- Methodology
- Open Access
Power estimation and sample size determination for replication studies of genome-wide association studies
- Wei Jiang^{1} and
- Weichuan Yu^{1}Email author
- Published: 11 January 2016
The Erratum to this article has been published in BMC Genomics 2017 18:73
Abstract
Background
Replication study is a commonly used verification method to filter out false positives in genome-wide association studies (GWAS). If an association can be confirmed in a replication study, it will have a high confidence to be true positive. To design a replication study, traditional approaches calculate power by treating replication study as another independent primary study. These approaches do not use the information given by primary study. Besides, they need to specify a minimum detectable effect size, which may be subjective. One may think to replace the minimum effect size with the observed effect sizes in the power calculation. However, this approach will make the designed replication study underpowered since we are only interested in the positive associations from the primary study and the problem of the “winner’s curse” will occur.
Results
An Empirical Bayes (EB) based method is proposed to estimate the power of replication study for each association. The corresponding credible interval is estimated in the proposed approach. Simulation experiments show that our method is better than other plug-in based estimators in terms of overcoming the winner’s curse and providing higher estimation accuracy. The coverage probability of given credible interval is well-calibrated in the simulation experiments. Weighted average method is used to estimate the average power of all underlying true associations. This is used to determine the sample size of replication study. Sample sizes are estimated on 6 diseases from Wellcome Trust Case Control Consortium (WTCCC) using our method. They are higher than sample sizes estimated by plugging observed effect sizes in power calculation.
Conclusions
Our new method can objectively determine replication study’s sample size by using information extracted from primary study. Also the winner’s curse is alleviated. Thus, it is a better choice when designing replication studies of GWAS. The R-package is available at: http://bioinformatics.ust.hk/RPower.html.
Keywords
- Replication study
- Power
- Empirical Bayes
Background
Genome-wide association studies (GWAS) are widely used to identify susceptibility variants of common diseases. Commonly, single nucleotide polymorphisms (SNPs) are genotyped across the whole genome in different individuals, and statistical methods are used to detect the associations between SNPs and disease status. According to the summary of GWAS catalog ([1], accessed [2015.05.28]), about 2000 GWAS reports related to 756 diseases/traits have been published so far, from which 14,609 associations show genome-wide significance (p-value ≤5×10^{−8}). More and more associations will be discovered from GWAS.
The basic statistical method used in GWAS analysis is hypothesis testing [2]. The possibilities of false positives cannot be completely removed in the analysis. Hence, all findings from GWAS need to be verified. Replication study is a commonly used approach to verifying positive findings [3, 4]. If an association between one specific SNP and a certain disease has been identified in the primary study and confirmed in the replication study, we usually treat this association as true positive with a high confidence. If an association identified in the primary study cannot be confirmed in the replication study, we often suspect that it is a false positive.
The power of replication study is crucial in this validation process. If the replication study is underpowered, then the positive findings will have a low chance to be replicated. It’s essential to design a replication study with enough statistical power.
How to estimate the power of a replication study in the design phase?
Traditionally, a replication study is regarded as another independent primary study. Thus, the same power calculation in the original primary study is used. For the associations identified in the primary study, a minimum effect size needs to be specified. Then, the underlying alternative distribution of test statistics is assumed to have specified effect size. The major limitation of this traditional power calculation method is that the specification of the effect size is subjective and may cause bias. Besides, no information from primary study has really been used.
One may think to plug the observed effect sizes from the primary study in the power calculation of the replication study. This power estimation approach doesn’t need to specify any parameters. Since only significant associations are considered in the replication study, the observed effect sizes for those associations will tend to be overestimated [5]. This phenomenon is known as the “winner’s curse” [6], which makes the estimated powers tend to have higher values.
A lot of methods have been proposed to overcome the winner’s curse in effect size estimation. An incomplete list includes conditional maximum likelihood estimation (CMLE, [7–9]), bootstrap [10], full Bayesian method [11] and Empirical Bayes method (EB, [12]). Since power function is usually not a linear function of effect size, the estimators obtained by simply plugging those bias-corrected effect sizes in power calculation may not achieve the best performance.
- 1.
Due to the nonlinear nature and restricted range (limited to [ 0,1]) of power function, the distribution of power is usually non-normal when effect size is normally distributed (illustrated in Fig. 1). The interval estimation of the power should consider the non-normality.
- 2.
Since the power values of different associations in the primary study are different, a summary value is needed to determine the sample size of replication study.
- 1.
For each association identified from the primary study, an EB based method is proposed to estimate its power in the replication study.
