Abstract
Background
The discriminative ability of a risk model is often measured by Harrell’s concordanceindex (cindex). The cindex estimates for two randomly chosen subjects the probability that the model predicts a higher risk for the subject with poorer outcome (concordance probability). When data are clustered, as in multicenter data, two types of concordance are distinguished: concordance in subjects from the same cluster (withincluster concordance probability) and concordance in subjects from different clusters (betweencluster concordance probability). We argue that the withincluster concordance probability is most relevant when a risk model supports decisions within clusters (e.g. who should be treated in a particular center). We aimed to explore different approaches to estimate the withincluster concordance probability in clustered data.
Methods
We used data of the CRASH trial (2,081 patients clustered in 35 centers) to develop a risk model for mortality after traumatic brain injury. To assess the discriminative ability of the risk model within centers we first calculated clusterspecific cindexes. We then pooled the clusterspecific cindexes into a summary estimate with different metaanalytical techniques. We considered fixed effect metaanalysis with different weights (equal; inverse variance; number of subjects, events or pairs) and random effects metaanalysis. We reflected on pooling the estimates on the logodds scale rather than the probability scale.
Results
The clusterspecific cindex varied substantially across centers (IQR = 0.700.81; I^{2} = 0.76 with 95% confidence interval 0.66 to 0.82). Summary estimates resulting from fixed effect metaanalysis ranged from 0.75 (equal weights) to 0.84 (inverse variance weights). With random effects metaanalysis – accounting for the observed heterogeneity in cindexes across clusters – we estimated a mean of 0.77, a betweencluster variance of 0.0072 and a 95% prediction interval of 0.60 to 0.95. The normality assumptions for derivation of a prediction interval were better met on the probability than on the logodds scale.
Conclusion
When assessing the discriminative ability of risk models used to support decisions at cluster level we recommend metaanalysis of clusterspecific cindexes. Particularly, random effects metaanalysis should be considered.
Keywords:
Clustered data; Concordance; Discrimination; Metaanalysis; Prediction; Risk modelBackground
Assessing the performance of a risk model is of great practical importance. An essential aspect of model performance is separating subjects with good outcome from subjects with poor outcome (discrimination) [1]. The concordance probability is a commonly used measure of discrimination reflecting the association between model predictions and true outcomes [2,3]. For binary outcome data it is the probability that a randomly chosen subject from the event group has a higher predicted probability of having an event than a randomly chosen subject from the nonevent group. For timetoevent outcome data it is the probability that, for a randomly chosen pair of subjects, the subject who experiences the event of interest earlier in time has a lower predicted value of the time to the occurrence of the event. For both kinds of outcome data the concordance probability is often estimated with Harrell’s concordance (c)index [2].
In risk modelling, clustered data are frequently used. A typical example is multicenter patient data, i.e. data of patients who are treated in different centers with similar inclusion criteria across the centers. Patients treated in the same center are nevertheless more alike than patients from different centers. A comparable type of clustering may occur in patients treated in different countries or in patients treated by different caregivers in the same center. Similarly, in public health research the study population is often clustered in geographical regions like countries, municipalities or neighbourhoods. It has been suggested that clustering should be taken into account in the development of risk models to obtain unbiased estimates of predictor effects [4]. This can be done by using a multilevel logistic regression model for binary outcomes or a frailty model for timetoevent outcomes [5,6].
It would be natural to take clustering also into account when measuring the performance of a risk model. For multilevel models, it has been proposed to consider the concordance probability of subjects within the same cluster (withincluster concordance probability) separately from the concordance probability of subjects in different clusters (betweencluster concordance probability) [7,8]. We propose using the withincluster concordance probability when risk models are used to support decisions within clusters, e.g. in clinical practice where decisions on interventions are commonly taken within centers. A valuable risk model should then be able to separate subjects within the same cluster into those with good outcome and poor outcome. We consider the withincluster concordance probability more relevant in this context than the betweencluster or overall concordance probability.
Here, we aimed to estimate the withincluster concordance probability from clustered data. We explored different metaanalytic methods for pooling clusterspecific concordance probability estimates with an illustration in predicting mortality among patients suffering from traumatic brain injury.
