Non-inferiority Test
4.1 Statistical Hypothesis and Test Statistics
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will be considered. It has been shown in Chapter 3 that once a test statistic satisfies the convexity condition, there is a great reduction in computation of a confidence-set 𝑝-value. The convexity of the two new-defined Wald test statistics will be justified in later section. On the other hand the estimated 𝑝-value will be applied for this problem. This chapter will end up with nu-merical studies on the type I error rate and power, as well as the sample size formulae of these proposed testing procedures.
4.1 Statistical Hypothesis and Test Statistics
Given some Δ0 > 0, consider the following hypothesis
⎧
⎨
⎩
𝐻03 : 𝜆1 ≤ 𝜆2− Δ0, 𝐻𝑎3 : 𝜆1 > 𝜆2− Δ0.
The null space corresponding 𝐻03 is Ω03= {𝜆1 ≤ 𝜆2 − Δ0}, see Figure 2.1.
The Wald test statistic with respect to the non-inferiority test can be easily derived and has the following form:
𝑍∗ =
𝛿 + Δˆ 0 𝑠𝑒(ˆ𝛿) ,
where ˆ𝛿 = ¯𝑌1 − ¯𝑌2 is the MLE of 𝛿 = 𝜆1 − 𝜆2, and 𝑠𝑒(ˆ𝛿) is obtained by plugging some consistent estimators of 𝜆1, 𝜆2 in the standard error of ˆ𝛿. In this study, two estimators of 𝜆1, 𝜆2 are considered: The unconstrained and constrained MLE. The test statistic with the unconstrained estimator of the standard error can be easily seen and given as
𝑍𝑈∗ =
𝑌¯1− ¯𝑌2+ Δ0
√𝜆ˆ1
𝑛1 +𝜆𝑛ˆ2
2
.
‧
On the other hand, the constrained MLE is solved by maximizing the likeli-hood
𝐿(𝜆1, 𝜆2) = 𝑌1ln 𝜆1 − 𝑛1𝜆1+ 𝑌2ln 𝜆2− 𝑛2𝜆2,
subject to 𝜆1 = 𝜆2 + Δ0. The restricted MLE(RMLE) of 𝜆2 and 𝜆1 can be found as follows (see Appendix A.7 for details),
𝜆˜2 = 1
Consequently, the Wald test statistic with the constrained estimator of the standard error is given as follows,
𝑍𝑅∗ =
Note that in previous chapters, the two Wald test statistics correspondent to the superiority test are shown exactly of the same form when 𝜌 = 1.
However, the property is no longer true with respect to 𝑍𝑅∗ and 𝑍𝑈∗.
4.2 Asymptotic 𝑝-values
First of all, consider the asymptotic testing procedures based on the two asymptotic 𝑝-values at some observed 𝑧𝑅∗, 𝑧𝑈∗,
𝑝𝐴,𝑅∗ = 1 − Φ(𝑧𝑅∗), 𝑝𝐴,𝑈∗ = 1 − Φ(𝑧𝑈∗).
The following theorem gives the asymptotic distributions of 𝑍𝑅∗ and 𝑍𝑈∗ in this Poisson problem. Subsequently, the correspondent asymptotic power
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function and the behavior of type I error rate of two asymptotic tests can be further investigated. Define 𝛿∗ = 𝜆1− 𝜆2 + Δ0 and 𝛿0∗ be the correspondent true value.
Theorem 7. As 𝑛1, 𝑛2 → ∞,
𝑍𝑅∗𝜎∗− 𝜇∗ → 𝑁 (0, 1),𝑑 𝑍𝑈∗ − 𝜇∗ → 𝑁 (0, 1),𝑑 where
𝜎∗2 = (1 + 𝜌)𝜆2− Δ0+ 𝜌𝛿0∗+
√
((1 + 𝜌)𝜆2+ Δ0+ 𝜌𝛿∗0)2− 4𝜆2Δ0(1 + 𝜌) 2((1 + 𝜌)𝜆2− Δ0+ 𝛿∗0) , and
𝜇∗ = 𝛿0∗
√𝜆2(1+𝜌)+𝛿0∗ 𝑛2𝜌
.
By Theorem 7, we can show that the asymptotic tests of 𝑍𝑅∗ and 𝑍𝑈∗ have their power functions as follows,
𝛽¯𝑍𝑅∗(𝛿∗0, 𝜆2, 𝑛2, 𝜌, Δ0) = 1 − Φ(𝑧𝛼𝜎∗− 𝜇∗), (4.1) and
𝛽¯𝑍𝑈 ∗(𝛿∗0, 𝜆2, 𝑛2, 𝜌, Δ0) = 1 − Φ(𝑧𝛼− 𝜇∗), (4.2) approximately.
By (4.1) and (4.2), under 𝛿∗0 = 0, 𝜎∗ = 1, 𝜇∗ = 0, and 𝛽¯𝑍𝑅∗ = ¯𝛽𝑍𝑈 ∗ = 1 − Φ(𝑧𝛼) = 𝛼.
That is, the type I error rates of both tests achieve the significance level 𝛼 at the boundary of the null parameter space. Further by (4.2), we can find
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that the maximal type I error rate of 𝑍𝑈∗ occurred at 𝛿0∗ = 0 and is equal to 𝛼. Hence the asymptotic test based on 𝑍𝑈∗ is a valid test asymptotically.
On the other hand, with a complicated component 𝜎∗ involved, the jus-tification of validity of 𝑍𝑅∗ is less straight forward. By simple algebra the following inequality about 𝜎∗ can be shown,
√
(1 + 𝜌)𝜆2− Δ0+ 𝜌𝛿0∗
(1 + 𝜌)𝜆2− Δ0+ 𝛿0∗ ≤ 𝜎∗. (4.3) When 𝜌 ≤ 1, from (4.3), then 𝜎∗ ≥ 1, and hence 𝑧𝛼𝜎∗ − 𝜇∗ ≥ 𝑧𝛼, for any 𝛿0∗ < 0. We can find that the maximum of ¯𝛽𝑍𝑅∗ occurred at 𝛿∗0 = 0 and is equal to 𝛼. Therefore, the type I error rate of 𝑍𝑅∗ is controlled at the significance level 𝛼, and the correspondent asymptotic 𝑝-value is asymptotically valid whenever 𝜌 ≤ 1. For example, Figure 4.1 gives the plots of the asymptotic type I error rate of 𝑍𝑅∗versus 𝛿0∗ at 𝜆2 = 0.2, 𝑛2 = 2, Δ0 = 0.2𝜆2and 𝛼 = 5%.
The three plots on the left panel are correspondent to 𝜌 = 0.2, 0.5, 0.8. One can see that the type I error rate increases with 𝛿∗0 and the maximum, equal to 𝛼, occurs at the boundary 𝛿∗0 = 0. We further find that as long as 𝜌 is not too unbalanced, the type I error rate can be still controlled. See the right panel of Figure 4.1 for 𝜌 = 1.2, 1.4, 1.6. In contrast, when 𝜌 > 1, the type I error rate can exceed the nominal level 𝛼 especially when 𝜌 is extremely large, and 𝑛2 is relatively small. See the left panel of Figure 4.2 for the type I error rate of 𝑍𝑅∗ with 𝜌 = 1.7, 3, 5 and 𝑛2 = 2. However, as the sample sizes are slightly increased, the inflation of the type I error rate can be successfully improved. In the previous example, if 𝑛2 is increased from 2 to 7, the type I error rates are then controlled within the level 𝛼, see the right panel of Figure 4.2. In summary, the asymptotic test based on 𝑍𝑅∗ is not always valid when the first group is extremely larger than the second group, 𝜌 >> 1, and the group sizes are small.