- 2.
Due to the non-normality of the estimated power and the inaccuracy of the hyperparameters estimation, a novel interval estimation method combining Monte Carlo sampling and Bootstrap is proposed to estimate the corresponding credible interval of each association’s power in the replication study.
- 3.
The average power of the discovered true associations is used for determining the sample size of replication study. An weighted average method is proposed to estimate the average power. Our proposed interval estimation method can also be used to construct the credible interval of the average power.
- 4.
Only the summary statistics of the primary study are needed when using our proposed method to design a replication study. This feature is helpful since summary statistics are more accessible than individual-level genotype data due to the privacy issue and other constraints.
The rest of this paper is organized as follows. In section ‘Methods’, we will introduce the Bayesian framework to estimate the power of replication studies. We will prove that Bayesian predictive power is immune to the winner’s curse. Then we will present how to estimate the power with two-component mixture prior under the Bayesian framework. We will also give the details about estimation of hyperparameters, interval estimation and the estimation of average power. In section ‘Results and discussion’, we will first use simulation results to demonstrate that our EB based method is better than other plug-in based estimators in terms of overcoming the winner’s curse and providing higher estimation accuracy. We will also demonstrate that the coverage probability of given credible interval is well-calibrated. Then we will show the sample sizes determined to replicate findings of 6 diseases from Wellcome Trust Case Control Consortium (WTCCC) [13], which are much higher than the sample sizes estimated by plugging observed effect sizes in the power calculation formula. The increased sample sizes are reasonable due to the winner’s curse. In the same section, we will discuss limitations of current modeling and estimation approach. Section ‘Conclusions’ concludes the paper.
Methods
We use parenthesized superscript “ (j)” to denote primary study (j=1) and replication study (j=2). For example, we denote the sample size in the primary study as n ^{(1)}. The sample size in the control group and case group are \(n_{0}^{(1)}\) and \(n_{1}^{(1)}\), respectively. The total number of SNPs genotyped in the primary study is m. Among those genotyped SNPs, the proportion of the SNPs having no association with the disease (null SNPs) is π _{0}(0≤π _{0}≤1).
Allele based contingency table of one SNP in primary/replication study. Please see the main text for explanation of the notations
Non-effect allele | Effect allele | Total | |
---|---|---|---|
Control | \(n_{00}^{(j)}\) | \(n_{01}^{(j)}\) | \(2n_{0}^{(j)}\) |
Case | \(n_{10}^{(j)}\) | \(n_{11}^{(j)}\) | \(2n_{1}^{(j)}\) |
Total | \(n_{00}^{(j)}+n_{10}^{(j)}\) | \(n_{01}^{(j)}+n_{11}^{(j)}\) | 2n ^{(j)} |
The true value of the log odds ratio μ is usually unknown. The asymptotic standard error of \(\widehat {\mu }^{(j)}\) can be approximated using Woolf’s method [14],
Wald test can be used to examine whether the null hypothesis should be rejected. The test statistic is \(z^{(j)}=\widehat {\mu }^{(j)}/\sigma ^{(j)}\). The significance levels in the primary study and the replication study are fixed to α _{1} and α _{2}, respectively.
Bayesian predictive power
A traditional power calculation method needs to specify a minimum detectable effect size μ _{ min } first. Then, the power of replication study is β ^{(2)}(μ _{ min }). Consequently, the power can be used to determine the sample size.
Please notice that, although the selection bias can be reduced using estimator which can adjust estimated effect size, no unbiased estimator exists [11]. With estimated effect size, the power of replication study can be obtained by using \(\beta ^{(2)}\left (\widehat {\mu }_{\textit {CMLE}}^{(1)}\right)\). The plug-in based power estimator is not optimized in terms of minimizing Bayes risk.
which takes the average of all power function values among all possible μ values given observed z ^{(1)}. We will provide a detailed formula of the Bayesian predictive power under one specific prior in the following subsection.
Two-component mixture prior
where δ _{0} is the distribution with point mass on zero and \({\sigma _{0}^{2}}\) is the variance of the effect sizes in associated SNPs.
There is an unknown hyperparameter \({\sigma _{0}^{2}}\) in the calculation of Bayesian predictive power. In the following subsection, we will present how to estimate \({\sigma _{0}^{2}}\) with Empirical Bayes approach.
Hyperparameter \({\sigma _{0}^{2}}\)
- 1.