Methods
Mortality in traumatic brain injury patients
We present a case study of predicting mortality after Traumatic Brain Injury (TBI). Risk models using baseline characteristics provide adequate discrimination between patients with good and poor 6month outcomes after TBI [9,10]. We used patients enrolled in the Medical Research Council Corticosteroid Randomisation after Significant Head Injury [11] trial (registration ISRCTN74459797, http://www.controlledtrials.com/ webcite), who were recruited between 1999 and 2004. This was a large international doubleblind, randomized placebocontrolled trial of the effect of early administration of a 48h infusion of methylprednisolone on outcome after head injury. The trial included 10,008 adults clustered in 239 centers with Glasgow Coma Scale (GCS) [12] Total Score ≤ 14, who were enrolled within 8 hours after injury. By design the patient inclusion criteria were equal in all 239 centers.
We considered patients with moderate or severe brain injury (GCS Total Score ≤ 12) and observed 6month Glasgow Outcome Scale (GOS) [13]. Patients who were treated in one of 35 European centers with more than 5 patients experiencing the event (n = 2,081), were used to assess the discriminative ability of a prediction model developed with data from 35 centers. Patients who were treated in one of 21 Asian centers with more than 5 patients experiencing the event (n = 1,421) were used to assess the discriminative ability at external validation.
We used a Cox proportional hazards model with age, GCS Motor Score and pupil reactivity as covariates similar to previously developed risk models [9,10]. We modelled center with a Gamma frailty (random effect) to account for heterogeneity in mortality among centers. We estimated parameters on the European selection of patients with the R package survival [14,15]. As center effect estimates are unavailable when using a risk model in new centers, we calculated individual risk predictions applying the Gamma frailty mean of 1 for each patient.
Clusterspecific concordance probabilities
We estimated the concordance probability within each cluster by Harrell’s cindex [2], i.e. the proportion of all usable pairs of subjects in which the predictions are concordant with the outcomes. A pair of subjects is usable if we can determine the ordering of their outcomes. For binary outcomes, pairs of subjects are usable if one of the subjects had an event and the other did not. For timetoevent outcomes, pairs of subjects are usable if their failure times are not equal and at least the smallest failure time is uncensored. For a usable subject pair the predictions are concordant with the outcomes if the ordering of the predictions is equal to the ordering of the outcomes. Values of the cindex close to 0.5 indicate that the model does not perform much better than a coinflip in predicting which subject of a randomly chosen pair will have a better outcome. Values of the cindex near 1 indicate that the model is almost perfectly able to predict which subject of a randomly chosen pair will have a favourable outcome. We estimated the variances of the clusterspecific cindexes with a method proposed by Quade [16]. Formulas are provided in Appendix 1.
Pooling clusterspecific concordance probability estimates
The withincluster concordance probability C_{w} can be estimated by pooling the clusterspecific concordance probability estimates into a weighted average. Previously, the clusterspecific concordance probability estimates were pooled with the number of usable subject pairs as weights [7,8]. Here, we define eight different ways for pooling of clusterspecific estimates – both on the probability scale and on the logodds scale – based on fixed effect metaanalysis and random effects metaanalysis.
We consider a dataset with subjects in K clusters. Let m_{k} be the number of subjects and e_{k} be the number of events in cluster k. We denote the number of usable subject pairs – pairs of subjects for whom we can determine the ordering of their outcomes – in cluster k by n_{k}. The clusterspecific concordance probability estimate for cluster k is denoted by with sampling variance estimate .
Fixed effect metaanalysis
Fixed effect metaanalysis assumes that one common withincluster concordance probability C_{W} exists that applies to all clusters. The observed clusterspecific estimates vary only because of chance created from sampling subjects. Fixed effect metaanalysis with cluster weights w_{k} results in:
The simplest approach would be to apply equal weights, w_{k} = 1/K for each cluster (method 1). This estimator is quite naive when the cluster size varies, because small clusters are given the same weight as large clusters and information about the precision of the clusterspecific estimates is ignored. Heuristic choices of weights taking the cluster size into account are the number of subjects, w_{k} = m_{k} (method 2), or the number of events, w_{k} = e_{k} (method 3). Analogous to the definition of the cindex a fourth option is the number of usable subject pairs as weights, w_{k} = n_{k} (method 4). The pooled estimate is then equal to the proportion of all usable withincluster subject pairs in which the predictions and outcomes are concordant. Another choice of metaanalysis weights are the inverse variances, (method 5). These weights express the precision of the clusterspecific estimates and are commonly used in metaanalysis of studyspecific treatment effects.
