# Global Hypothesis Test to Compare the Predictive Values of Diagnostic Tests Subject to a Case-Control Design

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Global Hypothesis Test

## 3. Simulation Experiments

#### 3.1. Type I Errors and Powers

#### 3.2. Effect of the Prevalence

## 4. Example

## 5. More Than Two BDTs

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## References

- Pepe, M.S. The Statistical Evaluation of Medical Tests for Classification and Prediction; Oxford University Press: New York, NY, USA, 2003. [Google Scholar]
- Bennett, B.M. On comparison of sensitivity, specificity and predictive value of a number of diagnostic procedures. Biometrics
**1972**, 28, 793–800. [Google Scholar] [CrossRef] [PubMed] - Bennett, B.M. On tests for equality of predictive values for t diagnostic procedures. Stat. Med.
**1985**, 4, 535–539. [Google Scholar] [CrossRef] [PubMed] - Leisenring, W.; Alonzo, T.; Pepe, M.S. Comparisons of predictive values of binary medical diagnostic tests for paired designs. Biometrics
**2000**, 56, 345–351. [Google Scholar] [CrossRef] [PubMed] - Wang, W.; Davis, C.S.; Soong, S.J. Comparison of predictive values of two diagnostic tests from the same sample of subjects using weighted least squares. Stat. Med.
**2006**, 25, 2215–2229. [Google Scholar] [CrossRef] [PubMed] - Kosinski, A.S. A weighted generalized score statistic for comparison of predictive values of diagnostic tests. Stat. Med.
**2013**, 32, 964–977. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Moskowitz, C.S.; Pepe, M.S. Comparing the predictive values of diagnostic tests: Sample size and analysis for paired study designs. Clin. Trials
**2006**, 3, 272–279. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Roldán-Nofuentes, J.A.; Luna del Castillo, J.D.; Montero-Alonso, M.A. Global hypothesis test to simultaneously compare the predictive values of two binary diagnostic tests. Comput. Stat. Data Anal.
**2012**, 56, 1161–1173. [Google Scholar] [CrossRef] - Mercaldo, N.D.; Kit, F.L.; Zhou, X.H. Confidence intervals for predictive values with an emphasis to case-control studies. Stat. Med.
**2007**, 26, 2170–2183. [Google Scholar] [CrossRef] [PubMed] - Vacek, P.M. The effect of conditional dependence on the evaluation of diagnostic tests. Biometrics
**1985**, 41, 959–968. [Google Scholar] [CrossRef] [PubMed] - Kocherlakota, S.; Kocherlakota, K. Bivariate Discrete Distributions; Marcel Dekker INC: New York, NY, USA, 1992. [Google Scholar]
- Bonferroni, C.E. Teoria statistica delle classi e calcolo delle probabilità. Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commerciali di Firenze
**1936**, 8, 3–62. [Google Scholar] - Holm, S. A simple sequential rejective multiple testing procedure. Scand. J. Stat.
**1979**, 6, 65–70. [Google Scholar] - RC Team. R: A Language and Environment for Statistical Computing; RC Team: Vienna, Austria, 2016; Available online: https://www.R-project.org/ (accessed on 17 March 2021).
- Leisch, F.; Weingessel, A.; Hornik, K. Bindata Package. Available online: https://cran.r-project.org/web/packages/bindata/ (accessed on 17 March 2021).
- Torrance-Rynard, V.L.; Walter, S.D. Effects of dependent errors in the assessment of diagnostic test performance. Stat. Med.
**1997**, 16, 2157–2175. [Google Scholar] [CrossRef]

**Figure 1.**Powers of the three methods when the negative predictive value (NPV) of a binary diagnostic test varies and the rest of the PVs are constant.

**Figure 2.**Powers of the three methods when the positive predictive value (PPV) of a binary diagnostic test varies and the rest of the PVs are constant.

Probabilities | |||||||

Case | Control | ||||||

${T}_{2}=1$ | ${T}_{2}=0$ | Total | ${T}_{2}=1$ | ${T}_{2}=0$ | Total | ||

${T}_{1}=1$ | ${\xi}_{111}$ | ${\xi}_{110}$ | $S{e}_{1}$ | ${T}_{1}=1$ | ${\xi}_{211}$ | ${\xi}_{210}$ | $1-S{p}_{1}$ |

${T}_{1}=0$ | ${\xi}_{101}$ | ${\xi}_{100}$ | $1-S{e}_{1}$ | ${T}_{1}=0$ | ${\xi}_{201}$ | ${\xi}_{200}$ | $S{p}_{1}$ |

Total | $S{e}_{2}$ | $1-S{e}_{2}$ | 1 | Total | $1-S{p}_{2}$ | $S{p}_{2}$ | 1 |

Observed Frequencies | |||||||

Case | Control | ||||||

${T}_{2}=1$ | ${T}_{2}=0$ | Total | ${T}_{2}=1$ | ${T}_{2}=0$ | Total | ||

${T}_{1}=1$ | ${n}_{111}$ | ${n}_{110}$ | ${n}_{11\xb7}$ | ${T}_{1}=1$ | ${n}_{211}$ | ${n}_{210}$ | ${n}_{21\xb7}$ |

${T}_{1}=0$ | ${n}_{101}$ | ${n}_{100}$ | ${n}_{10\xb7}$ | ${T}_{1}=0$ | ${n}_{201}$ | ${n}_{200}$ | ${n}_{20\xb7}$ |

Total | ${n}_{1\xb71}$ | ${n}_{1\xb70}$ | ${n}_{1}$ | Total | ${n}_{2\xb71}$ | ${n}_{2\xb70}$ | ${n}_{2}$ |

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.5385}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.9744}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.5385}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.9744}\text{}\mathbf{,}\text{}\mathit{p}=\mathbf{10}\mathbf{\%}$ | ||||||||||

