Coverage Probability of the Credible Interval and Credible Probability of the Conﬁdence Interval of the Hierarchical Normal Model

British Journal of Mathematics & Computer Science 20(4): 1-11, 2017, Article no.BJMCS.31816 ISSN: 2231-0851

SCIENCEDOMAIN international www.sciencedomain.org

∗ Ying-Ying Zhang1 and Teng-Zhong Rong1

1Department of Statistics and Actuarial Science, College of Mathematics and Statistics, Chongqing University, Chongqing, China.

Authors’ contributions

This work was carried out in collaboration between both authors. Author YYZ proved the two theorems, and wrote the ﬁrst draft of the manuscript. Author TZR did literature searches and revised the manuscript. Both authors read and approved the ﬁnal manuscript.

Article Information DOI: 10.9734/BJMCS/2017/31816 Editor(s): (1) H. M. Srivastava, Department of Mathematics and Statistics, University of Victoria, Canada. Reviewers: (1) Steven T. Garren, James Madison University, Harrisonburg, VA , USA. (2) Diana Bilkova, University of Economics, Prague, Czech Republic. Complete Peer review History: http://www.sciencedomain.org/review-history/17846

Received: 26th January 2017 Accepted: 9th February 2017 Original Research Article Published: 15th February 2017

Abstract

It is well known that the coverage probability of a given nominal level confidence interval and the credible probability of a given nominal level credible interval will attain the nominal level. Moreover, it is commonly believed that the two switching concepts probabilities, that is, the coverage probability of a given nominal level credible interval and the credible probability of a given nominal level confidence interval, can not attain the nominal level in general. For the hierarchical normal model, we show that the two switching concepts probabilities can attain the nominal level in the limit when a skillful classified variable is infinity. The numerical simulations illustrate the correctness of our findings.

*Corresponding author: E-mail: [email protected], [email protected]; Zhang and Rong; BJMCS, 20(4), 1-11, 2017; Article no.BJMCS.31816

Keywords: Hierarchical normal model; coverage probability of the credible interval; credible probability of the conﬁdence interval; limit. 2010 Mathematics Subject Classiﬁcation: 62Fxx, 62F15.

1 Introduction

Statistical inferences are covered in many classical textbooks [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. The main estimates of statistical inferences are point estimates and interval estimates. The usual interval estimates are confidence intervals and credible intervals. The confidence intervals are usually measured under the coverage probability, while the credible intervals are usually measured under the credible probability. There is no article research on the credible probability of the confidence interval. However, there are a lot of research on the coverage probability of the credible interval. [11] find that allowing for genotyping error yielded relative risk estimates that were approximately unbiased, together with 95% credible intervals giving approximately correct coverage probability. [12] find that the Bayesian credible intervals based on the same priors also have super frequentist coverage probabilities while some of the frequentist confidence intervals procedures have drastically poor coverage. [13] find that the impact of different choices of prior distributions on the coverage probability of credible intervals is unknown. [14] compare three Bayesian sample size criteria: the Average Coverage Criterion (ACC) which controls the coverage rate of fixed length credible intervals over the predictive distribution of the data, the Average Length Criterion (ALC) which controls the length of credible intervals with a fixed coverage rate, and the Worst Outcome Criterion (WOC) which ensures the desired coverage rate and interval length over all (or a subset of) possible datasets. [15] find that Bayesian interval estimates for the treatment effect are longer on average, though there is little improvement in coverage probability. [16] find that the 95% confidence/credible intervals also possess good coverage properties, given that the point estimates perform good. [17] find that the coverage probability of the credible interval close to the nominal value, with a small coverage asymmetry in some cases. [18] find that 95% credible intervals may not retain nominal coverage, and treatment rank probabilities may become distorted. [19] find that Bayesian inference provides reliable credible intervals in terms of bias and coverage probability. [20] carry out simulations and find that spatial capture-recapture models produced more accurate parameter estimates with better credible interval coverage than non-spatial capture-recapture models. [21] find that the coverage probability of given credible interval is well-calibrated in the simulation experiments.