Next we focus on the investigation on the power function of the two
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asymptotic testing procedures over the alternative parameter space. It can be easily shown that the power function of 𝑍𝑈∗ is always greater than or equal to 𝛼. That is, it is asymptotically unbiased. However, similar to previous results, the performance of 𝑍𝑅∗ is more complicated.
First, we examine the case that 𝜆2 → 0. When one considers that Δ0 is proportional to 𝜆2, the non-inferiority limit approaches to 0 as 𝜆2. Given 𝛿0∗, 𝑛2, we can find that 𝜇∗ →√𝑛2𝜌𝛿0∗, 𝜎∗ →√
𝜌 as 𝜆2 approaches to 0, then we have
𝜆lim2→0
𝛽¯𝑍𝑅∗ = 1 − Φ(𝑧𝛼
√𝜌 −√
𝑛2𝜌𝛿∗0).
As 𝜌 ≤ 1, the limit always exceeds 𝛼. But, it is not necessarily true when 𝜌 > 1. In Figure 4.3, all the power functions ¯𝛽𝑍𝑅∗ are above the level 𝛼 = 5%
when 𝜆2 = 0.02, 𝑛2 = 2 and 𝜌 = 0.2, 0.4, 0.6, 0.8, 1, 1.2. In the left panel of Figure 4.4, we see that the unbiasedness breaks down when 𝜌 exceeds 1.3.
Again, the problem can be improved with a slight increment in the sample size. In this example, the power function becomes no less than 𝛼 when 𝑛2 is increased from 2 to 7, see the right panel of Figure 4.4.
Next, we study the case that 𝜆2 → ∞. It follows that 𝜇∗ → 0, 𝜎∗ → 1 as 𝜆2 approaches to infinite given some 𝛿0∗, 𝑛2. Then, the power converges to
𝜆lim2→∞
𝛽¯𝑍𝑅∗ = 1 − Φ(𝑧𝛼) = 𝛼.
The limit is then independent with 𝜌 as 𝜆2 approaches to infinite. For 𝜆2 = 100, 200, 𝑛2 = 2 and 𝜌 = 0.5, 5, 50, the power is found decreasing as 𝜆2 increases. And, all the powers are above the nominal level and increase as 𝜌 increases, see Figure 4.5.
In summary, while the asymptotic test of 𝑍𝑈∗ is always unbiased, the power of the asymptotic test of 𝑍𝑅∗ may be below the nominal level when 𝜆2 is relatively small and 𝜌 is larger than one.
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Based on power function of a testing procedure, the necessary sample size for achievement of a prespecified power at some alternative setting at signif-icance level can be further determined. Given 𝜌, to achieve a prespecified power level 1 − 𝛽0 at 𝜆2, 𝛿0∗ > 0, the minimal sample size of the second group required for the 𝑍𝑅∗ and 𝑍𝑈∗ at significance level 𝛼 is given as
𝑛2,𝑍𝑅∗ ≥( 𝑧𝛼+ 𝑧𝛽 𝛿∗0
)2
{ 𝜆2(1 + 𝜌) − Δ0+ 𝛿0∗ 𝜌
}
. (4.4)
and,
𝑛2,𝑍𝑈 ∗ ≥( 𝑧𝛼+ 𝑧𝛽 𝛿∗0
)2{ 𝜆2(1 + 𝜌) − Δ0+ 𝛿0∗ 𝜌
}
. (4.5)
respectively. The size of the first group is fund as 𝑛1 = [𝑛2⋅ 𝜌] + 1.
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4.3 Exact 𝑝-values
In testing the superiority, we have found that the confidence-set 𝑝-value has advantage of validity, and the revised estimated 𝑝-value has benefit of con-venient use. Further both have satisfactory performances in numerical stud-ies. Therefore, we adopt the two exact 𝑝-value in testing the non-inferiority.
The exact testing procedures of 𝑍𝑈∗ and 𝑍𝑅∗ based on the correspondent confidence-set 𝑝-value and estimated 𝑝-value are proposed and studied. It is known that the null parameter space of a non-inferiority test is different from that of a superiority test. An exact 𝑝-value is defined as follows, given an observed 𝑧0,
𝑝∗(𝜆1,𝜆2)(𝑧0) = 𝑃 (𝑍 ≥ 𝑧0∣𝐻03: 0 < 𝜆1 ≤ 𝜆2− Δ0)
= ∑
𝑦1≥0
∑
𝑦2≥0
𝑝𝑜𝑖(𝑦1, 𝑛1𝜆1)𝑝𝑜𝑖(𝑦2, 𝑛2𝜆2)𝐼{𝑍≥𝑧0},
where 𝑝𝑜𝑖(𝑦, 𝜆′) is the probability function of Poisson distribution with mean 𝜆′, and 𝑦1, 𝑦2 are possible outcomes of 𝑌1, 𝑌2, respectively.
To solve for the computational difficulty brought by an infinite null pa-rameter space, a confidence-set p-value is considered. The confidence-set 𝑝-value of 𝑍𝑅∗ is presented as follows,
𝑝(𝛾)𝐶𝐼,𝑍
𝑅∗ = sup
(𝜆1,𝜆2)∈𝐶𝛾∗∗
𝑃 (𝑍𝑅∗ ≥ 𝑧𝑅∗∣𝜆1, 𝜆2) + 𝛾,
and the confidence-set 𝑝-value of 𝑍𝑈∗ is presented as follows, 𝑝(𝛾)𝐶𝐼,𝑍
𝑈 ∗ = sup
(𝜆1,𝜆2)∈𝐶∗∗𝛾
𝑃 (𝑍𝑈∗ ≥ 𝑧𝑈∗∣𝜆1, 𝜆2) + 𝛾,
where 𝐶𝛾∗∗ is a 100(1 − 𝛾)% confidence interval of 𝜆1 and 𝜆2 over the null parameter space Ω03. Following from Chapter 3, we first consider the cross product set 𝐶𝛾,0 = (𝐿1, 𝑈1) × (𝐿2, 𝑈2), where (𝐿1, 𝑈1) is the 100√(1 − 𝛾)%
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confidence interval of 𝜆1and (𝐿2, 𝑈2) is the 100√(1 − 𝛾)% confidence interval of 𝜆2. Here, the two exact interval estimates are applied,
(𝐿1, 𝑈1) = 1 2𝑛1
(
𝜒2(1−(1−√1−𝛾)/2, 2𝑌
1), 𝜒2((1−√1−𝛾)/2, 2(𝑌1+1))
) , and
(𝐿2, 𝑈2) = 1 2𝑛2
(
𝜒2(1−(1−√1−𝛾)/2, 2𝑌
2), 𝜒2((1−√1−𝛾)/2, 2(𝑌2+1))
) .
Then 𝐶𝛾,0 is a 100(1−𝛾)% confidence interval of 𝜆1 and 𝜆2in the unrestricted parameter space Ω. Subsequently, the confidence set 𝐶𝛾∗∗ is constructed as the intersection of the cross product set 𝐶𝛾,0 and Ω03. That is,
𝐶𝛾∗∗= 𝐶𝛾,0∩ Ω03= {𝐿1 ≤ 𝜆1 ≤ min(𝑈1, 𝜆2− Δ0), 𝐿2 ≤ 𝜆2 ≤ 𝑈2}.
Note that when the observed interval estimate 𝐶𝛾,0 is completely outside of Ω03, 𝐶𝛾∗∗ is empty. In this case, we define 𝑝𝐶𝐼 = 𝛾 < 𝛼, and reject the null hypothesis 𝐻03.