${\rho}_{1}=0.25{\rho}_{2}=0.25$ | ${\rho}_{1}=0.50{\rho}_{2}=0.50$ | ${\rho}_{1}=0.75{\rho}_{2}=0.75$ | ||||||||

${n}_{1}$ | ${n}_{2}$ | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. |

50 | 50 | 0.031 | 0.051 | 0.029 | 0.027 | 0.048 | 0.027 | 0.004 | 0.013 | 0.004 |

50 | 75 | 0.029 | 0.059 | 0.029 | 0.025 | 0.051 | 0.026 | 0.004 | 0.017 | 0.005 |

50 | 100 | 0.028 | 0.063 | 0.030 | 0.029 | 0.061 | 0.028 | 0.008 | 0.018 | 0.007 |

75 | 75 | 0.023 | 0.061 | 0.026 | 0.031 | 0.056 | 0.028 | 0.015 | 0.034 | 0.017 |

100 | 100 | 0.027 | 0.063 | 0.029 | 0.023 | 0.052 | 0.024 | 0.020 | 0.043 | 0.019 |

200 | 200 | 0.044 | 0.086 | 0.045 | 0.032 | 0.063 | 0.031 | 0.025 | 0.050 | 0.026 |

500 | 500 | 0.055 | 0.107 | 0.056 | 0.058 | 0.102 | 0.057 | 0.040 | 0.077 | 0.039 |

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.8615}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.8769}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.8615}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.8769}\text{}\mathbf{,}\text{}\mathit{p}=\mathbf{25}\mathbf{\%}$ | ||||||||||

${\rho}_{1}=0.25{\rho}_{2}=0.25$ | ${\rho}_{1}=0.50{\rho}_{2}=0.50$ | ${\rho}_{1}=0.75{\rho}_{2}=0.75$ | ||||||||

${n}_{1}$ | ${n}_{2}$ | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. |

50 | 50 | 0.048 | 0.094 | 0.046 | 0.018 | 0.047 | 0.018 | 0.001 | 0.007 | 0.002 |

50 | 75 | 0.053 | 0.100 | 0.051 | 0.025 | 0.063 | 0.026 | 0.002 | 0.012 | 0.003 |

50 | 100 | 0.053 | 0.106 | 0.057 | 0.034 | 0.076 | 0.032 | 0.008 | 0.023 | 0.008 |

75 | 75 | 0.059 | 0.105 | 0.055 | 0.039 | 0.087 | 0.037 | 0.007 | 0.016 | 0.006 |

100 | 100 | 0.059 | 0.117 | 0.059 | 0.056 | 0.102 | 0.054 | 0.011 | 0.040 | 0.010 |

200 | 200 | 0.058 | 0.099 | 0.057 | 0.048 | 0.094 | 0.049 | 0.044 | 0.090 | 0.042 |

500 | 500 | 0.052 | 0.098 | 0.053 | 0.051 | 0.101 | 0.052 | 0.049 | 0.090 | 0.048 |

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.9692}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.5846}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.9692}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.5846}\text{}\mathbf{,}\text{}\mathit{p}=\mathbf{50}\mathbf{\%}$ | ||||||||||

${\rho}_{1}=0.25{\rho}_{2}=0.25$ | ${\rho}_{1}=0.50{\rho}_{2}=0.50$ | ${\rho}_{1}=0.75{\rho}_{2}=0.75$ | ||||||||

${n}_{1}$ | ${n}_{2}$ | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. |

50 | 50 | 0.026 | 0.049 | 0.026 | 0.025 | 0.061 | 0.026 | 0.006 | 0.017 | 0.006 |

50 | 75 | 0.020 | 0.049 | 0.023 | 0.019 | 0.052 | 0.024 | 0.007 | 0.028 | 0.010 |

50 | 100 | 0.019 | 0.043 | 0.023 | 0.016 | 0.045 | 0.019 | 0.010 | 0.034 | 0.014 |

75 | 75 | 0.024 | 0.065 | 0.027 | 0.020 | 0.051 | 0.027 | 0.012 | 0.038 | 0.017 |

100 | 100 | 0.028 | 0.066 | 0.029 | 0.021 | 0.052 | 0.025 | 0.012 | 0.042 | 0.019 |

200 | 200 | 0.047 | 0.088 | 0.044 | 0.034 | 0.074 | 0.032 | 0.021 | 0.058 | 0.026 |

500 | 500 | 0.052 | 0.099 | 0.052 | 0.050 | 0.097 | 0.049 | 0.037 | 0.077 | 0.034 |

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.5312}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.9896}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.5312}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.9896}\text{}\mathbf{,}\text{}\mathit{p}=\mathbf{10}\mathbf{\%}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{1}}\le \mathbf{1}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{2}}\le \mathbf{1}$ | ||||||||||

${\rho}_{1}=0.25{\rho}_{2}=0.25$ | ${\rho}_{1}=0.50{\rho}_{2}=0.50$ | ${\rho}_{1}=0.75{\rho}_{2}=0.75$ | ||||||||

${n}_{1}$ | ${n}_{2}$ | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. |

50 | 50 | 0.033 | 0.056 | 0.034 | 0.020 | 0.051 | 0.024 | 0.004 | 0.014 | 0.004 |

50 | 75 | 0.024 | 0.049 | 0.024 | 0.026 | 0.050 | 0.025 | 0.005 | 0.019 | 0.006 |

50 | 100 | 0.032 | 0.057 | 0.033 | 0.030 | 0.056 | 0.030 | 0.004 | 0.016 | 0.004 |

75 | 75 | 0.034 | 0.054 | 0.033 | 0.025 | 0.052 | 0.026 | 0.014 | 0.036 | 0.015 |

