- Research Article
- Open Access
- Open Peer Review
Confidence intervals construction for difference of two means with incomplete correlated data
- Hui-Qiong Li^{1}Email author,
- Nian-Sheng Tang^{1} and
- Jie-Yi Yi^{2}
https://doi.org/10.1186/s12874-016-0125-3
© Li et al. 2016
- Received: 23 July 2015
- Accepted: 10 February 2016
- Published: 11 March 2016
Abstract
Background
Incomplete data often arise in various clinical trials such as crossover trials, equivalence trials, and pre and post-test comparative studies. Various methods have been developed to construct confidence interval (CI) of risk difference or risk ratio for incomplete paired binary data. But, there is little works done on incomplete continuous correlated data. To this end, this manuscript aims to develop several approaches to construct CI of the difference of two means for incomplete continuous correlated data.
Methods
Large sample method, hybrid method, simple Bootstrap-resampling method based on the maximum likelihood estimates (B _{1}) and Ekbohm’s unbiased estimator (B _{2}), and percentile Bootstrap-resampling method based on the maximum likelihood estimates (B _{3}) and Ekbohm’s unbiased estimator (B _{4}) are presented to construct CI of the difference of two means for incomplete continuous correlated data. Simulation studies are conducted to evaluate the performance of the proposed CIs in terms of empirical coverage probability, expected interval width, and mesial and distal non-coverage probabilities.
Results
Empirical results show that the Bootstrap-resampling-based CIs B _{1}, B _{2}, B _{4} behave satisfactorily for small to moderate sample sizes in the sense that their coverage probabilities could be well controlled around the pre-specified nominal confidence level and the ratio of their mesial non-coverage probabilities to the non-coverage probabilities could be well controlled in the interval [0.4, 0.6].
Conclusions
If one would like a CI with the shortest interval width, the Bootstrap-resampling-based CIs B _{1} is the optimal choice.
Keywords
- Bootstrap
- Confidence interval
- Correlated data
- Incomplete data
Background
FEV_{1} indicators of patients for 12 ug and 24 ug formoterol solution aerosol
12 ug(x _{1}) | 24 ug(x _{2}) |
---|---|
2.250 | 2.700 |
0.925 | 0.900 |
1.010 | 1.270 |
2.100 | 2.150 |
2.500 | 2.450 |
1.750 | 1.725 |
1.370 | 1.120 |
3.400 | |
2.250 | |
1.460 | |
1.480 | |
2.050 | |
3.500 | |
2.650 | |
2.190 | |
0.840 | |
1.750 | |
2.525 | |
1.080 | |
3.120 | |
3.100 | |
2.700 | |
1.870 | |
0.940 |
For the above crossover clinical trial, our main interest is to test the equivalence between 12 ug and 24 ug formoterol solution aerosols with respect to the FEV_{1} value. To this end, we can construct a (1−α)100 % confidence interval for the difference of two FEV_{1} values. If the resultant confidence interval (CI) lies entirely in the interval (−δ _{0},δ _{0}) with δ _{0}(>0) being some pre-specified clinical acceptable threshold, we thus could conclude the equivalence between two doses of formoterol solution aerosol at the α significance level. As a result, reliable CIs for the difference in the presence of incomplete data are necessary.
The problem of testing the equality and constructing CI for the difference of two correlated proportions in the presence of incomplete paired binary data has received considerable attention in past years. For example, ones can refer to [2–6] for the large sample method, and [7] for the corrected profile likelihood method. When sample size is small, [8] proposed the exact unconditional test procedure for testing equality of two correlated proportions with incomplete correlated data. Tang, Ling and Tian [9] developed the exact unconditional and approximate unconditional CIs for proportion difference in the presence of incomplete paired binary data. Lin et al. [10] presented a Bayesian method to test equality of two correlated proportions with incomplete correlated data. Li et al. [11] discussed the confidence interval construction for rate ratio in matched-pair studies with incomplete data. However, all the aforementioned methods were developed for incomplete paired binary data.
Statistical inference on the difference of two means with incomplete correlated data has received a limited attention. For example, [12] discussed the problem of testing the equality of two means with missing data on one response and recommended [13] statistic when the variances were not too different. Lin and Stivers [14] also gave a similar comparison. Lin and Stivers [15] and [12] suggested some test statistics for testing the equality of two means with incomplete data on both response. However, to our knowledge, little work has been done on CI construction for the difference of two means with incomplete correlated data under the MCAR assumption.
Inspired by [16–19], we develop several CIs for the difference of two means with incomplete correlated data under the MCAR assumption based on the large sample method, hybrid method and Bootstrap-resampling method. The presented Bootstrap-resampling CIs have not been considered in the literature related to missing observations.
The rest of this article is organized as follows. Several methods are presented to construct CIs for the difference of the two means with incomplete correlated data in Section “Methods”. Simulation studies and an example are conducted to evaluate the finite performance of the proposed CIs in terms of coverage probability, expected interval width, and mesial and distal non-coverage probabilities in Section “Results”. A brief discussion is given in Section “Discussion”. Some concluding remarks are given in Section “Conclusion”.
Methods
where (x _{1m },x _{2m }) is referred to as a paired observation, while x _{1,n+j } and x _{2,n+k } are referred to as incomplete or unpaired observations. Similar to [20, 21], throughout this article, it is assumed that the missing data mechanism is MCAR (i.e., independent of treatment and outcome). Based on these observations, we here want to construct reliable explicit CIs for the difference of two means δ=μ _{1}−μ _{2} under MCAR assumption.
Confidence interval based on the large sample method
To make a comparison with the following proposed methods, we assume that x follows a bivariate normal distribution in this subsection. In this case, if only variable x _{1} or x _{2} is subject to missingness (i.e., n _{1}=0 or n _{2}=0), one can obtain the closed forms of the maximum likelihood estimates (MLEs) of μ and Σ [22]. However, there are no closed forms of the MLEs for μ and Σ when variables x _{1} and x _{2} are simultaneously subject to missingness (i.e., n _{1}≠0 and n _{2}≠0), though one can find the MLEs of μ and Σ using an iterative algorithm [23]. To get the closed forms of MLEs for μ and Σ, [15] proposed the modified MLEs using a non-iterative procedure and provided several test statistics based on the obtained estimators of μ and Σ.
(i) Confidence interval based on Lin and Stivers’s test statistics
respectively, where \(\overline {x}_{1}^{(n)}=\frac {1}{n}\sum _{j=1}^{n}x_{1j}\), \(\overline {x}_{2}^{(n)}=\frac {1}{n}\sum _{j=1}^{n}x_{2j}\), \(\overline {x}_{1}^{(n_{1})}=\frac {1}{n_{1}}\sum _{j=1}^{n_{1}}x_{1,n+j}\), \(\overline {x}_{2}^{(n_{2})}=\frac {1}{n_{2}}\sum _{k=1}^{n_{2}}x_{2,n+k}\), a=nh(n+n _{2}+n _{1} β _{21}), b=nh(n+n _{1}+n _{2} β _{12}), β _{21}=ρσ _{2}/σ _{1}, β _{12}=ρσ _{1}/σ _{2}, h=1/{(n+n _{1})(n+n _{2})−n _{1} n _{2} ρ ^{2}}. An approximate 100(1−α) % CI of δ is given by \(\left (\hat {\delta }-\textit {z}_{\alpha /2}\sqrt {\text {Var}(\hat {\delta })}, \hat {\delta }+\textit {z}_{\alpha /2}\sqrt {\text {Var}(\hat {\delta })}\right)\), which is denoted as T _{ w1}-CI.
which is asymptotically distributed as t-distribution with n degrees of freedom under H _{0}, where V _{1}=[{A ^{2}/n+(1−A)^{2}/n _{1} }m _{1}+{ B ^{2}/n+(1−B)^{2}/n _{2} } m _{2}−2ABm _{12}/n]/(n−1), A={n(n+n _{2}+n _{1} m _{12}/m _{1}}/{ (n+n _{1})(n+n _{2})−n _{1} n _{2} r ^{2}}^{−1}, B={n(n+n _{1}+n _{2} m _{12}/ m _{2}} /{ (n + n _{1})(n + n _{2})−n _{1} n _{2} r ^{2}}^{−1}, \(m_{1}=\sum _{j=1}^{n} \left (x_{1j}-\overline {x}_{1}^{(n)}\right)^{2}\), \(m_{2}=\sum _{j=1}^{n}\left (x_{2j}-\overline {x}_{2}^{(n)}\right)^{2}\), \(m_{12}=\sum _{j=1}^{n}\left (x_{1j}\,-\,\overline {x}_{1}^{(n)}\right)\left (x_{2j}-\overline {x}_{2}^{(n)}\right)\), \(r=m_{12}/\sqrt {m_{1}m_{2}}\). Therefore, the approximate 100(1−α) % CI on the basis of T _{1} is given by (L, U), where \(L=A\left (\overline {x}_{1}^{(n)}-\overline {x}_{1}^{(n_{1})}\right)-B\left (\overline {x}_{2}^{(n)}-\overline {x}_{2}^{(n_{2})}\right)+\overline {x}_{1}^{(n_{1})}-\overline {x}_{2}^{(n_{2})}-t_{\alpha /2}(n)\sqrt {V_{1}}\), and \(U=A\left (\overline {x}_{1}^{(n)}-\overline {x}_{1}^{(n_{1})}\right)-B\left (\overline {x}_{2}^{(n)}-\overline {x}_{2}^{(n_{2})}\right)+\overline {x}_{1}^{(n_{1})}-\overline {x}_{2}^{(n_{2})}+t_{\alpha /2}(n)\sqrt {V_{1}}\), which is denoted as T _{1}-CI.
which is asymptotically distributed as t distribution with degrees ν of freedom, where \(\bar {x}_{1}^{(n+n_{1})}=(n+n_{1})^{-1}\sum _{j=1}^{n+n_{1}}x_{1j}\), \(\bar {x}_{2}^{(n+n_{2})}=(n+n_{2})^{-1}\sum _{j=1}^{n+n_{2}}x_{2j}\), h _{1}=n{(n+n _{2})m _{1}/(n+n _{1})+(n+n _{1})m _{2}/(n+n _{2})−2m _{12}}/{(n−1)(n+n _{1})(n+n _{2})}, h _{2}=n _{1} b _{1}/{(n _{1}−1)(n+n _{1})^{2}}, h _{3}=n _{2} b _{2}/{(n _{2}−1)(n+n _{2})^{2}}, \(b_{1}=\sum _{j=n+1}^{n+n_{1}}\left (x_{1j}-\overline {x}_{1}^{(n_{1})}\right)^{2}\), \(b_{2}=\sum _{j=n+1}^{n+n_{2}}\left (x_{2j}-\overline {x}_{2}^{(n_{1})}\right)^{2}\), and \(\nu =\left (h_{1}+h_{2}+h_{3}\right)^{2}/\{{h_{1}^{2}}/(n-1)+{h_{2}^{2}}/(n_{1}-1)+{h_{3}^{2}}/(n_{2}-1)\}\). Therefore, the approximate 100(1−α) % CI of δ for statistic T _{2} is denoted as T _{2}-CI.
which is asymptotically distribution as t-distribution with degrees n+n _{1}+n _{2}−4 of freedom. Note that when n _{2}>n _{1}, b _{1}+c _{2} should be replaced by b _{2}+c _{1}. Thus, the approximate 100(1−α) % CI of δ for T _{3} is denoted as T _{3}-CI, where \(c_{1}=\sum _{j=1}^{n+n_{1}}\left (x_{1j}-{n+n_{1}}\sum _{j=1}^{n+n_{1}}x_{1j}\right)^{2}\), and \(c_{2}=\sum _{j=1}^{n+n_{2}}\left (x_{2j}-\frac {1}{n+n_{2}}\sum _{j=1}^{n+n_{2}}x_{2j}\right)^{2}\).
Also, [12] presented the similar but simpler test statistics for testing the mean difference δ=μ _{1}−μ _{2}, which are adopted to construct CIs of δ as follows.
(ii) Confidence interval based on Ekbohm’s test statistics
Following [12], an unbiased estimator of δ is given by \(\hat {\delta }=\bar {x}_{1}^{(n+n_{1})}-\bar {x}_{2}^{(n+n_{2})}\), and its variance is given by \(\text {Var}(\hat {\delta })= \text {Var}(\hat {\mu })=\left \{(n+n_{2}){\sigma _{1}^{2}}+(n+n_{2}){\sigma _{2}^{2}}-2n\rho \sigma _{1}\sigma _{2}\right \}/\left \{(n+n_{1})(n+n_{2})\right \}\). An approximate 100(1 − α) % CI of δ can be obtained by \(\left (\hat {\delta }-\textit {z}_{\alpha /2}\sqrt {\text {Var}(\hat {\delta })}, \hat {\delta }+\right.\) \(\left.\textit {z}_{\alpha /2}\sqrt {\text {Var}(\hat {\delta })}\right),\) which is denoted as T _{ w2}-CI.
When σ _{1}=σ _{2}, Ekbothm (1976) proposed the following statistic for testing H _{0}: \(T_{4}=(\tilde {\delta }-\delta _{0})\sqrt {\!(n\,+\,\!n_{1})(n\,+\,\!n_{2})\,-\,n_{1}n_{2}\lambda ^{2}}/\) \( \left \{\hat \sigma \!\sqrt {2n(1-\lambda)+(n_{1}+n_{2})(1-\lambda ^{2})}\right \}\), where \(\tilde {\delta }=\left [n\left (n+n_{2}+n_{1}\lambda \right)\overline {x}_{1}^{(n)}\!-n\left (n+n_{1}+n_{2}\lambda \right)\overline {x}_{2}^{(n)}+n_{1}\!\left \{n\,+\,n_{2}\!\left (1\,-\,\lambda ^{2}\right)-\!n\lambda \right \}\overline {x}_{1}^{(n_{1})}-n_{2}\left \{n\,+\,n_{1}\!\left (1\!\,-\,\!\lambda \right)^{2}\!\,-\,n\lambda \right \}\overline {x}_{2}^{(n_{2})}\right ]\!\big /\!\!\left \{(n\,+\,n_{1})(n+n_{2})\,-\,n_{1}n_{2}\lambda ^{2}\right \}\), \(\hat {\sigma }^{2}\,=\,\left \{m_{1}\,+\,m_{2}\,+\,(1+\lambda ^{2})(b_{1}\,+\,b_{2})\right \}/\left \{2(n-\!1)+\!(1\,+\,\lambda ^{2})(n_{1}\,+\,n_{2}\,-\,2)\right \}\), and λ=2m _{12}/(m _{1}+m _{2}). Under H _{0}, T _{4} is asymptotically distributed as t-distribution with degrees n of freedom. Therefore, the approximate 100(1−α) % CI is denoted as T _{4}-CI.
