Cumulative Geographic Residual Test

1. Cumulative Geographic Residual Test Example: Taiwan Petrochemical Study Andrea Cook

2. Outline 1. Motivation • Petrochemical exposure in relation to childhood brain and leukemia cancers 2. Cumulative Geographic Residuals • Unconditional • Conditional 3. Application • Childhood Leukemia • Childhood Brain Cancer

3. Taiwan Petrochemical Study Matched Case-Control Study • 3 controls per case • Matched on Age and Gender • Resided in one of 26 of the overall 38 administrative districts of Kaohsiung County, Taiwan • Controls selected using national identity numbers (not dependent on location).

4. Study Population Due to dropout approximately 50% 3 to 1 matching, 40% 2 to 1 matching, and 10% 1 to 1 matching.

5. Map of Kaohsiung

6. Cumulative Residuals • Unconditional (Independence) • Model definition using logistic regression • Extension to Cluster Detection • Conditional (Matched Design) • Model definition using conditional logistic regression • Extension to Cluster Detection

7. Logistic Model Assume the logistic model where, and the link function,

8. Residual Formulation Then define a residual as, Assuming the model is correctly specified would imply there is no pattern in residuals. => Use Residuals to test for misspecification. Cumulative Residuals for Model Checking; Lin, Wei, Ying 2002

9. Hypothesis Test Hypothesis of interest, Geographic Location, (ri, ti ) Independent of Outcome, Yi|Xi  Cumulative Geographic Residual Moving Block Process is Patternless

10. Unconditional Cluster Detection Define the Cumulative Geographic Residual Moving Block Process as,

11. Asymptotic Distribution However, the distribution of, is hard to define analytically, but we have found another distribution that is asymptotically equivalent, which consists of a fixed component of data and random variables

12. Significance Test Testing the NULL • Simulate N realizations of by repeatedly simulating , while fixing the data at their observed values. • Calculate P-value

13. Conditional Logistic Model Type of Matching: 1 case to Ms controls Data Structure: Assume that conditional on , an unobserved stratum-specific intercept, and given the logit link, implies, The conditional likelihood, conditioning on is,

14. Conditional Residual Then define a residual as, => Use these correlated Residuals to test for patterns based on location.

15. Conditional Cumulative Residual However, the distribution of, is hard to define analytically, but we have found another distribution that is asymptotically equivalent, which consists of a fixed component of data and random variables

16. Testing the NULL Simulate N realizations of by repeatedly simulating , while fixing the data at their observed values. Calculate P-value Significance Test

17. Application • Study: Kaohsiung, Taiwan Matched Case-Control Study • Method: Conditional Cumulative Geographic Residual Test (Normal and Mixed Discrete)

18. Results Odds Ratio (p-values) Marginally Significant Clustering for both outcomes without adjusting for smoking history.

19. Childhood Leukemia

20. Childhood Brain Cancer

21. Discussion Cumulative Geographic Residuals • Unconditional and Conditional Methods for Binary Outcomes • Can find multiple significant hotspots holding type I error at appropriate levels. • Not computer intensive compared to other cluster detection methods Taiwan Study • Found a possible relationship between Childhood Leukemia and Petrochemical Exposure, but not with the outcome Childhood Brain Cancer.