100 | 100 | 0.027 | 0.055 | 0.026 | 0.027 | 0.055 | 0.026 | 0.017 | 0.041 | 0.017 |

200 | 200 | 0.033 | 0.059 | 0.031 | 0.025 | 0.050 | 0.024 | 0.022 | 0.055 | 0.021 |

500 | 500 | 0.046 | 0.087 | 0.049 | 0.031 | 0.068 | 0.033 | 0.018 | 0.050 | 0.024 |

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.85}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.95}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.85}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.95}\text{}\mathbf{,}\text{}\mathit{p}=\mathbf{25}\mathbf{\%}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{1}}\le \mathbf{1}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{2}}\le \mathbf{1}$ | ||||||||||

${\rho}_{1}=0.25{\rho}_{2}=0.25$ | ${\rho}_{1}=0.50{\rho}_{2}=0.50$ | ${\rho}_{1}=0.75{\rho}_{2}=0.75$ | ||||||||

${n}_{1}$ | ${n}_{2}$ | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. |

50 | 50 | 0.023 | 0.058 | 0.022 | 0.005 | 0.030 | 0.007 | 0.001 | 0.005 | 0.001 |

50 | 75 | 0.037 | 0.077 | 0.036 | 0.014 | 0.039 | 0.015 | 0.001 | 0.008 | 0.001 |

50 | 100 | 0.049 | 0.092 | 0.048 | 0.022 | 0.056 | 0.022 | 0.001 | 0.007 | 0.002 |

75 | 75 | 0.042 | 0.087 | 0.041 | 0.025 | 0.055 | 0.025 | 0.004 | 0.014 | 0.004 |

100 | 100 | 0.048 | 0.095 | 0.043 | 0.028 | 0.066 | 0.027 | 0.005 | 0.025 | 0.005 |

200 | 200 | 0.033 | 0.059 | 0.031 | 0.025 | 0.050 | 0.024 | 0.022 | 0.055 | 0.021 |

500 | 500 | 0.048 | 0.097 | 0.046 | 0.056 | 0.101 | 0.051 | 0.050 | 0.099 | 0.049 |

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.9562}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.8312}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.9562}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.8312}\text{}\mathbf{,}\text{}\mathit{p}=\mathbf{50}\mathbf{\%}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{1}}\le \mathbf{1}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{2}}\le \mathbf{1}$ | ||||||||||

${\rho}_{1}=0.25{\rho}_{2}=0.25$ | ${\rho}_{1}=0.50{\rho}_{2}=0.50$ | ${\rho}_{1}=0.75{\rho}_{2}=0.75$ | ||||||||

${n}_{1}$ | ${n}_{2}$ | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. |

50 | 50 | 0.031 | 0.072 | 0.031 | 0.014 | 0.041 | 0.015 | 0.001 | 0.007 | 0.001 |

50 | 75 | 0.032 | 0.069 | 0.033 | 0.022 | 0.049 | 0.022 | 0.005 | 0.015 | 0.005 |

50 | 100 | 0.025 | 0.057 | 0.026 | 0.025 | 0.064 | 0.026 | 0.008 | 0.025 | 0.008 |

75 | 75 | 0.038 | 0.081 | 0.037 | 0.027 | 0.054 | 0.025 | 0.006 | 0.017 | 0.006 |

100 | 100 | 0.039 | 0.084 | 0.038 | 0.031 | 0.073 | 0.030 | 0.008 | 0.030 | 0.009 |

200 | 200 | 0.033 | 0.059 | 0.031 | 0.025 | 0.050 | 0.024 | 0.022 | 0.055 | 0.021 |

500 | 500 | 0.051 | 0.099 | 0.049 | 0.050 | 0.097 | 0.047 | 0.043 | 0.087 | 0.042 |

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.5357}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.9802}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.5455}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.9596}\text{}\mathbf{,}\text{}\mathit{p}=\mathbf{10}\mathbf{\%}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{1}}\le \mathbf{0.9805}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{2}}\le \mathbf{0.6933}$ | ||||||||||

${\rho}_{1}=0.25{\rho}_{2}=0.17$ | ${\rho}_{1}=0.49{\rho}_{2}=0.35$ | ${\rho}_{1}=0.74{\rho}_{2}=0.52$ | ||||||||

${n}_{1}$ | ${n}_{2}$ | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. |

50 | 50 | 0.025 | 0.056 | 0.031 | 0.023 | 0.049 | 0.024 | 0.005 | 0.019 | 0.007 |

50 | 75 | 0.037 | 0.077 | 0.036 | 0.029 | 0.063 | 0.030 | 0.010 | 0.030 | 0.011 |

50 | 100 | 0.054 | 0.103 | 0.052 | 0.042 | 0.084 | 0.038 | 0.019 | 0.046 | 0.016 |

75 | 75 | 0.038 | 0.078 | 0.038 | 0.032 | 0.066 | 0.033 | 0.018 | 0.042 | 0.018 |

100 | 100 | 0.053 | 0.098 | 0.047 | 0.044 | 0.081 | 0.037 | 0.031 | 0.063 | 0.026 |

200 | 200 | 0.199 | 0.276 | 0.180 | 0.208 | 0.286 | 0.181 | 0.168 | 0.252 | 0.138 |

500 | 500 | 0.495 | 0.575 | 0.462 | 0.591 | 0.668 | 0.556 | 0.720 | 0.785 | 0.678 |

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.8571}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.9048}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.8727}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.8061}\text{}\mathbf{,}\text{}\mathit{p}=\mathbf{25}\mathbf{\%}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{1}}\le \mathbf{0.9354}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{2}}\le \mathbf{0.6614}$ | ||||||||||

${\rho}_{1}=0.23{\rho}_{2}=0.17$ | ${\rho}_{1}=0.47{\rho}_{2}=0.33$ | ${\rho}_{1}=0.70{\rho}_{2}=0.50$ | ||||||||

