Meta-analysis of external validation studies

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1 Meta-analysis of external validation studies Thomas Debray, PhD Julius Center for Health Sciences and Primary Care, University Medical Center Utrecht, The Netherlands Cochrane Netherlands, Utrecht, The Netherlands

2 Meta-analysis of external validation studies - Introduction Prediction ˆ Turn available information about individuals into a statement about the probability:... of having a particular disease diagnosis... of developing a particular event prognosis ˆ Use of multiple predictors Subject characteristics History and physical examination results Imaging results (Bio)markers ˆ Typical aims to inform patients and their families to guide treatment and other clinical decisions to create risk groups Vigo, July 10, 2017 TPA Debray 2/23

3 Meta-analysis of external validation studies - Introduction What is a good model? Accurate predictions CALIBRATION DISCRIMINATION IMPACT Improve patient outcomes Ability to distinguish between low and high risk patients Good and consistent performance across different settings and populations GENERALITY SUPPORT Influence decision making Vigo, July 10, 2017 TPA Debray 3/23

4 Meta-analysis of external validation studies - Introduction Numerous models for same target population + outcomes 1995 We believe that the main reasons why doctors reject published prognostic models are lack of clinical credibility and lack of evidence that a prognostic model can support decisions about patient care. Wyatt JC, Altman DG. Commentary: Prognostic models: clinically useful or quickly forgotten? BMJ. 1995;311(7019): Vigo, July 10, 2017 TPA Debray 4/23

5 Meta-analysis of external validation studies - Introduction Numerous models for same target population + outcomes Comparing risk prediction models should be routine when deriving a new model for the same purpose Collins GS, Moons KGM. Comparing risk prediction models. BMJ. 2012;344:e3186 e3186. Vigo, July 10, 2017 TPA Debray 5/23

6 Meta-analysis of external validation studies - Introduction Numerous models for same target population + outcomes There is an excess of models predicting incident CVD in the general population. The usefulness of most of the models remains unclear. Damen JAAG, Hooft L, et al. Prediction models for cardiovascular disease risk in the general population: systematic review. BMJ. 2016;353:i2416. Vigo, July 10, 2017 TPA Debray 6/23

7 Meta-analysis of external validation studies - Introduction Numerous models for same target population + outcomes Formal guidance for systematic review and meta-analysis Vigo, July 10, 2017 TPA Debray 7/23

8 Motivating example Previous guidance focused on meta-analysis of logistic regression models. Hence, focus of today is on survival models. Framingham Risk Score (Wilson et al. 1998) ˆ Model type: Cox regression ˆ Outcome: Fatal or non-fatal coronary heart disease (CHD) ˆ Timing: Initial CHD within 10 years ˆ Evidence: 24 validations in male populations Summarize estimates of model performance ˆ Concordance statistic (cstat) ˆ Ratio of observed versus expected events (OE) ˆ Calibration slope (slope) Vigo, July 10, 2017 TPA Debray 8/23

9 Statistical framework - data extraction Key problem: Poor and inconsistent reporting of prediction model performance. Standard error of the c-statistic [ ] c (1 c) 1 + n 1 c 2 c + m c 1+c SE(c) mn with c the reported c-statistic, n the number of observed events, m the total number of non-events and m = n = 1 2 (m + n) 1. Newcombe RG. Confidence intervals for an effect size measure based on the Mann-Whitney statistic. Part 2: asymptotic methods and evaluation. Stat Med. 2006;25(4): Vigo, July 10, 2017 TPA Debray 9/23

10 Estimated versus reported standard error of the c-statistic EuroSCORE II Framingham Reported standard error Estimated standard error Vigo, July 10, 2017 TPA Debray 10/23

11 Statistical framework - data extraction Ratio of observed versus expected events ˆ Reference value: 1 ˆ Observed survival probability (S KM,t ) ˆ Expected (predicted) event rate (P E,t ) (O:E) t = 1 S KM,t P E,t SE(O:E) t = 1 P E,t SE(S KM,t ) If unavailable, SE(S KM,t ) can be approximated as well: SE(S KM,t ) S KM,t (1 S KM,t ) N t Vigo, July 10, 2017 TPA Debray 11/23

12 Extraction of event rates: an example Vigo, July 10, 2017 TPA Debray 12/23

13 Statistical framework - data extraction Calibration slope ˆ Reference value: 1 ˆ The calibration slope β is calculated as follows using IPD: y k Bernoulli(p k ) logit(p k ) = α + β LP k ˆ When missing, we can derive β from reported event counts across j risk strata (e.g. as presented in calibration tables): O j Binom(N j, p O,j ) logit(p O,j ) = η + β logit(p E,j ) Vigo, July 10, 2017 TPA Debray 13/23

14 Extracted risk estimates for the Framingham Risk Score Observed risk Expected risk The diagonal line indicates perfect calibration. Risk estimates were reported for 5 years follow-up (dashed lines), 7.5 years follow-up (dotted lines) and 10 years follow-up (full lines).

