Deviance Residual

Deviance Residual in Cost Regressions and Survival Curves

Economic evaluations built on patient-level data lean on two kinds of fitted model, namely survival curves fitted to trial data and extrapolated beyond follow-up, and regressions of costs or resource use. In both, the deviance residual is the usual tool for finding the patients a model describes worst before its estimates feed a cost-effectiveness model. One version is used for generalised linear models and another, a rescaled martingale residual, for survival models, and the two rest on the same idea. Two illustrative examples are worked by hand.

Unit deviances and the deviance residual in a generalised linear model

A generalised linear model (GLM) is fitted by maximum likelihood, and its lack of fit is summarised by the deviance, which compares the fitted model with a saturated model that fits every observation exactly. For the Poisson family, whose dispersion is fixed at 1, the deviance is twice the gap in log-likelihood between the two; for families with an estimated dispersion, such as the gamma, twice that gap equals the deviance divided by the dispersion. The deviance is a sum of one non-negative contribution per observation, the unit deviance, and McCullagh and Nelder, the standard text on GLMs, define the deviance residual from it.

$$r_{D,i} = \mathrm{sign}(y_i - \hat{\mu}_i)\sqrt{d_i}, \qquad D = \sum_{i=1}^{n} d_i = \sum_{i=1}^{n} r_{D,i}^2$$

where $r_{D,i}$ is the deviance residual for observation $i$, $y_i$ is the observed outcome, $\hat{\mu}_i$ is the fitted mean, $d_i$ is the unit deviance of observation $i$, $D$ is the model deviance and $n$ is the number of observations. Because the squares add up to the deviance, the residuals show which observations contribute most to the lack of fit.

The unit deviance depends on the distribution assumed for the outcome. For the gamma family, used for costs, and the Poisson family, used for counts of resource use, Stata's GLM documentation gives the unit deviances below.

$$d_i^{\text{gamma}} = 2\left\{\frac{y_i - \hat{\mu}_i}{\hat{\mu}_i} - \log\frac{y_i}{\hat{\mu}_i}\right\}, \qquad d_i^{\text{Poisson}} = 2\left\{y_i \log\frac{y_i}{\hat{\mu}_i} - (y_i - \hat{\mu}_i)\right\}$$

where $\log$ is the natural logarithm and the other terms are as defined above; the Poisson term $y_i \log(y_i / \hat{\mu}_i)$ is taken as zero when $y_i = 0$, so a zero count contributes $d_i = 2\hat{\mu}_i$. The gamma unit deviance is undefined for a cost of zero, so zero costs are commonly handled in the first part of a two-part model, with the GLM fitted to positive costs in the second part.

Healthcare costs are positively skewed and often heavy tailed, so raw differences $y_i - \hat{\mu}_i$ are dominated by a few expensive patients. The Pearson residual divides the difference by the square root of the variance function, giving $(y_i - \hat{\mu}_i)/\hat{\mu}_i$ for the gamma family, but Stata's manual notes that Pearson residuals often have markedly skewed distributions for non-normal families. Deviance residuals are approximately normal when the model is correct; McCullagh and Nelder recommend them for checking GLM fit, and R returns them by default.

From the martingale residual to the survival deviance residual

Survival models need a different starting point, because a censored time cannot be compared with a fitted mean. The survival version works on the hazard scale instead. The chain starts with the Cox-Snell residual, the fitted cumulative hazard at each patient's observed time, and the martingale residual is the event indicator minus that value.

$$M_i = \delta_i - \hat{H}(t_i \mid x_i)$$

where $M_i$ is the martingale residual of patient $i$, $\delta_i$ is 1 if the patient had the event and 0 if censored, $t_i$ is the observed event or censoring time, $x_i$ are the patient's covariates and $\hat{H}$ is the fitted cumulative hazard. The martingale residual compares observed with expected events, but it cannot exceed 1 and has no lower bound, so it is skewed and its plots are sometimes difficult to interpret. Therneau, Grambsch and Fleming, borrowing the GLM idea, proposed in 1990 a transformation for the Cox model that makes it roughly symmetric about zero.

$$r_{D,i} = \mathrm{sign}(M_i)\sqrt{-2\left\{M_i + \delta_i \log(\delta_i - M_i)\right\}}$$

where $r_{D,i}$ is the survival deviance residual and the other terms are as defined above, with $\delta_i \log(\delta_i - M_i)$ taken as zero for a censored patient. For a censored patient the formula reduces to $-\sqrt{2\hat{H}(t_i \mid x_i)}$, so censored patients always have negative residuals, larger in size the more hazard they accumulated before censoring. A positive residual marks a patient who had the event before the model expected even one event, that is with $\hat{H}(t_i \mid x_i) < 1$.