If the null hypothesis is valid, then all SNPs follow a standard normal distribution with variance equal to one. When \(\frac {1}{m}\sum _{i=1}^{m} \left (z^{(1)}_{i}\right)^{2}\leq 1\), i.e., the sample variance is no bigger than one as in the null hypothesis case, we will have \(\widehat {\sigma }_{0}^{2}=0\). In this case, the result of our EB based power estimation method will degenerate to type I error rate, which is the probability that the identified association can be replicated even when the association doesn’t exist.
- 2.
When \(\frac {1}{m}\sum _{i=1}^{m} \left (z^{(1)}_{i}\right)^{2}>1\) but π _{0}=1, we will have \(\widehat {\sigma }_{0}^{2}=+\infty \). In this case, the above shrinkage coefficient will degenerate to λ=1. The shrinkage effect in our EB based method will disappear.
There is a bias-variance tradeoff in tuning γ. An automatic procedure is proposed in [18] without tuning γ: A natural cubic spline will fit to evaluated values with different γ, then \(\widehat {\pi }_{0}\) is the spline’s value at γ=1.
\(\widehat {\sigma }_{0}^{2}\) can be calculated by plugging \(\widehat {\pi }_{0}\) in Eq. (13). By plugging \(\widehat {\sigma }_{0}^{2}\) into Eq. (11), an EB based estimator of the replication study’s power can be obtained, which is denoted as \(\widehat {\eta }^{(2)}_{\textit {EB}}\). The corresponding credible interval can be constructed, which is presented in the following subsection.
Credible interval
From Eq. (10), the posterior distribution of log odds ratio μ under alternative hypothesis \(\mathcal {H}_{1}\) is a normal distribution. Figure 1 shows the histogram of power values when μ is normally distributed. The shape of the histogram indicates the non-normality of the calculated power. Hence, the asymptotic approach based on normal distribution theory is not appropriate in the interval estimation of the replication study’s power. The construction of the credible interval should consider the non-normality. We propose to use Monte Carlo sampling to construct the credible interval of β ^{(2)}(μ). The credible interval is constructed with known hyperparameters \({\sigma _{0}^{2}}\). Since estimation error will occur in estimating \({\sigma _{0}^{2}}\), the constructed credible interval will have smaller coverage probability than nominal level. To incorporate the variance of the estimator \(\widehat {\sigma }_{0}^{2}\), a method combining Monte Carlo sampling and Bootstrap is proposed. The test statistics from the primary study z ^{(1)} will be resampled N _{1} times with replacement. For each run, \({\sigma _{0}^{2}}\) is re-estimated. Monte Carlo sampling is used to generate N _{2} power values with each re-estimated \({\sigma _{0}^{2}}\). The credible interval is constructed among all N _{1} N _{2} sampled power values.
Average power
where S is the index set of the associated SNPs identified from primary study and |S| is the cardinality of S. The subscript i means that the quantity is evaluated for SNP i.
where ϕ(x) is the probability density function (pdf) of N(0,1).
By setting the estimated average power larger than a threshold, e.g. \(\bar {\eta }^{(2)}_{\textit {EB}}>80\,\%\), the sample size of replication study can be determined.
Credible interval of the average power
The proposed interval estimation method can also be used to construct the credible interval of the average power. We resample the test statistics from the primary study N _{1} times. In each run, a re-estimated variance of the effect sizes in the non-null SNPs \(\widehat {\sigma }_{0}^{2}\) can be obtained. For a fixed \(\widehat {\sigma }_{0}^{2}\) value, we first calculate the local true discovery rate of the primary study ltdr ^{(1)} with Eq. (17) for each association. Then Monte Carlo sampling is used to generate N _{2} sets of the power values, in each of which there are power values of the replication study for all associations identified from the primary study. In each set, an average power can be obtained by taking weighted average of those generated power values among all associations. Hence, N _{2} average power values can be generated in each run. The credible interval of average power can be constructed among all N _{1} N _{2} sampled average power values.
Results and discussion
Simulation experiments
- 1.
Can EB based power estimator \(\widehat {\eta }_{\textit {EB}}^{(2)}\) perform well in terms of overcoming the winner’s curse?
- 2.
Can \(\widehat {\eta }_{\textit {EB}}^{(2)}\) estimate power accurately?
- 3.
Is the corresponding credible interval well-calibrated?
- 4.
Can weighted average estimator \(\bar {\eta }_{\textit {EB}}^{(2)}\) estimate average power \(\bar {\beta }^{(2)}(\mu)\) accurately?