${n}_{1}$ | ${n}_{2}$ | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. |

50 | 50 | 0.259 | 0.335 | 0.230 | 0.254 | 0.345 | 0.230 | 0.195 | 0.334 | 0.210 |

50 | 75 | 0.409 | 0.496 | 0.378 | 0.454 | 0.543 | 0.424 | 0.467 | 0.606 | 0.470 |

50 | 100 | 0.505 | 0.584 | 0.462 | 0.598 | 0.677 | 0.556 | 0.683 | 0.776 | 0.675 |

75 | 75 | 0.416 | 0.498 | 0.382 | 0.469 | 0.557 | 0.436 | 0.501 | 0.608 | 0.476 |

100 | 100 | 0.528 | 0.606 | 0.488 | 0.625 | 0.699 | 0.579 | 0.718 | 0.793 | 0.685 |

200 | 200 | 0.822 | 0.862 | 0.790 | 0.891 | 0.923 | 0.873 | 0.974 | 0.983 | 0.964 |

500 | 500 | 0.996 | 0.999 | 0.996 | 1 | 1 | 1 | 1 | 1 | 1 |

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.9643}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.6786}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.9818}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.3455}\text{}\mathbf{,}\text{}\mathit{p}=\mathbf{50}\mathbf{\%}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{1}}\le \mathbf{0.7071}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{2}}\le \mathbf{0.50}$ | ||||||||||

${\rho}_{1}=0.18{\rho}_{2}=0.13$ | ${\rho}_{1}=0.35{\rho}_{2}=0.25$ | ${\rho}_{1}=0.53{\rho}_{2}=0.38$ | ||||||||

${n}_{1}$ | ${n}_{2}$ | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. |

50 | 50 | 0.890 | 0.939 | 0.893 | 0.935 | 0.969 | 0.941 | 0.977 | 0.989 | 0.978 |

50 | 75 | 0.978 | 0.990 | 0.977 | 0.995 | 0.997 | 0.993 | 0.999 | 0.999 | 0.999 |

50 | 100 | 0.995 | 0.998 | 0.995 | 0.999 | 0.998 | 0.999 | 1 | 1 | 1 |

75 | 75 | 0.984 | 0.992 | 0.983 | 0.995 | 0.999 | 0.994 | 0.999 | 1 | 0.999 |

100 | 100 | 0.998 | 0.999 | 0.998 | 1 | 1 | 0.999 | 1 | 1 | 1 |

200 | 200 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

500 | 500 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.5278}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.9969}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.5357}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.9802}\text{}\mathbf{,}\text{}\mathit{p}=\mathbf{10}\mathbf{\%}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{1}}\le \mathbf{0.9841}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{2}}\le \mathbf{0.3910}$ | ||||||||||

${\rho}_{1}=0.25{\rho}_{2}=0.10$ | ${\rho}_{1}=0.49{\rho}_{2}=0.19$ | ${\rho}_{1}=0.74{\rho}_{2}=0.29$ | ||||||||

${n}_{1}$ | ${n}_{2}$ | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. |

50 | 50 | 0.030 | 0.059 | 0.030 | 0.019 | 0.048 | 0.020 | 0.007 | 0.019 | 0.008 |

50 | 75 | 0.031 | 0.063 | 0.032 | 0.023 | 0.054 | 0.023 | 0.009 | 0.024 | 0.009 |

50 | 100 | 0.033 | 0.064 | 0.033 | 0.030 | 0.063 | 0.030 | 0.010 | 0.031 | 0.009 |

75 | 75 | 0.033 | 0.057 | 0.032 | 0.025 | 0.055 | 0.025 | 0.015 | 0.036 | 0.015 |

100 | 100 | 0.034 | 0.067 | 0.033 | 0.026 | 0.059 | 0.026 | 0.026 | 0.054 | 0.026 |

200 | 200 | 0.123 | 0.182 | 0.094 | 0.122 | 0.177 | 0.095 | 0.108 | 0.168 | 0.078 |

500 | 500 | 0.666 | 0.770 | 0.662 | 0.669 | 0.781 | 0.667 | 0.699 | 0.811 | 0.696 |

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.8444}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.9852}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.8571}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.9048}\text{}\mathbf{,}\text{}\mathit{p}=\mathbf{25}\mathbf{\%}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{1}}\le \mathbf{0.9511}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{2}}\le \mathbf{0.3779}$ | ||||||||||

${\rho}_{1}=0.24{\rho}_{2}=0.09$ | ${\rho}_{1}=0.48{\rho}_{2}=0.19$ | ${\rho}_{1}=0.71{\rho}_{2}=0.28$ | ||||||||

${n}_{1}$ | ${n}_{2}$ | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. |

50 | 50 | 0.172 | 0.247 | 0.142 | 0.154 | 0.234 | 0.127 | 0.107 | 0.198 | 0.097 |

50 | 75 | 0.422 | 0.537 | 0.396 | 0.398 | 0.521 | 0.378 | 0.365 | 0.541 | 0.367 |

50 | 100 | 0.627 | 0.734 | 0.615 | 0.641 | 0.755 | 0.638 | 0.674 | 0.779 | 0.653 |

75 | 75 | 0.434 | 0.549 | 0.400 | 0.432 | 0.555 | 0.410 | 0.402 | 0.552 | 0.391 |

100 | 100 | 0.635 | 0.753 | 0.634 | 0.655 | 0.774 | 0.656 | 0.666 | 0.796 | 0.683 |

200 | 200 | 0.965 | 0.981 | 0.964 | 0.977 | 0.987 | 0.974 | 0.989 | 0.994 | 0.988 |