Random effects metaanalysis
In our context a random effects metaanalysis considers that the clusterspecific estimates vary not only because of sampling variability but also because of differences in true concordance probabilities. This is appropriate for high values of I^{2}[17]. I^{2} measures the proportion of variability in clusterspecific estimates that is due to betweencluster heterogeneity rather than chance. Random effects metaanalysis assumes that clusterspecific concordance probabilities C_{k} are distributed about mean μ with betweencluster variance τ^{2}, with the observed normally distributed about C_{k} with sampling variance . The mean withincluster concordance probability estimate is the average of the clusterspecific estimates with the inverse variances as weights (method 6):
For estimation of the betweencluster variance τ^{2} we used the DerSimonian and Laird [18] method. Alternative estimators for τ^{2} can be found in DerSimonian and Kacker [19].
With the additional assumption of normally distributed C_{k} we can derive a prediction interval for the withincluster concordance probability C_{W} in a new or unspecified cluster [20]. If τ^{2} were known, then and C_{W} ~ N(μ, τ^{2}) imply (assuming independence of C_{W} and given μ ) that . Hence the withincluster concordance probability C_{W} in a new cluster is normally distributed, with mean and variance (Figure 1). Since τ^{2} is estimated, we assume to take a more conservative tdistribution with K  2 degrees of freedom instead of the standard normal distribution [20]. Thus, a 95% prediction interval of the withincluster concordance probability C_{W} in an unspecified cluster can be approximated by: with denoting the 97.5% percentile of the tdistribution with K  2 degrees of freedom.
Figure 1. Example of random effects metaanalysis of concordance probability estimates in 7 clusters. Clusterspecific estimates are in grey. Under the assumption of normally distributed clusterspecific concordance probabilities, the predictive distribution resulting from a random effects metaanalysis is in black. The mean, the mean ± one standard deviation and the 2.5 and 97.5 percentiles of the predictive distribution are plotted with vertical lines.
Metaanalysis scale
When calculating a prediction interval of the withincluster concordance probability C_{W}, Riley et al [21] advised to perform a random effects metaanalysis on a scale that helps meet the normality assumption for the random effects. When the normality assumption of the random effects model holds, the C_{k} are normally distributed with mean μ and variance . As a consequence, the standardized residuals z_{k} defined below should approximately have a standard normal distribution:
To consider if the normality assumption is valid we used a normal probability plot of z_{k} and applied the ShapiroWilk test to z_{k}[22]. In a normal probability plot z_{k} is plotted against a theoretical normal distribution in such a way that the points should form an approximate straight line. Departures from this straight line indicate departures from normality. The ShapiroWilk test returns the probability of obtaining the teststatistic as least as extreme as the observed one, under the nullhypothesis that z_{k} are normally distributed (pvalue). When the pvalue is above significance level α, say 5%, the null hypothesis that z_{k} is normally distributed is not rejected.
Since the concordance probability is restricted to [0, 1] the normality assumption of random effects metaanalysis may be violated. We considered inverse variance weighted metaanalysis on the logodds scale as an alternative approach (methods 7 and 8 for fixed effect and random effects metaanalysis respectively). The resulting estimators for the withincluster concordance probability are defined in Appendix 2. The normality assumption on logodds scale was again assessed by the normal probability plot and the ShapiroWilk test.
Table 1 contains a summary of the eight pooling methodologies described above. For all the metaanalyses we used the R package rmeta [14,23].
Table 1. Overview of the 8 methods for pooling of clusterspecific concordance probability estimates
Results
The European patients were slightly older in comparison with the Asian patients (median age 36 vs. 31 years) and were more likely to have the worst GCS Motor Score of 1, i.e. no motor response (21% versus 4%) compared to the Asian patients (Table 2). However, 6 month mortality was lower in the European patients (27%) than in the Asian patients (35%).