It is well known that the coverage probability of a given nominal level 1 − α confidence interval and the credible probability of a given nominal level 1 − α credible interval will attain the nominal level 1 − α. In Example 9.2.18, [4] tell us that for the hierarchical normal model, the two switching concepts probabilities, that is, the coverage probability of the 1−α credible interval and the credible probability of the 1−α confidence interval, can not attain the nominal level 1−α in general. Whether it is possible, under some circumstances, that the two switching concepts probabilities attain the nominal level 1 − α is not mentioned in the example and the relevant exercises in [4]. Nobody has considered this problem to the best of our knowledge. By inspecting√ the calculations in Example 9.2.18, we find that they use a specific configuration for τ (n) = σ/ n which is a known function of n. Inspired by this specific configuration, we show that the two switching concepts probabilities can attain the nominal level 1 − α in the limit by taking suitable configurations for τ (n). The numerical simulations illustrate the correctness of our findings.

The rest of the paper is organized as follows. In the next Section 2, we prove two theorems for the hierarchical normal model. Theorem 2.1 concerns the limiting behavior of the coverage probability of the 1−α credible interval. Theorem 2.2 concerns the limiting behavior of the credible probability of the 1 − α conﬁdence interval. In Section 3, the numerical simulations are carried out to illustrate the correctness of the two theorems. Section 4 concludes.

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2 Main Results ( ) 2 The hierarchical normal model of( Example) 9.2.18 in [4] is as follows. Let X1,...,Xn be iid N θ, σ , and let θ have the prior pdf N µ, τ 2 , where µ, σ, and τ are all known. For simplicity, we assume −∞ ∞ < µ < and σ > 0 are known constants. Furthermore, we assume that τ (n√) is a known function of n, which is inspired by Example 9.2.18 in [4]. There, they use τ (n) = σ/ n which is a known function of n. In fact, from theorems 1 and 2 below,√ we find that τ (n) as a function of n is critical to define the skillful classified variable L = lim nτ 2 (n). n→∞

For the coverage probability of the 1 − α credible interval of the hierarchical normal model Pn (θ), σ2 from Example 9.2.18 and by noting that γ = nτ 2(n) , we have ( ) √ γ (θ − µ) Pn (θ) = Pθ Z − √ ≤ z α 1 + γ σ/ n 2 ( √ )

σ (θ − µ) σ2 = Pθ Z − √ ≤ z α 1 + , nτ 2 (n) 2 nτ 2 (n)

α where Z ∼ N (0, 1) and z α is the upper quantile of Z. Let 2 2 √ σ (θ − µ) σ2 A (n) = √ and B (n) = 1 + . nτ 2 (n) nτ 2 (n)

Then ( )

Pn (θ) = Pθ |Z − A (n)| ≤ z α B (n) ( 2 )

= Pθ A (n) − z α B (n) ≤ Z ≤ A (n) + z α B (n) . (2.1) 2 2

We have the following theorem which concerns the limiting behavior√ of the coverage probability of the 1 − α credible interval. Note that the choice of L = lim nτ 2 (n) as the classiﬁed variable is n→∞ skillful. √ Theorem 2.1. Let L = lim nτ 2 (n) and assume θ ≠ µ. Then n→∞   1 − α, if L = ∞, 0, if L = 0, lim Pn (θ) = ( ) n→∞  σ(θ−µ) Pθ Z − ≤ z α , if L ∈ (0, ∞) . L 2

Proof. If L = ∞, then σ (θ − µ) σ (θ − µ) A (n) = √ → = 0, as n → ∞, nτ 2 (n) L √ √ σ2 σ2 B (n) = 1 + → 1 + = 1, as n → ∞. nτ 2 (n) ∞