500 | 500 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.95}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.95}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.9643}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.6786}\text{}\mathbf{,}\text{}\mathit{p}=\mathbf{50}\mathbf{\%}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}^{+}\le \mathbf{0.8388}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}^{-}\le \mathbf{0.3333}$ | ||||||||||

${\rho}_{1}=0.21{\rho}_{2}=0.08$ | ${\rho}_{1}=0.42{\rho}_{2}=0.17$ | ${\rho}_{1}=0.63{\rho}_{2}=0.25$ | ||||||||

${n}_{1}$ | ${n}_{2}$ | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. |

50 | 50 | 0.929 | 0.969 | 0.942 | 0.954 | 0.983 | 0.966 | 0.965 | 0.992 | 0.978 |

50 | 75 | 0.994 | 0.998 | 0.995 | 0.999 | 0.999 | 0.999 | 0.999 | 1 | 0.999 |

50 | 100 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

75 | 75 | 0.995 | 0.998 | 0.995 | 0.997 | 0.999 | 0.998 | 1 | 1 | 1 |

100 | 100 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

200 | 200 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

500 | 500 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

**Table 6.**Power of the test ${H}_{0}:PP{V}_{1}=PP{V}_{2}$ and type I error of the test ${H}_{0}:NP{V}_{1}=NP{V}_{2}$ when $PP{V}_{1}=0.75$, $NP{V}_{1}=0.95$, $PP{V}_{2}=0.60$ and $NP{V}_{2}=0.95$.

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.5357}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.9802}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.5455}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.9596}\text{}\mathbf{,}\text{}\mathit{p}=\mathbf{10}\mathbf{\%}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{1}}\le \mathbf{0.9805}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{2}}\le \mathbf{0.6933}$ | |||||||

${\rho}_{1}=0.25{\rho}_{2}=0.17$ | ${\rho}_{1}=0.49{\rho}_{2}=0.35$ | ${\rho}_{1}=0.74{\rho}_{2}=0.52$ | |||||

${n}_{1}$ | ${n}_{2}$ | Power | Type I error | Power | Type I error | Power | Type I error |

50 | 50 | 0.001 | 0.030 | 0.001 | 0.024 | 0.001 | 0.007 |

50 | 75 | 0.007 | 0.029 | 0.004 | 0.027 | 0.002 | 0.001 |

50 | 100 | 0.021 | 0.029 | 0.013 | 0.027 | 0.008 | 0.009 |

75 | 75 | 0.008 | 0.031 | 0.005 | 0.030 | 0.003 | 0.018 |

100 | 100 | 0.026 | 0.026 | 0.017 | 0.026 | 0.010 | 0.021 |

200 | 200 | 0.160 | 0.024 | 0.159 | 0.028 | 0.119 | 0.024 |

500 | 500 | 0.449 | 0.026 | 0.544 | 0.023 | 0.672 | 0.020 |

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.8571}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.9048}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.8727}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.8061}\text{}\mathbf{,}\text{}\mathit{p}=\mathbf{25}\mathbf{\%}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{1}}\le \mathbf{0.9354}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{2}}\le \mathbf{0.6614}$ | |||||||

${\rho}_{1}=0.23{\rho}_{2}=0.17$ | ${\rho}_{1}=0.47{\rho}_{2}=0.33$ | ${\rho}_{1}=0.70{\rho}_{2}=0.50$ | |||||

${n}_{1}$ | ${n}_{2}$ | Power | Type I error | Power | Type I error | Power | Type I error |

50 | 50 | 0.211 | 0.025 | 0.223 | 0.011 | 0.210 | 0.002 |

50 | 75 | 0.346 | 0.021 | 0.420 | 0.008 | 0.470 | 0.002 |

50 | 100 | 0.448 | 0.026 | 0.551 | 0.010 | 0.675 | 0.001 |

75 | 75 | 0.352 | 0.030 | 0.435 | 0.020 | 0.479 | 0.005 |

100 | 100 | 0.472 | 0.027 | 0.559 | 0.025 | 0.683 | 0.010 |

200 | 200 | 0.785 | 0.023 | 0.870 | 0.027 | 0.961 | 0.019 |

500 | 500 | 0.996 | 0.024 | 0.999 | 0.026 | 1 | 0.027 |

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.9643}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.6786}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.9818}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.3455}\text{}\mathbf{,}\text{}\mathit{p}=\mathbf{50}\mathbf{\%}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{1}}\le \mathbf{0.7071}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{2}}\le \mathbf{0.50}$ | |||||||

${\rho}_{1}=0.18{\rho}_{2}=0.13$ | ${\rho}_{1}=0.35{\rho}_{2}=0.25$ | ${\rho}_{1}=0.53{\rho}_{2}=0.38$ | |||||

${n}_{1}$ | ${n}_{2}$ | Power | Type I error | Power | Type I error | Power | Type I error |

50 | 50 | 0.893 | 0.003 | 0.941 | 0.002 | 0.980 | 0.001 |

50 | 75 | 0.977 | 0.003 | 0.991 | 0.002 | 0.999 | 0.001 |

50 | 100 | 0.990 | 0.002 | 0.999 | 0.002 | 1 | 0.001 |

75 | 75 | 0.983 | 0.005 | 0.993 | 0.002 | 1 | 0.003 |

100 | 100 | 0.997 | 0.005 | 1 | 0.004 | 1 | 0.002 |

200 | 200 | 1 | 0.012 | 1 | 0.006 | 1 | 0.007 |

500 | 500 | 1 | 0.024 | 1 | 0.024 | 1 | 0.019 |

**Table 7.**Power of the test ${H}_{0}:PP{V}_{1}=PP{V}_{2}$ and type I error of the test ${H}_{0}:NP{V}_{1}=NP{V}_{2}$ $PP{V}_{1}=0.95$, $NP{V}_{1}=0.95$, $PP{V}_{2}=0.75$ and $NP{V}_{2}=0.95$.