Following [12], when σ _{1}=σ _{2}, another statistic for testing H _{0} can be expressed as \(T_{5}= \left (\bar {x}_{1}^{(n+n_{1})}\!-\bar {x}_{2}^{(n+n_{2})}\,-\,\delta _{0}\!\right)\!\sqrt {(n\,+\,n_{1})(n\,+\,n_{2})/(R_{1}\!\,+\,R_{2})}\), which is asymptotically distributed as t distribution with degrees ν _{ σ } of freedom under H _{0}, where R _{1} = n(m _{1} + m _{2} − 2m _{12}) /(n − 1), R _{2} =(n _{1} + n _{2})(b _{1}+b _{2})/(n _{1}+n _{2}−2), and \(\nu _{\sigma }=\left (R_{1}+R_{2}\right)^{2}\!\!\!~/\left \{{R_{1}^{2}}/(n+1)+{R_{2}^{2}}/(n_{1}+n_{2})\right \}-2\). Thus, an approximate 100(1−α) % CI of δ for T _{5} is denoted as T _{5}-CI.
Confidence interval based on the generalized estimating equations(GEEs)
To relax the bivariate normality assumption of x, the method of the generalized estimating equations (GEEs) with exchangeable working correlation structure (e.g., [25]) can be adopted to make statistical inference on δ in the incomplete correlated data because the GEE approach have become one of the most widely used methods in dealing with correlated response data [26, 27]. Following [28], the GEEs with exchangeable working correlation structure can be used to estimate parameter vector μ; the so-called sandwich variance estimator can be used to consistently estimate the covariance matrix of μ; and the ML method under a bivariate normal assumption via available paired observations is used to estimate the correlation parameter. Thus, an approximate 100(1−α) % CI of δ based on GEE method is denoted as T _{ g }-CI.
Confidence interval based on the hybrid method
To obtain the above presented approximate 100(1−α) % hybrid CI for μ _{1}−μ _{2}, one requires evaluating the (1−α) 100 % CIs of θ _{1} = μ _{1} (denoted as (l _{1}, u _{1})) and θ _{2}=μ _{2} (denoted as (l _{2}, u _{2})), and estimating the correlation coefficient \(\widehat {\text {corr}}(\hat {\theta }_{1}, \hat \theta _{2})\). For the former, following [19], we consider the following two methods for getting the confidence limits (l _{1}, u _{1}) and (l _{2}, u _{2}) of θ _{1} and θ _{2}.
where N _{ i }=n+n _{ i } and \(\hat {\theta }_{i}=\frac {1}{N_{i}}\sum _{j=1}^{N_{i}}x_{ij}\) for i=1,2.
where N _{ i }=n+n _{ i } and \(\tilde {\theta }_{i}=\left (\sum _{j=1}^{N_{i}}x_{ij}+0.5z_{\alpha /2}^{2}\right)/\left (N_{i}+z_{\alpha /2}^{2}\right)\) for i=1,2.
To construct CI for δ=μ _{1}−μ _{2} via the above described hybrid method, we can simply set θ _{1}=μ _{1} and θ _{2}=μ _{2}. If Σ is known, the estimated correlation coefficient \(\widehat {\text {corr}}(\hat {\mu }_{1}, \hat {\mu }_{2})\) of \(\hat {\mu }_{1}\) and \(\hat {\mu }_{2}\) is given by \(\widehat {\text {corr}}(\hat {\mu }_{1}, \hat {\mu }_{2})=2n\rho /\sqrt {(n+n_{1})(n+n_{2})}\). If Σ is unknown, \(\widehat {\text {corr}}(\hat {\mu }_{1}, \hat {\mu }_{2})\) is given by \(\widehat {\text {corr}}(\hat {\mu }_{1}, \hat {\mu }_{2})=nr/\left \{(n+n_{1})(n+n_{2})-n_{1}n_{2}r^{2}\right \}\), where \(r=m_{12}/\sqrt {m_{1}m_{2}}\), \(m_{1}=\sum _{j=1}^{n}\left (x_{1j}-\overline {x}_{1}^{(n)}\right)^{2}\) and \(m_{2}=\sum _{j=1}^{n}\left (x_{2j}-\overline {x}_{2}^{(n)}\right)^{2}.\) Thus, using Eqs. (1) and (2) yields CIs of δ=μ _{1}−μ _{2}. When l _{ i } and u _{ i } are estimated by the Wilson score method, we denote the corresponding CI as W _{ s }-CI; when l _{ i } and u _{ i } are estimated by the Agresti-coull method, the corresponding CI is denoted as W _{ a }-CI.
Bootstrap-resampling-based confidence intervals
When the distribution of x is known, one can obtain the approximate CIs of δ based on the asymptotic distributions of the constructed test statistics under the null hypotheses H _{0}:δ=δ _{0}. However, when the distribution of x is unknown, the asymptotic distributions of the constructed test statistics may not be reliable, especially with small sample size. On the other hand, estimators of some nuisance parameters have not the closed-form solutions even if the approximate distribution is reliable, and they must be obtained by using some iterative algorithms, which are computationally intensive. In this case, the Bootstrap method is often adopted to construct CIs of parameter of interest. The Bootstrap CIs can be constructed via the following steps.
Step 2. For the above generated Bootstrap resampling sample \(D_{b}^{*}\), we first compute \(\hat {\mu }_{1}^{*}=(n+n_{1})^{-1}\sum _{j=1}^{n+n_{1}}x_{1j}^{*}\) and \(\hat {\mu }_{2}^{*}=(n+n_{2})^{-1}\sum _{j=1}^{n+n_{2}}x_{2j}^{*}\), and then calculate the estimated value \(\hat {\delta }^{*}\) of δ via \(\hat {\delta }^{*}=\hat {\mu }_{1}^{*}-\hat {\mu }_{2}^{*}\).
Step 3. Repeating the above steps 1 and 2 for a total of G times yields G Bootstrap estimates \(\left \{\hat {\delta }_{g}^{*}: g=1,2,\cdots,G\right \}\) of δ. Let \(\hat \delta _{(1)}^{*}<\hat \delta _{(2)}<\cdots <\hat \delta _{(G)}^{*}\) be the ordered values of \(\left \{\hat \delta _{g}^{*}: g=1,2,\cdots,G\right \}\).
Step 4. Based on the bootstrap estimates \(\left \{\hat {\delta }_{g}^{*}, g=1,2,\ldots,G\vphantom {\left \{\hat {\delta }_{g}^{*}, g=\right.}\right \}\), Bootstrap-resampling-based CIs for δ can be constructed as follows.
Generally, the standard error se \((\hat {\delta })\) of \(\hat {\delta }\) can be estimated by the sample standard deviation of the G replications, i.e., \(\hat {\text {se}}(\hat {\delta })=\sqrt {(G-1)^{-1}\sum _{g=1}^{G}\left (\hat {\delta }_{g}^{*}-\bar {\delta }_{B}^{*}\right)^{2}}\), where \(\bar \delta _{B}^{*}=\left (\hat {\delta }_{1}^{*}+\cdots +\hat {\delta }_{G}^{*}\right)/G\). If \(\left \{\hat {\delta }_{g}^{*}: g=1,\cdots, G\right \}\) is approximately normally distributed, an approximate 100(1−α) % Bootstrap CI for δ is given by \(\left (\hat {\delta }-z_{\alpha /2}\hat {\text {se}}(\hat {\delta }), \hat {\delta }+z_{\alpha /2}\hat {\text {se}}(\hat \delta)\right)\), where z _{ α/2} is the upper α/2-percentile of the standard normal distribution, which is referred as the simple Bootstrap confidence interval. When \(\hat {\delta }=a\overline {x}_{1}^{(n)}+(1-a)\overline {x}_{1}^{(n_{1})}-b\overline {x}_{2}^{(n)}-(1-b)\overline {x}_{2}^{(n_{2})}\), the corresponding simple Bootstrap CI is denoted as B _{1}. When \(\hat {\delta }=\bar {x}_{1}^{(n+n_{1})}-\bar {x}_{2}^{(n+n_{2})}\), the corresponding simple Bootstrap CI is denoted as B _{2}.
Alternatively, if \(\left \{\hat {\delta }_{g}^{*}: g=1,\cdots, G\right \}\) is not normally distributed, it follows from ([16] p.132) that the approximate 100(1−α) % Bootstrap-resampling-based percentile CI for δ is \(\left (\hat \delta _{\left ([G\alpha /2]\right)}^{*},\hat {\delta }_{([G(1-\alpha /2)])}^{*}\right)\), where [ a] represents the integer part of a, which is referred as the percentile Bootstrap CI. When \(\hat \delta =a\overline {x}_{1}^{(n)}+(1-a)\overline {x}_{1}^{(n_{1})}-b\overline {x}_{2}^{(n)}-\left (1-b\right)\overline {x}_{2}^{(n_{2})}\), the corresponding percentile Bootstrap CI is denoted as B _{3}. When \(\hat {\delta }=\bar {x}_{1}^{(n+n_{1})}-\bar {x}_{2}^{(n+n_{2})}\), the corresponding percentile Bootstrap CI is denoted as B _{4}.
Results
Simulation studies
Summary of various abbreviations
Abbreviation | Definition |
---|---|
T _{1} | CI based on T _{1} statistic |
T _{2} | CI based on T _{2} statistic |
T _{3} | CI based on T _{3} statistic |
T _{4} | CI based on T _{4} statistic |
T _{5} | CI based on T _{5} statistic |
T _{ g } | CI based on GEE method |
W _{ s } | CI based on Wilson score method |
W _{ a } | CI based on Agresti-coull method |
B _{1} | Simple Bootstrap CI based on |
\(\hat \delta =a\overline {x}_{1}^{(n)}+\left (1-a\right)\overline {x}_{1}^{(n_{1})}-b\overline {x}_{2}^{(n)}-(1-b)\overline {x}_{2}^{(n_{2})}\) | |
B _{2} | Simple Bootstrap CI based on \(\hat {\delta }=\bar {x}_{1}^{(n+n_{1})}-\bar {x}_{2}^{(n+n_{2})}\) |
B _{3} | Percentile Bootstrap CI based on |
\(\hat \delta =a\overline {x}_{1}^{(n)}+(1-a)\overline {x}_{1}^{(n_{1})}-b\overline {x}_{2}^{(n)}-(1-b)\overline {x}_{2}^{(n_{2})}\) | |
B _{4} | Percentile Bootstrap CI based on \(\hat {\delta }=\bar {x}_{1}^{(n+n_{1})}-\bar {x}_{2}^{(n+n_{2})}\) |
ECPs | Empirical coverage probabilities, is defined by Eq. (3) |
ECW | Empirical confidence widths, is defined by Eq. (3) |
RNCP | The ratio of the mesial non-coverage probabilities to the |