Therefore, ( )

lim Pn (θ) = Pθ |Z| ≤ z α = 1 − α. n→∞ 2 √ If L = 0, that is, nτ 2 (n) → 0, as n → ∞. Then

2 1 τ (n) , τ (n) , n 4 τ (n) → 0, as n → ∞. (2.2)

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When θ > µ, we have A (n) → ∞, as n → ∞. We want to show that the lower bound in (2.1)

A (n) − z α B (n) → ∞, as n → ∞. 2 But B (n) may also tend to ∞, as n → ∞. We have √ σ (θ − µ) σ2 A (n) − z α B (n) = √ − 1 + z α 2 nτ 2 (n) nτ 2 (n) 2 √ σ (θ − µ) nτ 2 (n) + σ2 = √ − √ z α nτ 2 (n) nτ (n) 2 [ ] 1 √ = √ σ (θ − µ) − τ (n) nτ 2 (n) + σ2z α . nτ 2 (n) 2

By (2.2), the limit [ √ ] [ √ ] 2 2 1 nτ (n) + σ lim τ (n) nτ 2 (n) + σ2z α = lim n 4 τ (n) z α →∞ 2 →∞ 1 2 n n n 4 [ √ ] √ [ ] 1 √ σ2 1 [√ ] σ2 = lim n 4 τ (n) nτ 2 (n) + √ z α = lim n 4 τ (n) lim nτ 2 (n) + lim √ z α n→∞ n 2 n→∞ n→∞ n→∞ n 2 √ = 0 · 0 + 0z α = 0. 2

√ 1 → ∞ → ∞ − Since nτ 2(n) , as n , and σ (θ µ) > 0, we therefore have [ ] lim A (n) − z α B (n) = ∞. n→∞ 2

Consequently, lim Pn (θ) = 0. When θ < µ, we have A (n) → −∞, as n → ∞. Similarly, we can n→∞ prove that the upper bound in (2.1)

A (n) + z α B (n) → −∞, as n → ∞. 2

Consequently, lim Pn (θ) = 0. n→∞ If L ∈ (0, ∞), then

σ (θ − µ) σ (θ − µ) A (n) = √ → , as n → ∞, nτ 2 (n) L √ √ σ2 σ2 B (n) = 1 + → 1 + = 1, as n → ∞. nτ 2 (n) ∞

Therefore, ( )

σ (θ − µ) lim Pn (θ) = Pθ Z − ≤ z α ∈ (0, 1 − α) . n→∞ L 2 The proof of the theorem is complete.

In Theorem 2.1, if L does not exist, then this case is complicated, and we do not pursue it. We have the following remarks for Theorem 2.1. 2 Remark 2.1. We can take specific configurations of the parameter τ (n), so that lim Pn (θ) can take n→∞ 2 different values√ in Theorem 2.1. For example, the following configurations of τ (n) will guarantee 2 L = lim nτ (n) = ∞, and thus lim Pn (θ) = 1 − α: n→∞ n→∞

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2 2 2 2 • τ (n) → c ∈ (0, ∞), as n → ∞. In particular, τ (n) = τ0 → τ0 ∈ (0, ∞), as n → ∞. • τ 2 (n) → ∞, as n → ∞. • lim τ 2 (n) does not exist, and there exist an ε > 0 and an N > 0, such that for any n ≥ N, n→∞

τ 2 (n) ≥ ε > 0,

that is, τ 2 (n) is bounded away from 0. In particular, { 1, n is odd, τ 2 (n) = 2, n is even. √ 2 2 2 − 1 • τ (n) → 0, as n → ∞, but nτ (n) → L = ∞, as n → ∞. For instance, τ (n) = n 4 . √ The conﬁguration τ (n) = σ/ n in [4] has

