1. Bootstrap Estimation of Disease Incidence Proportion with Measurement Errors Masataka Taguri (Univ. of Tokyo.) Hisayuki Tsukuma (Toho Univ.)

2. Contents Motivation Formulation of the Problem Regression Calibration Method Proposed Estimator Interval Estimation (Delta & Bootstrap) Numerical Examination

3. Motivation Measurement error problem –The measurement error of exposure variable will tend to give bias estimates of relative risk 〈 ex 〉 In nutritional studies, FFQ (food frequency questionnaire) is not exact measurement ★ How to obtain correct risk estimates?

4. Formulation of Problem (1) Data form –Main data : : binary response (Disease or not) : exposure measured with error : No. of main data-set –Validation data : : true exposure : No. of validation data-set

5. Formulation of Problem (2) Disease model Assume linear relationship of and ～ Naive model ( : observable) ★ Usual interest : estimation of

6. Regression Calibration (RC) method Model ～ Rosner, et al.1989 (i) estimate by maximum likelihood (ii) estimate by ordinary least squares (iii) estimate by

7. Asymptotic Variance of Apply the delta method, then asymptotic variance of is given by

8. Subject for investigation ★ RC estimator : might be unstable (heavy tailed) ・ ： asymptotically ratio of two normal variables MLE OLSE

9. Objective of our study ① To propose an estimator of –examination of the stability of estimators ② Interval estimation of –Bootstrap Method [Percentile & BCa] –Delta Method ③ Interval estimation of –Bootstrap Method [Percentile & BCa] –Delta Method

10. Inverse Regression Estimator To construct alternative estimator of, consider the inverse regression model; The ordinary least squares estimator of

11. Generalized Estimator Ridge-type Estimator for Generalized Inverse Regression Estimator ★ ⇒ RC estimator ★ The asymptotic distribution of has moments.

12. Interval Estimation of (by Delta method) [Algorithm] 1) Use and to construct the confidence intervals of 2) Estimate the asymptotic variance of and by delta method 3) Make confidence intervals by normal approximation Rosner, et al.1989

13. Interval Estimation of (by Bootstrap method) Estimator : [Algorithm] 1)Compute by resamples from main data-set. 2) Compute by resamples from validation data-set. 3) Compute for 4) Construct confidence intervals of by Bootstrap Method [Percentile & BCa]. (Efron & Tibshirani, 1993)

14. Interval Estimation of Estimator of p : : RC / Inverse Regression Est. Confidence Interval (C.I.) of p : by Delta or Bootstrap (Percentile & BCa) ★ p should exist between 0 and 1 ⇒ C.I. of logit(p) → C.I. of p

15. Numerical Examination – Set-up Validation sample 1°Generate from N(0,1) 2°Assume the model Generate Make validation sample for ★ The above model is a special case of

16. Main sample 3°Generate Generate from Bernoulli dist.( ). Combine with (paired value of ) ⇒ 4°Set Compute 95% C.I. of logit(p), then p by Normal approximation, Percentile & BCa. ★ Set ・ Computation : for all combination of

17. Table 1 Estimates of α 1 （ R=1000, B=2000, n=1000, m=100) Esti- mator MeanSD Skew- ness Kurto- sis MinMax Range α 1 ＝ 0.4050 (exp(α 1 )=1.5) ， λ 1 ＝ 0.5 α1Rα1R 0.4090.2140.2273.298-0.321.2491.573 α1Gα1G 0.4050.2120.2243.297-0.321.2381.556 α 1 ＝ 0.6931 (exp(α 1 )=2.0) ， λ 1 ＝ 0.5 α1Rα1R 0.6920.2170.2013.2200.0091.5281.519 α1Gα1G 0.6850.2140.1913.2080.0091.5031.494 α 1 ＝ 1.0986 (exp(α 1 )=3.0) ， λ 1 ＝ 0.5 α1Rα1R 1.0520.2170.4133.2350.5161.9241.408 α1Gα1G 1.0410.2130.4023.2130.5121.8921.380 RC GI

18. Examination on stability of α 1 (1) underestimates except : monotone increasing w.r.t. except for is smaller than for all cases. Biases of, : not so different on the whole. (2) SD, skewness, kurtosis, range : values for > values for ⇒ Tail of distribution is heavier than that of → Outliers may often appear.