It is known that once a test statistic satisfies the Barnard convexity con-dition, the computation of the correspondent confidence-set 𝑝-value can be further reduced due to the monotonic property of Poisson distribution. In the following, the two Wald test statistics are investigated to confirm whether they satisfy the Barnard convexity condition.
Theorem 8. 𝑍𝑅∗ satisfy the convexity condition.
The convexity of 𝑍𝑅∗ in Theorem 8 is shown from the monotonicity of 𝑍𝑅∗ with respect to 𝑌1 and 𝑌2, see Appendix A.9. As a consequence, from Theorem 8 and 5 of Chapter 3, the confidence-set 𝑝-value of 𝑍𝑅∗ is evaluated in the boundary of the confidence set 𝐶𝛾∗∗. That is,
𝑝(𝛾)𝐶𝐼,𝑍
𝑅∗ = sup
(𝜆1,𝜆2)∈∂𝐶𝛾∗∗
𝑃 (𝑍𝑅∗ ≥ 𝑧𝑅∗∣𝜆1, 𝜆2) + 𝛾,
‧
where ∂𝐶𝛾∗∗ is the boundary of 𝐶𝛾∗∗. The associated confidence-set 𝑝-value based on 𝑍𝑅∗ can has its computation dramatically reduced. Furthermore, the probabilities 𝑃 (𝑍𝑅∗ > 𝑧𝑅∗ ∣ 𝜆1, 𝜆2) can be shown to be increasing as 𝜆1 increases and 𝜆2 decreases, see proof of Theorem 5 of Chapter 3. Therefore, when 𝐶𝛾∗∗ is not empty, the supremum in 𝑝𝐶𝐼,𝑍𝑅∗ either occurs at the point (𝑈1, 𝐿2) or somewhere on the intersect of 𝐶𝛾,0 and the line 𝜆1 = 𝜆2− Δ0.
Next, to check the convexity condition on 𝑍𝑈∗, we consider the partial derivative of 𝑍𝑈∗ w.r.t. 𝑌1 and 𝑌2 respectively,
Since the numerator and denominator are both positive in (4.7), we can find that the partial derivative of 𝑍𝑈∗ w.r.t. 𝑌2 is negative. Then 𝑍𝑈∗ is decreasing in 𝑌2. But, (4.6) can not be showed always positive because
1 𝑛1(𝑛𝑌1
1 + 𝑌𝑛2
2 − Δ0) + 2𝑌𝑛22
2 < 0 may occurs at some 𝑌1, 𝑌2 in the numerator of (4.6). Hence, one can not conclude the monotonicity of 𝑍𝑈∗ in 𝑌1. Several contour plots of 𝑍𝑈∗ = 𝑘 for 𝑘 ranged from 2 to 10, are given in Figure 4.6-4.9 for 𝑛2 = 10, 𝜌 = 3/5, Δ0 = 0.2, 2. In Figure 4.6, the point marked symbol of star indicates a break down of monotonicity. One can see that the failure of the convexity condition 𝑍𝑈∗ is likely to occur for small 𝑌1, 𝑌2. Further, the content depends on the non-inferiority limit, Δ0 and 𝜌. When Δ0 = 0.2, 𝑍𝑈∗ satisfies the convexity condition in the full sample space, see Figure 4.7. On the other hand, Figure 4.8 and 4.9 are the contour plots for 𝜌 = 1, 5/3 and Δ0 = 2.
Since 𝑍𝑈∗ fails to satisfy the Barnard convexity, the supremum in 𝑝𝐶𝐼,𝑍𝑈 ∗
may occur somewhere outside the boundary of 𝐶𝛾∗∗. However, since it is
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observed that the break-down of convexity is not severe from the numerical study. For simplicity, we suggest to find the confidence-set 𝑝-value of 𝑍𝑈∗ at the boundary ∂𝐶𝛾∗∗,
𝑝(𝛾)𝐶𝐼,𝑍
𝑈 ∗ = sup
(𝜆1,𝜆2)∈∂𝐶∗∗𝛾
𝑃 (𝑍𝑅∗ ≥ 𝑧𝑅∗∣𝜆1, 𝜆2) + 𝛾.
Following Chapter 3.3, we propose a revised estimated 𝑝-value in testing the non-inferiority. The estimated exact 𝑝-values based on 𝑍𝑅∗ and 𝑍𝑈∗ are redefined as
𝑝𝐸,𝑍𝑅∗ = 𝑃 (𝑍𝑅∗ ≥ 𝑧𝑅∗∣˜𝜆13, ˜𝜆23), and,
𝑝𝐸,𝑍𝑈 ∗ = 𝑃 (𝑍𝑈∗ ≥ 𝑧𝑈∗∣˜𝜆13, ˜𝜆23),
respectively. In which, ˜𝜆13 and ˜𝜆23 are some estimators of 𝜆1, 𝜆2 under the restricted null parameter space Ω03. Again, similar to Chapter 3.3, we con-sider a revised RMLE. First, find the unrestricted MLE ˆ𝜆1 and ˆ𝜆2 of 𝜆1 and 𝜆2. If ˆ𝜆1 ≤ ˆ𝜆2− Δ0, then ˆ𝜆1, ˆ𝜆2 are exact the RMLEs under Ω03 and let (˜𝜆13, ˜𝜆23) = (ˆ𝜆1, ˆ𝜆2). If ˆ𝜆1 > ˆ𝜆2−Δ0, we consider the RMLE on the boundary 𝜆1 = 𝜆2− Δ0, that is, (˜𝜆13, ˜𝜆23) = (˜𝜆1, ˜𝜆2). In summary,
(˜𝜆13, ˜𝜆23) =
⎧
⎨
⎩
(ˆ𝜆1, ˆ𝜆2), if ˆ𝜆1 ≤ ˆ𝜆2− Δ0; (˜𝜆1, ˜𝜆2), if ˆ𝜆1 > ˆ𝜆2− Δ0.
In this chapter, the exact 𝑝-value bases on 𝑍𝑅∗ is increasing as 𝜆1 and de-creasing as 𝜆2. respectively. The exact 𝑝-value is an increasing function as (𝜆1, 𝜆2) moves toward at the down-right direction. Hence, adopting the MLE on the boundary leads to a more conservative conclusion. In next section, the performance of these proposed testing procedures will be compared through numerical studies.
As the Wald statistic depends on the data only through the two sufficient
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statistics (𝑌1, 𝑌2), the exact power of the test correspondent to an exact 𝑝-value, 𝑝, is given by
∑
𝑦1≥0
∑
𝑦2≥0
𝑝𝑜𝑖(𝑦1, 𝑛1𝜆1)𝑝𝑜𝑖(𝑦2, 𝑛2𝜆2)𝐼{𝑝≤𝛼},
where 𝑝 is either 𝑝𝐶𝐼, 𝑝𝐸 of 𝑍𝑅∗ or 𝑍𝑈∗. Given a predetermined power level 1 − 𝛽0, at some specific 𝜆2, Δ0, 𝛿0, the required sample size of the second group is the smallest integers such that the exact power achieves level,
𝑛2 = min{𝑛2 : ∑
𝑦1≥0
∑
𝑦2≥0
𝑝𝑜𝑖(𝑦1, 𝑛1𝜆1)𝑝𝑜𝑖(𝑦2, 𝑛2𝜆2)𝐼{𝑝≤𝛼} ≥ 1 − 𝛽0}, (4.8)
for some 𝜌 > 0. Further 𝑛1 = [𝑛2 ⋅ 𝜌] + 1.