√ √ σ2 σ2 nτ 2 (n) = n = √ → 0 = L, as n → ∞. n n

Thus, by Theorem 2.1, lim Pn (θ) = 0. n→∞ 2 √c ∈ ∞ If τ (n) = n , where c (0, ), then √ √ c nτ 2 (n) = n √ = c → c = L ∈ (0, ∞) , as n → ∞. n Thus, by Theorem 2.1, ( )

σ (θ − µ) lim Pn (θ) = Pθ Z − ≤ z α . n→∞ c 2 Remark 2.2. When θ = µ, it is easy to check that A (n) = 0. Therefore, ( √ ) ( √ ) σ2 σ2 Pn (θ) = Pθ |Z| ≤ z α 1 + = 2Φ z α 1 + − 1, 2 nτ 2 2 nτ 2

∼ σ2 ∈ ∞ where Φ (x) is the cdf of Z N (0, 1). Note that Pn (θ) does not depend on θ. Since nτ 2 [0, ], we have Pn (θ) ∈ [1 − α, 1]. For the credible probability of the 1 − α conﬁdence interval of the hierarchical normal model, from Example 9.2.18, we have ( ) √ γ (¯x − µ) Pn (¯x) = Px¯ Z − √ √ ≤ z α 1 + γ . 1 + γσ/ n 2 Let γ (¯x − µ) √ C (n) = √ √ and B (n) = 1 + γ. 1 + γσ/ n Then ( )

Pn (¯x) = Px¯ |Z − C (n)| ≤ z α B (n) ( 2 )

= Px¯ C (n) − z α B (n) ≤ Z ≤ C (n) + z α B (n) . (2.3) 2 2 ( ) Note that γ = σ2/ nτ 2 , and thus √ σ2 B (n) = 1 + nτ 2 (n)

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and

2 2 − σ (¯x − µ) σ (¯x − µ) 2 √γ (¯x µ√) √nτ 2 √nτ 2 σ − √ nτ C (n) = = = = 2 (¯x µ) 1 + γσ/ n σ2 σ nτ 2+σ2 nτ 2 2 1 + √ √ √σ nτ + σ σ nτ 2 n nτ n σ (¯x − µ) σ (¯x − µ) σ (¯x − µ) = √ = √ = √ . 2 2 4 2 2 √ τ nτ + σ nτ + σ τ ( nτ 2)2 + σ2τ 2

Analogous to Theorem 2.1, we have the following theorem which concerns the limiting behavior of the credible probability of the 1 − α conﬁdence interval. √ Theorem 2.2. Let L = lim nτ 2 (n) and assume x¯ ≠ µ. Then n→∞   1 − α, if L = ∞, 0, if L = 0, lim Pn (¯x) = ( ) n→∞  σ(¯x−µ) Px¯ Z − ≤ z α , if L ∈ (0, ∞) . L 2 Proof. The proof of Theorem 2.2 follows from that of Theorem 2.1. If L = ∞, then | − | | − | | − | | | √ σ (¯x µ) ≤ √σ (¯x µ) → σ (¯x µ) → ∞ C (n) = √ √ = 0, as n , 2 2 L ( nτ 2) + σ2τ 2 ( nτ 2) √ √ σ2 σ2 B (n) = 1 + → 1 + = 1, as n → ∞. nτ 2 (n) ∞

Therefore, ( )

lim Pn (¯x) = Px¯ |Z| ≤ z α = 1 − α. →∞ 2 √ n If L = 0, that is, nτ 2 (n) → 0, as n → ∞, then (2.2) is right. Whenx ¯ > µ, we have C (n) → ∞, as n → ∞. We want to show that the lower bound in (2.3)

C (n) − z α B (n) → ∞, as n → ∞. 2 But B (n) may also tend to ∞, as n → ∞. We have √ σ (¯x − µ) σ2 − α √ − α C (n) z B (n) = √ z 1 + 2 2 2 2 nτ ( nτ 2) + σ2τ 2 [ √ √ ] 2 ( ) 1 σ √ 2 √ − − α 2 2 2 = √ σ (¯x µ) z 1 + 2 nτ + σ τ . 2 2 nτ ( nτ 2) + σ2τ 2