For our hypothetical disease, its prevalence is 1 %. To test the marginal association between SNPs and the disease, log-odds ratio test is used. The significance levels are α _{1}=5×10^{−5} and α _{2}=5×10^{−3} in primary study and replication study, respectively.
Empirical biases of power estimators of the replication study in the simulation experiments. The settings of the experiments can be seen in the main text
\(\beta ^{(2)}\left (\widehat {\mu }^{(1)}\right)\) | \(\beta ^{(2)}\left (\widehat {\mu }_{\textit {CMLE}}^{(1)}\right)\) | \(\beta ^{(2)}\left (\widehat {\mu }_{BR2}^{(1)}\right)\) | \(\beta ^{(2)}\left (\widehat {\mu }_{\textit {EB}}^{(1)}\right)\) | \(\widehat {\eta }_{\textit {EB}}^{(2)}\) | |
---|---|---|---|---|---|
Run 1 | 0.142 | −0.113 | 0.038 | 0.058 | 0.032 |
Run 2 | 0.146 | −0.109 | 0.045 | 0.021 | 0.001 |
Run 3 | 0.144 | −0.068 | 0.045 | 0.047 | 0.021 |
Run 4 | 0.137 | −0.090 | 0.042 | 0.052 | 0.026 |
Run 5 | 0.144 | −0.126 | 0.026 | 0.038 | 0.016 |
Average | 0.142 | −0.101 | 0.039 | 0.043 | 0.019 |
Root mean square error (RMSE) of power estimators of the replication study in the simulation experiments. The settings of the experiments can be seen in the main text
\(\beta ^{(2)}\left (\widehat {\mu }^{(1)}\right)\) | \(\beta ^{(2)}\left (\widehat {\mu }_{\textit {CMLE}}^{(1)}\right)\) | \(\beta ^{(2)}\left (\widehat {\mu }_{BR2}^{(1)}\right)\) | \(\beta ^{(2)}\left (\widehat {\mu }_{\textit {EB}}^{(1)}\right)\) | \(\widehat {\eta }_{\textit {EB}}^{(2)}\) | |
---|---|---|---|---|---|
Run 1 | 0.246 | 0.334 | 0.201 | 0.202 | 0.195 |
Run 2 | 0.243 | 0.312 | 0.196 | 0.191 | 0.188 |
Run 3 | 0.247 | 0.303 | 0.203 | 0.198 | 0.192 |
Run 4 | 0.236 | 0.307 | 0.186 | 0.192 | 0.186 |
Run 5 | 0.249 | 0.317 | 0.198 | 0.196 | 0.194 |
Average | 0.244 | 0.315 | 0.197 | 0.196 | 0.191 |
Coverage probability of the 95 % credible intervals in simulation experiments. The simulation settings can be seen in the main text
Without Bootstrap | With Bootstrap | |
---|---|---|
Run 1 | 0.932 | 0.960 |
Run 2 | 0.947 | 0.960 |
Run 3 | 0.918 | 0.943 |
Run 4 | 0.914 | 0.949 |
Run 5 | 0.878 | 0.925 |
Average | 0.918 | 0.947 |
When effect sizes follow the distribution of Eq. (20), the average empirical bias and root mean square error (RMSE) of power estimators of the replication study in the simulation experiments
Average | \(\beta ^{(2)}\left (\widehat {\mu }^{(1)}\right)\) | \(\beta ^{(2)}\left (\widehat {\mu }_{\textit {CMLE}}^{(1)}\right)\) | \(\beta ^{(2)}\left (\widehat {\mu }_{BR2}^{(1)}\right)\) | \(\beta ^{(2)}\left (\widehat {\mu }_{\textit {EB}}^{(1)}\right)\) | \(\widehat {\eta }_{\textit {EB}}^{(2)}\) |
---|---|---|---|---|---|
Empirical Bias | 0.085 | −0.079 | 0.023 | 0.028 | 0.003 |
RMSE | 0.189 | 0.279 | 0.167 | 0.168 | 0.163 |
When effect sizes follow the distribution of Eq. (21), the average empirical bias and root mean square error (RMSE) of power estimators of the replication study in the simulation experiments
Average | \(\beta ^{(2)}\left (\widehat {\mu }^{(1)}\right)\) | \(\beta ^{(2)}\left (\widehat {\mu }_{\textit {CMLE}}^{(1)}\right)\) | \(\beta ^{(2)}\left (\widehat {\mu }_{BR2}^{(1)}\right)\) | \(\beta ^{(2)}\left (\widehat {\mu }_{\textit {EB}}^{(1)}\right)\) | \(\widehat {\eta }_{\textit {EB}}^{(2)}\) |
---|---|---|---|---|---|
Empirical Bias | 0.071 | −0.081 | 0.015 | 0.033 | 0.007 |
RMSE | 0.173 | 0.263 | 0.153 | 0.154 | 0.150 |
WTCCC datasets
- 1.