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4.4 Numerical Studies
Based on the two test statistics, 𝑍𝑈∗, 𝑍𝑅∗, the asymptotic test using the asymptotic 𝑝-value(denoted as 𝑝𝐴), and the two exact tests using the confidence-set 𝑝-value and the estimated 𝑝-value(denoted as 𝑝𝐸) are explored in this sec-tion. There are two confidence-set 𝑝-values constructed with 𝛾 = 0.001, 0.005, and denoted as 𝑝(𝛾=0.001)𝐶𝐼,⋅ , 𝑝(𝛾=0.001)𝐶𝐼,⋅ , respectively. Because the Wald statistics are function of the two sufficient statistics 𝑌1, 𝑌2, the powers of the associ-ated tests can be directly calculassoci-ated through their sampling distribution. In testing the non-inferiority, the maximal acceptable non-inferiority limit Δ0 is chosen as 0.2𝜆2. We consider 𝑛2 = 10, 𝜌 = 3/5, 1, 5/3, 𝛼 = 0.05. The type I error rate and power of four test procedures are computed at true difference 𝛿0∗ = 𝜆1− 𝜆2+ Δ0 ranged within -0.25 to 2 for 𝜆2 = 1, 2. Table 4.1 - 4.2 show the type I error rate and power calculated. On the other hand, the required sample sizes of the second group to achieve 80% power at 𝛿∗0 = 0.6, 1 are presented in Table 4.3 - 4.8. In which, only the results of the confidence-set 𝑝-value with 𝛾 = 0.001 are reported.
We first compare the two asymptotic tests in Table 4.1 and 4.2. The two tests have monotone increasing power with 𝛿0∗. As 𝛿0∗ ≤ 0, the maximal type I error rates of 𝑍𝑅∗, 𝑍𝑈∗ occur at 𝛿0∗ = 0, the boundary of the null parameter space for testing the non-inferiority. However, the finite sample results in Table 4.1 and 4.2 show that 𝑍𝑅∗ has more chance in rejecting the null hypothesis than 𝑍𝑈∗ when 𝜌 = 3/5 < 1. It is contrary when 𝜌 ≥ 1.
As 𝛿0∗ = 0, 𝜆2 = 1, the type I error rate of 𝑍𝑅∗(𝑍𝑈∗) exceeds the significance level 𝛼 = 0.05 for 𝜌 = 3/5(5/3). As the mean value increases, the inflation of the type I error rate is reduced, but the improvement is not significant. In summary, 𝑍𝑈∗ is liberal as 𝜌 = 5/3, 1, and 𝑍𝑅∗ is liberal as 𝜌 = 3/5, 1.
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Next the two exact 𝑝-values, 𝑝𝐶𝐼, 𝑝𝐸 are investigated. In last section, although we have justified numerically that due to the breakdown to the convexity condition, the supremum in 𝑝𝐶𝐼 of 𝑍𝑈∗ does not guarantee to occur at the boundary of the confidence-set. However, in this thesis, for simplicity we propose to search for the supremum over the boundary of the confidence-set. Here the supremums of the two confidence-set exact 𝑝-values are searched over 16 grids on the boundary of the truncated confidence-set in the null parameter space. From Table 4.1 to 4.2, we discover that the two exact 𝑝-values are always well-controlled at 𝛼 = 0.05. By Table 4.1 and 4.2, we find that the power of 𝑝𝐶𝐼 by using 𝑍𝑅∗ is greater than that of 𝑝𝐶𝐼 by using 𝑍𝑈∗. The trend is not in accordance with that of the asymptotic tests. On the other hand, in applying the estimated 𝑝-value, the two test statistics 𝑍𝑅∗ and 𝑍𝑈∗ generate indifferent performances.
Table 4.3 - 4.8 present the required sample size of the second group for 80% power at 𝛿0∗ = 0.6, 1.0. And, based on the required sample sizes, the type I error rate at 𝛿∗0=-0.2,-0.1,-0.05,0, and the power at 𝛿0∗=0.6 or 1 of these tests are also reported. The results for the two asymptotic tests are based on the asymptotic sample size formulae (4.4) and (4.5). For the two exact tests, the figures are the minimal integers such that the exact power achieves the level by (4.8). All the tests need less sample size when 𝛿0∗ increases, as expected.
Between the two asymptotic tests, 𝑍𝑈∗ needs a slightly smaller sample than 𝑍𝑅∗ for 𝜌 > 1. It is the contrary as 𝜌 < 1. On the other hand, we find that the exact type I error rate of two asymptotic tests often exceeds the nominal level 𝛼 = 5% as 𝜌 ≥ 1. The inflation is more severe in the application of 𝑍𝑈∗.
With the calculated sample size, every exact test achieves the prespecified power level and has a well-controlled type I error rate. Similarly, for the application of testing inferiority,the asymptotic sample sizes (4.4) and (4.5) can be regarded as an efficient alternative of (4.8) for the exact tests. A
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much quicker solution can be obtained and the result is found to be close to the exact sample size.