And the limit [√ √ ] [ √ √ ] σ2 (√ ) √ σ2 σ2 lim 1 + nτ 2 2 + σ2τ 2 = lim nτ 2 1 + 1 + n→∞ nτ 2 n→∞ nτ 2 nτ 2 [ ( )] [ ] √ σ2 √ σ2 = lim nτ 2 1 + = lim nτ 2 + √ = 0 + 0 = 0. n→∞ nτ 2 n→∞ n Since σ (¯x − µ) > 0 and

√ 1 → √ 1 ∞ → ∞ √ = , as n , ( nτ 2)2 + σ2τ 2 0 + 0

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we therefore have [ ] lim C (n) − z α B (n) = ∞. n→∞ 2

Consequently, lim Pn (¯x) = 0. Whenx ¯ < µ, we have C (n) → −∞, as n → ∞. Similarly, we can n→∞ prove that the upper bound in (2.3)

C (n) + z α B (n) → −∞, as n → ∞. 2

Consequently, lim Pn (¯x) = 0. n→∞ If L ∈ (0, ∞), then

− − − √ σ (¯x µ) → σ√(¯x µ) σ (¯x µ) → ∞ C (n) = √ = , as n , 2 2 L ( nτ 2) + σ2τ 2 L + 0 √ √ σ2 σ2 B (n) = 1 + → 1 + = 1, as n → ∞. nτ 2 (n) ∞

Therefore, ( )

σ (¯x − µ) lim Pn (¯x) = Px¯ Z − ≤ z α ∈ (0, 1 − α) . n→∞ L 2

The proof of the theorem is complete.

In Theorem 2.2, if L does not exist, then this case is complicated, and we do not pursue it. Note that Remark 2.1 is also suitable for Theorem 2.2. Moreover, we have the following remark for Theorem 2.2.

Remark 2.3. Whenx ¯ = µ, it is easy to check that C (n) = 0. Therefore, ( √ ) ( √ ) σ2 σ2 Pn (¯x) = Px¯ |Z| ≤ z α 1 + = 2Φ z α 1 + − 1. 2 nτ 2 2 nτ 2

σ2 ∈ ∞ ∈ − Note that Pn (¯x) does not depend onx ¯. Since nτ 2 [0, ], we have Pn (¯x) [1 α, 1].

3 Numerical Simulations

In this section, we will numerically illustrate the correctness of Theorems 2.1 and 2.2.

We ﬁrst illustrate Theorem 2.1. Since we are considering the limiting behavior of the coverage probability of the 1 − α credible interval Pn (θ), θ is ﬁxed and unknown. In our simulations, we assume that θ = 2. The other parameters are

µ = 1, σ = 3, α = 0.1, where µ is the prior mean, σ is the standard deviation of the normal model, and α is the significance level (1−α = 0.9 is the nominal level). We will use the configurations of the parameter τ (n) specified

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in Remark 2.1:

τ11 (n) = τ0 = 1 → 1 ∈ (0, ∞) , as n → ∞, τ (n) = n → ∞, as n → ∞, 12 { 1, n is odd, τ (n) = √ 13 2, n is even, − 1 τ14 (n) = n 8 → 0, as n → ∞, σ τ (n) = √ , 2 n √ √ c 2 τ3 (n) = 1 = 1 . n 4 n 4 It is easy to check that for the above conﬁgurations,  ∞ √  , for τ (n) = τ11 (n) , τ12 (n) , τ13 (n) , τ14 (n) , 2 L = lim nτ (n) = 0, for τ (n) = τ2 (n) , n→∞  c ∈ (0, ∞) , for τ (n) = τ3 (n) .