19. Table 2-1 95% Confidence Intervals of α 1 （ R=1000, B=2000, n=1000, m=100) α 1 ＝ 0.4050 (exp(α 1 )=1.5) ， λ 1 ＝ 0.5 MethodCov. Prob.LengthShape NR0.9560.82121.0000 NG0.9540.81161.0000 PR0.9420.84951.1323 PG0.9450.83771.1265 BR0.9580.88460.6939 BG0.9560.8729 0.6920 Nor.app. BCa Parcentile

20. Table 2-2 95% Confidence Intervals of α 1 （ R=1000, B=2000, n=1000, m=100) α 1 ＝ 0.6931 (exp(α 1 )=2.0) ， λ 1 ＝ 0.5 MethodCov. Prob.LengthShape NR0.9450.82141.0000 NG0.9430.81071.0000 PR0.9350.85161.2311 PG0.9350.83781.2209 BR0.9300.86990.7114 BG0.9270.8575 0.7082

21. Table 2-3 95% Confidence Intervals of α 1 （ R=1000, B=2000, n=1000, m=100) α 1 ＝ 1.0986 (exp(α 1 )=3.0) ， λ 1 ＝ 0.5 MethodCov. Prob.LengthShape NR0.9300.83761.0000 NG0.9220.82461.0000 PR0.9370.87221.3589 PG0.9350.85431.3431 BR0.8960.86830.7664 BG0.8900.8534 0.7605

22. Examination on confidence interval of α 1 (1) Cov. Prob. of NG < Cov. Prob. of NR. Cov. Prob. of normal approximation : monotone decreasing w.r.t. (odds ratio) (2) Length of C.I. by < Length of C.I. by (for all cases) ⇒ stable property of (3) Bootstrap : slightly worse than Normal approxim. （ from viewpoint of Cov. Prob. ） ★ Percentile is best in case of (4) BCa ： not better than other methods ⇔ The distribution of estimates is not so skew?

23. Table 3-1 95% Confidence Intervals of p （ R=1000, B=2000, n=1000, m=100 ; exp(α 1 )=2.0, λ 1 =0.5) MethodCov. Prob.LengthShape T=-1, Pr[D|T]=0.0256 NR0.9170.03421.7567 NG0.9190.03411.7481 PR0.9330.03181.2882 PG0.9330.03211.3169 BR0.9540.03140.8901 BG0.9540.03150.9193 T=0, Pr[D|T]=0.05 NR0.9050.03331.3310 NG0.9050.03321.3301 PR0.9190.03131.0245 PG0.9270.03141.0585 BR0.9600.03220.7160 BG0.9520.03210.7567

24. Table 3-2 95% Confidence Intervals of p （ R=1000, B=2000, n=1000, m=100 ; exp(α 1 )=2.0, λ 1 =0.5) MethodCov. Prob.LengthShape T=1, Pr[D|T]=0.0952 NR0.9500.08281.4061 NG0.9500.08131.4020 PR0.9230.08451.4280 PG0.9260.08181.3663 BR0.9610.08170.8939 BG0.9590.08000.8449 T=2, Pr[D|T]=0.1739 NR0.9560.24071.5871 NG0.9560.23551.5845 PR0.9380.25461.7572 PG0.9420.24701.7196 BR0.9430.23561.0770 BG0.9420.22941.0500

25. Examination on confidence interval of p (1) Cov. Prob. for Percentile : does not keep the nominal level, especially in case of Cov. Prob. for BCa : quite satisfactory for almost all cases. (2) Length of C.I. is minimum when T=0. Ｉ t becomes large rapidly with increase of T. (3) Length for NG is always shorter than that for NR.

26. References [1] Carroll, R.J., Ruppert, D., Stefanski, L.A.: Measurement Error in Nonlinear Models, Chapman & Hall, New York (1995). [2] Efron, B., Tibshirani, R.J.: An Introduction to the Bootstrap, Chapman & Hall, New York (1993). [3] Rosner, B, Willett, W.C., Spiegelman D.: Correction of logistic regression relative risk estimates and confidence intervals for systematic within-person measurement error, Statistics in Medicine, 8, 1051--1069 (1989).