dropout Links?

A dropout model describes how the chance of becoming missing varies with covariates and outcome information. Even when two candidate models use the same outcome model and the same predictors, a logistic link and a probit link impose different shapes on that relationship. They can therefore fit observed dropout patterns similarly while implying different dropout probabilities, especially at predictor values beyond those well represented in the data.

The comparison is meaningful only when the models are matched in their other components. Keep the outcome distribution, the dropout predictors, the at-risk definition, and the data contribution the same; change the link and its coefficients. Then differences in fitted predictions can be attributed to the link choice rather than to a change in outcome history or predictor set. The earlier material on observed-data fit and fit agreement versus extrapolation provides useful context for interpreting this comparison.

Definition: Let $D_{it}=1$ indicate that subject $i$, while at risk at visit $t$, drops out before the outcome at that visit is observed; let $D_{it}=0$ indicate that the outcome is observed. Let $H_{it}$ denote the fixed, available history and covariates used as predictors, and let $Y_{it}$ be the outcome whose observation is at issue. A matched logistic dropout model has $$ P(D_{it}=1\mid H_{it},Y_{it})=\operatorname{expit}(\eta_{it}), \qquad \operatorname{expit}(u)=\frac{1}{1+\exp(-u)}, $$ and a matched probit dropout model has $$ P(D_{it}=1\mid H_{it},Y_{it})=\Phi(\eta_{it}), $$ where $\Phi$ is the standard normal cumulative distribution function. In both cases, $\eta_{it}$ is the same specified linear predictor form, though its fitted coefficients differ by link.

What Must Be Held Fixed?

In a selection model, the distribution of outcomes and the dropout mechanism jointly determine the observed-data distribution. Write the outcome model as $f_\theta(Y_i\mid X_i)$, where $X_i$ denotes the covariates in that model, and the dropout model as $P_\gamma(D_i\mid Y_i,H_i)$. A matched link comparison holds the outcome model and the dropout predictors fixed, replacing only the link in the dropout probability. The parameters are refitted under each candidate link; coefficients on a logit scale and a probit scale are not directly comparable in magnitude.

For a subject with missing outcomes, the observed-data likelihood contribution sums or integrates the joint outcome-and-dropout contribution over the unobserved outcome values. Thus a link comparison should consider both quantities that the observed data inform, such as dropout rates across observed histories, and quantities that require extrapolation, such as the distribution of outcomes among those who dropped out. Similar observed-data fit does not by itself establish that the two missing-outcome predictions are similar or that either is correct.

The standard comparison assumes a correctly specified common outcome model, the same predictor definitions and risk sets, and a correctly computed likelihood for each link. It also requires that the fitting procedure converges to a usable solution. In an MNAR analysis, predictions for unobserved outcomes depend on the assumed outcome and dropout models; link choice alone does not turn those outcomes into observed information.

Link Scale and Probability Scale

A coefficient gives a change on the link scale, not a constant change in dropout probability. Under the logit link, a one-unit increase in a predictor with coefficient $\beta$ multiplies the odds of dropout by $\exp(\beta)$, conditional on the other predictors. Under the probit link, the same increase shifts the standard-normal index by $\beta$; it does not multiply the odds by a fixed amount. To compare the links, calculate predicted probabilities at meaningful predictor values.

The links are similar near their centers after rescaling. The logistic function has derivative $1/4$ at zero, while $\Phi$ has derivative $\phi(0)=1/\sqrt{2\pi}$, where $\phi$ is the standard normal density. If $c$ rescales a logistic index to match the probit curve locally at zero, equating derivatives gives

\frac{c}{4}=\frac{1}{\sqrt{2\pi}} \quad\Longrightarrow\quad c=\frac{4}{\sqrt{2\pi}}\approx1.596.

This local relationship helps explain why fitted probabilities can be close over a central range. It is not a guarantee of agreement in the tails, nor a universal conversion between fitted coefficients: the predictors, their range, and the fitted intercept all matter.

Worked Example: Comparing Dropout Probabilities Across a Predictor Range

Suppose a hypothetical matched comparison uses one predictor $x$, with the outcome model and all other specifications held fixed. For illustration, choose coefficients so both links give dropout probabilities of 0.20 at $x=0$ and 0.40 at $x=1$. This is a calibration example, not a claim that these coefficients were estimated from a study.