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Table 4.1: Type I error rate and power of asymptotic 𝑝-value and exact 𝑝-value at 𝜆2 = 1, 𝑛2 = 10, these 𝑝-values are based on test statistics 𝑍𝑅∗, 𝑍𝑈∗ respectively.
Test 𝛿∗0
𝜌 Statistic 𝑝-value -0.25 -0.15 -0.1 -0.05 0.0 0.5 1.0 1.5 2.0
3/5 𝑍𝑅∗ 𝑝𝐴,𝑅∗ 0.0140 0.0262 0.0344 0.0442 0.0557 0.2542 0.5301 0.7630 0.9024 𝑝(𝛾=0.001)𝐶𝐼,𝑅∗ 0.0098 0.0185 0.0245 0.0319 0.0406 0.2182 0.5010 0.7474 0.8951
𝑝(𝛾=0.005)𝐶𝐼,𝑅∗ 0.0096 0.0179 0.0237 0.0306 0.0388 0.2062 0.4841 0.7352 0.8886 𝑝𝐸,𝑅∗ 0.0103 0.0198 0.0264 0.0345 0.0441 0.2318 0.5136 0.7531 0.8968
𝑍𝑈 ∗ 𝑝𝐴,𝑈 ∗ 0.0096 0.0177 0.0233 0.0300 0.0380 0.1978 0.4684 0.7223 0.8817 𝑝(𝛾=0.001)𝐶𝐼,𝑈 ∗ 0.0098 0.0185 0.0245 0.0319 0.0406 0.2182 0.5010 0.7474 0.8951
𝑝(𝛾=0.005)𝐶𝐼,𝑈 ∗ 0.0096 0.0177 0.0233 0.0300 0.0380 0.1979 0.4692 0.7242 0.8838 𝑝𝐸,𝑈 ∗ 0.0103 0.0198 0.0264 0.0345 0.0441 0.2318 0.5136 0.7531 0.8968
1 𝑍𝑅∗ 𝑝𝐴,𝑅∗ 0.0121 0.0233 0.0311 0.0405 0.0517 0.2726 0.6027 0.8465 0.9559
𝑝(𝛾=0.001)
𝐶𝐼,𝑅∗ 0.0112 0.0212 0.0282 0.0368 0.0471 0.2614 0.5876 0.8361 0.9523 𝑝(𝛾=0.005)
𝐶𝐼,𝑅∗ 0.0095 0.0185 0.0249 0.0329 0.0425 0.2482 0.5750 0.8279 0.9486 𝑝𝐸,𝑅∗ 0.0113 0.0212 0.0282 0.0368 0.0471 0.2617 0.5905 0.8404 0.9545
𝑍𝑈 ∗ 𝑝𝐴,𝑈 ∗ 0.0134 0.0245 0.0322 0.0415 0.0526 0.2727 0.6027 0.8465 0.9559
𝑝(𝛾=0.001)
𝐶𝐼,𝑈 ∗ 0.0095 0.0185 0.0249 0.0329 0.0425 0.2490 0.5802 0.8346 0.9521 𝑝(𝛾=0.005)𝐶𝐼,𝑈 ∗ 0.0080 0.0161 0.0219 0.0292 0.0383 0.2381 0.5627 0.8228 0.9477 𝑝𝐸,𝑈 ∗ 0.0113 0.0213 0.0282 0.0368 0.0472 0.2659 0.5983 0.8420 0.9537
5/3 𝑍𝑅∗ 𝑝𝐴,𝑅∗ 0.0090 0.0185 0.0254 0.0342 0.0449 0.2833 0.6603 0.9006 0.9807 𝑝(𝛾=0.001)𝐶𝐼,𝑅∗ 0.0092 0.0186 0.0256 0.0343 0.0449 0.2832 0.6588 0.8958 0.9776
𝑝(𝛾=0.005)𝐶𝐼,𝑅∗ 0.0087 0.0174 0.0238 0.0317 0.0416 0.2774 0.6451 0.8854 0.9756 𝑝𝐸,𝑅 0.0106 0.0210 0.0285 0.0379 0.0493 0.2911 0.6614 0.8997 0.9797
𝑍𝑈 ∗ 𝑝𝐴,𝑈 ∗ 0.0155 0.0294 0.0390 0.0508 0.0648 0.3346 0.7035 0.9187 0.9851 𝑝(𝛾=0.001)𝐶𝐼,𝑈 ∗ 0.0064 0.0140 0.0198 0.0273 0.0369 0.2774 0.6583 0.8958 0.9776
𝑝(𝛾=0.005)𝐶𝐼,𝑈 ∗ 0.0064 0.0140 0.0198 0.0273 0.0368 0.2676 0.6281 0.8810 0.9753 𝑝𝐸,𝑈 ∗ 0.0106 0.0210 0.0285 0.0379 0.0493 0.2911 0.6615 0.9006 0.9806
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Table 4.2: Type I error rate and power of asymptotic 𝑝-value and exact 𝑝-value at 𝜆2 = 2, 𝑛2 = 10, these 𝑝-values are based on test statistics 𝑍𝑅∗, 𝑍𝑈∗ respectively.
Test 𝛿
𝜌 Statistic 𝑝-value -0.25 -0.15 -0.1 -0.05 0.0 0.5 1 1.5 2
3/5 𝑍𝑅∗ 𝑝𝐴,𝑅∗ 0.0228 0.0327 0.0387 0.0454 0.0529 0.1759 0.3738 0.5917 0.7712 𝑝(𝛾=0.001)𝐶𝐼,𝑅∗ 0.0170 0.0251 0.0302 0.0360 0.0425 0.1575 0.3535 0.5740 0.7584
𝑝(𝛾=0.005)𝐶𝐼,𝑅∗ 0.0165 0.0244 0.0292 0.0348 0.0410 0.1510 0.3413 0.5600 0.7466 𝑝𝐸,𝑅∗ 0.0209 0.0302 0.0358 0.0422 0.0493 0.1667 0.3598 0.5768 0.7593 𝑍𝑈 ∗ 𝑝𝐴,𝑈 ∗ 0.0176 0.0257 0.0307 0.0363 0.0427 0.1510 0.3364 0.5530 0.7408 𝑝(𝛾=0.001)𝐶𝐼,𝑈 ∗ 0.0170 0.0251 0.0302 0.0360 0.0425 0.1575 0.3535 0.5740 0.7584
𝑝(𝛾=0.005)𝐶𝐼,𝑈 ∗ 0.0153 0.0225 0.0270 0.0321 0.0379 0.1417 0.3282 0.5491 0.7403 𝑝𝐸,𝑈 ∗ 0.0209 0.0302 0.0358 0.0422 0.0493 0.1667 0.3598 0.5768 0.7593 1 𝑍𝑅∗ 𝑝𝐴,𝑅∗ 0.0183 0.0277 0.0336 0.0404 0.0482 0.1895 0.4297 0.6782 0.8539
𝑝(𝛾=0.001)
𝐶𝐼,𝑅∗ 0.0178 0.0271 0.0329 0.0395 0.0471 0.1837 0.4184 0.6673 0.8473 𝑝(𝛾=0.005)
𝐶𝐼,𝑅∗ 0.0159 0.0243 0.0296 0.0357 0.0427 0.1730 0.4046 0.6554 0.8396 𝑝𝐸,𝑅∗ 0.0183 0.0277 0.0336 0.0404 0.0481 0.1884 0.4256 0.6727 0.8506 𝑍𝑈 ∗ 𝑝𝐴,𝑈 ∗ 0.0202 0.0303 0.0366 0.0438 0.0520 0.1951 0.4327 0.6791 0.8543 𝑝(𝛾=0.001)𝐶𝐼,𝑈 ∗ 0.0168 0.0259 0.0317 0.0383 0.0459 0.1831 0.4183 0.6673 0.8473
𝑝(𝛾=0.005)
𝐶𝐼,𝑈 ∗ 0.0157 0.0241 0.0294 0.0356 0.0426 0.1730 0.4046 0.6554 0.8396 𝑝𝐸,𝑈 ∗ 0.0183 0.0277 0.0336 0.0404 0.0481 0.1884 0.4256 0.6727 0.8506 5/3 𝑍𝑅∗ 𝑝𝐴,𝑅∗ 0.0159 0.0251 0.0311 0.0381 0.0462 0.2029 0.4733 0.7366 0.9008 𝑝(𝛾=0.001)𝐶𝐼,𝑅∗ 0.0164 0.0256 0.0315 0.0385 0.0466 0.2030 0.4733 0.7368 0.9014
𝑝(𝛾=0.005)𝐶𝐼,𝑅∗ 0.0153 0.0243 0.0300 0.0366 0.0442 0.1854 0.4437 0.7195 0.8960 𝑝𝐸,𝑅∗ 0.0171 0.0264 0.0324 0.0394 0.0474 0.2032 0.4735 0.7382 0.9037 𝑍𝑈 ∗ 𝑝𝐴,𝑈 ∗ 0.0220 0.0333 0.0404 0.0487 0.0583 0.2314 0.5096 0.7644 0.9153
𝑝(𝛾=0.001)
𝐶𝐼,𝑈 ∗ 0.0155 0.0248 0.0308 0.0379 0.0461 0.2029 0.4733 0.7368 0.9014 𝑝(𝛾=0.005)𝐶𝐼,𝑈 ∗ 0.0143 0.0227 0.0280 0.0343 0.0415 0.1809 0.4384 0.7103 0.8899 𝑝𝐸,𝑈 ∗ 0.0164 0.0256 0.0315 0.0385 0.0466 0.2030 0.4735 0.7382 0.9037