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.5278}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.9969}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.5357}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.9802}\text{}\mathbf{,}\text{}\mathit{p}=\mathbf{10}\mathbf{\%}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{1}}\le \mathbf{0.9841}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{2}}\le \mathbf{0.3910}$ | |||||||

${\rho}_{1}=0.25{\rho}_{2}=0.10$ | ${\rho}_{1}=0.49{\rho}_{2}=0.35$ | ${\rho}_{1}=0.74{\rho}_{2}=0.29$ | |||||

${n}_{1}$ | ${n}_{2}$ | Power | Type I error | Power | Type I error | Power | Type I error |

50 | 50 | 0.001 | 0.030 | 0.001 | 0.024 | 0 | 0.006 |

50 | 75 | 0.002 | 0.032 | 0.002 | 0.027 | 0.001 | 0.008 |

50 | 100 | 0.005 | 0.031 | 0.003 | 0.027 | 0.002 | 0.008 |

75 | 75 | 0.002 | 0.032 | 0.001 | 0.026 | 0.001 | 0.017 |

100 | 100 | 0.010 | 0.030 | 0.006 | 0.022 | 0.005 | 0.023 |

200 | 200 | 0.071 | 0.027 | 0.068 | 0.028 | 0.056 | 0.022 |

500 | 500 | 0.654 | 0.025 | 0.678 | 0.028 | 0.693 | 0.025 |

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.8444}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.9852}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.8571}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.9048}\text{}\mathbf{,}\text{}\mathit{p}=25\mathbf{\%}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{1}}\le \mathbf{0.9511}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{2}}\le \mathbf{0.3779}$ | |||||||

${\rho}_{1}=0.24{\rho}_{2}=0.09$ | ${\rho}_{1}=0.48{\rho}_{2}=0.19$ | ${\rho}_{1}=0.71{\rho}_{2}=0.28$ | |||||

${n}_{1}$ | ${n}_{2}$ | Power | Type I error | Power | Type I error | Power | Type I error |

50 | 50 | 0.120 | 0.025 | 0.118 | 0.012 | 0.097 | 0.001 |

50 | 75 | 0.378 | 0.028 | 0.371 | 0.012 | 0.376 | 0.001 |

50 | 100 | 0.604 | 0.026 | 0.634 | 0.012 | 0.652 | 0.001 |

75 | 75 | 0.382 | 0.027 | 0.396 | 0.022 | 0.388 | 0.005 |

100 | 100 | 0.622 | 0.031 | 0.644 | 0.026 | 0.679 | 0.012 |

200 | 200 | 0.963 | 0.025 | 0.974 | 0.024 | 0.987 | 0.026 |

500 | 500 | 1 | 0.028 | 1 | 0.024 | 1 | 0.024 |

$\mathit{S}{\mathit{e}}_{\mathbf{1}}=\mathbf{0.95}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{1}}=\mathbf{0.95}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{e}}_{\mathbf{2}}=\mathbf{0.9643}\text{}\mathbf{,}\text{}\mathit{S}{\mathit{p}}_{\mathbf{2}}=\mathbf{0.6786}\text{}\mathbf{,}\text{}\mathit{p}=\mathbf{50}\mathbf{\%}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{1}}\le \mathbf{0.8388}\text{}\mathbf{,}\text{}\mathbf{0}\le {\mathit{\rho}}_{\mathbf{2}}\le \mathbf{0.3333}$ | |||||||

${\rho}_{1}=0.21{\rho}_{2}=0.08$ | ${\rho}_{1}=0.42{\rho}_{2}=0.17$ | ${\rho}_{1}=0.63{\rho}_{2}=0.25$ | |||||

${n}_{1}$ | ${n}_{2}$ | Power | Type I error | Power | Type I error | Power | Type I error |

50 | 50 | 0.942 | 0.002 | 0.965 | 0.001 | 0.978 | 0 |

50 | 75 | 0.995 | 0.003 | 0.999 | 0 | 0.997 | 0 |

50 | 100 | 1 | 0.002 | 1 | 0.002 | 1 | 0 |

75 | 75 | 0.996 | 0.006 | 0.997 | 0.003 | 0.999 | 0 |

100 | 100 | 1 | 0.010 | 1 | 0.006 | 1 | 0.002 |

200 | 200 | 1 | 0.029 | 1 | 0.015 | 1 | 0.010 |

500 | 500 | 1 | 0.026 | 1 | 0.024 | 1 | 0.024 |

Type I errors | |||||||||||

$\begin{array}{c}PP{V}_{1}=PP{V}_{2}=0.90\text{},\text{}NP{V}_{1}=NP{V}_{2}=0.80\\ S{e}_{1}=0.2571\text{},\text{}S{p}_{1}=0.9905\text{},\text{}S{e}_{2}=0.2571\text{},\text{}S{p}_{2}=0.9905\text{},\text{}p=25\%\text{},\text{}{\rho}_{1}=0.75\text{},\text{}{\rho}_{2}=0.75\end{array}$ | |||||||||||

$p{}^{\prime}=p=25\%$ | $p{}^{\prime}=20\%$ | $p{}^{\prime}=22.50\%$ | $p{}^{\prime}=27.50\%$ | $p{}^{\prime}=30\%$ | |||||||

${n}_{1}$ | ${n}_{2}$ | Global | Bonf. | Global | Bonf. | Global | Bonf. | Global | Bonf. | Global | Bonf. |

50 | 50 | 0.001 | 0.001 | 0.001 | 0.002 | 0.001 | 0.002 | 0.001 | 0.001 | 0.001 | 0.001 |

50 | 75 | 0.001 | 0.002 | 0.001 | 0.002 | 0.001 | 0.002 | 0.001 | 0.002 | 0.001 | 0.002 |