Missing data control: Chiamo score is used as genotype calling accuracy in the WTCCC data. The genotypes with Chiamo score <0.95 are regarded as missing values. The SNPs with more than 10 % missing entries are removed.
- 2.
Minor allele frequency control: Among all samples, the SNPs with minor allele frequency <0.05 are removed.
- 3.
Hardy-Weinberg equilibrium control: The SNPs with p-values <0.001 in the Hardy-Weinberg equilibrium test are removed.
The estimated hyperparameters π _{0} and \({\sigma _{0}^{2}}\) for 6 diseases of WTCCC dataset
\(\widehat {\pi }_{0}\) | \(\widehat {\sigma }_{0}^{2}\) | |
---|---|---|
Coronary artery disease | 0.949 | 0.004 |
Crohn’s disease | 0.840 | 0.006 |
Hypertension | 0.966 | 0.007 |
Rheumatoid arthritis | 0.947 | 0.008 |
Type 1 diabetes | 0.967 | 0.014 |
Type 2 diabetes | 0.940 | 0.005 |
Sample size of the replication study needed for 6 diseases of WTCCC dataset when average power is estimated by EB based method. The Control-to-Case ratio of the replication study is set to 1. The significance levels used in the primary study and the replication study are α _{1}=5×10^{−8} and α _{2}=5×10^{−6}, respectively
50 % | 60 % | 70 % | 80 % | 90 % | |
---|---|---|---|---|---|
Coronary artery disease | 4095 | 4784 | 5652 | 6885 | 9121 |
Crohn’s disease | 4405 | 5252 | 6376 | 8092 | 11,552 |
Hypertension | 5993 | 6992 | 8244 | 10,014 | 13,215 |
Rheumatoid arthritis | 2666 | 3329 | 4147 | 5291 | 7357 |
Type 1 diabetes | 2146 | 2640 | 3249 | 4094 | 5588 |
Type 2 diabetes | 4027 | 4726 | 5633 | 6988 | 9721 |
Sample size of the replication study needed for 6 diseases of WTCCC dataset when average power is estimated by plugging in observed effect sizes. The Control-to-Case ratio of the replication study is set to 1. The significance levels used in the primary study and the replication study are α _{1}=5×10^{−8} and α _{2}=5×10^{−6}, respectively
50 % | 60 % | 70 % | 80 % | 90 % | |
---|---|---|---|---|---|
Coronary artery disease | 2019 | 2290 | 2608 | 3023 | 3675 |
Crohn’s disease | 2270 | 2572 | 2922 | 3369 | 4058 |
Hypertension | 2672 | 2988 | 3345 | 3788 | 4448 |
Rheumatoid arthritis | 1553 | 1892 | 2273 | 2748 | 3465 |
Type 1 diabetes | 1532 | 1856 | 2229 | 2706 | 3443 |
Type 2 diabetes | 2085 | 2357 | 2676 | 3095 | 3775 |
For coronary artery disease and type 2 diabetes, we obtained the publicly available summary statistics of the meta-analysis from two consortiums: CARDIoGRAMplusC4D Consortium [22] and DIAGRAM Consortium [23], respectively. CARDIoGRAM GWAS is a meta-analysis of 22 GWAS studies of European descent involving 22,233 cases and 64,762 controls. The odds ratio calculated from high power CARDIoGRAM GWAS will be used as underlying true odds ratio to calculate the average power of the replication study for coronary artery disease in WTCCC. The average power obtained in this manner is denoted as \(\bar {\beta }^{(2)}\left (\hat {\mu }_{\textit {meta}}\right)\). Figure 5 a plots the relationship between \(\bar {\beta }^{(2)}\left (\hat {\mu }_{\textit {meta}}\right)\) and n ^{(2)}, which is the sample size needed in the replication study. The figure shows that our EB based power estimator \(\bar {\eta }_{\textit {EB}}^{(2)}\) is very close to the power calculated using the results of CARDIoGRAM GWAS. Also it can be shown that \(\bar {\beta }^{(2)}\left (\hat {\mu }_{\textit {meta}}\right)\) is in the credible interval we estimated. DIAGRAM GWAS is a meta-analysis consisting of 12,171 type 2 diabetes cases and 56,862 controls across 12 GWAS from European descent populations. Similar to CARDIoGRAM GWAS, the allele based odds ratio calculated from DIAGRAM GWAS is used for calculating the average power of the replication study for type 2 diabetes in WTCCC. Figure 5 f plots the relationship between \(\bar {\beta }^{(2)}\left (\hat {\mu }_{\textit {meta}}\right)\) and n ^{(2)}. It can be shown that the result estimated by our EB based method \(\bar {\eta }_{\textit {EB}}^{(2)}\) is close to the power calculated using the results of DIAGRAM GWAS.