ECPs of various confidence intervals under bivariate normal distribution with different ρ and δ, μ _{1}, \(\mu _{2} {\sigma _{1}^{2}}\) and (n,n _{1},n _{2})=(5,2,2) and \({\sigma _{2}^{2}}=4\)
ρ | \({\sigma _{1}^{2}}\) | δ | μ _{1} | μ _{2} | T _{1} | T _{2} | T _{ g } | W _{ s } | W _{ a } | B _{1} | B _{2} | B _{3} | B _{4} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
-0.9 | 1 | -0.25 | 0 | 0.25 | 0.9390 | 0.9590 | 0.9370 | 0.9350 | 0.8800 | 0.9520 | 0.9560 | 0.9370 | 0.9570 |
0 | 1 | 1 | 0.9440 | 0.9580 | 0.9470 | 0.9220 | 0.8760 | 0.9470 | 0.9490 | 0.9300 | 0.9490 | ||
0.5 | 2 | 1.5 | 0.9430 | 0.9670 | 0.9530 | 0.9400 | 0.8860 | 0.9470 | 0.9480 | 0.9310 | 0.9480 | ||
8 | -0.25 | 0 | 0.25 | 0.9410 | 0.9630 | 0.9450 | 0.9180 | 0.8600 | 0.9430 | 0.9490 | 0.9340 | 0.9490 | |
0 | 1 | 1 | 0.9370 | 0.9580 | 0.9350 | 0.9240 | 0.8640 | 0.9510 | 0.9500 | 0.9390 | 0.9500 | ||
0.5 | 2 | 1.5 | 0.9380 | 0.9570 | 0.9410 | 0.9240 | 0.8750 | 0.9570 | 0.9560 | 0.9470 | 0.9560 | ||
-0.5 | 1 | -0.25 | 0 | 0.25 | 0.9440 | 0.9610 | 0.9530 | 0.9200 | 0.8570 | 0.9490 | 0.9430 | 0.9330 | 0.9420 |
0 | 1 | 1 | 0.9420 | 0.9660 | 0.9230 | 0.9270 | 0.8660 | 0.9570 | 0.9560 | 0.9460 | 0.9550 | ||
0.5 | 2 | 1.5 | 0.9460 | 0.9660 | 0.9380 | 0.9250 | 0.8640 | 0.9480 | 0.9560 | 0.9430 | 0.9540 | ||
8 | -0.25 | 0 | 0.25 | 0.9290 | 0.9590 | 0.9480 | 0.9230 | 0.8730 | 0.9470 | 0.9450 | 0.9390 | 0.9440 | |
0 | 1 | 1 | 0.9290 | 0.9560 | 0.9420 | 0.9210 | 0.8790 | 0.9460 | 0.9430 | 0.9380 | 0.9440 | ||
0.5 | 2 | 1.5 | 0.9350 | 0.9690 | 0.9410 | 0.9330 | 0.8880 | 0.9520 | 0.9540 | 0.9470 | 0.9520 | ||
-0.1 | 1 | -0.25 | 0 | 0.25 | 0.9300 | 0.9570 | 0.9500 | 0.9170 | 0.8630 | 0.9550 | 0.9500 | 0.9450 | 0.9470 |
0 | 1 | 1 | 0.9380 | 0.9590 | 0.9450 | 0.9170 | 0.8600 | 0.9540 | 0.9500 | 0.9450 | 0.9520 | ||
0.5 | 2 | 1.5 | 0.9400 | 0.9620 | 0.9440 | 0.9140 | 0.8560 | 0.9510 | 0.9460 | 0.9420 | 0.9460 | ||
8 | -0.25 | 0 | 0.25 | 0.9460 | 0.9600 | 0.9310 | 0.9050 | 0.8490 | 0.9460 | 0.9470 | 0.9440 | 0.9470 | |
0 | 1 | 1 | 0.9450 | 0.9670 | 0.9440 | 0.9150 | 0.8590 | 0.9560 | 0.9500 | 0.9480 | 0.9510 | ||
0.5 | 2 | 1.5 | 0.9350 | 0.9610 | 0.9360 | 0.9150 | 0.8570 | 0.9500 | 0.9520 | 0.9440 | 0.9490 | ||
0 | 1 | -0.25 | 0 | 0.25 | 0.9380 | 0.9610 | 0.9400 | 0.9330 | 0.8860 | 0.9550 | 0.9550 | 0.9530 | 0.9530 |
0 | 1 | 1 | 0.9290 | 0.9610 | 0.9280 | 0.9200 | 0.8680 | 0.9470 | 0.9480 | 0.9470 | 0.9470 | ||
0.5 | 2 | 1.5 | 0.9300 | 0.9580 | 0.9420 | 0.9230 | 0.8800 | 0.9520 | 0.9510 | 0.9500 | 0.9510 | ||
8 | -0.25 | 0 | 0.25 | 0.9210 | 0.9590 | 0.9390 | 0.9090 | 0.8400 | 0.9430 | 0.9450 | 0.9450 | 0.9450 | |
0 | 1 | 1 | 0.9240 | 0.9570 | 0.9400 | 0.9050 | 0.8520 | 0.9430 | 0.9440 | 0.9430 | 0.9430 | ||
0.5 | 2 | 1.5 | 0.9360 | 0.9680 | 0.9380 | 0.9140 | 0.8540 | 0.9530 | 0.9530 | 0.9530 | 0.9520 | ||
0.1 | 1 | -0.25 | 0 | 0.25 | 0.9310 | 0.9690 | 0.9480 | 0.9150 | 0.8530 | 0.9510 | 0.9510 | 0.9490 | 0.9490 |
0 | 1 | 1 | 0.9330 | 0.9670 | 0.9440 | 0.9150 | 0.8550 | 0.9500 | 0.9500 | 0.9490 | 0.9510 | ||
0.5 | 2 | 1.5 | 0.9310 | 0.9570 | 0.9490 | 0.9150 | 0.8630 | 0.9520 | 0.9520 | 0.9510 | 0.9520 | ||
8 | -0.25 | 0 | 0.25 | 0.9220 | 0.9520 | 0.9420 | 0.9190 | 0.8700 | 0.9510 | 0.9510 | 0.9520 | 0.9520 | |
0 | 1 | 1 | 0.9290 | 0.9540 | 0.9360 | 0.9210 | 0.8690 | 0.9490 | 0.9490 | 0.9470 | 0.9470 | ||
0.5 | 2 | 1.5 | 0.9180 | 0.9530 | 0.9350 | 0.9340 | 0.8860 | 0.9520 | 0.9520 | 0.9500 | 0.9500 | ||
0.5 | 1 | -0.25 | 0 | 0.25 | 0.9230 | 0.9530 | 0.9470 | 0.8980 | 0.8470 | 0.9540 | 0.9540 | 0.9530 | 0.9530 |
0 | 1 | 1 | 0.9330 | 0.9620 | 0.9390 | 0.9050 | 0.8510 | 0.9440 | 0.9440 | 0.9440 | 0.9440 | ||
0.5 | 2 | 1.5 | 0.9280 | 0.9640 | 0.9330 | 0.9140 | 0.8640 | 0.9520 | 0.9520 | 0.9500 | 0.9500 | ||
8 | -0.25 | 0 | 0.25 | 0.9360 | 0.9660 | 0.9420 | 0.9030 | 0.8450 | 0.9470 | 0.9470 | 0.9460 | 0.9460 | |
0 | 1 | 1 | 0.9220 | 0.9600 | 0.9350 | 0.9060 | 0.8410 | 0.9500 | 0.9500 | 0.9480 | 0.9480 | ||
0.5 | 2 | 1.5 | 0.9300 | 0.9650 | 0.9500 | 0.9140 | 0.8570 | 0.9580 | 0.9580 | 0.9570 | 0.9570 | ||
0.9 | 1 | -0.25 | 0 | 0.25 | 0.9190 | 0.9540 | 0.9400 | 0.9300 | 0.8710 | 0.9450 | 0.9450 | 0.9440 | 0.9430 |
0 | 1 | 1 | 0.9390 | 0.9640 | 0.9460 | 0.9360 | 0.8870 | 0.9590 | 0.9580 | 0.9570 | 0.9580 | ||
0.5 | 2 | 1.5 | 0.9240 | 0.9610 | 0.9310 | 0.9220 | 0.8760 | 0.9470 | 0.9460 | 0.9470 | 0.9470 | ||
8 | -0.25 | 0 | 0.25 | 0.9200 | 0.9590 | 0.9440 | 0.9050 | 0.8440 | 0.9440 | 0.9430 | 0.9430 | 0.9450 | |
0 | 1 | 1 | 0.9310 | 0.9620 | 0.9430 | 0.9040 | 0.8390 | 0.9450 | 0.9450 | 0.9460 | 0.9460 | ||
0.5 | 2 | 1.5 | 0.9310 | 0.9620 | 0.9400 | 0.9190 | 0.8610 | 0.9530 | 0.9520 | 0.9520 | 0.9530 |
ECW of various confidence intervals under bivariate normal distribution with different ρ and δ, μ _{1}, μ _{2}, \({\sigma _{1}^{2}}\) and (n,n _{1},n _{2})=(5,2,2) and \({\sigma _{2}^{2}}=4\)
ρ | \({\sigma _{1}^{2}}\) | δ | μ _{1} | μ _{2} | T _{1} | T _{2} | T _{ g } | W _{ s } | W _{ a } | B _{1} | B _{2} | B _{3} | B _{4} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
-0.9 | 1 | -0.25 | 0 | 0.25 | 8.0510 | 9.8480 | 7.6040 | 4.9790 | 4.0830 | 6.5400 | 6.9700 | 6.5380 | 6.9700 |
0 | 1 | 1 | 8.0980 | 9.8440 | 7.6290 | 4.9880 | 4.0930 | 6.5410 | 6.9710 | 6.5410 | 6.9710 | ||
0.5 | 2 | 1.5 | 8.1690 | 9.7210 | 7.6410 | 5.0880 | 4.2070 | 6.5420 | 6.9700 | 6.5410 | 6.9680 | ||
8 | -0.25 | 0 | 0.25 | 10.8170 | 12.0750 | 9.6020 | 6.5090 | 5.2840 | 8.8020 | 9.1950 | 8.8010 | 9.1960 | |
0 | 1 | 1 | 10.8350 | 12.1090 | 9.5830 | 6.5080 | 5.2840 | 8.8030 | 9.1950 | 8.8050 | 9.1940 | ||
0.5 | 2 | 1.5 | 10.8310 | 12.0670 | 9.5720 | 6.5610 | 5.3560 | 8.8080 | 9.2000 | 8.8070 | 9.1960 | ||
-0.5 | 1 | -0.25 | 0 | 0.25 | 12.7390 | 14.0040 | 11.0300 | 7.6080 | 6.1620 | 10.2980 | 10.7370 | 10.2990 | 10.7370 |
0 | 1 | 1 | 12.7510 | 14.0800 | 11.0500 | 7.6310 | 6.1810 | 10.3020 | 10.7410 | 10.3000 | 10.7380 | ||
0.5 | 2 | 1.5 | 12.7460 | 14.0150 | 11.0120 | 7.6540 | 6.2220 | 10.3070 | 10.7470 | 10.3080 | 10.7450 | ||
8 | -0.25 | 0 | 0.25 | 7.9520 | 9.4420 | 7.3030 | 4.7520 | 3.8910 | 6.4600 | 6.5990 | 6.4620 | 6.6000 | |
0 | 1 | 1 | 7.9990 | 9.4880 | 7.3300 | 4.7760 | 3.9140 | 6.4630 | 6.6000 | 6.4630 | 6.6030 | ||
0.5 | 2 | 1.5 | 7.9410 | 9.4190 | 7.3300 | 4.8830 | 4.0460 | 6.4650 | 6.6040 | 6.4630 | 6.6010 | ||
-0.1 | 1 | -0.25 | 0 | 0.25 | 10.1230 | 11.1210 | 8.9290 | 6.0060 | 4.8870 | 8.2480 | 8.3910 | 8.2510 | 8.3940 |
0 | 1 | 1 | 10.1150 | 11.2600 | 9.9060 | 6.0040 | 4.8850 | 8.2490 | 8.3920 | 8.2490 | 8.3930 | ||
0.5 | 2 | 1.5 | 10.0550 | 11.1650 | 9.8830 | 6.0750 | 4.9860 | 8.2460 | 8.3880 | 8.2480 | 8.3890 | ||
8 | -0.25 | 0 | 0.25 | 11.8990 | 12.9260 | 10.2600 | 7.0330 | 5.7080 | 9.6020 | 9.7660 | 9.6030 | 9.7670 | |
0 | 1 | 1 | 11.9170 | 12.9540 | 10.2910 | 7.0500 | 5.7240 | 9.6030 | 9.7670 | 9.6020 | 9.7670 | ||
0.5 | 2 | 1.5 | 11.9290 | 13.0050 | 10.2490 | 7.1130 | 5.8070 | 9.5990 | 9.7620 | 9.5980 | 9.7610 | ||
0 | 1 | -0.25 | 0 | 0.25 | 7.4380 | 8.7970 | 6.9270 | 4.4570 | 3.6460 | 6.2020 | 6.2080 | 6.1980 | 6.2060 |
0 | 1 | 1 | 7.4070 | 9.0290 | 6.9140 | 4.4570 | 3.6480 | 6.2100 | 6.2160 | 6.2100 | 6.2160 | ||
0.5 | 2 | 1.5 | 7.4750 | 9.0040 | 6.9620 | 4.6380 | 3.8580 | 6.2020 | 6.2080 | 6.2000 | 6.2060 | ||
8 | -0.25 | 0 | 0.25 | 9.0700 | 10.2520 | 8.2140 | 5.4680 | 4.4610 | 7.4900 | 7.4970 | 7.4910 | 7.4960 | |
0 | 1 | 1 | 9.0480 | 10.0050 | 8.1310 | 5.4190 | 4.4290 | 7.4910 | 7.4980 | 7.4880 | 7.4960 | ||
0.5 | 2 | 1.5 | 9.1370 | 10.2100 | 8.2110 | 5.5930 | 4.6170 | 7.4920 | 7.5000 | 7.4910 | 7.4970 | ||
0.1 | 1 | -0.25 | 0 | 0.25 | 10.5430 | 11.8910 | 9.3750 | 6.3650 | 5.1880 | 8.6680 | 8.6760 | 8.6700 | 8.6770 |
0 | 1 | 1 | 10.5330 | 11.7900 | 9.3610 | 6.3410 | 5.1710 | 8.6680 | 8.6760 | 8.6660 | 8.6740 | ||
0.5 | 2 | 1.5 | 10.6010 | 11.7180 | 9.3710 | 6.4860 | 5.3310 | 8.6700 | 8.6780 | 8.6680 | 8.6770 | ||
8 | -0.25 | 0 | 0.25 | 7.3190 | 8.8790 | 6.8430 | 4.3920 | 3.5910 | 6.1080 | 6.1080 | 6.1070 | 6.1070 | |
0 | 1 | 1 | 7.2750 | 8.7620 | 6.8270 | 4.3840 | 3.5900 | 6.1090 | 6.1090 | 6.1090 | 6.1090 | ||
0.5 | 2 | 1.5 | 7.3480 | 8.7970 | 6.8640 | 4.5800 | 3.8160 | 6.1070 | 6.1070 | 6.1040 | 6.1040 | ||
0.5 | 1 | -0.25 | 0 | 0.25 | 8.7070 | 9.8380 | 7.9460 | 5.2650 | 4.3050 | 7.2590 | 7.2590 | 7.2570 | 7.2570 |
0 | 1 | 1 | 8.7510 | 9.9250 | 7.9940 | 5.3100 | 4.3450 | 7.2570 | 7.2570 | 7.2540 | 7.2540 | ||
0.5 | 2 | 1.5 | 8.8320 | 10.0890 | 8.0490 | 5.4970 | 4.5480 | 7.2590 | 7.2590 | 7.2590 | 7.2590 | ||