50 | 100 | 0.002 | 0.003 | 0.002 | 0.003 | 0.002 | 0.003 | 0.002 | 0.003 | 0.002 | 0.003 |

75 | 75 | 0.003 | 0.007 | 0.003 | 0.008 | 0.003 | 0.008 | 0.003 | 0.007 | 0.003 | 0.007 |

100 | 100 | 0.006 | 0.010 | 0.006 | 0.011 | 0.006 | 0.011 | 0.006 | 0.009 | 0.006 | 0.009 |

200 | 200 | 0.010 | 0.020 | 0.010 | 0.020 | 0.010 | 0.020 | 0.010 | 0.019 | 0.010 | 0.019 |

500 | 500 | 0.020 | 0.024 | 0.020 | 0.024 | 0.020 | 0.024 | 0.020 | 0.023 | 0.020 | 0.023 |

Powers | |||||||||||

$\begin{array}{c}PP{V}_{1}=0.95\text{},\text{}PP{V}_{2}=0.75\text{},\text{}NP{V}_{1}=0.95\text{},\text{}NP{V}_{2}=0.95\\ S{e}_{1}=0.8444\text{},\text{}S{p}_{1}=0.9852\text{},\text{}S{e}_{2}=0.8571\text{},\text{}S{p}_{2}=0.9048\text{},\text{}p=25\%\text{},\text{}{\rho}_{1}=0.71\text{},\text{}{\rho}_{2}=0.28\end{array}$ | |||||||||||

$p{}^{\prime}=p=25\%$ | $p{}^{\prime}=20\%$ | $p{}^{\prime}=22.50\%$ | $p{}^{\prime}=27.50\%$ | $p{}^{\prime}=30\%$ | |||||||

${n}_{1}$ | ${n}_{2}$ | Global | Bonf. | Global | Bonf. | Global | Bonf. | Global | Bonf. | Global | Bonf. |

50 | 50 | 0.172 | 0.142 | 0.145 | 0.139 | 0.143 | 0.135 | 0.142 | 0.121 | 0.139 | 0.117 |

50 | 75 | 0.422 | 0.396 | 0.381 | 0.382 | 0.385 | 0.382 | 0.374 | 0.380 | 0.375 | 0.379 |

50 | 100 | 0.647 | 0.615 | 0.600 | 0.587 | 0.627 | 0.604 | 0.616 | 0.614 | 0.598 | 0.596 |

75 | 75 | 0.434 | 0.400 | 0.414 | 0.396 | 0.415 | 0.396 | 0.397 | 0.394 | 0.390 | 0.390 |

100 | 100 | 0.635 | 0.634 | 0.618 | 0.604 | 0.632 | 0.612 | 0.639 | 0.620 | 0.623 | 0.615 |

200 | 200 | 0.965 | 0.964 | 0.941 | 0.931 | 0.951 | 0.952 | 0.958 | 0.956 | 0.942 | 0.933 |

500 | 500 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

RRMSEs of the estimators of PVs of Test 1 | |||||||||||

$\begin{array}{c}PP{V}_{1}=PP{V}_{2}=0.90\text{},\text{}NP{V}_{1}=NP{V}_{2}=0.80\\ S{e}_{1}=0.2571\text{},\text{}S{p}_{1}=0.9905\text{},\text{}S{e}_{2}=0.2571\text{},\text{}S{p}_{2}=0.9905\text{},\text{}p=25\%\text{},\text{}{\rho}_{1}=0.75\text{},\text{}{\rho}_{2}=0.75\end{array}$ | |||||||||||

$p{}^{\prime}=p=25\%$ | $p{}^{\prime}=20\%$ | $p{}^{\prime}=22.50\%$ | $p{}^{\prime}=27.50\%$ | $p{}^{\prime}=30\%$ | |||||||

${n}_{1}$ | ${n}_{2}$ | ${\widehat{PPV}}_{i}$ | ${\widehat{NPV}}_{i}$ | ${\widehat{PPV}}_{i}$ | ${\widehat{NPV}}_{i}$ | ${\widehat{PPV}}_{i}$ | ${\widehat{NPV}}_{i}$ | ${\widehat{PPV}}_{i}$ | ${\widehat{NPV}}_{i}$ | ${\widehat{PPV}}_{i}$ | ${\widehat{NPV}}_{i}$ |

