1. PSI 2014 Conference The Tower Hotel, Tower Bridge, London Risk Calculation in Screening with Biomarkers Nick Cowans Statistical Programmer Veramed Limited 1

2. Biomarkers Something we measure in order to predict risk of a disease/syndrome in a subject 2 Cholesterol: coronary/vascular disease CA-125: ovarian cancer PAPP-A, Free hCGβ: Carrying a Trisomy 21 fetus

3. Characteristic of Biomarkers Concentrations bound at zero No upper limit 0  “  ” Log transform 3

4. 4 Mean (  ) Variance (2)(2)  = [ ,  2 ]

5. Unaffected Population N(  U,  2 U ) Affected Population N(  A,  2 A ) AA UU 5

6. Calculating Risk Likelihood affected: 2 l Likelihood unaffected: l X = x U/l 6 Likelihood ratio = 2 TWICE as likely to be from affected population

7. Estimating parameters,  We cannot know , population parameters A conventional method : Collect some data in each population, Z Use Maximum Likelihood Estimation to estimate , call it Ignore uncertainty Use in risk calculation 7

8. Making predictions p(X=x| A ) p(X=x| U ) new data parameter estimates 8 Your prediction is conditional on your parameter estimates, which are uncertain Likelihood Ratio =

9. Estimating parameters Not a problem when sample size is large If sample size is small, repeating experiments can result in very different estimates: 10 samples (or experiments) True mean = 10 n = 10,000: n = 10: more uncertainty 9

10. Predictive approach Rather than predict conditional on uncertain estimated parameters as in the estimative approach Obtain predictive density function Bayesian concept; parameters given own distributions parameter estimates from data Z p(X|  ) estimative approach: ^ Just data Z 10 predictive density function: p(X|Z)

11. Predictive approach With Normally distributed data, the predictive density function becomes a scaled and shifted central t-distribution Number of samples in original data, Z, is no longer ignored, and is incorporated via degrees of freedom 11

12. Predictive approach Estimative Method 12 2l2l l

13. Predictive approach Estimative Method Predictive Method 13 2l2l l l 1.6 l

14. Case study Screening for trisomy 21 In trisomy 21 pregnancies: PAPP-A decreased Free hCGβ increased Bivariate Normal distribution Risk calculated based upon maternal age, biochemistry and nuchal fold thickness 14

15. Marker distribution 15

16. Results 16 Risk ROC Curves

17. Summary Fallacy of estimative approach: takes no account of sampling variability of the estimator Predictive approach weighs possible distributions of new data according to the various plausible values for  More likely to see differences in risks with small sample sizes 17

18. Acknowledgments Professor Kevin Spencer (Barking Havering and Redbridge University NHS Trust) Professor David Wright (University of Plymouth) Dr Kevin Walters (University of Sheffield) 18

19. Thank you 19