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of the second group 𝑛2 of the asymptotic 𝑝-values and exact 𝑝-value which are conducted at 𝑍𝑅∗, 𝑍𝑈∗. Based on the required samples 𝑛2, the power and the type I error rate (in parentheses) are given at various 𝛿∗0 in Ω03 .Test 𝜆2
Statistic 𝑝-value 𝛿∗0 0.3 0.4 0.6 1 2
𝑍𝑅∗ 𝑝𝐴,𝑅∗ 𝑛2 25 29 37 53 93
0.6 0.8132 0.8125 0.8076 0.7969 0.7960
0 (0.0485) (0.0527) (0.0519) (0.0507) (0.0506) -0.05 (0.0237) (0.0284) (0.0291) (0.0300) (0.0312) -0.1 (0.0093) (0.0134) (0.0148) (0.0166) (0.0182) -0.2 (0.0004) (0.0016) (0.0026) (0.0039) (0.0052)
𝑃𝐶𝐼,𝑅∗(𝛾=0.005) 27 32 39 58 100
0.6 0.8166 0.8084 0.8003 0.8103 0.8067
0 (0.0439) (0.0437) (0.0444) (0.0445) (0.0446) -0.05 (0.0207) (0.0221) (0.0239) (0.0253) (0.0266) -0.1 (0.0077) (0.0094) (0.0116) (0.0133) (0.0149) -0.2 (0.0003) (0.0007) (0.0018) (0.0028) (0.0039)
𝑃𝐸,𝑅∗ 25 29 39 55 98
0.6 0.8039 0.8009 0.8113 0.8099 0.8125
0 (0.0485) (0.0474) (0.0487) (0.0500) (0.0501) -0.05 (0.0237) (0.0245) (0.0266) (0.0291) (0.0304) -0.1 (0.0093) (0.0108) (0.0130) (0.0157) (0.0174) -0.2 (0.0007) (0.0011) (0.0020) (0.0035) (0.0047)
𝑍𝑈 ∗ 𝑝𝐴,𝑈 ∗ 𝑛2 29 33 41 57 97
0.6 0.8298 0.8191 0.8125 0.8096 0.8035
0 (0.0398) (0.0408) (0.0418) (0.0455) (0.0477) -0.05 (0.0182) (0.0205) (0.0222) (0.0260) (0.0289) -0.1 (0.0064) (0.0089) (0.0105) (0.0137) (0.0165) -0.2 (0.0002) (0.0008) (0.0015) (0.0029) (0.0045)
𝑃𝐶𝐼,𝑈 ∗(𝛾=0.005) 27 32 40 58 100
0.6 0.8087 0.8157 0.8131 0.8079 0.8067
0 (0.0439) (0.0416) (0.0432) (0.0443) (0.0446) -0.05 (0.0207) (0.0203) (0.0230) (0.0252) (0.0266) -0.1 (0.0077) (0.0082) (0.0109) (0.0133) (0.0149) -0.2 (0.0003) (0.0006) (0.0016) (0.0028) (0.0039)
𝑃𝐸,𝑈 ∗ 25 29 39 55 97
0.6 0.8039 0.8009 0.8113 0.8099 0.8105
0 (0.0484) (0.0474) (0.0487) (0.0500) (0.0501) -0.05 (0.0236) (0.0245) (0.0266) (0.0291) (0.0304) -0.1 (0.0091) (0.0108) (0.0129) (0.0157) (0.0175) -0.2 (0.0002) (0.0011) (0.0020) (0.0035) (0.0048)
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of the second group 𝑛2 of the asymptotic 𝑝-values and exact 𝑝-value which are conducted at 𝑍𝑅∗, 𝑍𝑈∗. Based on the required samples 𝑛2, the power and the type I error rate (in parentheses) are given at various 𝛿∗0 in Ω03.Test 𝜆2
Statistic 𝑝-value 𝛿∗0 0.3 0.4 0.6 1 2
𝑍𝑅∗ 𝑝𝐴,𝑅∗ 𝑛2 19 23 29 41 72
0.6 0.8103 0.8152 0.8038 0.7976 0.7975
0 (0.0501) (0.0472) (0.0509) (0.0494) (0.0500) -0.05 (0.0260) (0.0247) (0.0287) (0.0293) (0.0308) -0.1 (0.0116) (0.0115) (0.0147) (0.0162) (0.0180) -0.2 (0.0007) (0.0016) (0.0026) (0.0039) (0.0052)
𝑃𝐶𝐼,𝑅∗(𝛾=0.005) 20 24 30 44 79
0.6 0.8020 0.8102 0.8009 0.8081 0.8161
0 (0.0412) (0.0439) (0.0440) (0.0444) (0.0448) -0.05 (0.0201) (0.0228) (0.0242) (0.0256) (0.0266) -0.1 (0.0079) (0.0103) (0.0121) (0.0137) (0.0149) -0.2 (0.0005) (0.0012) (0.0021) (0.0030) (0.0039)
𝑃𝐸,𝑅∗ 20 23 32 44 78
0.6 0.8154 0.8118 0.8389 0.8225 0.8253
0 (0.0480) (0.0458) (0.0498) (0.0492) (0.0498) -0.05 (0.0244) (0.0243) (0.0273) (0.0286) (0.0300) -0.1 (0.0106) (0.0113) (0.0135) (0.0154) (0.0171) -0.2 (0.0017) (0.0014) (0.0022) (0.0035) (0.0046)
𝑍𝑈 ∗ 𝑝𝐴,𝑈 ∗ 𝑛2 19 22 28 41 72
0.6 0.8103 0.8045 0.7967 0.8023 0.8002
0 (0.0503) (0.0537) (0.0539) (0.0517) (0.0507) -0.05 (0.0263) (0.0301) (0.0313) (0.0308) (0.0313) -0.1 (0.0122) (0.0149) (0.0167) (0.0171) (0.0183) -0.2 (0.0019) (0.0021) (0.0034) (0.0042) (0.0053)
𝑃𝐶𝐼,𝑈 ∗(𝛾=0.005) 21 24 30 44 78
0.6 0.8122 0.8089 0.8008 0.8081 0.8115
0 (0.0221) (0.0356) (0.0416) (0.0435) (0.0448) -0.05 (0.0087) (0.0175) (0.0226) (0.0249) (0.0267) -0.1 (0.0031) (0.0074) (0.0111) (0.0133) (0.0150) -0.2 (0.0004) (0.0007) (0.0018) (0.0029) (0.0040)
𝑃𝐸,𝑈 ∗ 20 23 29 42 75
0.6 0.8154 0.8118 0.8035 0.8051 0.8123
0 (0.0451) (0.0454) (0.0485) (0.0494) (0.0498) -0.05 (0.0210) (0.0236) (0.0275) (0.0290) (0.0303) -0.1 (0.0076) (0.0106) (0.0142) (0.0159) (0.0175) -0.2 (0.0001) (0.0010) (0.0026) (0.0038) (0.0049)
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of the second group 𝑛2 of the asymptotic 𝑝-values and exact 𝑝-value which are conducted at 𝑍𝑅∗, 𝑍𝑈∗. Based on the required samples 𝑛2, the power and the type I error rate (in parentheses) are given at various 𝛿∗0 in Ω03.Test 𝜆2
Statistic 𝑝-value 𝛿∗0 0.3 0.4 0.6 1 2
𝑍𝑅∗ 𝑝𝐴,𝑅∗ 𝑛2 16 19 24 34 60
0.6 0.8026 0.8076 0.8034 0.7949 0.8009
0 (0.0389) (0.0444) (0.0472) (0.0504) (0.0489) -0.05 (0.0198) (0.0234) (0.0267) (0.0302) (0.0301) -0.1 (0.0093) (0.0108) (0.0138) (0.0169) (0.0176) -0.2 (0.0007) (0.0014) (0.0028) (0.0042) (0.0051)