8 | -0.25 | 0 | 0.25 | 10.2360 | 11.4530 | 9.1100 | 6.1750 | 5.0390 | 8.3820 | 8.3820 | 8.3810 | 8.3810 | |
0 | 1 | 1 | 10.1380 | 11.2610 | 9.0610 | 6.1540 | 5.0260 | 8.3810 | 8.3810 | 8.3850 | 8.3850 | ||
0.5 | 2 | 1.5 | 10.1020 | 11.3160 | 9.0800 | 6.2300 | 5.1320 | 8.3830 | 8.3830 | 8.3830 | 8.3830 | ||
0.9 | 1 | -0.25 | 0 | 0.25 | 7.2300 | 8.9110 | 6.8140 | 4.3740 | 3.5750 | 6.0000 | 6.0070 | 6.0020 | 6.0090 |
0 | 1 | 1 | 7.3030 | 8.6810 | 6.7940 | 4.3620 | 3.5700 | 5.9960 | 6.0020 | 5.9950 | 6.0020 | ||
0.5 | 2 | 1.5 | 7.2340 | 8.8310 | 6.7930 | 4.5270 | 3.7720 | 5.9990 | 6.0060 | 5.9990 | 6.0050 | ||
8 | -0.25 | 0 | 0.25 | 8.4830 | 9.7340 | 7.8050 | 5.1900 | 4.2510 | 7.0030 | 7.0110 | 6.9980 | 7.0050 | |
0 | 1 | 1 | 8.4410 | 9.6700 | 7.7630 | 5.1400 | 4.2100 | 7.0010 | 7.0080 | 6.9970 | 7.0050 | ||
0.5 | 2 | 1.5 | 8.4160 | 9.8250 | 7.8290 | 5.3240 | 4.4150 | 7.0000 | 7.0080 | 7.0020 | 7.0100 |
RNCP of various confidence intervals under bivariate normal distribution with different ρ and δ, μ _{1}, μ _{2}, \({\sigma _{1}^{2}}\) and (n,n _{1},n _{2})=(5,2,2) and \({\sigma _{2}^{2}}=4\)
ρ | \({\sigma _{1}^{2}}\) | δ | μ _{1} | μ _{2} | T _{1} | T _{2} | T _{ g } | W _{ s } | W _{ a } | B _{1} | B _{2} | B _{3} | B _{4} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
-0.9 | 1 | -0.25 | 0 | 0.25 | 0.4754 | 0.4805 | 0.4731 | 0.4769 | 0.4660 | 0.5000 | 0.4091 | 0.4921 | 0.4186 |
0 | 1 | 1 | 0.4286 | 0.5286 | 0.4563 | 0.3846 | 0.4892 | 0.4528 | 0.4314 | 0.4286 | 0.4706 | ||
0.5 | 2 | 1.5 | 0.4737 | 0.5909 | 0.4839 | 0.4667 | 0.4590 | 0.4906 | 0.4231 | 0.4638 | 0.4038 | ||
8 | -0.25 | 0 | 0.25 | 0.4237 | 0.5108 | 0.5048 | 0.4268 | 0.4574 | 0.5088 | 0.5686 | 0.5303 | 0.5686 | |
0 | 1 | 1 | 0.4603 | 0.5143 | 0.4857 | 0.4474 | 0.5000 | 0.5102 | 0.5000 | 0.5082 | 0.5000 | ||
0.5 | 2 | 1.5 | 0.4677 | 0.5744 | 0.4545 | 0.5395 | 0.4983 | 0.4186 | 0.4545 | 0.4717 | 0.4773 | ||
-0.5 | 1 | -0.25 | 0 | 0.25 | 0.5536 | 0.5436 | 0.5234 | 0.5375 | 0.5289 | 0.5686 | 0.5789 | 0.5821 | 0.5862 |
0 | 1 | 1 | 0.5000 | 0.5235 | 0.4948 | 0.4795 | 0.5389 | 0.4651 | 0.4773 | 0.4815 | 0.4667 | ||
0.5 | 2 | 1.5 | 0.5741 | 0.5176 | 0.5294 | 0.6533 | 0.5266 | 0.5577 | 0.6591 | 0.6140 | 0.6304 | ||
8 | -0.25 | 0 | 0.25 | 0.5070 | 0.5829 | 0.5098 | 0.5195 | 0.5481 | 0.5472 | 0.5273 | 0.5410 | 0.5357 | |
0 | 1 | 1 | 0.5352 | 0.5364 | 0.5306 | 0.4684 | 0.5585 | 0.5370 | 0.5263 | 0.5645 | 0.5536 | ||
0.5 | 2 | 1.5 | 0.4769 | 0.5355 | 0.4719 | 0.3731 | 0.5256 | 0.5208 | 0.4348 | 0.4717 | 0.4375 | ||
-0.1 | 1 | -0.25 | 0 | 0.25 | 0.5000 | 0.5744 | 0.5300 | 0.4699 | 0.6086 | 0.5333 | 0.5000 | 0.4727 | 0.4717 |
0 | 1 | 1 | 0.4839 | 0.5585 | 0.4842 | 0.4458 | 0.5714 | 0.5000 | 0.5400 | 0.5091 | 0.5417 | ||
0.5 | 2 | 1.5 | 0.5333 | 0.5632 | 0.5000 | 0.5116 | 0.5000 | 0.5102 | 0.5185 | 0.5000 | 0.5000 | ||
8 | -0.25 | 0 | 0.25 | 0.4630 | 0.5750 | 0.4848 | 0.4526 | 0.5176 | 0.4444 | 0.4151 | 0.4464 | 0.4528 | |
0 | 1 | 1 | 0.5091 | 0.5879 | 0.5104 | 0.5059 | 0.5119 | 0.5455 | 0.4800 | 0.5000 | 0.4898 | ||
0.5 | 2 | 1.5 | 0.5385 | 0.5179 | 0.5288 | 0.5529 | 0.5248 | 0.5200 | 0.5208 | 0.5179 | 0.4902 | ||
0 | 1 | -0.25 | 0 | 0.25 | 0.5484 | 0.5641 | 0.5667 | 0.6119 | 0.4800 | 0.4889 | 0.5333 | 0.5319 | 0.5319 |
0 | 1 | 1 | 0.4789 | 0.5923 | 0.5000 | 0.4000 | 0.4996 | 0.4906 | 0.4808 | 0.4906 | 0.4906 | ||
0.5 | 2 | 1.5 | 0.4286 | 0.5714 | 0.5000 | 0.2857 | 0.5097 | 0.5000 | 0.5102 | 0.5200 | 0.5306 | ||
8 | -0.25 | 0 | 0.25 | 0.4684 | 0.5829 | 0.5149 | 0.4835 | 0.5397 | 0.4912 | 0.5091 | 0.5091 | 0.5091 | |
0 | 1 | 1 | 0.5789 | 0.5977 | 0.4700 | 0.4737 | 0.5028 | 0.4561 | 0.4464 | 0.4912 | 0.4561 | ||
0.5 | 2 | 1.5 | 0.5313 | 0.5500 | 0.5000 | 0.5233 | 0.5100 | 0.4894 | 0.5106 | 0.5106 | 0.5000 | ||
0.1 | 1 | -0.25 | 0 | 0.25 | 0.5217 | 0.5065 | 0.5488 | 0.5176 | 0.5566 | 0.5102 | 0.5102 | 0.4902 | 0.4902 |
0 | 1 | 1 | 0.5224 | 0.5788 | 0.4651 | 0.5176 | 0.5212 | 0.4200 | 0.4200 | 0.4314 | 0.4286 | ||
0.5 | 2 | 1.5 | 0.5362 | 0.5116 | 0.5824 | 0.6235 | 0.5852 | 0.5417 | 0.5417 | 0.5714 | 0.5417 | ||
8 | -0.25 | 0 | 0.25 | 0.4359 | 0.5417 | 0.4490 | 0.5309 | 0.5833 | 0.4490 | 0.4490 | 0.4583 | 0.4583 | |
0 | 1 | 1 | 0.4789 | 0.5304 | 0.4904 | 0.3544 | 0.4914 | 0.4118 | 0.4118 | 0.4528 | 0.4528 | ||
0.5 | 2 | 1.5 | 0.4878 | 0.5170 | 0.5053 | 0.2879 | 0.5314 | 0.4167 | 0.4167 | 0.4200 | 0.4200 | ||
0.5 | 1 | -0.25 | 0 | 0.25 | 0.4935 | 0.5106 | 0.4563 | 0.4510 | 0.5125 | 0.5000 | 0.5000 | 0.5106 | 0.5106 |
0 | 1 | 1 | 0.5522 | 0.5947 | 0.4505 | 0.4211 | 0.5085 | 0.3929 | 0.3929 | 0.4107 | 0.4107 | ||
0.5 | 2 | 1.5 | 0.4861 | 0.5944 | 0.4943 | 0.5000 | 0.4692 | 0.5417 | 0.5417 | 0.5000 | 0.5000 | ||
8 | -0.25 | 0 | 0.25 | 0.4688 | 0.5647 | 0.4592 | 0.4227 | 0.5081 | 0.5472 | 0.5472 | 0.5185 | 0.5185 | |
0 | 1 | 1 | 0.5256 | 0.5750 | 0.5474 | 0.5426 | 0.5008 | 0.5200 | 0.5200 | 0.5577 | 0.5577 | ||
0.5 | 2 | 1.5 | 0.5286 | 0.5000 | 0.4875 | 0.5233 | 0.5093 | 0.5238 | 0.5238 | 0.5349 | 0.5349 | ||
0.9 | 1 | -0.25 | 0 | 0.25 | 0.5062 | 0.5652 | 0.5000 | 0.5714 | 0.4861 | 0.5273 | 0.5273 | 0.5357 | 0.5263 |
0 | 1 | 1 | 0.5246 | 0.5111 | 0.5238 | 0.3281 | 0.5100 | 0.5122 | 0.5000 | 0.4884 | 0.5000 | ||
0.5 | 2 | 1.5 | 0.4605 | 0.5692 | 0.4141 | 0.2179 | 0.2217 | 0.4528 | 0.4444 | 0.4340 | 0.4340 | ||
8 | -0.25 | 0 | 0.25 | 0.5250 | 0.5341 | 0.5104 | 0.5053 | 0.5045 | 0.5179 | 0.5088 | 0.5439 | 0.5455 | |
0 | 1 | 1 | 0.5362 | 0.5579 | 0.5155 | 0.4688 | 0.6133 | 0.5273 | 0.5273 | 0.5370 | 0.5370 | ||
0.5 | 2 | 1.5 | 0.5217 | 0.5579 | 0.4778 | 0.4938 | 0.4672 | 0.4681 | 0.4583 | 0.5000 | 0.4681 |
Following [17, 30], an interval can be regarded as satisfactory if (i) its ECP is close to the pre-specified 95 % confidence level, (ii) it possesses shorter interval width, and (iii) its RNCP lies in the interval [0.4,0.6]; too mesially located if its RNCP is less than 0.4; and too distally if its RNCP is greater than 0.6.
ECPs of various confidence intervals under bivariate t-distribution with different ρ and δ, μ _{1}, μ _{2}, \({\sigma _{1}^{2}}\) and (n,n _{1},n _{2})=(5,5,5) and \({\sigma _{2}^{2}}=4\)
ρ | \({\sigma _{1}^{2}}\) | δ | μ _{1} | μ _{2} | T _{1} | T _{2} | T _{ g } | W _{ s } | W _{ a } | B _{1} | B _{2} | B _{3} | B _{4} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
-0.9 | 1 | -0.25 | 0 | 0.25 | 0.9260 | 0.9750 | 0.9460 | 0.9510 | 0.9020 | 0.9470 | 0.9470 | 0.9500 | 0.9500 |
0 | 1 | 1 | 0.9060 | 0.9590 | 0.9490 | 0.9340 | 0.8820 | 0.9450 | 0.9450 | 0.9510 | 0.9510 | ||
0.5 | 2 | 1.5 | 0.9160 | 0.9710 | 0.9370 | 0.9480 | 0.8930 | 0.9490 | 0.9490 | 0.9530 | 0.9530 | ||
8 | -0.25 | 0 | 0.25 | 0.8950 | 0.9630 | 0.9380 | 0.9460 | 0.8920 | 0.9490 | 0.9380 | 0.9410 | 0.9410 | |
0 | 1 | 1 | 0.9030 | 0.9580 | 0.9430 | 0.9450 | 0.9020 | 0.9400 | 0.9410 | 0.9410 | 0.9410 | ||
0.5 | 2 | 1.5 | 0.9080 | 0.9640 | 0.9370 | 0.9490 | 0.9070 | 0.9500 | 0.9480 | 0.9520 | 0.9520 | ||
-0.5 | 1 | -0.25 | 0 | 0.25 | 0.9160 | 0.9700 | 0.9460 | 0.9380 | 0.8810 | 0.9440 | 0.9410 | 0.9430 | 0.9420 |
0 | 1 | 1 | 0.9150 | 0.9670 | 0.9510 | 0.9380 | 0.8970 | 0.9470 | 0.9480 | 0.9480 | 0.9480 | ||
0.5 | 2 | 1.5 | 0.9190 | 0.9650 | 0.9440 | 0.9440 | 0.8940 | 0.9480 | 0.9520 | 0.9540 | 0.9540 | ||
8 | -0.25 | 0 | 0.25 | 0.9160 | 0.9680 | 0.9490 | 0.9580 | 0.9160 | 0.9530 | 0.9480 | 0.9440 | 0.9510 | |
0 | 1 | 1 | 0.9080 | 0.9690 | 0.9510 | 0.9590 | 0.9200 | 0.9460 | 0.9450 | 0.9400 | 0.9480 | ||
0.5 | 2 | 1.5 | 0.9130 | 0.9750 | 0.9400 | 0.9630 | 0.9200 | 0.9410 | 0.9410 | 0.9230 | 0.9460 | ||
-0.1 | 1 | -0.25 | 0 | 0.25 | 0.9230 | 0.9660 | 0.9480 | 0.9500 | 0.9020 | 0.9530 | 0.9470 | 0.9410 | 0.9490 |
0 | 1 | 1 | 0.9060 | 0.9600 | 0.9380 | 0.9370 | 0.8920 | 0.9430 | 0.9450 | 0.9390 | 0.9500 | ||
0.5 | 2 | 1.5 | 0.9020 | 0.9660 | 0.9410 | 0.9400 | 0.8910 | 0.9530 | 0.9460 | 0.9350 | 0.9460 | ||
8 | -0.25 | 0 | 0.25 | 0.9110 | 0.9670 | 0.9450 | 0.9650 | 0.9290 | 0.9440 | 0.9420 | 0.8800 | 0.9470 | |
0 | 1 | 1 | 0.9190 | 0.9720 | 0.9360 | 0.9650 | 0.9270 | 0.9510 | 0.9450 | 0.8810 | 0.9470 | ||
0.5 | 2 | 1.5 | 0.9140 | 0.9700 | 0.9390 | 0.9630 | 0.9270 | 0.9480 | 0.9440 | 0.8890 | 0.9470 | ||
0 | 1 | -0.25 | 0 | 0.25 | 0.9180 | 0.9580 | 0.9430 | 0.9500 | 0.8980 | 0.9470 | 0.9390 | 0.7900 | 0.9420 |
0 | 1 | 1 | 0.9150 | 0.9710 | 0.9550 | 0.9550 | 0.9130 | 0.9490 | 0.9500 | 0.8030 | 0.9500 | ||
