Optimality

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260 G. Zheng, B. Freidlin and J. L. Gastwirth Type I errors for all tests are expected to be close to the nominal level α = 0.05 and the powers of all tests are comparable. When HWE holds, the probabilities (p0, p1, p2) for cases and (q0, q1, q2) for controls can be calculated using (3.1) under the null and alternative hypotheses, where (g0, g1, g2) = (q 2 ,2pq, q 2 ) and (f1, f2) are specified by the null or alternative hypotheses and D = � figi. After calculating (p0, p1, p2) and (q0, q1, q2) under the null hypothesis, we first simulated the genotype distributions (r0, r1, r2)∼mul(R; p0, p1, p2) and (s0, s1, s2)∼mul(S;q0, q1, q2) for cases and controls, respectively (see Table 1). When HWE does not hold, we assumed a mixture of two populations with two different allele frequencies p1 and p2. Hence, we simulated two independent samples with different allele frequencies for cases (and controls) and combined these two samples for cases (and for controls). Thus, cases (controls) contain samples from a mixture of two populations with different allele frequencies. When p is small, some counts can be zero. Therefore, we added 1/2 to the count of each genotype in cases and controls in all simulations. To obtain the critical values, a simulation under the null hypothesis was done with 200,000 replicates. For each replicate, we calculated the test statistics. For each test statistic, we used its empirical distribution function based on 200,000 replicates to calculate the critical value for α = 0.05. The alternatives were chosen so that the power of the optimal test Z0, Z 1/2, Z1 was near 80% for the recessive, additive, dominant models, respectively. To determine the empirical power, 10,000 replicates were simulated using multinomial distributions with the above probabilities. To calculate ZMERT, the correlation ρ0,1 was estimated using the simulated data. In Table 3, we present the mean of correlation matrix using 10,000 replicates when r = s = 250. The three correlations ρ 0,1/2, ρ0,1, ρ 1/2,1 were estimated by replacing pi with ni/n, i = 0, 1,2 using the data simulated under the null and alternatives and various models. The null and alternative hypotheses used in Table 3 were also used to simulate critical values and powers (Table 4). Note that the minimum correlation ρ0,1 is less than .50. Hence, the ZMAX3 should have greater efficiency robustness than ZMERT. However, when the dominant model can be eliminated based on prior scientific knowledge (e.g. the disease often skips generations), the correlation between Z0 and Z 1/2 optimal for the recessive and additive models would be greater than .75. Thus, for these two models, ZMERT should have comparable power to ZMAX2 = max(|Z0|,|Z 1/2|) and is easier to use. The correlation matrices used in Table 4 for r�= s and Table 5 for mixed samples are not presented as they did not differ very much from those given in Table 3. Tables 4 and 5 present simulation results where all three genetic models are plausible. When HWE holds Table 4 shows that the Type I error is indeed close to the α = 0.05 level. Since the model is not known, the minimum power across three genetic models is written in bold type. A test with the maximum of the minimum power among all test statistic has the most power robustness. Our comparison fo- Table 3 The mean correlation matrices of three optimal test statistics based on 10,000 replicates when HWE holds p .1 .3 .5 Model ρ 0,1/2 ρ0,1 ρ 1/2,1 ρ 0,1/2 ρ0,1 ρ 1/2,1 ρ 0,1/2 ρ0,1 ρ 1/2,1 null .97 .22 .45 .91 .31 .68 .82 .33 .82 rec .95 .34 .63 .89 .36 .74 .81 .37 .84 add .96 .23 .48 .89 .32 .71 .79 .33 .84 dom .97 .21 .44 .90 .29 .69 .79 .30 .83 The same models (rec,add,dom) are used in Table 4 when r = s = 250.