50 | 50 | 27.4 | 1.8 | 36.6 | 5.5 | 31.8 | 2.6 | 29.3 | 3.8 | 34.5 | 6.4 |

50 | 75 | 20.5 | 1.7 | 27.0 | 5.1 | 23.5 | 2.8 | 21.9 | 3.6 | 25.5 | 6.1 |

50 | 100 | 16.1 | 1.7 | 21.9 | 5.2 | 18.8 | 2.9 | 17.8 | 3.5 | 21.1 | 6.0 |

75 | 75 | 20.1 | 1.4 | 26.5 | 5.1 | 23.1 | 2.6 | 21.4 | 3.5 | 24.1 | 6.1 |

100 | 100 | 15.4 | 1.2 | 19.2 | 4.9 | 18.1 | 2.5 | 16.1 | 3.3 | 18.6 | 5.9 |

200 | 200 | 8.0 | 0.9 | 11.7 | 4.7 | 9.9 | 2.4 | 8.7 | 3.0 | 10.2 | 5.6 |

500 | 500 | 4.4 | 0.5 | 7.1 | 4.1 | 5.4 | 2.2 | 4.9 | 2.8 | 5.7 | 5.5 |

Observed Frequencies | |||||||
---|---|---|---|---|---|---|---|

Case | Control | ||||||

${T}_{2}=1$ | ${T}_{2}=0$ | Total | ${T}_{2}=1$ | ${T}_{2}=0$ | Total | ||

${T}_{1}=1$ | 77 | 10 | 87 | ${T}_{1}=1$ | 4 | 2 | 6 |

${T}_{1}=0$ | 6 | 12 | 18 | ${T}_{1}=0$ | 13 | 101 | 114 |

Total | 83 | 22 | 105 | Total | 17 | 103 | 120 |

**Table 10.**Type I errors and powers when 0.5 is added to the samples in which ${n}_{i10}={n}_{i01}=0$.

Type I Errors | ||||||||||||

$\begin{array}{c}PP{V}_{1}=PP{V}_{2}=0.70\text{},\text{}NP{V}_{1}=NP{V}_{2}=0.95\\ S{e}_{1}=0.5385\text{},\text{}S{p}_{1}=0.9744\text{},\text{}S{e}_{2}=0.5385\text{},\text{}S{p}_{2}=0.9744\text{},\text{}p=10\%\end{array}$ | ||||||||||||

${\rho}_{1}=0.25{\rho}_{2}=0.25$ | ${\rho}_{1}=0.50{\rho}_{2}=0.50$ | ${\rho}_{1}=0.75{\rho}_{2}=0.75$ | ||||||||||

${n}_{1}$ | ${n}_{2}$ | ${\overline{P}}_{1}$ | ${\overline{P}}_{2}$ | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. |

50 | 50 | 3.3 | 78.1 | 0.028 | 0.065 | 0.030 | 0.023 | 0.056 | 0.025 | 0.012 | 0.040 | 0.015 |

50 | 75 | 3.1 | 63.7 | 0.038 | 0.075 | 0.039 | 0.026 | 0.060 | 0.026 | 0.012 | 0.041 | 0.014 |

50 | 100 | 2.7 | 51.9 | 0.047 | 0.096 | 0.046 | 0.035 | 0.074 | 0.037 | 0.011 | 0.041 | 0.015 |

75 | 75 | 0.4 | 64.3 | 0.037 | 0.074 | 0.037 | 0.026 | 0.062 | 0.027 | 0.015 | 0.045 | 0.019 |

100 | 100 | 0.1 | 51.8 | 0.040 | 0.085 | 0.038 | 0.032 | 0.090 | 0.034 | 0.021 | 0.048 | 0.023 |

200 | 200 | 0 | 22.7 | 0.061 | 0.104 | 0.060 | 0.045 | 0.088 | 0.042 | 0.024 | 0.055 | 0.025 |

500 | 500 | 0 | 2.8 | 0.052 | 0.099 | 0.0524 | 0.056 | 0.101 | 0.048 | 0.044 | 0.094 | 0.045 |

Powers | ||||||||||||

$\begin{array}{c}PP{V}_{1}=0.75\text{},\text{}PP{V}_{2}=0.60\text{},\text{}NP{V}_{1}=0.95\text{},\text{}NP{V}_{2}=0.95\\ S{e}_{1}=0.8571\text{},\text{}S{p}_{1}=0.9048\text{},\text{}S{e}_{2}=0.8727\text{},\text{}S{p}_{2}=0.8061\text{},\text{}p=25\%\text{},\text{}0\le {\rho}_{1}\le 0.9354\text{},\text{}0\le {\rho}_{2}\le 0.6614\end{array}$ | ||||||||||||

${\rho}_{1}=0.25{\rho}_{2}=0.25$ | ${\rho}_{1}=0.50{\rho}_{2}=0.50$ | ${\rho}_{1}=0.75{\rho}_{2}=0.75$ | ||||||||||

${n}_{1}$ | ${n}_{2}$ | ${\overline{P}}_{1}$ | ${\overline{P}}_{2}$ | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. | Global | $\alpha =5\%$ | Bonf. |

50 | 50 | 14.8 | 18.1 | 0.297 | 0.369 | 0.269 | 0.342 | 0.427 | 0.306 | 0.369 | 0.484 | 0.340 |

50 | 75 | 14.5 | 9.6 | 0.417 | 0.501 | 0.379 | 0.492 | 0.576 | 0.449 | 0.604 | 0.688 | 0.568 |

50 | 100 | 14.2 | 5.5 | 0.515 | 0.590 | 0.476 | 0.629 | 0.693 | 0.576 | 0.739 | 0.798 | 0.704 |

75 | 75 | 5.8 | 9.7 | 0.424 | 0.506 | 0.392 | 0.502 | 0.584 | 0.456 | 0.622 | 0.705 | 0.579 |

100 | 100 | 2.5 | 5.5 | 0.525 | 0.603 | 0.486 | 0.624 | 0.693 | 0.583 | 0.761 | 0.819 | 0.728 |

200 | 200 | 0.1 | 0.7 | 0.817 | 0.863 | 0.786 | 0.910 | 0.930 | 0.876 | 0.975 | 0.983 | 0.965 |

500 | 500 | 0 | 0 | 0.996 | 0.999 | 0.997 | 1 | 1 | 1 | 1 | 1 | 1 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Regad, S.B.; Roldán-Nofuentes, J.A.
Global Hypothesis Test to Compare the Predictive Values of Diagnostic Tests Subject to a Case-Control Design. *Mathematics* **2021**, *9*, 658.
https://doi.org/10.3390/math9060658

**AMA Style**

Regad SB, Roldán-Nofuentes JA.
Global Hypothesis Test to Compare the Predictive Values of Diagnostic Tests Subject to a Case-Control Design. *Mathematics*. 2021; 9(6):658.
https://doi.org/10.3390/math9060658

**Chicago/Turabian Style**

Regad, Saad Bouh, and José Antonio Roldán-Nofuentes.
2021. "Global Hypothesis Test to Compare the Predictive Values of Diagnostic Tests Subject to a Case-Control Design" *Mathematics* 9, no. 6: 658.
https://doi.org/10.3390/math9060658