𝑃𝐶𝐼,𝑅∗(𝛾=0.005) 17 20 25 36 64
0.6 0.8187 0.8171 0.8062 0.8052 0.8102
0 (0.0394) (0.0434) (0.0421) (0.0422) (0.0448) -0.05 (0.0192) (0.0227) (0.0238) (0.0243) (0.0269) -0.1 (0.0084) (0.0103) (0.0124) (0.0131) (0.0152) -0.2 (0.0006) (0.0012) (0.0023) (0.0031) (0.0041)
𝑃𝐸,𝑅∗ 16 19 24 35 63
0.6 0.8109 0.8146 0.8119 0.8089 0.8197
0 (0.0462) (0.0460) (0.0499) (0.0499) (0.0499) -0.05 (0.0232) (0.0254) (0.0286) (0.0296) (0.0304) -0.1 (0.0103) (0.0126) (0.0150) (0.0164) (0.0175) -0.2 (0.0036) (0.0017) (0.0030) (0.0040) (0.0049)
𝑍𝑈 ∗ 𝑝𝐴,𝑈 ∗ 𝑛2 13 16 21 31 57
0.6 0.7797 0.7871 0.7866 0.7846 0.7960
0 (0.0891) (0.0733) (0.0630) (0.0595) (0.0543) -0.05 (0.0574) (0.0469) (0.0388) (0.0372) (0.0342) -0.1 (0.0347) (0.0279) (0.0222) (0.0219) (0.0205) -0.2 (0.0116) (0.0071) (0.0056) (0.0062) (0.0063)
𝑃𝐶𝐼,𝑈 ∗(𝛾=0.005) 17 20 25 36 63
0.6 0.8064 0.8153 0.8062 0.8052 0.8054
0 (0.0203) (0.0347) (0.0388) (0.0421) (0.0444) -0.05 (0.0095) (0.0165) (0.0213) (0.0242) (0.0267) -0.1 (0.0059) (0.0062) (0.0105) (0.0130) (0.0153) -0.2 (0.0019) (0.0003) (0.0017) (0.0029) (0.0042)
𝑃𝐸,𝑈 ∗ 16 19 24 35 62
0.6 0.8027 0.8146 0.8119 0.8089 0.8140
0 (0.0382) (0.0460) (0.0486) (0.0499) (0.0497) -0.05 (0.0186) (0.0253) (0.0273) (0.0296) (0.0304) -0.1 (0.0076) (0.0126) (0.0140) (0.0164) (0.0176) -0.2 (0.0002) (0.0015) (0.0028) (0.0040) (0.0050)
‧
of the second group 𝑛2 of the asymptotic 𝑝-values and exact 𝑝-value which are conducted at 𝑍𝑅∗, 𝑍𝑈∗. Based on the required samples 𝑛2, the power and the type I error rate (in parentheses) are given at various 𝛿∗0 in Ω03.Test 𝜆2
Statistic 𝑝-value 𝛿∗0 0.3 0.4 0.6 1 2
𝑍𝑅∗ 𝑝𝐴,𝑅∗ 𝑛2 12 13 16 22 36
1.0 0.8492 0.7810 0.7918 0.7993 0.7883
0 (0.0608) (0.0466) (0.0572) (0.0509) (0.0512) -0.05 (0.0364) (0.0309) (0.0408) (0.0364) (0.0381) -0.1 (0.0185) (0.0191) (0.0278) (0.0252) (0.0277) -0.2 (0.0012) (0.0057) (0.0108) (0.0108) (0.0136)
𝑃𝐶𝐼,𝑅∗(𝛾=0.005) 14 15 18 24 40
1.0 0.8347 0.8286 0.8001 0.8050 0.8143
0 (0.0371) (0.0373) (0.0437) (0.0444) (0.0444) -0.05 (0.0228) (0.0243) (0.0293) (0.0310) (0.0321) -0.1 (0.0122) (0.0146) (0.0184) (0.0208) (0.0226) -0.2 (0.0000) (0.0033) (0.0056) (0.0082) (0.0104)
𝑃𝐸,𝑅∗ 13 14 17 24 38
1.0 0.8120 0.8050 0.8098 0.8189 0.8007
0 (0.0407) (0.0451) (0.0510) (0.0493) (0.0502) -0.05 (0.0230) (0.0289) (0.0340) (0.0349) (0.0370) -0.1 (0.0106) (0.0167) (0.0212) (0.0238) (0.0266) -0.2 (0.0003) (0.0031) (0.0062) (0.0098) (0.0128)
𝑍𝑈 ∗ 𝑝𝐴,𝑈 ∗ 𝑛2 14 16 18 24 39
1.0 0.8124 0.8317 0.7960 0.8037 0.8042
0 (0.0262) (0.0451) (0.0359) (0.0424) (0.0453) -0.05 (0.0147) (0.0281) (0.0230) (0.0295) (0.0330) -0.1 (0.0073) (0.0159) (0.0138) (0.0198) (0.0234) -0.2 (0.0006) (0.0034) (0.0038) (0.0079) (0.0109)
𝑃𝐶𝐼,𝑈 ∗(𝛾=0.005) 13 15 18 24 40
1.0 0.8046 0.8285 0.8083 0.8042 0.8129
0 (0.0365) (0.0268) (0.0412) (0.0424) (0.0444) -0.05 (0.0225) (0.0154) (0.0269) (0.0295) (0.0321) -0.1 (0.0120) (0.0080) (0.0165) (0.0198) (0.0226) -0.2 (0.0009) (0.0014) (0.0048) (0.0079) (0.0104)
𝑃𝐸,𝑈 ∗ 12 14 17 24 38
1.0 0.8059 0.8095 0.8098 0.8189 0.8007
0 (0.0226) (0.0420) (0.0510) (0.0493) (0.0502) -0.05 (0.0114) (0.0264) (0.0340) (0.0349) (0.0370) -0.1 (0.0046) (0.0152) (0.0212) (0.0238) (0.0266) -0.2 (0.0001) (0.0029) (0.0062) (0.0098) (0.0128)
‧
of the second group 𝑛2 of the asymptotic 𝑝-values and exact 𝑝-value which are conducted at 𝑍𝑅∗, 𝑍𝑈∗. Based on the required samples 𝑛2, the power and the type I error rate (in parentheses) are given at various 𝛿∗0 in Ω03.Test 𝜆2
Statistic 𝑝-value 𝛿∗0 0.3 0.4 0.6 1 2
𝑍𝑅∗ 𝑝𝐴,𝑅∗ 𝑛2 9 10 13 17 28
1.0 0.8152 0.8030 0.8149 0.7985 0.7936
0 (0.0391) (0.0504) (0.0513) (0.0478) (0.0498) -0.05 (0.0225) (0.0335) (0.0351) (0.0342) (0.0370) -0.1 (0.0108) (0.0212) (0.0227) (0.0238) (0.0269) -0.2 (0.0004) (0.0070) (0.0077) (0.0103) (0.0133)
𝑃𝐶𝐼,𝑅∗(𝛾=0.005) 10 11 14 18 30
1.0 0.8318 0.8083 0.8246 0.8009 0.8038
0 (0.0386) (0.0369) (0.0423) (0.0440) (0.0446) -0.05 (0.0221) (0.0227) (0.0287) (0.0311) (0.0326) -0.1 (0.0106) (0.0128) (0.0186) (0.0212) (0.0232) -0.2 (0.0004) (0.0025) (0.0063) (0.0089) (0.0110)
𝑃𝐸,𝑅∗ 10 10 13 18 29
1.0 0.8446 0.8010 0.8141 0.8174 0.8064
0 (0.0287) (0.0377) (0.0465) (0.0492) (0.0496) -0.05 (0.0138) (0.0237) (0.0320) (0.0352) (0.0366) -0.1 (0.0050) (0.0137) (0.0210) (0.0243) (0.0264) -0.2 (0.0000) (0.0028) (0.0074) (0.0105) (0.0128)