Robust tests for genetic association 261 Table 4 Power comparison when HWE holds in cases and controls under three genetic models with α = .05 Test statistics p Model Z0 Z 1/2 Z1 ZMERT ZMAX2 ZMAX3 TMAX TP χ 2 2 r = 250, s = 250 .1 null .058 .048 .050 .048 .047 .049 .049 .049 .053 rec .813 .364 .138 .606 .725 .732 .941 .862 .692 add .223 .813 .802 .733 .782 .800 .705 .424 .752 dom .108 .796 .813 .635 .786 .795 .676 .556 .763 .3 null .051 .052 .051 .054 .052 .052 .053 .049 .050 rec .793 .537 .178 .623 .714 .726 .833 .846 .691 add .433 .812 .768 .786 .742 .773 .735 .447 .733 dom .133 .717 .809 .621 .737 .746 .722 .750 .719 .5 null .049 .047 .051 .047 .047 .047 .050 .051 .050 rec .810 .662 .177 .644 .738 .738 .772 .813 .709 add .575 .802 .684 .807 .729 .760 .719 .450 .714 dom .131 .574 .787 .597 .714 .713 .747 .802 .698 r = 50, s = 250 .1 null .035 .052 .049 .052 .051 .053 .045 .051 .052 rec .826 .553 .230 .757 .797 .802 .779 .823 .803 add .250 .859 .842 .773 .784 .795 .718 .423 .789 dom .114 .807 .814 .658 .730 .734 .636 .447 .727 .3 null .048 .048 .048 .048 .050 .050 .050 .048 .049 rec .836 .616 .190 .715 .787 .787 .733 .752 .771 add .507 .844 .794 .821 .786 .813 .789 .500 .778 dom .171 .728 .812 .633 .746 .749 .696 .684 .729 .5 null .049 .046 .046 .046 .047 .047 .046 .049 .046 rec .838 .692 .150 .682 .771 .765 .705 .676 .748 add .615 .818 .697 .820 .744 .780 .743 .493 .728 dom .151 .556 .799 .565 .710 .708 .662 .746 .684 cuses on the test statistics: ZMAX3, TMAX, TP and χ 2 2. Our results show that TMAX has greater efficiency robustness than TP while ZMAX3, TMAX and χ 2 2 have similar minimum powers. Notice that ZMAX3 is preferable to χ 2 2 although the difference in minimum powers depends on the allele frequency. When HWE does not hold, ZMAX3 still possesses its efficiency robustness, but TMAX and TP do not perform as well. Thus, population stratification affects their performance. ZMAX3 also remains more robust than χ 2 2 even when HWE does not hold. From both Table 4 and Table 5, χ 2 2 is more powerful than ZMERT except for the additive model. However, when the genetic model is known, the corresponding optimal CA trend test is more powerful than χ 2 2 with 2 df. From Tables 4 and 5, one sees that the robust test ZMAX3 tends to be more powerful than χ 2 2 under the various scenarios we simulated. Further comparisons of these two test statistics using p-values and the same data given in Table 4 (when r = s = 250) are reported in Table 6. Following Zheng et al. [25], which reported a matched study, where both tests are applied to the same data, the p-values for each test are grouped as < .01, (.01, .05), (.05, .10), and > .10. Cross classification of the p-values are given in Table 6 for allele frequencies p = .1 and .3 under all three genetic models. Table 6 is consistent with the results of Tables 4 and 5, i.e., ZMAX3 is more powerful than the chi-squared test with 2 degrees of freedom when the genetic model is unknown. This is seen by comparing the counts at the upper right corner with the counts at lower left corner. When the counts at the upper right corner are greater than the corresponding counts at the lower left corner, ZMAX3 usually has smaller p-values than χ 2 2. In particular, we compare two tests with p-values < .01 versus p-values in (.01, .05). Notice that in most situations
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Institute of Mathematical Statistic
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Institute of Mathematical Statistic
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iv Contents Copulas and Decoupling
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vi Finally, thanks go the Statistic
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SCIENTIFIC PROGRAM The Second Erich
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x Recent Advances in Longitudinal D
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The Second Lehmann Symposium—Opti
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xiv Richard Davis Colorado State Un
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xvi Matthias Matheas Rice Universit
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xviii Hui Zhao University of Texas
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IMS Lecture Notes-Monograph Series
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Proof. The maximum LR test rejects
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4. Location-scale families On likel
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6. Conclusions On likelihood ratio
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Student’s t-test for scale mixtur
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Optimality in multiple testing 17 w
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Optimality in multiple testing 19 I
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power 0.02 0.03 0.04 0.05 0.06 0.07
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Optimality in multiple testing 23 i
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Optimality in multiple testing 25 (
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Optimality in multiple testing 27 M
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6. Summary Optimality in multiple t
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Optimality in multiple testing 31 [
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On the false discovery proportion 3
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≤ N� � N� P m=1 On the fals
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Since j≤ s, it follows that On th
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0.025 0.02 0.015 0.01 0.005 On the
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Massive multiple hypotheses testing
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P value EDF qdf 0.0 0.2 0.4 0.6 0.8
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Proof. See Appendix. Massive multip
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X1 Massive multiple hypotheses test
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Then π0α ∗ cal π0α ∗ cal +
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Frequentist statistics: theory of i
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2.3. Failure and confirmation Frequ
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together with the probabilistic ass
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Statistical models: problem of spec
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Modeling inequality, spread in mult
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6.3. Part (ii) Put (6.1) (6.2) ˙Γ
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Bayesian transformation hazard mode
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Cumulative Hazard 0.0 0.2 0.4 0.6 0
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Hazard function 0.0 0.5 1.0 1.5 2.0
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Linear prediction after model selec
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Then (3.5) Local asymptotic minimax
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Moment-density estimation 323 may n
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Moment-density estimation 327 Corol
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Moment-density estimation 329 Suppo
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Asymptotics of the MDPDE 337 Lemma
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