15 Statistical framework - data extraction #validations analysed reported study authors approximated cstat SE.cstat OE slope For 10 studies, calibration performance was only available for < 10 years follow-up. Vigo, July 10, 2017 TPA Debray 15/23

16 Statistical framework - meta-analysis models Meta-analysis of the c-statistic ˆ Need to apply logit transformation logit(c i ) N ( µ discr, Var (logit(c i )) + τdiscr 2 ) ˆ Use delta method to derive Var (logit(c i )) from SE(c i ) ˆ For Bayesian models, we propose weakly informative priors: µ discr N (0, 10 6 ) τdiscr Unif(0, 2) τdiscr Student-t(0, 0.5 2, 3) T [0, 10] based on empirical data from 26 meta-analyses (with ˆτ discr ranging from 0 to 0.50). Vigo, July 10, 2017 TPA Debray 16/23

17 Statistical framework - meta-analysis models Meta-analysis of the c-statistic Estimation K Summary 95% CI 95% PI REML Bayesian (Unif) Bayesian (Student-t) For 3 studies, we did not have information on c i but could nevertheless approximate SE(c i ). Vigo, July 10, 2017 TPA Debray 17/23

18 Statistical framework - meta-analysis models Meta-analysis of the total O:E ratio We can use different models to account for sampling variability: Option 1 ln(o:e) i N ( µ cal.oe, Var (ln(o:e) i ) + τcal.oe 2 ) Option 2 O i Binom (N i, p O,i ) E i Binom (N i, p E,i ) ln (p O,i /p E,i ) N ( µ cal.oe, τcal.oe 2 ) Option 3 O i Poisson (E i exp(η i )) η i N ( µ cal.oe, τcal.oe 2 ) For all models, the interpretation of µ cal.oe and τ cal.oe is the same. Vigo, July 10, 2017 TPA Debray 18/23

19 Statistical framework - meta-analysis models Meta-analysis of the total O:E ratio (c ed) ˆ For Bayesian models, we propose weakly informative priors: µ cal.oe N (0, 10 6 ) τ cal.oe Unif(0, 2) τcal.oe Student-t(0, 1.5 2, 3) T [0, 10] based on empirical data from 16 meta-analyses (with ˆτ cal.oe ranging from 0 to 1.39). ˆ Possible to extrapolate event rates at time l to time t using: ( ) t ln(1 pl ) p t = 1 S KM,t = 1 exp l Straightforward to integrate in Bayesian estimation framework Vigo, July 10, 2017 TPA Debray 19/23

20 Statistical framework - meta-analysis models Meta-analysis of the total O:E ratio Estimation K Summary 95% CI 95% PI REML Bayesian 1 (Unif) Bayesian 1 (Student-t) ML Bayesian 3 (Unif) Bayesian 3 (Student-t) When applying extrapolation, we have 10 additional studies for meta-analysis (similar results). Vigo, July 10, 2017 TPA Debray 20/23

21 Statistical framework - meta-analysis models Meta-analysis of the calibration slope ˆ No transformations needed ˆ Rely on binomial approximation O ij Binom(N ij, p O,ij ) logit(p O,ij ) = α i + β i logit(p E,ij ) β i N (µ cal.slope, τ 2 cal.slope ) ˆ For Bayesian models, we propose weakly informative priors: µcal.slope N (0, 10 6 ) τcal.slope Unif(0, 2) τ cal.slope Student-t(0, 1.5 2, 3) T [0, 10] Vigo, July 10, 2017 TPA Debray 21/23

22 Statistical framework - meta-analysis models Meta-analysis of the calibration slope Estimation K Summary 95% CI 95% PI ML Bayesian Bayesian When applying extrapolation, we have 8 additional studies for meta-analysis (similar results but smaller intervals). Vigo, July 10, 2017 TPA Debray 22/23

23 Meta-analysis of external validation studies - Conclusion Final remarks ˆ Substantial efforts often needed to restore missing information ˆ Bayesian estimation methods recommended to fully propagate uncertainty arising from data restoration ˆ Development of R package metamisc to assist in data preparation and meta-analysis ˆ Presence of statistical heterogeneity most likely ˆ Straightforward extension to meta-regression and multivariate meta-analysis Vigo, July 10, 2017 TPA Debray 23/23