0.5 | 2 | 1.5 | 0.9180 | 0.9670 | 0.9500 | 0.9590 | 0.9200 | 0.9450 | 0.9510 | 0.7940 | 0.9540 | ||
8 | -0.25 | 0 | 0.25 | 0.9380 | 0.9660 | 0.9380 | 0.9560 | 0.9280 | 0.9510 | 0.9510 | 0.9380 | 0.9530 | |
0 | 1 | 1 | 0.9360 | 0.9650 | 0.9340 | 0.9530 | 0.9220 | 0.9560 | 0.9520 | 0.9370 | 0.9540 | ||
0.5 | 2 | 1.5 | 0.9310 | 0.9540 | 0.9340 | 0.9510 | 0.9230 | 0.9450 | 0.9530 | 0.9400 | 0.9540 | ||
0.1 | 1 | -0.25 | 0 | 0.25 | 0.9360 | 0.9640 | 0.9420 | 0.9530 | 0.9210 | 0.9480 | 0.9510 | 0.9430 | 0.9550 |
0 | 1 | 1 | 0.9350 | 0.9620 | 0.9340 | 0.9520 | 0.9190 | 0.9560 | 0.9520 | 0.9400 | 0.9520 | ||
0.5 | 2 | 1.5 | 0.9290 | 0.9600 | 0.9340 | 0.9440 | 0.9160 | 0.9440 | 0.9470 | 0.9340 | 0.9480 | ||
8 | -0.25 | 0 | 0.25 | 0.9300 | 0.9530 | 0.9330 | 0.9470 | 0.9190 | 0.9400 | 0.9380 | 0.9350 | 0.9400 | |
0 | 1 | 1 | 0.9340 | 0.9590 | 0.9310 | 0.9520 | 0.9160 | 0.9410 | 0.9410 | 0.9360 | 0.9420 | ||
0.5 | 2 | 1.5 | 0.9390 | 0.9660 | 0.9330 | 0.9520 | 0.9210 | 0.9530 | 0.9500 | 0.9490 | 0.9530 | ||
0.5 | 1 | -0.25 | 0 | 0.25 | 0.9370 | 0.9640 | 0.9370 | 0.9490 | 0.9120 | 0.9450 | 0.9440 | 0.9430 | 0.9470 |
0 | 1 | 1 | 0.9450 | 0.9590 | 0.9360 | 0.9450 | 0.9080 | 0.9460 | 0.9420 | 0.9380 | 0.9440 | ||
0.5 | 2 | 1.5 | 0.9430 | 0.9680 | 0.9400 | 0.9520 | 0.9200 | 0.9540 | 0.9480 | 0.9490 | 0.9540 | ||
8 | -0.25 | 0 | 0.25 | 0.9340 | 0.9580 | 0.9460 | 0.9520 | 0.9190 | 0.9420 | 0.9450 | 0.9470 | 0.9480 | |
0 | 1 | 1 | 0.9400 | 0.9630 | 0.9470 | 0.9530 | 0.9210 | 0.9550 | 0.9560 | 0.9580 | 0.9580 | ||
0.5 | 2 | 1.5 | 0.9270 | 0.9610 | 0.9330 | 0.9470 | 0.9230 | 0.9420 | 0.9420 | 0.9470 | 0.9460 | ||
0.9 | 1 | -0.25 | 0 | 0.25 | 0.9430 | 0.9660 | 0.9410 | 0.9500 | 0.9140 | 0.9470 | 0.9470 | 0.9480 | 0.9480 |
0 | 1 | 1 | 0.9410 | 0.9530 | 0.9440 | 0.9400 | 0.9040 | 0.9470 | 0.9460 | 0.9510 | 0.9500 | ||
0.5 | 2 | 1.5 | 0.9430 | 0.9660 | 0.9480 | 0.9490 | 0.9160 | 0.9540 | 0.9560 | 0.9550 | 0.9560 | ||
8 | -0.25 | 0 | 0.25 | 0.9320 | 0.9540 | 0.9520 | 0.9450 | 0.9200 | 0.9460 | 0.9460 | 0.9490 | 0.9490 | |
0 | 1 | 1 | 0.9460 | 0.9660 | 0.9470 | 0.9590 | 0.9300 | 0.9470 | 0.9470 | 0.9490 | 0.9490 | ||
0.5 | 2 | 1.5 | 0.9410 | 0.9580 | 0.9460 | 0.9510 | 0.9200 | 0.9550 | 0.9550 | 0.9580 | 0.9580 |
ECW of various confidence intervals under bivariate t-distribution with different ρ and δ, μ _{1}, μ _{2}, \({\sigma _{1}^{2}}\) and (n,n _{1},n _{2})=(5,5,5) and \({\sigma _{2}^{2}}=4\)
ρ | \({\sigma _{1}^{2}}\) | δ | μ _{1} | μ _{2} | T _{1} | T _{2} | T _{ g } | W _{ s } | W _{ a } | B _{1} | B _{2} | B _{3} | B _{4} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
-0.9 | 1 | -0.25 | 0 | 0.25 | 39.9860 | 40.8450 | 35.0080 | 27.6870 | 23.6020 | 35.9410 | 35.9410 | 36.4110 | 36.4110 |
0 | 1 | 1 | 39.5890 | 40.3210 | 34.6710 | 27.5260 | 23.4670 | 35.9280 | 35.9280 | 36.4080 | 36.4080 | ||
0.5 | 2 | 1.5 | 39.2290 | 40.6160 | 34.8600 | 27.6570 | 23.5830 | 35.9050 | 35.9050 | 36.3930 | 36.3930 | ||
8 | -0.25 | 0 | 0.25 | 32.6680 | 34.3430 | 29.0000 | 22.6520 | 19.2790 | 29.8250 | 29.8650 | 30.3280 | 30.3510 | |
0 | 1 | 1 | 32.7540 | 34.2080 | 28.8030 | 22.5610 | 19.2030 | 29.8370 | 29.8760 | 30.3360 | 30.3630 | ||
0.5 | 2 | 1.5 | 32.3510 | 34.3530 | 28.9420 | 22.6560 | 19.2890 | 29.8380 | 29.8770 | 30.3350 | 30.3580 | ||
-0.5 | 1 | -0.25 | 0 | 0.25 | 38.5200 | 39.5450 | 34.4200 | 27.2240 | 23.2190 | 35.0120 | 35.0610 | 35.4930 | 35.5290 |
0 | 1 | 1 | 37.8690 | 39.0530 | 33.9970 | 26.9350 | 22.9710 | 35.0060 | 35.0530 | 35.4800 | 35.5150 | ||
0.5 | 2 | 1.5 | 38.7140 | 39.7290 | 34.1600 | 27.1930 | 23.1930 | 35.0190 | 35.0680 | 35.4960 | 35.5300 | ||
8 | -0.25 | 0 | 0.25 | 29.3790 | 32.8700 | 27.8810 | 21.7710 | 18.5360 | 27.1550 | 28.3790 | 27.6280 | 28.8530 | |
0 | 1 | 1 | 28.9240 | 32.3540 | 27.5350 | 21.5080 | 18.3100 | 27.1670 | 28.4000 | 27.6550 | 28.8590 | ||
0.5 | 2 | 1.5 | 30.1060 | 33.6320 | 28.5080 | 22.3350 | 19.0220 | 27.1880 | 28.4140 | 27.6570 | 28.8780 | ||
-0.1 | 1 | -0.25 | 0 | 0.25 | 31.3890 | 36.1820 | 31.7890 | 25.3610 | 21.6710 | 29.6610 | 31.4040 | 30.1280 | 31.7960 |
0 | 1 | 1 | 30.9880 | 35.1900 | 30.9350 | 24.6920 | 21.0960 | 29.6870 | 31.4300 | 30.1620 | 31.8160 | ||
0.5 | 2 | 1.5 | 31.1860 | 35.2420 | 31.2250 | 24.8410 | 21.2380 | 29.6730 | 31.4220 | 30.1440 | 31.8080 | ||
8 | -0.25 | 0 | 0.25 | 23.8340 | 31.2500 | 26.6610 | 20.8190 | 17.7300 | 20.9570 | 26.8340 | 21.2360 | 27.2880 | |
0 | 1 | 1 | 23.4990 | 31.1470 | 26.6370 | 20.8520 | 17.7590 | 20.9660 | 26.8530 | 21.2550 | 27.3120 | ||
0.5 | 2 | 1.5 | 23.1750 | 30.4390 | 26.1330 | 20.3920 | 17.3770 | 20.9520 | 26.8280 | 21.2470 | 27.2670 | ||
0 | 1 | -0.25 | 0 | 0.25 | 16.7960 | 30.7840 | 27.3290 | 21.9590 | 18.8280 | 16.6250 | 27.2590 | 16.9850 | 27.6450 |
0 | 1 | 1 | 17.1650 | 30.5510 | 27.3190 | 21.8760 | 18.7550 | 16.6250 | 27.2600 | 16.9750 | 27.6580 | ||
0.5 | 2 | 1.5 | 16.9980 | 30.6500 | 27.2430 | 22.0410 | 18.9120 | 16.6160 | 27.2610 | 16.9700 | 27.6440 | ||
8 | -0.25 | 0 | 0.25 | 27.2420 | 29.7100 | 27.2280 | 22.7000 | 20.2600 | 26.0380 | 27.9560 | 26.2890 | 28.2960 | |
0 | 1 | 1 | 27.6030 | 29.9040 | 27.3850 | 22.8460 | 20.3900 | 26.0420 | 27.9600 | 26.2820 | 28.2960 | ||
0.5 | 2 | 1.5 | 27.4440 | 29.6420 | 27.2230 | 22.6840 | 20.2480 | 26.0420 | 27.9660 | 26.2770 | 28.2950 | ||
0.1 | 1 | -0.25 | 0 | 0.25 | 36.7630 | 38.5960 | 35.2540 | 29.7190 | 26.5230 | 35.0020 | 36.7010 | 35.3140 | 37.1130 |
0 | 1 | 1 | 36.9580 | 38.9090 | 35.4500 | 29.9490 | 26.7290 | 34.9960 | 36.6930 | 35.3230 | 37.1390 | ||
0.5 | 2 | 1.5 | 36.6820 | 38.7640 | 35.2490 | 29.7940 | 26.5910 | 34.9890 | 36.6840 | 35.3050 | 37.1090 | ||
8 | -0.25 | 0 | 0.25 | 26.8170 | 28.2480 | 25.9750 | 21.6000 | 19.2790 | 25.9390 | 26.5530 | 26.1980 | 26.8650 | |
0 | 1 | 1 | 27.0150 | 28.2910 | 26.0250 | 21.6540 | 19.3270 | 25.9380 | 26.5470 | 26.1960 | 26.8510 | ||
0.5 | 2 | 1.5 | 27.1980 | 28.6990 | 26.3160 | 21.9060 | 19.5540 | 25.9400 | 26.5480 | 26.1920 | 26.8500 | ||
0.5 | 1 | -0.25 | 0 | 0.25 | 35.0610 | 35.8660 | 32.9600 | 27.7210 | 24.7450 | 33.0080 | 33.6310 | 33.3300 | 34.0030 |
0 | 1 | 1 | 35.2510 | 35.8450 | 32.9690 | 27.7590 | 24.7790 | 32.9910 | 33.6170 | 33.3000 | 33.9910 | ||
0.5 | 2 | 1.5 | 34.6160 | 35.6320 | 32.7980 | 27.5930 | 24.6330 | 32.9950 | 33.6200 | 33.3110 | 33.9920 | ||
8 | -0.25 | 0 | 0.25 | 26.0830 | 26.9080 | 24.8210 | 20.5850 | 18.3740 | 25.0540 | 25.0810 | 25.3290 | 25.3590 | |
0 | 1 | 1 | 25.6840 | 26.7000 | 24.6450 | 20.4400 | 18.2450 | 25.0390 | 25.0650 | 25.3160 | 25.3480 | ||
0.5 | 2 | 1.5 | 25.9800 | 26.9400 | 24.8400 | 20.5900 | 18.3810 | 25.0410 | 25.0680 | 25.3280 | 25.3580 | ||
0.9 | 1 | -0.25 | 0 | 0.25 | 31.7980 | 32.2880 | 29.9420 | 25.1170 | 22.4300 | 30.2210 | 30.2530 | 30.5230 | 30.5690 |
0 | 1 | 1 | 31.8710 | 32.0900 | 29.8060 | 24.9940 | 22.3200 | 30.1980 | 30.2290 | 30.5050 | 30.5500 | ||
0.5 | 2 | 1.5 | 31.3990 | 32.0560 | 29.7450 | 24.9700 | 22.3010 | 30.2140 | 30.2440 | 30.5180 | 30.5600 | ||
8 | -0.25 | 0 | 0.25 | 25.4700 | 26.4860 | 24.4770 | 20.2660 | 18.0900 | 24.6850 | 24.6850 | 24.9600 | 24.9600 | |
0 | 1 | 1 | 25.5190 | 26.3770 | 24.3630 | 20.1840 | 18.0160 | 24.6740 | 24.6740 | 24.9450 | 24.9450 | ||
0.5 | 2 | 1.5 | 25.4630 | 26.4500 | 24.4990 | 20.2850 | 18.1100 | 24.6930 | 24.6930 | 24.9760 | 24.9760 |
RNCP of various confidence intervals under bivariate t-distribution with different ρ and δ, μ _{1}, μ _{2}, \({\sigma _{1}^{2}}\) and (n,n _{1},n _{2})=(5,5,5) and \({\sigma _{2}^{2}}=4\)
ρ | \({\sigma _{1}^{2}}\) | δ | μ _{1} | μ _{2} | T _{1} | T _{2} | T _{ g } | W _{ s } | W _{ a } | B _{1} | B _{2} | B _{3} | B _{4} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
-0.9 | 1 | -0.25 | 0 | 0.25 | 0.4324 | 0.5200 | 0.5000 | 0.5918 | 0.5102 | 0.4717 | 0.4717 | 0.4800 | 0.4800 |
0 | 1 | 1 | 0.4574 | 0.4634 | 0.5062 | 0.4848 | 0.5000 | 0.4727 | 0.4727 | 0.5102 | 0.5102 | ||
0.5 | 2 | 1.5 | 0.4524 | 0.5862 | 0.5238 | 0.5385 | 0.5047 | 0.4118 | 0.4118 | 0.4255 | 0.4255 | ||
8 | -0.25 | 0 | 0.25 | 0.4762 | 0.4865 | 0.4878 | 0.4815 | 0.4815 | 0.4754 | 0.4677 | 0.4746 | 0.4746 | |
0 | 1 | 1 | 0.5361 | 0.5238 | 0.4675 | 0.5091 | 0.5000 | 0.4833 | 0.4746 | 0.4746 | 0.4746 | ||
0.5 | 2 | 1.5 | 0.4783 | 0.5278 | 0.4795 | 0.5098 | 0.5484 | 0.5400 | 0.5000 | 0.5208 | 0.5208 | ||
-0.5 | 1 | -0.25 | 0 | 0.25 | 0.4524 | 0.4000 | 0.4595 | 0.4839 | 0.4538 | 0.4464 | 0.4237 | 0.4211 | 0.4138 |
0 | 1 | 1 | 0.6000 | 0.6061 | 0.5797 | 0.5645 | 0.5534 | 0.5283 | 0.5385 | 0.5577 | 0.5385 | ||