𝑍𝑈 ∗ 𝑝𝐴,𝑈 ∗ 𝑛2 9 10 12 17 28
1.0 0.8163 0.8032 0.7950 0.8038 0.7969
0 (0.0572) (0.0658) (0.0577) (0.0532) (0.0515) -0.05 (0.0403) (0.0478) (0.0417) (0.0387) (0.0384) -0.1 (0.0259) (0.0336) (0.0291) (0.0273) (0.0280) -0.2 (0.0034) (0.0155) (0.0126) (0.0124) (0.0139)
𝑃𝐶𝐼,𝑈 ∗(𝛾=0.005) 10 11 14 18 30
1.0 0.8154 0.8025 0.8246 0.8008 0.8038
0 (0.0385) (0.0314) (0.0373) (0.0416) (0.0446) -0.05 (0.0221) (0.0217) (0.0238) (0.0291) (0.0326) -0.1 (0.0106) (0.0145) (0.0141) (0.0198) (0.0232) -0.2 (0.0004) (0.0053) (0.0036) (0.0081) (0.0110)
𝑃𝐸,𝑈 ∗ 10 10 13 18 29
1.0 0.8446 0.8010 0.8091 0.8174 0.8064
0 (0.0287) (0.0377) (0.0455) (0.0488) (0.0496) -0.05 (0.0138) (0.0237) (0.0316) (0.0347) (0.0366) -0.1 (0.0050) (0.0137) (0.0208) (0.0239) (0.0264) -0.2 (0.0000) (0.0028) (0.0073) (0.0100) (0.0128)
‧
of the second group 𝑛2 of the asymptotic 𝑝-values and exact 𝑝-value which are conducted at 𝑍𝑅∗, 𝑍𝑈∗. Based on the required samples 𝑛2, the power and the type I error rate (in parentheses) are given at various 𝛿∗0 in Ω03.Test 𝜆2
Statistic 𝑝-value 𝛿∗0 0.3 0.4 0.6 1 2
𝑍𝑅∗ 𝑝𝐴,𝑅∗ 𝑛2 8 9 10 14 23
1.0 0.8221 0.8264 0.7806 0.7906 0.7896
0 (0.0380) (0.0446) (0.0427) (0.0465) (0.0479) -0.05 (0.0223) (0.0288) (0.0301) (0.0334) (0.0359) -0.1 (0.0103) (0.0169) (0.0206) (0.0233) (0.0263) -0.2 (0.0002) (0.0032) (0.0084) (0.0104) (0.0133)
𝑃𝐶𝐼,𝑅∗(𝛾=0.005) 8 9 11 15 25
1.0 0.8136 0.8086 0.8082 0.8011 0.8072
0 (0.0378) (0.0352) (0.0420) (0.0426) (0.0443) -0.05 (0.0223) (0.0227) (0.0289) (0.0304) (0.0323) -0.1 (0.0103) (0.0138) (0.0191) (0.0210) (0.0230) -0.2 (0.0002) (0.0030) (0.0073) (0.0091) (0.0109)
𝑃𝐸,𝑅∗ 8 9 11 15 24
1.0 0.8348 0.8296 0.8242 0.8242 0.8080
0 (0.0380) (0.0446) (0.0489) (0.0485) (0.0497) -0.05 (0.0223) (0.0288) (0.0336) (0.0345) (0.0366) -0.1 (0.0103) (0.0169) (0.0220) (0.0239) (0.0263) -0.2 (0.0002) (0.0032) (0.0078) (0.0104) (0.0127)
𝑍𝑈 ∗ 𝑝𝐴,𝑈 ∗ 𝑛2 6 7 9 12 22
1.0 0.7863 0.7829 0.7941 0.7721 0.7916
0 (0.1250) (0.0864) (0.0728) (0.0631) (0.0567) -0.05 (0.0975) (0.0724) (0.0546) (0.0482) (0.0432) -0.1 (0.0684) (0.0619) (0.0399) (0.0361) (0.0323) -0.2 (0.0102) (0.0450) (0.0192) (0.0188) (0.0171)
𝑃𝐶𝐼,𝑈 ∗(𝛾=0.005) 9 10 12 16 25
1.0 0.8324 0.8326 0.8323 0.8151 0.8040
0 (0.0467) (0.0284) (0.0329) (0.0403) (0.0443) -0.05 (0.0377) (0.0204) (0.0205) (0.0278) (0.0323) -0.1 (0.0291) (0.0146) (0.0116) (0.0186) (0.0230) -0.2 (0.0069) (0.0056) (0.0025) (0.0075) (0.0109)
𝑃𝐸,𝑈 ∗ 8 9 11 15 24
1.0 0.8262 0.8296 0.8242 0.8243 0.8080
0 (0.0378) (0.0446) (0.0489) (0.0485) (0.0497) -0.05 (0.0223) (0.0288) (0.0336) (0.0345) (0.0366) -0.1 (0.0103) (0.0169) (0.0219) (0.0239) (0.0263) -0.2 (0.0002) (0.0032) (0.0076) (0.0104) (0.0127)
‧
國立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
Figure 4.1: As 𝑛2 = 2, 𝜆2 = 0.2, Δ0 = 0.2𝜆2, 𝜌 = 0.2, 0.5, 0.8, 1.2, 1.4, 1.6, 𝛿0 =
−0.16 : 0.001 : 0, the asymptotic type I error rate of the 𝑍𝑅∗(solid line).
Figure 4.2: As 𝑛2 = 2, 7, 𝜆2 = 0.2, Δ0 = 0.2𝜆2, 𝜌 = 1.7, 3, 5, 𝛿0∗ = −0.16 : 0.001 : 0, the asymptotic type I error rate of the 𝑍𝑅∗(solid line).
‧
國立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
Figure 4.3: As 𝑛2 = 2, 𝜆2 = 0.02, Δ0 = 0.2𝜆2, 𝜌 = 0.2, 0.4, 0.6, 0.8, 1, 1.2, 𝛿∗0 = 0 : 0.001 : 0.05, the asymptotic power of the 𝑍𝑅∗(solid line).
Figure 4.4: As 𝑛2 = 2, 7, 𝜆2 = 0.02, Δ0 = 0.2𝜆2, 𝜌 = 1.3, 1.6, 2, 𝛿0∗ = 0 : 0.001 : 0.05, the asymptotic power of the 𝑍𝑅∗(solid line).
‧
國立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
Figure 4.5: As 𝑛2 = 2, 𝜆2 = 100, 200, Δ0 = 0.2𝜆2, 𝜌 = 0.5, 5, 50, 𝛿0∗ = 0 : 1 : 10, the asymptotic power of the 𝑍𝑅∗(solid line).
Figure 4.6: As 𝑛2 = 10, Δ0 = 2, 𝜌 = 0.6, a contour map of 𝑍𝑈∗ = 2, 3, 4, 5, 6, 7, 8, 9, 10.
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國立 政 治 大 學
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N a tio na
l C h engchi U ni ve rs it y
Figure 4.7: As 𝑛2 = 10; Δ0 = 0.2; 𝜌 = 0.6, a contour map of 𝑍𝑈∗ = 𝑘 for 𝑘 = 2, 3, 4.
Figure 4.8: As 𝑛2 = 10; Δ0 = 2; 𝜌 = 1, a contour map of 𝑍𝑈∗ = 𝑘 for 𝑘 = 2, 3, 4, 5, 6, 7, 8, 9, 10.
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國立 政 治 大 學
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N a tio na
l C h engchi U ni ve rs it y
Figure 4.9: As 𝑛2 = 10; Δ0 = 2; 𝜌 = 5/3, a some contour map of 𝑍𝑈∗ = 𝑘 for 𝑘 = 2, 3, 4, 5, 6, 7, 8, 9, 10.
‧
國立 政 治 大 學