Deviance Residuals For Subject-Level Survival Fit

Why Transform Martingale Residuals?

A martingale residual for a subject’s event history is the observed event indicator minus the fitted cumulative hazard at that subject’s observed follow-up time. It is useful because its sign distinguishes observed event experience from fitted event experience. But the residual is bounded above by 1, while its negative values can extend far below zero. This asymmetry makes a large negative residual harder to compare directly with a positive one.

A deviance residual applies a signed square-root transformation to the martingale residual. The transformation is intended to make unusually poorly fitted event histories easier to see and compare on a common diagnostic scale. It does not turn the residual into a normal score, a hypothesis test, or a measure of causal effect.

Definition and Conditions

For subject \(i\), let \(T_i\) be the observed follow-up time and let \(\delta_i\) equal 1 if the event was observed by \(T_i\), and 0 if the subject was right-censored at \(T_i\). Let \(\hat h_i(t)\) be the fitted subject-level event hazard, and define the fitted cumulative hazard by integrating it through follow-up:

In a joint model, \(\hat h_i(t)\) may depend on baseline covariates and on the fitted longitudinal trajectory or subject-specific random effects. The calculation must use a clearly specified subject-level fit consistently across subjects—for example, a plug-in fit using the same rule for estimating the latent subject-specific quantities.

The formula requires \(\hat H_i(T_i)\geq 0\), as a cumulative hazard must be. For the usual single-event right-censored setting, \(\delta_i\) is either 0 or 1. The survival-model interpretation also relies on a suitable fitted hazard model and on censoring being non-informative conditional on the variables and latent quantities included in the model. If censoring depends on unaccounted-for event risk, residual patterns can reflect that problem rather than only hazard-model fit.

What the Transformation Does

Since \(M_i=\delta_i-\hat H_i(T_i)\), the quantity \(\delta_i-M_i\) in the logarithm is exactly \(\hat H_i(T_i)\). Substitution gives a form that is convenient for calculation:

When \(\delta_i=0\), the logarithmic term is taken as zero, so \(D_i=-\sqrt{2\hat H_i(T_i)}\). A censored subject’s residual is therefore negative under this definition. When \(\delta_i=1\), the result is \(D_i=\operatorname{sign}\{1-\hat H_i(T_i)\}\sqrt{2\{\hat H_i(T_i)-1-\log\hat H_i(T_i)\}}\). An observed event with fitted cumulative hazard below 1 has a positive residual; an event with fitted cumulative hazard above 1 has a negative residual.

Justification and Key Properties

The expression inside the square root is nonnegative. For an observed event, put \(x=\hat H_i(T_i)>0\). The unsigned quantity is \(2(x-1-\log x)\). The function \(g(x)=x-1-\log x\) has derivative \(g'(x)=1-1/x\), which is negative for \(0<x<1\), zero at \(x=1\), and positive for \(x>1\). Thus \(g\) has its minimum \(g(1)=0\), so \(g(x)\geq 0\). For a censored subject, the unsigned quantity is \(2\hat H_i(T_i)\geq 0\).

The same expression can be written in terms of \(M_i\). It has the form of a signed square root of a Poisson-style deviance, with the fitted cumulative hazard playing the role of an expected count and \(\delta_i\) the observed event indicator. This motivates the transformation; it does not mean that a censored survival record is literally an independent Poisson observation or that the resulting residuals have an exact standard normal distribution.

The sign preserves whether the observed event indicator is above or below the fitted cumulative hazard. The square root changes the scale nonlinearly, so residual magnitudes are often easier to compare than raw martingale residual magnitudes. The transformation does not remove dependence among residuals created by estimating shared model parameters or by shared latent structure.

Using Residuals as Diagnostics

A deviance residual is most useful as a subject-level screening display: examine its distribution and inspect subjects with relatively large absolute values against their event status, follow-up time, covariates, and fitted trajectory. A pattern across subjects can be more informative than a single extreme value. For example, residuals that vary systematically with a covariate or with follow-up may indicate a feature of the fitted event model that warrants further checking.

The earlier material on martingale residuals for the event component covers the raw residual’s observed-minus-fitted construction. The deviance residual is an alternative display of that information: it changes the scale but does not add new event-history data. A raw martingale residual is direct to interpret in its original units of event indicator minus cumulative hazard; the deviance residual is often more convenient for comparing unusual values. Neither should be treated as an independent standard normal sample.

Common Mistakes and Limitations

• Calling the residual a z-score. A deviance residual is not generally standard normal, so a rule such as “absolute value above 2 means a significant outlier” is not justified without a separately justified reference procedure.
• Dropping the sign. The square-root magnitude alone loses whether the observed event indicator was above or below the fitted cumulative hazard. Retain the sign in the definition.
• Using the wrong cumulative hazard. Substituting a baseline cumulative hazard, or a marginal quantity, when the intended fit is subject-specific changes the residual. State which fitted hazard is being evaluated.
• Ignoring censoring. A censored subject has \(\delta_i=0\), not an observed event. Its residual is \(-\sqrt{2\hat H_i(T_i)}\); it must not be calculated with the event formula.
• Over-interpreting an extreme value. A large residual can prompt data review and model checks, but it is not by itself evidence of a recording error, a particular cause, or a failed model.

The plug-in calculation also treats estimated latent quantities as fixed when forming the residual. In a joint model, uncertainty in the longitudinal trajectory, random effects, baseline hazard, and other estimated parameters can affect residual comparisons. Deviance residuals provide a compact diagnostic view, not a complete account of that uncertainty. Their interpretation can also be misleading if the event model is poorly specified or the censoring assumptions fail.