Choosing Links For Asymmetric Risk Patterns

Reading the Shape of a Risk Curve

For a binary response, the event probability changes with the predictor while staying between zero and one. A plot of observed event rates against a predictor can reveal whether that change is approximately symmetric, leans toward one tail, or approaches a boundary unusually slowly. Those features can help identify plausible links, provided the plot is read with its sampling uncertainty in mind.

Let \(Y\) be a binary response and let \(p(x)=P(Y=1\mid X=x)\). In a model with one predictor and a linear predictor \(\eta(x)=\beta_0+\beta_1x\), a candidate link \(g\) describes the curve through \(g(p(x))=\eta(x)\). Thus, if the link is suitable and the linear predictor is adequate, plotting \(g\) of the underlying probability against \(x\) should give an approximately straight line. The observed rate in a group is only an estimate of that probability, not the probability itself.

Definition: A binary link is a one-to-one transformation \(g:(0,1)\to\mathbb{R}\) that relates an event probability \(p\) to a linear predictor \(\eta\) by \(g(p)=\eta\). Its inverse \(g^{-1}\) maps linear-predictor values back to probabilities.

The four candidates considered here have these forms. In the table, \(\Phi\) is the standard normal distribution function and \(\arctan\) takes values in \((-\pi/2,\pi/2)\).

Link	Transformation \(g(p)\)	Inverse curve \(g^{-1}(\eta)\)
Probit	\(\Phi^{-1}(p)\)	\(\Phi(\eta)\)
Complementary log-log	\(\log[-\log(1-p)]\)	\(1-\exp[-\exp(\eta)]\)
Log-log	\(-\log[-\log(p)]\)	\(\exp[-\exp(-\eta)]\)
Cauchit	\(\tan[\pi(p-\tfrac12)]\)	\(\tfrac12+\arctan(\eta)/\pi\)

A candidate link is not selected merely because its transformed rates look straight. Grouping a continuous predictor can discard information, and the uncertainty in each rate can be substantial. The useful question is narrower: does the observed curve show a shape or tail pattern that is more compatible with one family than another?

Symmetry and Direction of Asymmetry

A probability curve is symmetric about its midpoint in the following sense: moving the linear predictor by equal amounts above and below a central value gives probabilities equally far from one half, with the two probabilities summing to one. Probit and cauchit inverse links have this property. Complementary log-log and log-log curves instead have opposite orientations of asymmetry.

Theorem: The probit and cauchit inverse links satisfy \(F(-\eta)=1-F(\eta)\) for every real \(\eta\). Consequently, \(F(0)=1/2\), and the probabilities at \(-a\) and \(a\) are equally distant from \(1/2\) for every \(a>0\).

Proof. For the probit link, the standard normal density is even: \(\phi(t)=\phi(-t)\). Therefore the standard normal distribution is symmetric, so \(\Phi(-\eta)=1-\Phi(\eta)\). Setting \(\eta=0\) gives \(\Phi(0)=1/2\). For the cauchit inverse, the identity \(\arctan(-\eta)=-\arctan(\eta)\) gives \(F(-\eta)=\tfrac12-\arctan(\eta)/\pi=1-F(\eta)\). Finally, for either inverse link, this identity implies \(|F(a)-1/2|=|F(-a)-1/2|\). \(\square\)

Symmetry does not mean that probit and cauchit curves have the same steepness or tails. The cauchit curve has heavier tails: it approaches zero and one more slowly than the probit curve. A roughly symmetric observed transition is therefore not enough, by itself, to distinguish these two links.

Theorem: For the complementary log-log inverse link \(F_{\mathrm{cloglog}}(\eta)=1-\exp[-\exp(\eta)]\), as \(\eta\to-\infty\), \(F_{\mathrm{cloglog}}(\eta)\sim \exp(\eta)\), while as \(\eta\to+\infty\), \(1-F_{\mathrm{cloglog}}(\eta)=\exp[-\exp(\eta)]\). For the log-log inverse link \(F_{\mathrm{loglog}}(\eta)=\exp[-\exp(-\eta)]\), as \(\eta\to+\infty\), \(1-F_{\mathrm{loglog}}(\eta)\sim \exp(-\eta)\), while as \(\eta\to-\infty\), \(F_{\mathrm{loglog}}(\eta)=\exp[-\exp(-\eta)]\).