Table 2. Patient characteristics in selected European and Asian centers
We found that 6month mortality was clearly associated with higher age, worse GCS Motor Score and less pupil reactivity (Table 3). Heterogeneity in mortality among European centers was substantial as indicated by the hazard ratio of 1.7 for the 75 percentile versus the 25 percentile of the random center effect, based on the quartiles of the Gamma frailty distribution with mean 1 and variance estimate 0.146.
Table 3. Associations between predictors and 6month mortality in European centers
Among European centers (overall cindex 0.80) the cindexes varied substantially with an interquartile range of 0.70 to 0.81 (Figure 2). Pooled concordance probability estimates resulting from fixed effect metaanalysis ranged from 0.75 (equal weights) to 0.84 (inverse variance weights). Random effects metaanalysis (method 6) led to a mean concordance probability estimate , a betweencluster variance estimate and a wide 95% prediction interval (0.60 to 0.95) reflecting the strong heterogeneity in the clusterspecific concordance probabilities (I^{2} = 0.76 with 95% confidence interval 0.66 to 0.82). Random effects metaanalysis on logodds scale (method 8) led to similar results, but with a somewhat smaller asymmetric prediction interval (0.58 to 0.89).
Figure 2. Centerspecific and pooled concordance probability estimates with 95% confidence intervals for European centers. For pooled estimates based on random effects metaanalysis a 95% prediction interval for the concordance probability is presented by a horizontal line. 1 Equal = Fixed effect metaanalysis with equal weights; 2 Patients = Fixed effect metaanalysis with number of patients as weights; 3 Events = Fixed effect metaanalysis with number of events as weights; 4 Pairs = Fixed effect metaanalysis with number of usable patient pairs as weights; 5 Inv var FE = Fixed effect metaanalysis with inverse variance weights; 6 Inv var RE = Random effects metaanalysis with inverse variance weights; 7 Inv var FE logit = Fixed effect metaanalysis with inverse variance weights on logodds scale; 8 Inv var RE logit = Random effects metaanalysis with inverse variance weights on logodds scale.
Large differences in pooling weights, together with heterogeneity in the clusterspecific concordance probabilities, led to very different pooled estimates. We analysed the pooling weights to explain the differences in pooled estimates (Figure 3). The patientweighted estimate was dominated by center 2 with 494 of the 2,081 patients. The eventweighted estimate was dominated by center 12 with 107 out of 553 events. The patientpairweighted estimate was heavily determined by both center 2 and center 12 as the number of usable patient pairs is related to the number of patients times the number of events. The fixed effect inversevariance weighted estimate was also strongly influenced by centers with high number of patients or events, because the standard errors of the clusterspecific estimates depend heavily on the number of patients and events. Furthermore, the fixed effect inversevariance weighted estimate was upwardly influenced by center 1 as a result of the small standard error relative to the small number of patients and events. The random effects inversevariance weighted estimate was much less dominated by particular centers and close to the equally weighted estimate because of the large amount of heterogeneity. The standard error on the logodds scale increased with increasing cindex according to Equation 10 in Appendix 2 and therefore put less weight on the centers with a high concordance probability estimate resulting in lower pooled estimates. The large standard errors for centers with high cindex also decreased the heterogeneity (I^{2} = 0.61 with 95% confidence interval 0.44 to 0.73) on the logodds scale resulting in more similar weights for fixed effect and random effects metaanalysis.
Figure 3. Metaanalysis pooling weights for European centers. For methods 1 to 8 the weights are represented by the height of the bars on the right hand side of the Figure. Cindexes are printed in the bars if the cluster weight was at least equal to the average weight. 1 Equal = Fixed effect metaanalysis with equal weights; 2 Patients = Fixed effect metaanalysis with number of patients as weights; 3 Events = Fixed effect metaanalysis with number of events as weights; 4 Pairs = Fixed effect metaanalysis with number of usable patient pairs as weights; 5 Inv var FE = Fixed effect metaanalysis with inverse variance weights; 6 Inv var RE = Random effects metaanalysis with inverse variance weights; 7 Inv var FE logit = Fixed effect metaanalysis with inverse variance weights on logodds scale; 8 Inv var RE logit = Random effects metaanalysis with inverse variance weights on logodds scale.
