1. Homogeneity test for correlated data in ophthalmologic studies Chang-Xing Ma University at Buffalo 1

2. Outline Introduction Introduction Methods Methods Monte Carlo simulation studies A work example Concluding remarks 2

3. Introduction In ophthalmologic studies, measurements obtained from both eyes of an individual often are highly correlated. In ophthalmologic studies, measurements obtained from both eyes of an individual often are highly correlated. For example, in clinical trial, all patients are randomized into two treatment groups and the same treatment is applied to two eyes of patients from the same group. For example, in clinical trial, all patients are randomized into two treatment groups and the same treatment is applied to two eyes of patients from the same group. 3

4. Data We wish to test whether the response rates of the g groups are identical. We wish to test whether the response rates of the g groups are identical. 4

5. Existing methods A parametric model proposed by Rosner (1982) is considered where Z ijk =1 if the kth eye of jth individual in the ith group has a response at the end of the study, and 0 otherwise A parametric model proposed by Rosner (1982) is considered where Z ijk =1 if the kth eye of jth individual in the ith group has a response at the end of the study, and 0 otherwise The constant R is a measure of dependence between two eyes of the same person The constant R is a measure of dependence between two eyes of the same person 5

6. Existing methods Rosner (1982) proposed Z statistic for the model Rosner (1982) proposed Z statistic for the model N.-S. Tang, et al (2008) proposed Score Test for g=2 N.-S. Tang, et al (2008) proposed Score Test for g=2 6

7. Methods Based on the observed data Based on the observed data Log-likelihood Log-likelihood 7

8. MLE Solve following 3rd order polynomial Solve following 3rd order polynomial R can be updated by Fisher scoring method R can be updated by Fisher scoring method 8

9. Three tests Likelihood ratio test Wald-type test Wald-type test Score test Score test 9

10. Monte Carlo simulation studies 50,000 samples are generated based on null hypothesis and empirical type-I error rates are computed as the number of rejections/50000 10

11. Type-I error control 11

12. Type-I error control Generally, score tests produce satisfactory type-I error controls for any configuration likelihood ratio tests and Wald tests are liberal, especially for small samples and larger number of groups (g). When G>2, Wald tests are more liberal than LR tests and there tests get closer when sample size goes larger 12

13. power performance 13

14. power performance of proposed methods LR and Wald tests are generally more powerful than score tests Ronser (1982)’s T is generally with less power. For moderate or larger sample size, the powers of proposed three methods are close Score test procedure is highly recommended as it has also satisfactory type-I error control. 14

15. A work example N=216 with retinitis pigmentosa (RP) were classified into the genetic types of autosomal dominant RP (DOM), autosomal recessive RP (AR), sex- linked RP (SL), and isolate RP (ISO) for a study of differences between these four groups on the Snellen visual acuity (VA). 15

16. Distribution of the number of affected eyes genetic type number of affected eyes DOMARSLISO 0 157367 1 65224 2 791457 16

17. Results: difference between the proportion of affected eyes in the four groups is not significant by proposed method BUT significant by Rosner (1982) method (p=0.01) Note: T is the test statistic in Rosner (1982). 17

18. Concluding remarks We proposed three test procedures for testing homogeneity for correlated data in ophthalmologic studies. Simulation results show that the score test has satisfactorily type-I error control regardless of number of groups, sample sizes and parameter configurations 18