Why the Mean Boundaries Matter
For a binary outcome, the conditional mean is a probability \(p\), so it must lie between zero and one. A generalized linear model instead represents that mean through a linear predictor \(\eta\), which may take any real value. The link function connects these scales. How it behaves near \(p=0\) and \(p=1\) determines what values of \(\eta\) correspond to very small or very large probabilities, and whether a finite predictor can represent either boundary exactly.
This is important even when no fitted probability is exactly zero or one. Data can include only failures or only successes in a covariate pattern, and software can display a probability rounded to 0 or 1. Neither fact by itself establishes that the model has a finite predictor whose probability is exactly at the boundary.
The Three Links at the Boundaries
The link formulas make the relevant limits explicit. Here \(\Phi\) is the cumulative distribution function of the standard normal distribution, and \(\Phi^{-1}\) is its quantile function.
For all three links, the limit as \(p\) approaches zero is negative infinity, and the limit as \(p\) approaches one is positive infinity. Thus the mean boundaries correspond to predictors that diverge, rather than to finite values of \(\eta\). The links do not behave identically at interior probabilities: in particular, the complementary log-log link is not symmetric about \(p=0.5\). The focus here is the shared boundary fact, not a detailed comparison of their tail behavior.
Assume \(\eta\) is a real number, not positive or negative infinity.
For logit, \(1/(1+\exp(-\eta))\) has a positive numerator and a denominator greater than the numerator, so it lies strictly between zero and one. For probit, \(\Phi(\eta)\) is strictly between zero and one at every finite \(\eta\). For complementary log-log, \(\exp(\eta)\) is positive and finite, so \(\exp[-\exp(\eta)]\) is strictly between zero and one; subtracting it from one also gives a value strictly between zero and one.
As \(\eta\) tends to negative infinity, each inverse link tends to zero; as \(\eta\) tends to positive infinity, each tends to one. The boundaries are approached as limits, not reached for finite \(\eta\).
In a GLM, a predictor such as \(\eta=\beta_0+\beta_1x\) is finite whenever its coefficients and covariate values are finite. Under that condition, any of these three inverse links gives an interior probability. This statement concerns the model's mathematical prediction. A computer may round a very small probability to 0 or a probability extremely close to 1 to 1.
Worked Examples
Worked Example: Transforming Means Near Both Boundaries
Consider hypothetical binary-response means \(p=0.02\) and \(p=0.98\). Calculate their values on each link scale. These calculations show how the predictor changes as the mean moves near either boundary.
Logit. For \(p=0.02\), the odds are \(0.02/0.98=0.020408\), so \(g_{\mathrm{logit}}(0.02)=\log(0.020408)=-3.892\). For \(p=0.98\), the odds are \(0.98/0.02=49\), so \(g_{\mathrm{logit}}(0.98)=\log(49)=3.892\).
Probit. The standard normal quantiles are \(g_{\mathrm{probit}}(0.02)=\Phi^{-1}(0.02)\approx-2.054\) and \(g_{\mathrm{probit}}(0.98)=\Phi^{-1}(0.98)\approx2.054\).
Complementary log-log. At \(p=0.02\), \(-\log(1-p)=-\log(0.98)\approx0.020203\), and therefore \(g_{\mathrm{cloglog}}(0.02)=\log(0.020203)\approx-3.902\). At \(p=0.98\), \(-\log(1-p)=-\log(0.02)\approx3.9120\), and therefore \(g_{\mathrm{cloglog}}(0.98)=\log(3.9120)\approx1.364\).
All six link values are finite because both means are strictly between zero and one. The results also show why the links should not be treated as interchangeable: a given mean can correspond to different predictor values under different links.
Worked Example: Finite Predictors and Rounded Predictions
For a hypothetical model, suppose the predictor is \(\eta=4\). Calculate the three response-scale predictions and decide whether any is exactly one.
Logit: \(1/[1+\exp(-4)]=1/(1+0.018316)=0.9820\). Probit: \(\Phi(4)\approx0.9999683\). Complementary log-log: \(1-\exp[-\exp(4)]=1-\exp(-54.598)\approx1-1.934\times10^{-24}\).
Each mathematical prediction is less than one, because \(\eta=4\) is finite. If the last result is displayed to four decimal places, it appears as \(1.0000\); this is rounding, not an exact probability of one. The probit result is also very close to one, but its displayed value depends on the number of digits retained.