The link with the GLM version is exact. Treating $\delta_i$ as a Poisson count with mean $\hat{H}(t_i \mid x_i)$, and substituting $y_i = \delta_i$ and $\hat{\mu}_i = \delta_i - M_i$ into the Poisson unit deviance, gives the term under the square root, because $\delta_i \log \delta_i$ is zero whether $\delta_i$ is 0 or 1. Stata computes the same residual after Cox proportional hazards models and extends it to parametric survival models, where it calls the underlying residuals martingale-like.

Worked example with four patients and an exponential survival curve

The figures in this example are illustrative. Four patients in one trial arm are followed for 1, 3, 5 and 11 months; the patient followed for 5 months is censored and the other three die. An exponential model is fitted, and its maximum likelihood estimate of the constant hazard is the number of deaths divided by the total time at risk.

$$\hat{\lambda} = \frac{3}{1 + 3 + 5 + 11} = \frac{3}{20} = 0.15$$

where $\hat{\lambda}$ is the estimated monthly hazard. Under this model $\hat{H}(t) = 0.15\,t$, so each martingale residual is $\delta_i - 0.15\,t_i$. For the patient who died at month 1, $M = 0.85$ and the deviance residual follows.

$$r_D = +\sqrt{-2 \times \{0.85 + \log(0.15)\}} = +\sqrt{-2 \times (0.85 - 1.8971)} = +\sqrt{2.0942} = 1.447$$

where $\log(0.15) = -1.8971$. The table repeats the steps for all four patients.

1. Check the martingale residuals. They sum to zero, ${0.85 + 0.55 - 0.75 - 0.65 = 0}$, a property of the exponential maximum likelihood fit, in which the deaths equal $\hat{\lambda}$ times the total time at risk.

2. Read the signs. Patients 1 and 2 died while their fitted cumulative hazards were still below 1 (0.15 and 0.45), so their residuals are positive; patient 1, who died when the model expected only 0.15 events, has the largest. Patient 4 died only after accumulating a cumulative hazard of 1.65, so the residual is negative, and patient 3 is negative because censored patients always are.

3. Compare the two scales. On the martingale scale patient 4 (-0.65) lies further from zero than patient 2 (+0.55). On the deviance scale the order reverses, with the long survivor pulled in to -0.546 and the early death pushed out to +0.705, which is the symmetrising effect.

4. Add up the squares. The squared residuals sum to about 4.39, ${2.0942 + 0.4970 + 1.5000 + 0.2984 = 4.3896}$, the deviance of the exponential model against a model giving each patient a separate hazard. With four patients nothing here signals misfit; the example shows the mechanics only.

Worked example with three annual costs under a gamma model

The figures here are also illustrative. Three patients have annual costs of £1,000, £2,000 and £6,000, and a gamma GLM with a log link and only an intercept is fitted, so the fitted mean is the sample mean, £3,000, for every patient. For the patient costing £6,000 the unit deviance is calculated as follows.

$$d = 2 \times \left\{\frac{6000 - 3000}{3000} - \log\frac{6000}{3000}\right\} = 2 \times (1 - 0.6931) = 0.6137$$

where $d$ is the gamma unit deviance and $\log 2 = 0.6931$, so the deviance residual is $+\sqrt{0.6137} = +0.783$. The same steps give the other two rows of the table.

The Pearson residuals rank the expensive patient C as the most unusual, while the deviance residuals rank patient A first. The Pearson residual measures the gap in proportion to the mean, so twice the mean (+1.000) looks further out than one third of it (-0.667). The $-\log(y_i/\hat{\mu}_i)$ term in the gamma deviance judges the ratio of observed to fitted cost on roughly a log scale, so a factor of 3 below the mean ($\log 3 = 1.10$) outweighs a factor of 2 above it ($\log 2 = 0.69$). The Pearson residuals sum to zero, as they must for an intercept-only fit, whereas the deviance residuals sum to about -0.526, so a deviance residual plot need not centre exactly on zero. The model deviance is 1.622, the sum of the three unit deviances.

Reading deviance residual plots

In practice the residuals are plotted, and the horizontal axis decides what a plot can show. Stata's survival documentation suggests plotting them against the linear predictor, survival time, rank order of survival or observation number to find aberrant observations; with uncensored data a well-fitting model gives residuals that resemble random noise, while censored patients appear as clumps near zero.