To check the validity of the normality assumption in the random effects metaanalyses, we calculated standardized residuals (Equation 3), both on the probability and the logodds scale. The standardized residuals better fitted to the standard normal distribution on the probability scale than on the logodds scale (Figure 4, pvalues for rejection of the normality null hypothesis of 0.666 on probability scale and of 0.030 on logodds scale).
Figure 4. Normal probability plot of standardized residuals on probability scale and logodds scale (European centers). ShapiroWilk test results are printed at the bottom of each plot.
To illustrate the comparison in an external validation setting, we repeated the analysis of the withincluster concordance probability in Asian centers with the same risk model (Figure 5). Among Asian clusters (overall cindex 0.74) the cindexes varied less (IQR 0.710.78), which was reflected in a lower proportion of variation among clusters that is due to heterogeneity rather than chance (I^{2} = 0.32 with 95% confidence interval 0 to 0.60). As a result, different pooling methodologies led to more similar pooled estimates, because differences in cluster weights have less impact when clusterspecific estimates are more alike. Based on random effects metaanalysis, estimates of the mean withincluster concordance probability and the betweencluster variance were and respectively. The resulting prediction interval (0.67 to 0.83) was much smaller than for the European clusters. The heterogeneity disappeared on the logodds scale (I^{2} = 0) leading to equal estimates by fixed effect and random effects metaanalysis.
Figure 5. Centerspecific and pooled concordance probability estimates with 95% confidence intervals for Asian centers (external validation). For pooled estimates based on random effects metaanalysis (methods 6 and 8) a 95% prediction interval for the concordance probability is presented by a horizontal line. 1 Equal = Fixed effect metaanalysis with equal weights; 2 Patients = Fixed effect metaanalysis with number of patients as weights; 3 Events = Fixed effect metaanalysis with number of events as weights; 4 Pairs = Fixed effect metaanalysis with number of usable patient pairs as weights; 5 Inv var FE = Fixed effect metaanalysis with inverse variance weights; 6 Inv var RE = Random effects metaanalysis with inverse variance weights; 7 Inv var FE logit = Fixed effect metaanalysis with inverse variance weights on logodds scale; 8 Inv var RE logit = Random effects metaanalysis with inverse variance weights on logodds scale.
Discussion
We studied how to assess the discriminative ability of risk models in clustered data. The withincluster concordance probability is an important measure for risk models when these models are used to support decisions on interventions within the clusters. The withincluster concordance probability can be estimated by pooling clusterspecific concordance probability estimates (e.g. cindexes) with a metaanalysis, similar to pooling of studyspecific treatment effect estimates. We considered different pooling strategies (Table 1) and recommend random effects metaanalysis in case of substantial variability – beyond chance – of the concordance probability across clusters [20,21]. To decide if the metaanalysis should be undertaken on the probability scale or the logodds scale we suggest considering the normality assumptions on both scales by normal probability plots and ShapiroWilk tests of the standardized residuals.
The illustration of predicting 6month mortality after TBI prompted the use of random effects metaanalysis because of the strong difference – beyond chance – in concordance probability among centers. This was clearly visualized by the forest plot in Figure 2. Random effects metaanalysis results can be summarized by the mean concordance probability and a 95% prediction interval for possible values of the concordance probability. By definition, these results give insight into the variation of the discriminative ability among centers as opposed to fixed effect metaanalysis results [20,21]. By comparing normal probability plots and ShapiroWilk test results based on the standardized residuals we concluded the random effects metaanalysis results on probability scale to be the most appropriate (Figure 4). Although the methodology is illustrated with timetoevent outcomes of traumatic brain injury patients, it is also applicable to binary outcomes.