Proof. Put \(t=\exp(\eta)\). For the complementary log-log curve as \(\eta\to-\infty\), \(t\to0\), and \[ \frac{F_{\mathrm{cloglog}}(\eta)}{\exp(\eta)} =\frac{1-\exp(-t)}{t}\longrightarrow 1, \] using the standard limit \((1-\exp(-t))/t\to1\) as \(t\to0\). This proves the lower-tail equivalence. Its upper-tail identity follows directly by subtracting its inverse link from one: \(1-F_{\mathrm{cloglog}}(\eta)=\exp[-\exp(\eta)]\). For log-log, set \(s=\exp(-\eta)\). As \(\eta\to+\infty\), \(s\to0\), and \[ \frac{1-F_{\mathrm{loglog}}(\eta)}{\exp(-\eta)} =\frac{1-\exp(-s)}{s}\longrightarrow1. \] Its lower-tail identity \(F_{\mathrm{loglog}}(\eta)=\exp[-\exp(-\eta)]\) is the definition of its inverse. Thus the two links put the slowly decaying, approximately exponential tail on opposite sides of the transition. \(\square\)

This orientation is important when reading a plot in which risk rises with \(x\). A lower tail that rises gradually from near zero but an upper tail that becomes extremely close to one quickly is compatible with complementary log-log asymmetry. The reversed pattern is compatible with log-log asymmetry. A curve that is broadly balanced around its midpoint is more compatible with a symmetric family, though tail evidence may favor probit over cauchit or vice versa.

Worked Examples of Link Diagnostics

Worked Example: A Complementary Log-Log Pattern

Suppose a risk curve follows \(p(x)=1-\exp[-\exp(-1+x)]\). Evaluate its event rates at \(x=0,1,2\), and check whether the complementary log-log transformation is linear in \(x\).

At \(x=0\), \(\eta=-1\), so \(p(0)=1-\exp[-\exp(-1)]=1-\exp(-0.367879)\approx0.3078\). At \(x=1\), \(\eta=0\), giving \(p(1)=1-\exp(-1)\approx0.6321\). At \(x=2\), \(\eta=1\), giving \(p(2)=1-\exp(-2.718282)\approx0.9340\).

Apply \(g(p)=\log[-\log(1-p)]\). At \(x=1\), for example, \(1-p=\exp(-1)\), so \(g(p)=\log[-\log(\exp(-1))]=\log(1)=0\). In general, substituting the curve into the link gives \(g(p(x))=\log[-\log(\exp[-\exp(-1+x)])]=-1+x\). The transformed values are therefore \(-1,0,1\), exactly linear in \(x\). The increasingly compressed upper end of the probability scale is consistent with the complementary log-log curve.

A numerical check at \(p=0.37\) also illustrates why signs matter: \(-\log[-\log(0.37)]=-\log(0.994252\ldots)\approx+0.005764\), not a negative value. Transformations near the midpoint can be close to zero, so a sign error can materially change how a plotted pattern is read.

Worked Example: Telling Opposite Asymmetries Apart

Consider rates \(0.10, 0.50, 0.90\) at three equally spaced predictor values. Transform each rate under complementary log-log and log-log links. Which side of the transition is stretched on each transformed scale?

For complementary log-log, at \(p=0.10\), \(g(p)=\log[-\log(0.90)]=\log(0.105361)\approx-2.2504\). At \(p=0.50\), \(g(p)=\log[-\log(0.50)]=\log(0.693147)\approx-0.3665\). At \(p=0.90\), \(g(p)=\log[-\log(0.10)]=\log(2.302585)\approx0.8340\).

For log-log, at \(p=0.10\), \(g(p)=-\log[-\log(0.10)]=-\log(2.302585)\approx-0.8340\). At \(p=0.50\), \(g(p)=-\log(0.693147)\approx0.3665\). At \(p=0.90\), \(g(p)=-\log(0.105361)\approx2.2504\).

The transformations place the rates at opposite positions: complementary log-log stretches the lower-risk side more, whereas log-log stretches the higher-risk side more. If a plotted rise is slow near zero and sharp near one, that direction is more compatible with complementary log-log; if it is sharp near zero and gradual near one, log-log is the more natural asymmetric candidate. These three rates alone do not establish either model: the predictor spacing, denominators, and other covariates still matter.