0.5 | 2 | 1.5 | 0.5062 | 0.5429 | 0.5455 | 0.5357 | 0.5660 | 0.5000 | 0.5000 | 0.5435 | 0.5435 | ||
8 | -0.25 | 0 | 0.25 | 0.4762 | 0.5000 | 0.5070 | 0.5952 | 0.5119 | 0.5106 | 0.5385 | 0.5179 | 0.5306 | |
0 | 1 | 1 | 0.5217 | 0.5806 | 0.5085 | 0.5854 | 0.5500 | 0.4815 | 0.5273 | 0.5167 | 0.5000 | ||
0.5 | 2 | 1.5 | 0.4943 | 0.4000 | 0.4000 | 0.4595 | 0.5125 | 0.5316 | 0.5217 | 0.5195 | 0.5313 | ||
-0.1 | 1 | -0.25 | 0 | 0.25 | 0.5584 | 0.4706 | 0.5323 | 0.5800 | 0.4796 | 0.5319 | 0.5660 | 0.5932 | 0.5686 |
0 | 1 | 1 | 0.5532 | 0.5500 | 0.5278 | 0.5714 | 0.5463 | 0.4737 | 0.4727 | 0.4754 | 0.5000 | ||
0.5 | 2 | 1.5 | 0.4490 | 0.4706 | 0.4348 | 0.4333 | 0.4679 | 0.4255 | 0.4815 | 0.4769 | 0.4630 | ||
8 | -0.25 | 0 | 0.25 | 0.4831 | 0.4545 | 0.5231 | 0.4000 | 0.4648 | 0.5714 | 0.5000 | 0.4917 | 0.5283 | |
0 | 1 | 1 | 0.5062 | 0.5000 | 0.4844 | 0.4571 | 0.5068 | 0.4898 | 0.4727 | 0.4958 | 0.4717 | ||
0.5 | 2 | 1.5 | 0.4651 | 0.5000 | 0.4590 | 0.5676 | 0.5479 | 0.4423 | 0.4464 | 0.4595 | 0.4717 | ||
0 | 1 | -0.25 | 0 | 0.25 | 0.5244 | 0.5714 | 0.6140 | 0.5800 | 0.5196 | 0.5094 | 0.5246 | 0.4857 | 0.5000 |
0 | 1 | 1 | 0.5059 | 0.4483 | 0.4444 | 0.4444 | 0.4828 | 0.5490 | 0.5600 | 0.5228 | 0.5600 | ||
0.5 | 2 | 1.5 | 0.5366 | 0.3939 | 0.4800 | 0.4146 | 0.4875 | 0.4545 | 0.5306 | 0.5097 | 0.5217 | ||
8 | -0.25 | 0 | 0.25 | 0.5161 | 0.6176 | 0.5968 | 0.6136 | 0.5972 | 0.5714 | 0.6122 | 0.5968 | 0.5957 | |
0 | 1 | 1 | 0.4844 | 0.4571 | 0.4697 | 0.4468 | 0.4744 | 0.5682 | 0.5208 | 0.5079 | 0.5000 | ||
0.5 | 2 | 1.5 | 0.4928 | 0.4130 | 0.4545 | 0.4286 | 0.4545 | 0.4545 | 0.4255 | 0.4833 | 0.4348 | ||
0.1 | 1 | -0.25 | 0 | 0.25 | 0.5469 | 0.5833 | 0.5862 | 0.6170 | 0.5696 | 0.5385 | 0.5102 | 0.5088 | 0.5333 |
0 | 1 | 1 | 0.5692 | 0.4737 | 0.5303 | 0.5208 | 0.5309 | 0.5227 | 0.5208 | 0.5000 | 0.5000 | ||
0.5 | 2 | 1.5 | 0.4507 | 0.4500 | 0.4394 | 0.4107 | 0.4286 | 0.4464 | 0.4717 | 0.4545 | 0.4808 | ||
8 | -0.25 | 0 | 0.25 | 0.5000 | 0.5319 | 0.5373 | 0.5472 | 0.5062 | 0.5000 | 0.5161 | 0.5077 | 0.5000 | |
0 | 1 | 1 | 0.5303 | 0.5366 | 0.5072 | 0.5417 | 0.4762 | 0.4746 | 0.4915 | 0.4844 | 0.5000 | ||
0.5 | 2 | 1.5 | 0.5246 | 0.5294 | 0.5373 | 0.5417 | 0.5443 | 0.5532 | 0.5400 | 0.5294 | 0.5319 | ||
0.5 | 1 | -0.25 | 0 | 0.25 | 0.6190 | 0.5833 | 0.5397 | 0.6078 | 0.5341 | 0.5091 | 0.5714 | 0.5614 | 0.5660 |
0 | 1 | 1 | 0.4545 | 0.4878 | 0.4844 | 0.5091 | 0.4565 | 0.4444 | 0.4828 | 0.4677 | 0.4643 | ||
0.5 | 2 | 1.5 | 0.5088 | 0.5625 | 0.5000 | 0.5208 | 0.5125 | 0.5000 | 0.4615 | 0.4510 | 0.4565 | ||
8 | -0.25 | 0 | 0.25 | 0.5303 | 0.4762 | 0.5000 | 0.4583 | 0.4815 | 0.5172 | 0.5273 | 0.5283 | 0.5385 | |
0 | 1 | 1 | 0.5500 | 0.5676 | 0.5714 | 0.5532 | 0.6076 | 0.5333 | 0.5455 | 0.5238 | 0.5238 | ||
0.5 | 2 | 1.5 | 0.5479 | 0.5385 | 0.5224 | 0.5660 | 0.5584 | 0.4655 | 0.4828 | 0.4717 | 0.4630 | ||
0.9 | 1 | -0.25 | 0 | 0.25 | 0.5088 | 0.5000 | 0.4746 | 0.4800 | 0.4884 | 0.4906 | 0.4717 | 0.4808 | 0.4808 |
0 | 1 | 1 | 0.4915 | 0.5106 | 0.4848 | 0.5000 | 0.4479 | 0.4151 | 0.4259 | 0.4286 | 0.4200 | ||
0.5 | 2 | 1.5 | 0.5789 | 0.5294 | 0.5161 | 0.5098 | 0.5595 | 0.4783 | 0.4773 | 0.5333 | 0.5455 | ||
8 | -0.25 | 0 | 0.25 | 0.4559 | 0.5435 | 0.5147 | 0.4909 | 0.4875 | 0.5000 | 0.5000 | 0.4902 | 0.4902 | |
0 | 1 | 1 | 0.4815 | 0.5294 | 0.5283 | 0.5366 | 0.5429 | 0.6038 | 0.6038 | 0.5686 | 0.5686 | ||
0.5 | 2 | 1.5 | 0.4407 | 0.4524 | 0.5000 | 0.5102 | 0.5250 | 0.4222 | 0.4222 | 0.4048 | 0.4048 |
To investigate powers for the proposed CIs, we calculated the power in both the first and second simulation study. The results are shown in Tables 12 and 13. There is very little power in both the first and second simulation study to exclude a difference of zero.
Results of simulation studies
ECPs of various confidence intervals with different ρ and δ, μ _{1}, μ _{2}, (n,n _{1},n _{2})=(5,5,2), when \({\sigma _{1}^{2}}={\sigma _{2}^{2}}=4\)
Bivariate normal distribution | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
ρ | δ | μ _{1} | μ _{2} | T _{3} | T _{4} | T _{5} | W _{ s } | W _{ a } | B _{1} | B _{2} | B _{3} | B _{4} |
-0.9 | -0.25 | 0 | 0.25 | 0.935 | 0.960 | 0.906 | 0.920 | 0.880 | 0.952 | 0.954 | 0.947 | 0.954 |
0 | 1 | 1 | 0.944 | 0.956 | 0.894 | 0.920 | 0.869 | 0.946 | 0.947 | 0.933 | 0.947 | |
0.5 | 2 | 1.5 | 0.944 | 0.967 | 0.902 | 0.931 | 0.883 | 0.951 | 0.953 | 0.942 | 0.951 | |
-0.5 | -0.25 | 0 | 0.25 | 0.941 | 0.961 | 0.903 | 0.910 | 0.861 | 0.942 | 0.943 | 0.939 | 0.943 |
0 | 1 | 1 | 0.937 | 0.958 | 0.900 | 0.915 | 0.862 | 0.950 | 0.952 | 0.949 | 0.951 | |
0.5 | 2 | 1.5 | 0.941 | 0.962 | 0.898 | 0.925 | 0.882 | 0.952 | 0.957 | 0.952 | 0.957 | |
-0.1 | -0.25 | 0 | 0.25 | 0.933 | 0.958 | 0.900 | 0.903 | 0.838 | 0.944 | 0.945 | 0.945 | 0.946 |
0 | 1 | 1 | 0.939 | 0.966 | 0.907 | 0.912 | 0.853 | 0.952 | 0.951 | 0.954 | 0.953 | |
0.5 | 2 | 1.5 | 0.943 | 0.975 | 0.924 | 0.943 | 0.892 | 0.961 | 0.959 | 0.960 | 0.959 | |
0 | -0.25 | 0 | 0.25 | 0.936 | 0.964 | 0.914 | 0.913 | 0.860 | 0.949 | 0.949 | 0.950 | 0.950 |
0 | 1 | 1 | 0.925 | 0.959 | 0.906 | 0.908 | 0.861 | 0.941 | 0.941 | 0.940 | 0.940 | |
0.5 | 2 | 1.5 | 0.932 | 0.968 | 0.913 | 0.924 | 0.887 | 0.952 | 0.952 | 0.951 | 0.951 | |
0.1 | -0.25 | 0 | 0.25 | 0.922 | 0.960 | 0.918 | 0.911 | 0.858 | 0.948 | 0.948 | 0.948 | 0.947 |
0 | 1 | 1 | 0.923 | 0.963 | 0.909 | 0.906 | 0.859 | 0.944 | 0.946 | 0.944 | 0.944 | |
0.5 | 2 | 1.5 | 0.928 | 0.969 | 0.913 | 0.935 | 0.889 | 0.946 | 0.947 | 0.947 | 0.946 | |
0.5 | -0.25 | 0 | 0.25 | 0.927 | 0.968 | 0.923 | 0.904 | 0.843 | 0.950 | 0.947 | 0.934 | 0.947 |
0 | 1 | 1 | 0.928 | 0.964 | 0.923 | 0.913 | 0.857 | 0.942 | 0.944 | 0.935 | 0.947 | |
0.5 | 2 | 1.5 | 0.924 | 0.978 | 0.933 | 0.947 | 0.901 | 0.960 | 0.958 | 0.943 | 0.960 | |
0.9 | -0.25 | 0 | 0.25 | 0.913 | 0.947 | 0.974 | 0.929 | 0.880 | 0.951 | 0.951 | 0.777 | 0.951 |
0 | 1 | 1 | 0.908 | 0.952 | 0.976 | 0.930 | 0.883 | 0.947 | 0.955 | 0.781 | 0.951 | |
0.5 | 2 | 1.5 | 0.913 | 0.942 | 0.974 | 0.974 | 0.944 | 0.946 | 0.953 | 0.778 | 0.954 | |
Bivariate t-distribution | ||||||||||||
-0.9 | -0.25 | 0 | 0.25 | 0.922 | 0.972 | 0.908 | 0.929 | 0.870 | 0.952 | 0.953 | 0.946 | 0.956 |
0 | 1 | 1 | 0.915 | 0.973 | 0.914 | 0.935 | 0.868 | 0.948 | 0.943 | 0.937 | 0.948 | |
0.5 | 2 | 1.5 | 0.930 | 0.978 | 0.914 | 0.937 | 0.873 | 0.948 | 0.950 | 0.941 | 0.951 | |
-0.5 | -0.25 | 0 | 0.25 | 0.929 | 0.976 | 0.921 | 0.939 | 0.869 | 0.942 | 0.941 | 0.940 | 0.945 |
0 | 1 | 1 | 0.931 | 0.975 | 0.925 | 0.935 | 0.872 | 0.943 | 0.942 | 0.943 | 0.946 | |
0.5 | 2 | 1.5 | 0.922 | 0.971 | 0.910 | 0.924 | 0.868 | 0.953 | 0.951 | 0.950 | 0.955 | |
-0.1 | -0.25 | 0 | 0.25 | 0.932 | 0.973 | 0.922 | 0.925 | 0.856 | 0.951 | 0.951 | 0.955 | 0.954 |
0 | 1 | 1 | 0.926 | 0.971 | 0.924 | 0.923 | 0.859 | 0.941 | 0.942 | 0.946 | 0.947 | |
0.5 | 2 | 1.5 | 0.924 | 0.972 | 0.918 | 0.921 | 0.859 | 0.950 | 0.948 | 0.954 | 0.955 | |
0 | -0.25 | 0 | 0.25 | 0.919 | 0.973 | 0.921 | 0.918 | 0.852 | 0.944 | 0.944 | 0.949 | 0.949 |
0 | 1 | 1 | 0.925 | 0.972 | 0.923 | 0.925 | 0.864 | 0.940 | 0.940 | 0.947 | 0.947 | |
0.5 | 2 | 1.5 | 0.939 | 0.977 | 0.924 | 0.926 | 0.857 | 0.950 | 0.950 | 0.954 | 0.954 | |
0.1 | -0.25 | 0 | 0.25 | 0.930 | 0.971 | 0.929 | 0.928 | 0.857 | 0.954 | 0.954 | 0.956 | 0.956 |
0 | 1 | 1 | 0.929 | 0.982 | 0.927 | 0.928 | 0.857 | 0.949 | 0.949 | 0.950 | 0.951 | |
0.5 | 2 | 1.5 | 0.934 | 0.979 | 0.924 | 0.930 | 0.859 | 0.952 | 0.953 | 0.957 | 0.957 | |
0.5 | -0.25 | 0 | 0.25 | 0.929 | 0.973 | 0.947 | 0.940 | 0.864 | 0.944 | 0.950 | 0.942 | 0.951 |
0 | 1 | 1 | 0.920 | 0.976 | 0.937 | 0.928 | 0.861 | 0.943 | 0.944 | 0.936 | 0.946 | |
0.5 | 2 | 1.5 | 0.939 | 0.970 | 0.942 | 0.930 | 0.868 | 0.945 | 0.947 | 0.942 | 0.951 | |
0.9 | -0.25 | 0 | 0.25 | 0.923 | 0.969 | 0.978 | 0.943 | 0.880 | 0.939 | 0.938 | 0.797 | 0.939 |
0 | 1 | 1 | 0.920 | 0.966 | 0.977 | 0.952 | 0.887 | 0.939 | 0.942 | 0.795 | 0.949 | |
0.5 | 2 | 1.5 | 0.931 | 0.965 | 0.979 | 0.944 | 0.878 | 0.953 | 0.944 | 0.804 | 0.947 |
ECW of various confidence interals with different ρ and δ, μ _{1}, μ _{2}, (n,n _{1},n _{2})=(5,5,2), when \({\sigma _{1}^{2}}={\sigma _{2}^{2}}=4\)
Bivariate normal distribution | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
ρ | δ | μ _{1} | μ _{2} | T _{3} | T _{4} | T _{5} | W _{ s } | W _{ a } | B _{1} | B _{2} | B _{3} | B _{4} |