If the values of the local true discovery rates ltdr ^{(1)} have nearly the same level for all identified associations in the primary study, the variance of the average power will be inversely proportional to the number of the associations. When the identified number is small in the primary study, the credible interval for the average power is rather wide. This can be illustrated in the study of hypertension, where there is only 1 association showing genome-wide significance. From Fig. 5 c, we can see that the credible interval is rather wide. If we want to consider the credible interval for this situation, then the sample size can drastically increased.
Discussion
We propose to design replication study under the case-control setting where log-odds ratio test is used. The method can also be generalized to other tests within z-test scheme, such as regression slope test used for quantitative trait.
As described in [7], the winner’s curse depends strongly on the power of primary study. For a high power primary study, most non-null SNPs will result in significant associations after random draws from the population. Hence, the bias will be small in this case. There are more and more high power studies conducted for common diseases by using pooling strategy or meta-analysis strategy, but the high power studies for rare diseases are limited. Hence, it is still helpful and necessary to propose a designing procedure for the replication study with the consideration of winner’s curse.
With the development of the cost-effective sequencing technique, the targets of association studies extend from common variations to rare variants. A commonly used strategy to discover associations with rare variants is the collapsing method [24], in which several rare variants in a certain group are pooled together to enrich the signal. For each group, a “super variant” is constructed. If log-odds ratio test is adopted in testing the association between “super variant” and the disease, our method can be used directly for designing the replication study.
- 1.
The assumption of our approach is that all SNPs’ effect sizes are drawn independently from a two-component mixture distribution. Linkage disequilibrium widely exists in SNPs. Correlated genotype patterns can also introduce correlation between their effect sizes. The power estimation can be further improved by using correlation information in the prior set-up.
- 2.
Our proposed method assumes the effect sizes of associated SNPs are normally distributed. This thin tail distribution may not be realistic. How to design of replication study with other heavy-tail prior needs to be discussed.
Conclusions
Replication study is commonly used to verify findings discovered from GWAS. Power analysis is essential in designing a replication study. Traditional approach will not extract information from primary study. Also it will need users to specify a parameter μ _{ min }, which is subjective. Power estimation approach may address this problem, but there are several challenges in power estimation: the winner’s curse, credible interval and summarization.
In this paper, we propose an EB based power estimation method to resolve these challenges. Simulation experiments show our approach is better than other plug-in based approaches in terms of overcoming the winner’s curse and providing higher estimation accuracy. We also use simulation experiments to demonstrate the well calibration of the constructed credible interval. As an application example, we use our approach to determine the sample size needed in the WTCCC datasets of 6 diseases. Our approach gives an objective way to design replication study using information extracted from primary study.
Appendix
Appendix 1 — η ^{(2)} is the minimizer of \(R(\widehat {\theta })\)
The last equality is hold by Fubini’s theorem.
From the last equality, it can be seen that the Bayesian predictive power η ^{(2)} is the minimizer of the expression in the brace for each value of z ^{(1)}. Hence η ^{(2)} is also the minimizer of \(R(\widehat {\theta })\).
Appendix 2 — Derivation of η ^{(2)} under two-component mixture prior
The following property of multivariate Gaussian distribution is proved in the Chapter 2 of [25], which can be used to derive η ^{(2)}.
Property 1.
where W=Σ(Σ _{0}+Σ)^{−1}
where Φ(x) is the cumulative density function (cdf) of N(0,1).
Appendix 3 — Derivation of the \({\sigma _{0}^{2}}\) estimator
Appendix 4 — Derivation of ltdr ^{(1)} under a two-component mixture prior
where ϕ(x) is the probability density function (pdf) of N(0,1).
Notes
Declarations
Acknowledgements
This paper was partially supported by a theme-based research project T12-402/13N of the Hong Kong Research Grant Council (RGC). The publication costs for this article were partly funded by the Hong Kong Research Grant Council, grant number T12-402/13N.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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