For the logistic model, the intercept is $\operatorname{logit}(0.20)=-1.3863$, and the slope is $\operatorname{logit}(0.40)-\operatorname{logit}(0.20)=0.9808$. At $x=2$, $$ \eta=-1.3863+2(0.9808)=0.5753, \qquad P(D=1\mid x=2)=\frac{1}{1+\exp(-0.5753)}\approx0.6400. $$

For the probit model, the intercept is $\Phi^{-1}(0.20)=-0.8416$, and the slope is $\Phi^{-1}(0.40)-\Phi^{-1}(0.20)=-0.2533-(-0.8416)=0.5883$. At $x=2$, $$ \eta=-0.8416+2(0.5883)=0.3349, \qquad P(D=1\mid x=2)=\Phi(0.3349)\approx0.6311. $$

The models agree by construction at $x=0$ and $x=1$, but differ by about 0.0089, or 0.89 percentage points, at $x=2$. That difference is modest in this illustration. Its importance in an analysis depends on where the data lie and how the probabilities feed into the missing-outcome predictions.

From Dropout Probabilities to Missing-Outcome Predictions

For a discrete outcome with values $y_1,\ldots,y_K$, let $q_k=P(Y=y_k)$ under the outcome model and let $r_k=P(D=1\mid Y=y_k)$ under a chosen dropout link. The joint probability of outcome $y_k$ and dropout is $q_k r_k$. Consequently, among those who drop out, the implied outcome probability and mean are

P(Y=y_k\mid D=1)=\frac{q_k r_k}{\sum_{j=1}^{K}q_jr_j}, \qquad E(Y\mid D=1)=\sum_{k=1}^{K}y_k\frac{q_kr_k}{\sum_{j=1}^{K}q_jr_j}.

These are model-implied predictions, not direct summaries of the missing outcomes. They combine the outcome distribution with the link-specific dropout probabilities. If the outcome distribution or the predictors vary across subjects, the same calculation must be carried out conditional on the relevant covariates and then combined according to the target population.

Worked Example: Comparing Implied Missing-Outcome Distributions

Continue the hypothetical calibration above. Suppose $Y$ takes values 0, 1, and 2 with outcome-model probabilities 0.30, 0.40, and 0.30. Use $Y$ as the predictor $x$. The dropout probabilities at $Y=0$ and $Y=1$ are 0.20 and 0.40 for both links. At $Y=2$, they are 0.6400 for the logistic link and 0.6311 for the probit link.

Under the logistic model, the joint probabilities of dropout with $Y=0,1,2$ are $0.30(0.20)=0.06000$, $0.40(0.40)=0.1600$, and $0.30(0.6400)=0.1920$. Their sum is $P(D=1)=0.4120$. Dividing each joint probability by this total gives $$ P(Y=0,1,2\mid D=1) = \left(\frac{0.06000}{0.4120},\frac{0.1600}{0.4120},\frac{0.1920}{0.4120}\right) \approx(0.1456,\,0.3883,\,0.4660). $$ The predicted mean among dropouts is $0(0.1456)+1(0.3883)+2(0.4660)\approx1.320$.

Under the probit model, the corresponding joint probabilities are $0.06000$, $0.1600$, and $0.30(0.6311)=0.18933$. Their sum is $0.40933$, so $$ P(Y=0,1,2\mid D=1) \approx \left(\frac{0.06000}{0.40933},\frac{0.1600}{0.40933},\frac{0.18933}{0.40933}\right) \approx(0.1466,\,0.3909,\,0.4625). $$ The predicted mean is $0(0.1466)+1(0.3909)+2(0.4625)\approx1.316$. The distributions and means are close because the dropout probabilities are close over these three outcome values. The calculation makes explicit which assumptions produce the predictions: the stated outcome probabilities and the link-specific dropout probabilities.

Observed-Data Comparison and a Useful Limitation

A simple observed-data comparison examines whether each fitted model reproduces dropout frequencies across the observed predictor values or histories. This is not a complete test of an MNAR model: an outcome-dependent dropout probability involves an outcome that may be missing precisely when dropout occurs. Still, comparing predictions for quantities recorded for everyone in the risk set is a useful check on whether the candidate links describe the observed dropout pattern.

Worked Example: Two Links With the Same Observed-Data Fit

In a hypothetical dataset, there are 100 at-risk records at $z=0$, with 20 dropouts, and 100 at-risk records at $z=1$, with 35 dropouts. Fit a two-parameter dropout model with an intercept and a linear effect of $z$. Assume independent Bernoulli dropout indicators conditional on $z$, and that these records are correctly defined as at risk.

For the logistic link, the fitted probabilities can match the two observed proportions exactly: 0.20 and 0.35. The coefficients are $$ \hat\alpha=\operatorname{logit}(0.20)=-1.3863, \qquad \hat\beta=\operatorname{logit}(0.35)-\operatorname{logit}(0.20) =-0.6190-(-1.3863)=0.7673. $$ At $z=2$, the model predicts $\operatorname{expit}(-1.3863+2(0.7673))\approx0.5370$.