-0.9 | -0.25 | 0 | 0.25 | 6.350 | 7.032 | 5.019 | 3.821 | 3.148 | 5.150 | 5.370 | 5.149 | 5.368 |
0 | 1 | 1 | 6.389 | 7.038 | 5.047 | 3.833 | 3.162 | 5.151 | 5.370 | 5.151 | 5.370 | |
0.5 | 2 | 1.5 | 6.447 | 7.052 | 5.060 | 3.947 | 3.290 | 5.152 | 5.370 | 5.152 | 5.370 | |
-0.5 | -0.25 | 0 | 0.25 | 5.883 | 6.473 | 4.610 | 3.503 | 2.894 | 4.800 | 4.881 | 4.799 | 4.880 |
0 | 1 | 1 | 5.885 | 6.436 | 4.606 | 3.510 | 2.903 | 4.800 | 4.881 | 4.799 | 4.879 | |
0.5 | 2 | 1.5 | 5.877 | 6.413 | 4.606 | 3.655 | 3.078 | 4.802 | 4.883 | 4.802 | 4.882 | |
-0.1 | -0.25 | 0 | 0.25 | 5.282 | 5.891 | 4.187 | 3.198 | 2.651 | 4.333 | 4.337 | 4.333 | 4.338 |
0 | 1 | 1 | 5.318 | 5.898 | 4.186 | 3.213 | 2.670 | 4.335 | 4.340 | 4.334 | 4.338 | |
0.5 | 2 | 1.5 | 5.270 | 5.888 | 4.183 | 3.397 | 2.893 | 4.336 | 4.340 | 4.336 | 4.339 | |
0 | -0.25 | 0 | 0.25 | 5.114 | 5.733 | 4.046 | 3.096 | 2.571 | 4.190 | 4.190 | 4.189 | 4.189 |
0 | 1 | 1 | 5.147 | 5.729 | 4.076 | 3.139 | 2.614 | 4.190 | 4.190 | 4.190 | 4.190 | |
0.5 | 2 | 1.5 | 5.123 | 5.763 | 4.069 | 3.337 | 2.849 | 4.191 | 4.191 | 4.189 | 4.189 | |
0.1 | -0.25 | 0 | 0.25 | 4.869 | 5.519 | 3.921 | 3.004 | 2.500 | 4.033 | 4.037 | 4.032 | 4.037 |
0 | 1 | 1 | 4.870 | 5.550 | 3.899 | 3.004 | 2.504 | 4.032 | 4.037 | 4.033 | 4.038 | |
0.5 | 2 | 1.5 | 4.849 | 5.636 | 3.926 | 3.254 | 2.795 | 4.031 | 4.036 | 4.033 | 4.037 | |
0.5 | -0.25 | 0 | 0.25 | 3.805 | 5.050 | 3.412 | 2.608 | 2.188 | 3.202 | 3.360 | 3.202 | 3.360 |
0 | 1 | 1 | 3.811 | 5.019 | 3.398 | 2.624 | 2.213 | 3.201 | 3.360 | 3.199 | 3.357 | |
0.5 | 2 | 1.5 | 3.857 | 5.211 | 3.401 | 2.955 | 2.583 | 3.200 | 3.359 | 3.200 | 3.360 | |
0.9 | -0.25 | 0 | 0.25 | 1.776 | 5.606 | 2.702 | 2.133 | 1.832 | 1.537 | 2.505 | 1.537 | 2.505 |
0 | 1 | 1 | 1.766 | 5.561 | 2.676 | 2.147 | 1.853 | 1.539 | 2.503 | 1.538 | 2.503 | |
0.5 | 2 | 1.5 | 1.784 | 5.548 | 2.689 | 2.554 | 2.303 | 1.537 | 2.505 | 1.536 | 2.504 | |
Bivariate t-distribution | ||||||||||||
-0.9 | -0.25 | 0 | 0.25 | 35.039 | 42.148 | 28.140 | 21.360 | 17.207 | 30.479 | 31.779 | 31.062 | 32.486 |
0 | 1 | 1 | 35.226 | 42.660 | 28.523 | 21.569 | 17.374 | 30.470 | 31.763 | 31.048 | 32.470 | |
0.5 | 2 | 1.5 | 34.854 | 42.020 | 28.032 | 21.260 | 17.135 | 30.472 | 31.771 | 31.038 | 32.484 | |
-0.5 | -0.25 | 0 | 0.25 | 32.156 | 38.993 | 25.809 | 19.534 | 15.765 | 28.402 | 28.881 | 28.936 | 29.495 |
0 | 1 | 1 | 33.177 | 39.103 | 26.338 | 19.953 | 16.106 | 28.417 | 28.901 | 28.961 | 29.518 | |
0.5 | 2 | 1.5 | 31.999 | 38.876 | 25.558 | 19.403 | 15.677 | 28.393 | 28.870 | 28.941 | 29.480 | |
-0.1 | -0.25 | 0 | 0.25 | 28.753 | 36.668 | 23.542 | 17.849 | 14.456 | 25.621 | 25.643 | 26.126 | 26.164 |
0 | 1 | 1 | 28.672 | 36.649 | 23.652 | 17.809 | 14.435 | 25.637 | 25.661 | 26.146 | 26.184 | |
0.5 | 2 | 1.5 | 29.087 | 35.900 | 23.651 | 17.894 | 14.523 | 25.622 | 25.645 | 26.140 | 26.175 | |
0 | -0.25 | 0 | 0.25 | 27.123 | 35.382 | 22.633 | 17.113 | 13.892 | 24.786 | 24.786 | 25.284 | 25.284 |
0 | 1 | 1 | 27.852 | 35.371 | 23.033 | 17.424 | 14.146 | 24.797 | 24.797 | 25.292 | 25.292 | |
0.5 | 2 | 1.5 | 27.607 | 34.434 | 22.581 | 17.116 | 13.919 | 24.786 | 24.786 | 25.288 | 25.288 | |
0.1 | -0.25 | 0 | 0.25 | 26.299 | 34.969 | 22.037 | 16.679 | 13.565 | 23.842 | 23.869 | 24.322 | 24.332 |
0 | 1 | 1 | 26.797 | 35.384 | 22.411 | 16.960 | 13.787 | 23.854 | 23.882 | 24.349 | 24.365 | |
0.5 | 2 | 1.5 | 26.420 | 34.911 | 22.164 | 16.798 | 13.679 | 23.864 | 23.891 | 24.357 | 24.372 | |
0.5 | -0.25 | 0 | 0.25 | 20.192 | 32.428 | 19.137 | 14.443 | 11.860 | 18.938 | 19.877 | 19.369 | 20.262 |
0 | 1 | 1 | 20.217 | 32.478 | 19.118 | 14.526 | 11.942 | 18.950 | 19.891 | 19.385 | 20.271 | |
0.5 | 2 | 1.5 | 20.314 | 30.975 | 18.783 | 14.325 | 11.783 | 18.928 | 19.869 | 19.361 | 20.257 | |
0.9 | -0.25 | 0 | 0.25 | 9.426 | 36.100 | 15.345 | 11.627 | 9.744 | 9.094 | 14.818 | 9.355 | 15.174 |
0 | 1 | 1 | 9.491 | 34.843 | 15.055 | 11.622 | 9.750 | 9.090 | 14.804 | 9.352 | 15.167 | |
0.5 | 2 | 1.5 | 9.569 | 35.234 | 15.210 | 11.735 | 9.875 | 9.098 | 14.813 | 9.353 | 15.176 |
RNCP of various confidence intervals with different ρ and δ, μ _{1}, μ _{2}, (n,n _{1},n _{2})=(5,5,2), when \({\sigma _{1}^{2}}={\sigma _{2}^{2}}=4\)
Bivariate normal distribution | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
ρ | δ | μ _{1} | μ _{2} | T _{3} | T _{4} | T _{5} | W _{ s } | W _{ a } | B _{1} | B _{2} | B _{3} | B _{4} |
-0.9 | -0.25 | 0 | 0.25 | 0.4697 | 0.5652 | 0.4787 | 0.4000 | 0.5187 | 0.4583 | 0.5217 | 0.5185 | 0.5217 |
0 | 1 | 1 | 0.4464 | 0.5968 | 0.4190 | 0.4304 | 0.4151 | 0.4815 | 0.4340 | 0.4478 | 0.4340 | |
0.5 | 2 | 1.5 | 0.4386 | 0.6170 | 0.4796 | 0.6324 | 0.4796 | 0.4898 | 0.4792 | 0.5000 | 0.4800 | |
-0.5 | -0.25 | 0 | 0.25 | 0.4915 | 0.5577 | 0.5258 | 0.4396 | 0.5258 | 0.5000 | 0.5088 | 0.5246 | 0.5088 |
0 | 1 | 1 | 0.4444 | 0.5577 | 0.4800 | 0.4824 | 0.4800 | 0.4706 | 0.4286 | 0.4423 | 0.4490 | |
0.5 | 2 | 1.5 | 0.4915 | 0.5814 | 0.4950 | 0.6081 | 0.4902 | 0.4286 | 0.4545 | 0.4898 | 0.4773 | |
-0.1 | -0.25 | 0 | 0.25 | 0.4776 | 0.6042 | 0.4800 | 0.4330 | 0.4800 | 0.4912 | 0.4727 | 0.4630 | 0.4630 |
0 | 1 | 1 | 0.4918 | 0.5714 | 0.4839 | 0.4773 | 0.4839 | 0.4583 | 0.4490 | 0.4783 | 0.4681 | |
0.5 | 2 | 1.5 | 0.5862 | 0.6563 | 0.5200 | 0.6724 | 0.5132 | 0.4750 | 0.4878 | 0.4878 | 0.4878 | |
0 | -0.25 | 0 | 0.25 | 0.5077 | 0.5641 | 0.5233 | 0.4598 | 0.5233 | 0.5385 | 0.5385 | 0.5600 | 0.5600 |
0 | 1 | 1 | 0.5333 | 0.5769 | 0.5000 | 0.4891 | 0.5000 | 0.5085 | 0.5085 | 0.5000 | 0.5000 | |
0.5 | 2 | 1.5 | 0.5000 | 0.5957 | 0.5116 | 0.6053 | 0.5057 | 0.4167 | 0.4167 | 0.4286 | 0.4286 | |
0.1 | -0.25 | 0 | 0.25 | 0.5256 | 0.5652 | 0.5000 | 0.4205 | 0.5000 | 0.5000 | 0.5192 | 0.5000 | 0.5094 |
0 | 1 | 1 | 0.4545 | 0.5625 | 0.4778 | 0.4681 | 0.4725 | 0.5179 | 0.5273 | 0.5179 | 0.5000 | |
0.5 | 2 | 1.5 | 0.5694 | 0.6486 | 0.5057 | 0.6212 | 0.5057 | 0.5741 | 0.5660 | 0.5556 | 0.5556 | |
0.5 | -0.25 | 0 | 0.25 | 0.5139 | 0.6604 | 0.4805 | 0.4167 | 0.4805 | 0.4510 | 0.4630 | 0.4615 | 0.4815 |
0 | 1 | 1 | 0.4930 | 0.6667 | 0.5513 | 0.5057 | 0.5584 | 0.4746 | 0.5088 | 0.5077 | 0.5283 | |
0.5 | 2 | 1.5 | 0.5067 | 0.7027 | 0.5455 | 0.6604 | 0.5373 | 0.4878 | 0.5238 | 0.5439 | 0.5250 | |
0.9 | -0.25 | 0 | 0.25 | 0.5057 | 0.8286 | 0.5556 | 0.4028 | 0.5769 | 0.5000 | 0.4694 | 0.4798 | 0.4800 |
0 | 1 | 1 | 0.4624 | 0.8333 | 0.5000 | 0.5000 | 0.5000 | 0.5185 | 0.4565 | 0.5227 | 0.5000 | |
0.5 | 2 | 1.5 | 0.4943 | 0.7733 | 0.4074 | 0.6538 | 0.4231 | 0.4630 | 0.5319 | 0.4775 | 0.5435 | |
Bivariate t-distribution | ||||||||||||
-0.9 | -0.25 | 0 | 0.25 | 0.5195 | 0.6977 | 0.4891 | 0.5000 | 0.4930 | 0.4750 | 0.4375 | 0.4444 | 0.4318 |
0 | 1 | 1 | 0.4706 | 0.6905 | 0.5349 | 0.5152 | 0.5231 | 0.4717 | 0.5690 | 0.5469 | 0.5769 | |
0.5 | 2 | 1.5 | 0.5362 | 0.7436 | 0.5000 | 0.5469 | 0.5556 | 0.5192 | 0.4800 | 0.5085 | 0.4898 | |
-0.5 | -0.25 | 0 | 0.25 | 0.5915 | 0.6818 | 0.4684 | 0.4426 | 0.4426 | 0.4915 | 0.5085 | 0.5000 | 0.5000 |
0 | 1 | 1 | 0.4928 | 0.7143 | 0.4800 | 0.4531 | 0.4462 | 0.4912 | 0.4576 | 0.4737 | 0.4815 | |
0.5 | 2 | 1.5 | 0.5256 | 0.7021 | 0.5056 | 0.5526 | 0.5526 | 0.4167 | 0.3878 | 0.3529 | 0.3696 | |
-0.1 | -0.25 | 0 | 0.25 | 0.3971 | 0.5526 | 0.4937 | 0.4667 | 0.4667 | 0.5102 | 0.5000 | 0.5333 | 0.5217 |
0 | 1 | 1 | 0.5270 | 0.7250 | 0.5395 | 0.5325 | 0.5325 | 0.4667 | 0.4655 | 0.4630 | 0.4717 | |
0.5 | 2 | 1.5 | 0.4605 | 0.5750 | 0.4444 | 0.4810 | 0.4810 | 0.5000 | 0.4717 | 0.5106 | 0.5000 | |
0 | -0.25 | 0 | 0.25 | 0.5309 | 0.6341 | 0.5000 | 0.4819 | 0.4878 | 0.5088 | 0.5088 | 0.4902 | 0.4902 |
0 | 1 | 1 | 0.5067 | 0.6389 | 0.4805 | 0.4865 | 0.4800 | 0.5667 | 0.5667 | 0.5660 | 0.5660 | |
0.5 | 2 | 1.5 | 0.5574 | 0.7097 | 0.5132 | 0.5068 | 0.5000 | 0.5200 | 0.5200 | 0.5435 | 0.5435 | |
0.1 | -0.25 | 0 | 0.25 | 0.5714 | 0.5294 | 0.5556 | 0.5139 | 0.5139 | 0.5532 | 0.5652 | 0.5814 | 0.5814 |
0 | 1 | 1 | 0.5211 | 0.7813 | 0.5833 | 0.5833 | 0.5833 | 0.5098 | 0.4902 | 0.5000 | 0.4898 | |
0.5 | 2 | 1.5 | 0.4925 | 0.6563 | 0.4800 | 0.5000 | 0.5000 | 0.5208 | 0.5106 | 0.5116 | 0.5116 | |
0.5 | -0.25 | 0 | 0.25 | 0.5493 | 0.6744 | 0.4717 | 0.4833 | 0.4833 | 0.4821 | 0.5000 | 0.5000 | 0.4800 |
0 | 1 | 1 | 0.4625 | 0.7083 | 0.4444 | 0.4861 | 0.4861 | 0.4386 | 0.4912 | 0.4688 | 0.4630 | |
0.5 | 2 | 1.5 | 0.5161 | 0.6744 | 0.5172 | 0.5286 | 0.5286 | 0.5455 | 0.5283 | 0.5085 | 0.5306 | |
0.9 | -0.25 | 0 | 0.25 | 0.5455 | 0.8803 | 0.4348 | 0.5088 | 0.5088 | 0.4677 | 0.4677 | 0.4926 | 0.4754 |