The limiting behaviors of the coverage probabilities of the 1 − α credible intervals Pn (θ) for the six configurations of τ (n) are reported in Fig. 1. From Fig. 1, we clearly see that for the first four configurations of τ (n), Pn (θ) → 1 − α = 0.9, as n → ∞. For τ (n) = τ2 (n), Pn (θ) → 0, as n → ∞. For τ (n) = τ3 (n), ( )

σ (θ − µ) Pn (θ) → Pθ Z − ≤ z α = 0.557 ∈ (0, 0.9) = (0, 1 − α) , as n → ∞. c 2 The results of Fig. 1 illustrate the correctness of Theorem 2.1.

Coverage Probability of 1 − α = 0.9 Credible Interval 1.0 0.8 0.6 ) θ ( n P 0.4

τ(n) = τ11(n) τ(n) = τ12(n) τ(n) = τ (n) 0.2 13 τ(n) = τ14(n) τ(n) = τ2(n) τ(n) = τ3(n) 0.0

0 200 400 600 800 1000

Fig. 1. The limiting behaviors of the coverage probabilities of the 1 − α credible intervals

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Now we illustrate Theorem 2.2. Since we are considering the limiting behavior of the credible probability of the 1 − α confidence interval Pn (¯x),x ¯ is fixed. In our simulations, we assume that x¯ = 4. The other parameters are µ = 1, σ = 3, α = 0.1, the same as those in the previous simulation. We will use the six configurations of the parameter τ (n) specified√ in the previous simulation, while c is changed to 5. For the six configurations, L = lim nτ 2 (n) is the same as those in the previous simulation. The limiting behaviors of the n→∞ credible probabilities of the 1 − α confidence intervals Pn (¯x) for the six configurations of τ (n) are reported in Fig. 2. From Fig. 2, we clearly see that for the first four configurations of τ (n), Pn (¯x) → 1 − α = 0.9, as n → ∞. We see that for τ (n) = τ14 (n), Pn (¯x) increases very slowly. Therefore, we let n increase from 10 to 10000 (note that in Fig. 1, this number is 1000). For τ (n) = τ2 (n), Pn (¯x) → 0, as n → ∞. For τ (n) = τ3 (n), ( )

σ (¯x − µ) Pn (¯x) → Px¯ Z − ≤ z α = 0.438 ∈ (0, 0.9) = (0, 1 − α) , as n → ∞. c 2 The results of Fig. 2 illustrate the correctness of Theorem 2.2.

Credible Probability of 1 − α = 0.9 Confidence Interval 1.0 0.8 0.6 ) x ( n P 0.4

τ(n) = τ11(n) τ(n) = τ12(n) τ(n) = τ (n) 0.2 13 τ(n) = τ14(n) τ(n) = τ2(n) τ(n) = τ3(n) 0.0

0 2000 4000 6000 8000 10000

Fig. 2. The limiting behaviors of the credible probabilities of the 1 − α conﬁdence intervals

4 Conclusion

We prove two theorems for the hierarchical normal model. Theorem 2.1 concerns the limiting behavior of the coverage probability of the 1 − α credible interval Pn (θ). Theorem 2.2 concerns − the limiting√ behavior of the credible probability of the 1 α confidence interval Pn (¯x). When 2 L = lim nτ (n) = ∞, the two limiting probabilities lim Pn (θ) = lim Pn (¯x) = 1−α, that is, the n→∞ n→∞ n→∞ − − coverage probability of the 1 α credible interval and the credible probability of the 1 α confidence√ interval attain the nominal level 1 − α in the limit. The specific configuration τ (n) = σ/ n in [4]

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corresponds to L = 0 in Theorems 2.1 and 2.2, and thus lim Pn (θ) = lim Pn (¯x) = 0, that is, the n→∞ n→∞ two probabilities can not attain the nominal level 1 − α. The numerical simulations illustrate the correctness of Theorems 2.1 and 2.2. In summary, we can not blindly assume that a 1 − α credible interval will always be a 1 − α conﬁdence interval, or vice versa. The choice of prior matters.