Worked Example: Symmetric Shape, Different Tails

Compare probit and cauchit probabilities at linear-predictor values \(\eta=-2,0,2\). What can a roughly symmetric observed curve tell us?

For probit, \(\Phi(-2)\approx0.0228\), \(\Phi(0)=0.5\), and \(\Phi(2)\approx0.9772\). For cauchit, \(F(0)=0.5\), while \(F(2)=0.5+\arctan(2)/\pi\approx0.8524\) and \(F(-2)=1-F(2)\approx0.1476\).

Both pairs are symmetric around one half: the probabilities at \(-2\) and \(2\) sum to one. But at the same linear-predictor magnitude, the cauchit probabilities are less extreme. Its heavier tails retain more probability away from zero and one. Thus, symmetry supports considering both families; observed tail rates, if measured precisely enough, may help distinguish their shapes. It does not follow that the cauchit is preferable whenever observed rates look less extreme, since the fitted slope also controls how quickly \(\eta\) changes with \(x\).

Using Observed Rates Responsibly

When the predictor is continuous, a rate plot often uses intervals or groups of predictor values. If group \(j\) contains \(m_j\) independent binary observations with a common event probability \(p_j\), its observed rate is \(\hat p_j=y_j/m_j\), where \(y_j\) is the event count. Under these assumptions, its standard error is \(\sqrt{p_j(1-p_j)/m_j}\), estimated in practice by substituting \(\hat p_j\). Rates from small groups are much noisier than rates from large groups, even when they look equally far from a smooth curve.

A transformed rate can be especially unstable near zero or one. For example, the probit, complementary log-log, log-log, and cauchit transformations are defined for rates strictly between zero and one; an observed rate of exactly zero or one maps to an infinite transformed value. Do not silently add an arbitrary constant to make it finite and then treat the result as ordinary evidence. Instead, report the boundary rate with its denominator, use methods appropriate to the underlying binomial observations, and avoid over-interpreting a transformed plot at that boundary.

A practical diagnostic proceeds in stages:

Plot rates with their denominators.
Show the predictor locations and group sizes, and use uncertainty intervals when appropriate. Look first for the broad shape rather than isolated extreme rates.

Describe the asymmetry in the original probability scale.
Ask whether the rise looks roughly balanced, has a gradual lower-risk tail, or has a gradual upper-risk tail.

Check candidate transformed patterns.
For rates away from the boundaries, examine whether a candidate link makes the pattern plausibly linear in the predictor. Account for the different uncertainty of the rates.

Keep the conclusion provisional.
Consider the scientific setting and other predictors, then assess the fitted model with appropriate diagnostics. A visual pattern suggests candidates; it does not prove a link.

In a multiple-predictor model, a raw plot against one covariate can be misleading if other covariates vary systematically with it. The linearity check concerns the conditional probability at fixed values of the other predictors, or an appropriately adjusted display—not necessarily the unadjusted marginal rates. This is one reason to treat plots as diagnostic evidence rather than a link-selection rule.

Common Pitfalls and Practical Interpretation

Calling every asymmetric curve complementary log-log. The direction matters: log-log has the opposite tail orientation. State which side of the transition is gradual.
Equating symmetry with probit. Cauchit is symmetric too. Its heavier tails make probabilities less extreme for a given \(|\eta|\), but the slope and predictor scale also affect the plotted curve.
Reading noisy tail rates as genuine tail behavior. A rate based on a small denominator can be zero or one by chance. Check its sample size and uncertainty before drawing a conclusion.
Treating a straight transformed plot as proof. Grouping, covariate adjustment, sampling variability, and model misspecification can all affect the display. The plot provides a check, not a theorem about the data-generating process.

Key takeaway: Use curve shape to narrow the possibilities: probit and cauchit are symmetric, complementary log-log has a gradual lower-risk tail, and log-log has a gradual upper-risk tail. Tail observations are informative only when their uncertainty and the predictor scale are taken into account.

Check Your Understanding

Use the shape and tail principles from this tutorial to answer the following.

Which two inverse-link curves are symmetric about probability one half, and what does that symmetry mean?
A rising risk curve appears gradual near zero and very compressed near one. Which asymmetric link is the more natural candidate, and why?
What opposite pattern would make log-log asymmetry more plausible than complementary log-log asymmetry?
Why can a zero observed event rate not be directly transformed by the probit link?
Why is a visually straight transformed rate plot evidence for, but not proof of, a particular link model?