For the probit link, the fitted probabilities at the two observed values can also match exactly. Its coefficients are $$ \hat\alpha=\Phi^{-1}(0.20)=-0.8416, \qquad \hat\beta=\Phi^{-1}(0.35)-\Phi^{-1}(0.20) =-0.3853-(-0.8416)=0.4563. $$ At $z=2$, it predicts $\Phi(-0.8416+2(0.4563))\approx0.5283$.

Both models have the same fitted Bernoulli log likelihood on the observed records: $$ 20\log(0.20)+80\log(0.80)+35\log(0.35)+65\log(0.65)\approx-114.8. $$ There is no observed-data fit advantage here: each model has two parameters for two distinct predictor values and exactly reproduces both proportions. Their predictions at $z=2$, which is outside the observed predictor range in this example, differ. This illustrates why close fit at observed values does not validate extrapolation.

Diagnostics, Alternatives, and Common Pitfalls

A careful comparison asks whether the fitted probabilities are sensible over the predictor range that matters, whether the optimization converged, and whether the estimated coefficients or predictions are unusually sensitive to a small number of records. Plotting both fitted probability curves against the same predictor values can reveal divergences hidden by a coefficient table. Check observed dropout rates in meaningful strata when those rates can be computed from the observed data, and distinguish those checks from assessment of missing-outcome predictions, which cannot generally be verified directly.

The principal alternative is not automatically a third link. It may be to retain the common link but change a clearly justified feature of the model, or to use a different model class. Such comparisons answer different questions. Here the aim is specifically to isolate the consequences of logistic versus probit response shapes, so altering outcome history or predictors at the same time would confound that comparison. No link should be selected solely because its coefficients are numerically smaller or because its predictions look more convenient.

Comparing raw coefficients. A logit coefficient and a probit coefficient use different scales. Compare predicted probabilities at common covariate values instead.
Treating similar observed fit as proof of equivalent MNAR predictions. The likelihood may be similar while the models extrapolate differently where outcomes are unobserved.
Calling an unobserved prediction an empirical dropout rate. The distribution among dropouts depends on the assumed outcome model as well as the dropout link.
Changing several specifications at once. If predictors, outcome distributions, or risk sets also change, the source of disagreement is no longer isolated to the link.

The comparison has limits. Logistic and probit links are both smooth, monotone links, and neither is guaranteed to represent the actual dropout process. With sparse data, limited predictor range, or few dropout events, fitted curves may be imprecise; different links can also yield nearly indistinguishable predictions over the observed range. Conversely, a small link-driven difference in dropout probabilities may have larger consequences for a target summary when it changes which outcome values are assigned to missing cases. Report the comparison on the probability and outcome scales relevant to the analysis, and state that the missing-outcome predictions remain conditional on the model assumptions.

Key takeaway: To assess logit versus probit dropout links, hold the rest of the model fixed, refit both candidates, and compare predicted dropout probabilities and implied missing-outcome distributions on shared, meaningful scales. Agreement in observed-data fit does not establish agreement about unobserved outcomes.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In a matched comparison, which parts of a selection model should remain fixed when changing from logit to probit?
Why should you compare predicted probabilities rather than compare the numerical values of a logit and probit coefficient?
Suppose $Y$ has values 0 and 1, with $P(Y=1)=0.40$. A model gives dropout probabilities 0.20 for $Y=0$ and 0.40 for $Y=1$. Calculate $P(Y=1\mid D=1)$.
Two links reproduce the observed dropout proportions at every observed covariate value but give different predictions outside that range. What can and cannot be concluded from this result?

Solutions.

Keep the outcome model, dropout predictors and their definitions, at-risk set, and likelihood construction fixed. Refit the coefficients under each link so the comparison isolates the link choice.
The links use different scales and interpretations. A logit coefficient changes log odds, whereas a probit coefficient shifts a standard-normal index. Predicted probabilities at common predictor values are directly comparable.
The joint probability of $Y=1$ and dropout is $0.40(0.40)=0.1600$. The dropout probability is $0.60(0.20)+0.40(0.40)=0.1200+0.1600=0.2800$. Thus $P(Y=1\mid D=1)=0.1600/0.2800\approx0.5714$.
The observed-data fit at those covariate values is the same, so those observations do not distinguish the links by fitted dropout proportions. Their different predictions outside the observed range show link-dependent extrapolation; they do not show which prediction is correct or establish either model as true.