Even if a risk model contains regression coefficients that are optimal for the data in each cluster, differences in case mix may lead to different concordance probabilities across clusters [24]. Furthermore, predictor effects may vary because of clusterspecific circumstances, also leading to different clusterspecific concordance probabilities. Given the variability beyond chance in our case study, we consider a random effects metaanalysis of the clusterspecific cindexes as most appropriate.
The assumption of random effects metaanalysis is that underlying concordance probabilities among clusters are exchangeable, i.e. clusterspecific concordance probabilities are expected to be nonidentical, yet identically distributed [20]. If part of the variation can be explained by cluster characteristics, a metaregression – assuming partial exchangeability – of the concordance probability estimates with cluster characteristics as covariates is preferable.
We chose to analyse the concordance probability as it is the most commonly used measure of discriminative ability of a risk model. However, the same logic of pooling clusterspecific performance measure estimates can be applied to any other performance measure, like the discrimination slope, the explained variation (R^{2}) or the Brier score [25].
We used Harrell’s cindex to estimate clusterspecific concordance probabilities together with Quade’s formula for the clusterspecific variances of the cindex [2,16]. The same methodology of pooling clusterspecific performance measure estimates can be applied to other concordance probability estimators and its variances. Other estimators for the concordance probability in timetoevent data can be found in Gönen and Heller [26] and Uno et al [27]. These estimators are especially favourable when censoring varies by cluster as they are shown to be less sensitive to censoring distributions. Other variance estimators are described by Hanley and McNeil [28], and DeLong et al [29] for binary outcome data and by Nam and D'Agostino [30] and Pencina and D'Agostino [3] for timetoevent outcome data. The variance of the concordance probability estimate can also be estimated with a bootstrap procedure [31].
Conclusion
We recommend metaanalysis of clusterspecific cindexes when assessing discriminative ability of risk models used to support decisions at cluster level. Particularly, random effects metaanalysis should be considered as it allows for and provides insight into the variability of the concordance probability among clusters.
Appendix 1
The concordance probability is defined as the probability that a randomly chosen subject pair with different outcomes is concordant. For a randomly chosen subject pair (i, j) with outcomes Y_{i} and Y_{j} and model predictions and the concordance probability C is:
Harrell’s cindex [2] estimates the concordance probability by the proportion of all usable pairs of subjects (n_{u}) in which the predictions and outcomes are concordant (n_{c}), with tied predictions (n_{t}) counted as 1/2:
For binary outcomes y, pairs of subjects are usable if one of the subjects had an event and the other did not. The number of usable subject pairs n_{u}, the number of concordant subject pairs n_{c} and the number of tied subject pairs n_{t} are:
For timetoevent outcomes y, pairs of subjects are usable if their survival times are not equal and at least the smallest survival time is uncensored. We have to add the restriction that the smallest observation y_{i} of each subject pair is uncensored, denoted by δ_{i} = 1:
The variance of the cindex can be estimated according to Quade [16]:
All summations over i with n_{u,i} and n_{cd,i} the number of usable and the number of concordant minus discordant subject pairs of which subject i is one:
Appendix 2
Based on the delta method, a variance estimator for the logit of the cindex is:
We used this variance estimator to perform a metaanalysis on logodds scale. The pooling weights (method 7) for a fixed effect inverse variance metaanalysis on logodds scale are:
The pooling weights (method 8) for a random effects inverse variance metaanalysis on logodds scale are:
The resulting pooled estimates together with confidence and prediction intervals are transformed back to probability scale.
Abbreviations
cindex: Concordanceindex; CRASH: Corticosteroid randomisation after significant head injury; GCS: Glasgow coma scale; GOS: Glasgow outcome scale; IQR: Interquartile range.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
DK, ES and YV designed the study. PP participated in the collection of data and organisation of the databases from which this manuscript was developed. DK and YV analysed the data and wrote the first draft of the manuscript. All authors contributed to writing the manuscript and read and approved the final manuscript.
Funding
This work was supported by the Netherlands Organisation for Scientific Research (grant 917.11.383).
Acknowledgements
The authors express their gratitude to all of the principal investigators of the CRASH trial for providing the data. We thank Prof. Emmanuel Lesaffre (Department of Biostatistics, Erasmus MC, Rotterdam, The Netherlands) for helpful comments.
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