Marios Georgiadis, Faculty of Veterinary Medicine, Aristotle University of Thessaloniki, Greece

1. Sample size determination for estimation of the accuracy of two conditionally independent diagnostic tests Marios Georgiadis, Faculty of Veterinary Medicine, Aristotle University of Thessaloniki, Greece

2. this work was done by • Wes Johnson, University of California, Davis • Ian Gardner, University of California, Davis • Marios Georgiadis, Aristotle University of Thessaloniki

3. Hui-Walter model (Biometrics, 1980)

4. Assumptions • Validity of the assumptions is critical and should be given careful consideration • 1# 2 • Each test has the same Se-Sp in the two populations • Conditional independence (Vacek, 1985)

5. sample size estimation using HW • data from 2-tests applied on 2-populations • goal is to estimate Se1, Se2, Sp1, Sp2, 1, 2 • minimum sample size to achieve a desired level of precision • the method provides sample sizes to obtain CI’s of a specified maximum width for one or more of the 6 parameters • alternatively, we can specify CI widths for the difference in sensitivities (Se1-Se2) and specificities (Sp1-Sp2)

6. spreadsheet 1

7. HW estimates and CI’s • HW provided closed-form formulas for the ML estimates for the two Se’s, the two Sp’s and the two prevalences (6 parameters) • using these formulas with our 2-table data we get ML point estimates for the six parameters of interest • these point estimates are the points of the (6-dimensional) parameter space for which the likelihood function is maximized

9. spreadsheet 3

10. HW formulas for the Fisher Information Matrix (FIM)

11. once we get the FIM we can invert it to obtain the estimated variance-covariance matrix • the diagonal elements of this matrix are the standard large-sample estimates of the variances of the respective parameter estimates • The square roots of the diagonals are the usual s.e.’s • off-diagonal elements are the corresponding estimated covariances

12. excel spreadsheet 2

13. once we have the standard errors we can calculate CI’s • we need the assumption of asymptotic normality of the ML estimates - large sample sizes • rule of thumb: ML estimate  3*s.e. should not cover 0 or 1 • if the assumption does not hold we cannot calculate CI’s in the usual way

14. estimation of the differences: Se1-Se2 and Sp1 –Sp2 • an objective of the study might be to compare the sensitivities or the specificities of the tests • the point estimate of the differences is the difference of the point estimates • the estimated variances of the difference estimates are:

15. all the necessary estimated variances and covariances can be obtained from the estimated variance-covariance matrix • the standard error of the difference is the square root its estimated variance • if the asymptotic normality assumption holds we can create CI’s as before

16. calculation of sample size • if the sampling distribution of an estimator is approximately normal then the (1-)*100% CI is

17. the width (w) of this CI is • solving for N, we get: • to calculate the sample size, N, we need an estimate of s

18. spreadsheet 1

19. if the largest sample size is picked, all the CI widths will be as specified or smaller • estimation of only a subset of parameters might be of interest • prevalence estimates are not usually of interest • some performance estimates might be known • information on these is used in the spreadsheet but their CI widths are set arbitrarily large

20. for some combinations of parameter values the diagonals of C and can be negative • this is because these parameter values result in a singular information matrix • we have to make sure that we do not have negative diagonals or very large pairwise correlation values (close to or over 1 or -1) • another indication is that the sample sizes will become very large • in these situations, the usual ML method cannot be used to obtain s.e.’s and therefore our sample size calculations are not applicable

21. it’s a good idea to try some combinations of parameter guesses to make sure you are not near a problematic area of the parameter space • the same potential problems and warnings can be found in spreadsheet 2

22. initial parameter guesses • guesses of the 6 parameters of interest are necessary • since the sample size calculation is strongly dependent on those they have to be realistic • expert opinion - be careful: • sensitivity can vary with severity of infection and stage of disease process • sensitivity of a test with experimental samples might be higher than with real field samples • specificity can vary according to geographic distribution of cross-reacting microorganisms

23. best to do a pilot study • calculate sample sizes for a range of possible parameter values

24. if you wanted to conduct an evaluation study • if you want to use the HW model: first make sure that the assumptions hold • tests conditionally independent • populations have different prevalences • test performance the same in both populations • sample size calculations – precision and cost considerations • specify up front how much precision we need

25. formulate educated guesses for the parameters of interest (expert opinion and/or pilot study) • use spreadsheet 1 to get sample sizes • check to see if the large-sample approximation is reasonable by calculating the initial estimate/guess ±3*s.e. to determine if the interval obtained includes 0 or 1 • if it does, the sample is likely not large enough to justify large-sample normality

26. during the calculation process we should monitor the diagonals of matrix C and the pairwise correlations and be careful about the “singular information matrix” problem

27. conduct the study • insert raw data into spreadsheet 3 to get parameter estimates • use parameter estimates in spreadsheet 2 to get standard errors • if large sample theory holds, we can calculate CI’s for the parameters of interest • again, monitor information matrix diagonals and pairwise correlations

28. dependent tests • if the tests are conditionally dependent, we can still use the HW setup but we will need different methods of analysis of our results • since there are no sample-size calculation methods for such tests, we can still use our method, knowing that to obtain comparable precision we will probably need larger sample sizes • the calculated sizes can be used as an absolutely minimum value

29. HW data example