For comparison, at \(\eta=-4\), the logit prediction is \(1/[1+\exp(4)]\approx0.01799\), the probit prediction is \(\Phi(-4)\approx0.00003167\), and the complementary log-log prediction is \(1-\exp[-\exp(-4)]\approx0.01815\). Each is still positive. These calculations illustrate the finite-predictor property; they are not a substitute for choosing a link that suits the scientific setting.
Worked Example: All Failures in a Hypothetical Group
Suppose a hypothetical group contains 20 independent binary trials with a common success probability \(p\), and all 20 outcomes are failures. Under the binomial model, the likelihood for \(p\) is \(L(p)=(1-p)^{20}\), for \(0\leq p\leq1\). This likelihood is largest at \(p=0\), since \((1-p)^{20}\leq1\), with equality at zero.
At that boundary, the logit is \(\log[0/(1-0)]=-\infty\), the probit is \(\Phi^{-1}(0)=-\infty\), and the complementary log-log is \(\log[-\log(1-0)]=\log(0)=-\infty\). None gives a finite link value for the boundary mean.
The same point can be seen by approaching the boundary. If the predictor is \(\eta=-k\) for positive \(k\), then the logit inverse gives \(1/[1+\exp(k)]\), which tends to zero as \(k\) grows. The probit inverse gives \(\Phi(-k)\), which also tends to zero. The complementary log-log inverse gives \(1-\exp[-\exp(-k)]\), which tends to zero as well. At every finite \(k\), however, each prediction remains positive. Thus a boundary value that maximizes this group's likelihood is distinct from a prediction produced by a finite predictor.
The all-failure observation does not, by itself, prove that a larger regression model has no finite estimates: other covariate patterns can also contribute information. It does illustrate why groups with only one outcome can be a warning sign when a model's fitted predictors become very large in magnitude.
Assumptions, Diagnostics, and Limitations
The boundary arguments above require only that the response mean is a probability and that the stated link and inverse-link formulas are used. To interpret an example as a binomial likelihood calculation, additional assumptions are needed: the trials have a common success probability and are independent conditional on that probability. In a regression model, the mean may vary with covariates; the finite-predictor conclusion applies at a covariate pattern whose predictor is finite.
When fitting a model, inspect extreme fitted values and estimates rather than relying only on printed probabilities. A probability displayed as zero or one may simply be rounded. Useful checks include printing more digits, inspecting the corresponding linear predictors, and checking whether coefficient estimates or standard errors are unusually large. Patterns containing only successes or only failures can be associated with estimates that move toward an infinite predictor, but they are a diagnostic clue, not proof on their own.
The three links share the same open probability range for finite predictors, but their scales and resulting predictions differ. Link choice therefore cannot be justified by the boundary limits alone. Also, the mathematical result assumes finite inputs and coefficients; it does not rule out numerical overflow, underflow, rounding, or an estimation problem that drives fitted predictors toward very large magnitudes. These issues should be distinguished from an inverse link actually assigning a boundary probability at a finite predictor.
A common mistake is to say, “All three links map the real line onto \((0,1)\), so their predictions are interchangeable.” This confuses the link with its inverse and then draws an invalid conclusion. The links map \((0,1)\) onto the real line; their inverse links map the real line onto \((0,1)\). Even though all three inverse links have that range, they can give different probabilities at the same finite predictor.
Check Your Understanding
Answer each question, then check the short solutions below.
- For \(p=0.5\), calculate the logit and probit link values. What does the complementary log-log value show about symmetry around \(p=0.5\)?
- Can a finite predictor give an exact probability of zero under the probit inverse link? Explain.
- A software output reports a complementary log-log fitted probability of \(1.000\). Give two distinct explanations consistent with the link's finite-predictor property.
- For 12 independent trials with a common success probability and zero successes, where is the binomial likelihood maximized? What does each of the three links assign to that boundary mean?
Solutions. (1) The logit is \(\log(0.5/0.5)=0\), and the probit is \(\Phi^{-1}(0.5)=0\). The complementary log-log value is \(\log[-\log(0.5)]=\log(0.6931)\approx-0.3665\), so its value at \(0.5\) is not zero; the link is not symmetric about that mean. (2) No. For every finite \(\eta\), \(\Phi(\eta)>0\); zero is approached only as \(\eta\) tends to negative infinity. (3) The underlying prediction may be less than one but rounded to three decimal places, or numerical computation may have rounded it because the remaining difference from one is too small to represent at the displayed precision. (4) The likelihood \((1-p)^{12}\) is maximized at \(p=0\). Each link is negative infinity at that boundary, not a finite predictor.