A point far from zero marks an outlier, a patient the model fits poorly. For a curve feeding a partitioned survival model, a systematic trend in the residuals against time is informative, since positive residuals concentrated among early deaths suggest the curve understates early mortality. Large negative values among censored long survivors carry little information on their own, because a censored residual is $-\sqrt{2\hat{H}}$ and grows in size with follow-up by construction. Xue and Schifano's review of Cox model diagnostics uses ±1.96 as a rough threshold while warning that normal-based thresholds may not suit highly censored data. Poor fit also differs from influence. Stata lists likelihood displacement values, LMAX values and DFBETAs as measures of influence, so a residual plot is read alongside one of them.

Where residual checks sit in NICE survival guidance

NICE Decision Support Unit Technical Support Document 14 (TSD 14), written by Latimer in 2011 and last updated in March 2013, guides the choice of survival models for economic evaluations using patient-level trial data, and its place in that process is set out on the Cox-Snell residual page. TSD 14 does not mention deviance residuals; its only use of the word deviance refers to the deviance information criterion, a different quantity. Its review conclusions and the first step of its model selection process accept suitable residual plots in place of log-cumulative hazard plots, and its review of past appraisals records martingale residuals among the tests used in one appraisal (TA174).

TSD 14 also bears on what follows when a residual flags a patient. Its review conclusions state that data points should be excluded only when they can be clearly shown to be erroneous outliers, and that unless a very clear rationale is offered all data should be included. A large deviance residual is therefore a reason to check a record and the model, and does not by itself justify removing a patient.

Limits and common misreadings

Three properties of the deviance residual limit how far a plot can be trusted. They matter most for immature survival data and skewed costs.

The normal approximation fails with heavy censoring. Xue and Schifano report Therneau and colleagues' finding that the survival deviance residual is close to normal when censoring is light, but that with heavy censoring too many values lie near zero for normality, although the residuals remain symmetrised, and they cite simulations by Nardi and Schemper in which censoring often made the distribution bimodal. Normal cut-offs are therefore unreliable when survival data are immature. Stata's manual adds that the residuals are expected to be symmetric about zero but need not sum to zero.

The residual is designed to find outliers and says little about model form. Stata's documentation assigns deviance residuals to model accuracy and outliers, and martingale residuals to the functional form of covariates, while the proportional hazards assumption and overall goodness of fit have their own residual checks. A clean deviance residual plot does not show that hazards are proportional or that the chosen distribution is right.

Good fit within the data is a separate question from good decisions. Mihaylova and colleagues, reviewing methods for cost and resource use data, note that further parametric models outside the GLM family aim to fit better, for example in terms of deviance, without studies confirming more reliable inferences, possibly because they overfit extreme observations. In the same way, TSD 14 stresses that checks of fit to trial data address internal validity only, so a survival model with small residuals throughout follow-up can still extrapolate implausibly.

Sources

• Latimer N. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2011, last updated March 2013. Sections 3.4, 3.5, 4.2, 4.6, 5 (conclusions 4 and 8) and 6. https://doi.org/10.15131/shef.data.33745843
• McCullagh P, Nelder JA. Generalized Linear Models. 2nd ed. London: Chapman and Hall; 1989.
• Mihaylova B, Briggs A, O'Hagan A, Thompson SG. Review of statistical methods for analysing healthcare resources and costs. Health Economics. 2011;20(8):897-916. https://doi.org/10.1002/hec.1653
• R Core Team. glm.summaries: Accessing generalized linear model fits. R package stats, version 4.6.0. Accessed 2 October 2026. https://stat.ethz.ch/R-manual/R-devel/library/stats/html/glm.summaries.html
• StataCorp. Stata Base Reference Manual, Release 19. Entries glm and glm postestimation. College Station, TX: Stata Press; 2025. https://www.stata.com/manuals/rglm.pdf, https://www.stata.com/manuals/rglmpostestimation.pdf
• StataCorp. Stata Survival Analysis Reference Manual, Release 19. Entries stcox postestimation and streg postestimation. College Station, TX: Stata Press; 2025. https://www.stata.com/manuals/ststcoxpostestimation.pdf, https://www.stata.com/manuals/ststregpostestimation.pdf
• Therneau TM, Grambsch PM, Fleming TR. Martingale-based residuals for survival models. Biometrika. 1990;77(1):147-160. https://doi.org/10.1093/biomet/77.1.147
• Xue Y, Schifano ED. Diagnostics for the Cox model. Communications for Statistical Applications and Methods. 2017;24(6):583-604. https://doi.org/10.29220/CSAM.2017.24.6.583