0 | 1 | 1 | 0.5570 | 0.8534 | 0.5652 | 0.6042 | 0.6042 | 0.5000 | 0.5000 | 0.5194 | 0.4706 | |
0.5 | 2 | 1.5 | 0.4348 | 0.8333 | 0.5714 | 0.4821 | 0.4821 | 0.5000 | 0.5357 | 0.4898 | 0.5370 |
Power of various confidence intervals with different ρ and δ, μ _{1}, \(\mu _{2}, {\sigma _{1}^{2}}\) and (n,n _{1},n _{2})=(5,2,2) and \({\sigma _{2}^{2}}=4\)
ρ | \({\sigma _{1}^{2}}\) | δ | μ _{1} | μ _{2} | T _{1} | T _{2} | T _{ g } | W _{ s } | W _{ a } | B _{1} | B _{2} | B _{3} | B _{4} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
-0.9 | 1 | -0.25 | 0 | 0.25 | 6.40 | 4.35 | 5.10 | 7.30 | 12.60 | 5.20 | 5.10 | 6.40 | 5.05 |
0.5 | 2 | 1.5 | 7.25 | 5.20 | 5.50 | 9.10 | 13.70 | 6.50 | 6.35 | 7.80 | 6.35 | ||
8 | -0.25 | 0 | 0.25 | 5.80 | 3.30 | 5.25 | 7.60 | 13.30 | 5.50 | 5.10 | 6.10 | 4.95 | |
0.5 | 2 | 1.5 | 6.20 | 4.00 | 7.65 | 8.25 | 14.40 | 5.90 | 6.30 | 7.65 | 6.40 | ||
-0.5 | 1 | -0.25 | 0 | 0.25 | 6.50 | 4.00 | 6.60 | 7.75 | 13.10 | 4.70 | 4.65 | 5.45 | 4.80 |
0.5 | 2 | 1.5 | 7.75 | 4.90 | 7.60 | 10.15 | 15.85 | 6.10 | 5.95 | 6.40 | 5.70 | ||
8 | -0.25 | 0 | 0.25 | 6.60 | 4.55 | 7.00 | 9.55 | 15.30 | 5.80 | 5.40 | 6.20 | 5.70 | |
0.5 | 2 | 1.5 | 5.80 | 4.10 | 6.05 | 8.25 | 13.85 | 5.50 | 5.70 | 5.85 | 5.45 | ||
-0.1 | 1 | -0.25 | 0 | 0.25 | 6.95 | 3.55 | 8.45 | 6.90 | 12.70 | 4.65 | 4.50 | 4.70 | 4.65 |
0.5 | 2 | 1.5 | 7.70 | 4.85 | 7.55 | 9.90 | 15.75 | 6.80 | 6.80 | 6.85 | 6.65 | ||
8 | -0.25 | 0 | 0.25 | 7.25 | 3.95 | 7.90 | 9.75 | 15.60 | 6.25 | 6.15 | 6.25 | 6.20 | |
0.5 | 2 | 1.5 | 6.60 | 3.50 | 7.25 | 8.75 | 15.20 | 5.25 | 5.35 | 5.15 | 5.10 | ||
0 | 1 | -0.25 | 0 | 0.25 | 8.10 | 4.60 | 7.10 | 8.20 | 13.40 | 5.45 | 5.45 | 5.45 | 5.45 |
0.5 | 2 | 1.5 | 8.35 | 4.70 | 8.50 | 11.50 | 17.90 | 6.55 | 6.55 | 6.65 | 6.65 | ||
8 | -0.25 | 0 | 0.25 | 7.45 | 3.50 | 8.90 | 9.10 | 15.25 | 5.45 | 5.45 | 5.40 | 5.40 | |
0.5 | 2 | 1.5 | 7.30 | 3.65 | 7.45 | 10.55 | 16.80 | 6.10 | 6.10 | 6.10 | 6.10 | ||
0.1 | 1 | -0.25 | 0 | 0.25 | 7.05 | 3.95 | 9.85 | 8.40 | 13.85 | 5.45 | 5.60 | 5.60 | 5.70 |
0.5 | 2 | 1.5 | 7.55 | 4.45 | 8.45 | 11.55 | 16.90 | 5.85 | 6.15 | 5.90 | 5.95 | ||
8 | -0.25 | 0 | 0.25 | 6.30 | 3.85 | 8.70 | 8.05 | 14.20 | 4.75 | 4.85 | 5.00 | 5.05 | |
0.5 | 2 | 1.5 | 7.65 | 4.05 | 9.60 | 9.70 | 16.40 | 5.85 | 6.00 | 6.25 | 6.30 | ||
0.5 | 1 | -0.25 | 0 | 0.25 | 7.30 | 4.15 | 9.35 | 6.95 | 12.90 | 5.10 | 4.85 | 6.15 | 4.90 |
0.5 | 2 | 1.5 | 8.40 | 4.75 | 8.15 | 12.70 | 19.35 | 6.00 | 5.95 | 7.10 | 6.15 | ||
8 | -0.25 | 0 | 0.25 | 8.80 | 4.20 | 7.80 | 9.80 | 15.40 | 5.30 | 5.15 | 6.80 | 5.30 | |
0.5 | 2 | 1.5 | 9.10 | 4.05 | 8.40 | 11.55 | 16.45 | 6.65 | 6.95 | 8.50 | 7.15 | ||
0.9 | 1 | -0.25 | 0 | 0.25 | 7.30 | 5.25 | 8.10 | 7.50 | 13.60 | 5.10 | 5.35 | 7.20 | 5.40 |
0.5 | 2 | 1.5 | 8.45 | 5.35 | 8.55 | 18.00 | 26.95 | 7.55 | 7.70 | 8.25 | 7.75 | ||
8 | -0.25 | 0 | 0.25 | 8.95 | 5.40 | 5.35 | 7.25 | 13.45 | 5.80 | 5.90 | 7.10 | 6.10 | |
0.5 | 2 | 1.5 | 11.45 | 5.30 | 6.25 | 12.30 | 18.20 | 10.05 | 8.00 | 9.60 | 7.95 |
Power of various confidence intervals with different ρ and δ, μ _{1}, μ _{2}, (n,n _{1},n _{2})=(5,5,2), when \({\sigma _{1}^{2}}={\sigma _{2}^{2}}=4\)
Bivariate normal distribution | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
ρ | δ | μ _{1} | μ _{2} | T _{3} | T _{4} | T _{5} | W _{ s } | W _{ a } | B _{1} | B _{2} | B _{3} | B _{4} |
-0.9 | -0.25 | 0 | 0.25 | 1.5 | 2.5 | 4.3 | 7.5 | 12.4 | 5.2 | 4.8 | 6.6 | 5.0 |
0.5 | 2 | 1.5 | 3.2 | 4.1 | 6.4 | 10.8 | 15.6 | 7.5 | 7.4 | 9.4 | 7.4 | |
-0.5 | -0.25 | 0 | 0.25 | 3.9 | 3.0 | 5.7 | 8.5 | 12.9 | 5.6 | 5.1 | 5.6 | 5.2 |
0.5 | 2 | 1.5 | 4.0 | 3.0 | 6.5 | 9.9 | 14.4 | 6.8 | 6.9 | 7.3 | 6.8 | |
-0.1 | -0.25 | 0 | 0.25 | 3.6 | 2.9 | 6.2 | 9.5 | 14.8 | 5.8 | 5.8 | 5.9 | 6.0 |
0.5 | 2 | 1.5 | 5.5 | 4.9 | 8.7 | 11.3 | 16.4 | 8.3 | 8.2 | 7.9 | 7.9 | |
0 | -0.25 | 0 | 0.25 | 4.4 | 3.3 | 6.9 | 9.8 | 14.7 | 5.7 | 5.7 | 5.9 | 5.9 |
0.5 | 2 | 1.5 | 4.7 | 4.0 | 7.6 | 10.8 | 16.7 | 7.9 | 7.9 | 7.6 | 7.6 | |
0.1 | -0.25 | 0 | 0.25 | 3.5 | 2.9 | 5.5 | 8.2 | 13.3 | 5.8 | 5.7 | 5.7 | 5.7 |
0.5 | 2 | 1.5 | 5.1 | 4.3 | 8.1 | 11.6 | 16.2 | 7.6 | 7.3 | 7.5 | 7.4 | |
0.5 | -0.25 | 0 | 0.25 | 4.7 | 3.3 | 5.9 | 9.6 | 14.7 | 6.7 | 6.5 | 8.5 | 6.3 |
0.5 | 2 | 1.5 | 5.3 | 5.1 | 8.4 | 13.1 | 17.9 | 11.1 | 10.8 | 13.2 | 10.6 | |
0.9 | -0.25 | 0 | 0.25 | 3.9 | 3.5 | 4.7 | 9.7 | 15.4 | 10.7 | 6.5 | 27.5 | 6.4 |
0.5 | 2 | 1.5 | 9.1 | 6.0 | 8.2 | 13.7 | 18.0 | 27.9 | 11.4 | 27.3 | 11.2 | |
Bivariate t-distribution | ||||||||||||
-0.9 | -0.25 | 0 | 0.25 | 1.2 | 2.1 | 4.0 | 6.7 | 11.6 | 4.9 | 5.1 | 5.9 | 4.7 |
0.5 | 2 | 1.5 | 1.5 | 2.0 | 4.0 | 6.1 | 11.4 | 4.9 | 5.0 | 6.1 | 4.3 | |
-0.5 | -0.25 | 0 | 0.25 | 2.0 | 1.5 | 4.2 | 6.2 | 12.2 | 4.8 | 5.1 | 5.1 | 4.9 |
0.5 | 2 | 1.5 | 2.0 | 1.8 | 5.0 | 6.8 | 12.7 | 6.3 | 6.3 | 6.4 | 5.9 | |
-0.1 | -0.25 | 0 | 0.25 | 2.9 | 2.8 | 6.0 | 8.3 | 15.2 | 7.1 | 7.0 | 6.7 | 6.4 |
0.5 | 2 | 1.5 | 2.0 | 1.9 | 5.0 | 7.0 | 12.7 | 4.4 | 4.4 | 4.1 | 4.0 | |
0 | -0.25 | 0 | 0.25 | 2.5 | 2.0 | 4.1 | 6.7 | 12.4 | 5.0 | 5.0 | 4.5 | 4.5 |
0.5 | 2 | 1.5 | 2.2 | 1.9 | 4.6 | 6.5 | 12.8 | 6.1 | 6.1 | 5.9 | 5.9 | |
0.1 | -0.25 | 0 | 0.25 | 2.4 | 2.1 | 4.4 | 7.0 | 12.0 | 5.2 | 5.1 | 5.0 | 5.0 |
0.5 | 2 | 1.5 | 2.9 | 2.7 | 5.6 | 7.4 | 13.2 | 5.3 | 5.1 | 4.9 | 5.0 | |
0.5 | -0.25 | 0 | 0.25 | 1.3 | 2.0 | 4.4 | 6.1 | 11.4 | 5.0 | 5.2 | 6.4 | 5.2 |
0.5 | 2 | 1.5 | 1.7 | 2.0 | 4.7 | 6.1 | 11.4 | 4.9 | 5.1 | 5.9 | 4.7 | |
0.9 | -0.25 | 0 | 0.25 | 1.3 | 2.8 | 3.4 | 5.0 | 10.4 | 5.0 | 4.8 | 5.4 | 4.4 |
0.5 | 2 | 1.5 | 2.1 | 2.2 | 2.7 | 5.1 | 11.7 | 5.7 | 5.8 | 5.8 | 5.2 |
An worked example
Various 95 % confidence intervals for δ=μ _{1}−μ _{2} based on formoterol solution aerosol
T _{1} | T _{2} | T _{3} | T _{4} | T _{5} | T _{ g } | |
Lower | -0.2751 | -0.4764 | -0.472 | -0.5542 | -0.4431 | -0.4883 |
Upper | 0.1071 | 0.5220 | 0.3741 | 0.5999 | 0.4888 | 0.5039 |
Width | 0.3822 | 0.9984 | 0.8461 | 1.1541 | 0.9319 | 0.9922 |
W _{ s } | W _{ a } | B _{1} | B _{2} | B _{3} | B _{4} | |
Lower | -0.5940 | -0.5787 | -0.5408 | -0.5938 | -0.5259 | -0.5681 |
Upper | 0.6495 | 0.6334 | 0.3995 | 0.4394 | 0.4309 | 0.4058 |
Width | 1.2435 | 1.2121 | 0.9403 | 1.0332 | 0.9568 | 0.9739 |
Discussion
Although testing equivalence of two correlated means with incomplete data has been studied, there is little work done on their interval estimators. To address the issue, this paper proposes various interval estimators of the difference of two correlated means for Σ known and unknown cases based on the large sample method, hybrid method and Bootstrap-resampling method. Extensive simulation studies are conducted to evaluate the finite performance of the proposed CIs in terms of the empirical coverage probability, empirical interval width and ratio of the mesial non-coverage probability to the non-coverage probability (RNCP). Empirical results evidence that the Bootstrap-resampling-based CIs B _{1}, B _{2}, B _{4} behave satisfactorily for small to moderate sample sizes in the sense that their coverage probabilities could be well controlled around the pre-specified nominal confidence level and their RNCPs almost lie in the interval [0.4, 0.6]. However, confidence intervals based on the large sample method and hybrid method behave unsatisfactory for small sample sizes because the distributions of statistics T _{1},⋯,T _{5} are asymptotical, and these asymptotical distributions are proper only when N _{ i }→∞. When Σ is unknown, using GEE method to estimate variance is less efficient.
It is interesting to investigate confidence interval construction of the difference of two means with incomplete correlated data under missing at random and non-ignorable missing data mechanism assumptions of bivariate variables. We are working on the topics.
Conclusion
According to the aforementioned findings, we can draw the following conclusions. The Bootstrap-resampling-based CI B _{1} is a desirable interval estimator for the difference of two means with incomplete correlated data.
Declarations
Acknowledgements
The research of Hui-Qiong LI was supported by the Natural Science Foundation of China (11201412, 11561075). The work of the second author was partially supported by the grants from the National Science Foundation of China (11225103).
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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