Acknowledgement The authors gratefully acknowledge the constructive comments oﬀered by the referees. Their comments improve the quality of the paper signiﬁcantly. The research was supported by the Fundamental Research Funds for the Central Universities (CQDXWL-2012-004 and CDJRC10100010), China Scholarship Council (201606055028), and the MOE project of Humanities and Social Sciences on the west and the border area (14XJC910001). Competing Interests

Authors have declared that no competing interests exist.

References

[1] Ferguson TS. Mathematical Statistics [M]. Academic Press, New York; 1967. [2] Bickel PJ, Doksum KA. Mathematical statistics [M]. Holden Day, San Francisco; 1977. [3] Stuart A, Ord JK, Arnold S. Advanced theory of statistics, volume 2A: Classical Inference and the Linear Model [M]. Oxford University Press, London. 6th edition; 1999. [4] Casella G, Berger RL. Statistical inference [M]. Duxbury, USA. 2nd edition; 2002. [5] Shao J. Mathematical Statistics [M]. Springer, New York. 2nd edition; 2003. [6] Lehmann EL, Romano JP. Testing statistical hypotheses [M]. Springer, New York. 3rd edition; 2005. [7] Mao SS, Wang JL, Pu XL. Advanced mathematical statistics [M]. Higher Education Press, Beijing. 2nd edition; 2006. [8] Shi NZ, Tao J. Statistical hypothesis testing: Theory and methods [M]. World Scientiﬁc Publishing, Singapore; 2008. [9] Chen XR. Advanced mathematical statistics [M]. Press of University of Science and Technology of China, Hefei; 2009. [10] Zheng ZG, Tong XW, Zhao H. Advanced statistics [M]. Peking University Press, Beijing; 2012. [11] Bernardinelli L, Berzuini C, Seaman S, Holmans P. Bayesian trio models for association in the presence of genotyping errors [J]. Genetic Epidemiology. 2004;26:70-80. [12] Wang XY, He CZ, Sun DC. Bayesian inference on the patient population size given list mismatches [J]. Statistics in Medicine. 2005;24:249-267. [13] McCandless LC, Gustafson P, Levy A. Bayesian sensitivity analysis for unmeasured confounding in observational studies [J]. Statistics in Medicine. 2007;26:2331-2347. [14] Cao J, Lee JJ, Alber S. Comparison of bayesian sample size criteria: Acc, alc, and woc [J]. Journal of Statistical Planning and Inference. 2009;139:4111-4122. [15] McCandless LC, Gustafson P, Austin PC. Bayesian propensity score analysis for observational data [J]. Statistics in Medicine. 2009;28:94-112. [16] Oirbeek RV, Lesaﬀre E. An application of harrells c-index to ph frailty models [J]. Statistics in Medicine. 2010;29:3160-3171.

10 Zhang and Rong; BJMCS, 20(4), 1-11, 2017; Article no.BJMCS.31816

[17] Subtil F, Rabilloud M. A bayesian method to estimate the optimal threshold of a longitudinal biomarker [J]. Biometrical Journal. 2010;52:333-347. [18] Thorlund K, Thabane L, Mills EJ. Modelling heterogeneity variances in multiple treatment comparison meta-analysis - are informative priors the better solution [J]? BMC Medical Research Methodology. 2013;13:1-14. [19] Subtil F, Rabilloud M. Estimating the optimal threshold for a diagnostic biomarker in case of complex biomarker distributions [J]. BMC Medical Informatics and Decision Making. 2014;14:1-11. [20] Whittington J, Sawaya MA. A comparison of grizzly bear demographic parameters estimated from non-spatial and spatial open population capture-recapture models [J]. PLOS ONE. 2015;10:1-17. [21] Jiang W, Yu WC. Power estimation and sample size determination for replication studies of genome-wide association studies [J]. BMC Genomics. 2016;17:19-32.

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