# Computation of the Mann–Whitney Effect under Parametric Survival Copula Models

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## Abstract

**:**

## 1. Introduction

## 2. Proposed Method

#### 2.1. Survival Copula Models for Dependent Survival Time

- 1.
- The independence copula:$$C(u,v)=uv.$$
- 2.
- The Clayton copula [28]:$${C}_{\theta}(u,v)=max\left({({u}^{-\theta}+{v}^{-\theta}-1)}^{-1/\theta},0\right),\phantom{\rule{1.em}{0ex}}\theta \in [-1,\infty )\setminus \left\{0\right\}.$$
- 3.
- The Gumbel copula [29]:$${C}_{\theta}(u,v)=exp\left\{-{[{(-logu)}^{\theta +1}+{(-logv)}^{\theta +1}]}^{\frac{1}{\theta +1}}\right\},\phantom{\rule{1.em}{0ex}}\theta \in [0,\infty ).$$
- 4.
- The Frank copula [30]:$${C}_{\theta}(u,v)=-\frac{1}{\theta}log\left(1+\frac{({e}^{-\theta u}-1)({e}^{-\theta v}-1)}{{e}^{-\theta}-1}\right),\phantom{\rule{1.em}{0ex}}\theta \in (-\infty ,\infty )\setminus \left\{0\right\}.$$
- 5.
- The Farlie–Gumbel–Morgenstern (FGM) copula [31]:$${C}_{\theta}(u,v)=uv+\theta uv(1-u)(1-v),\phantom{\rule{1.em}{0ex}}\theta \in [-1,1].$$
- 6.

#### 2.2. Proposed Method for Computing p

#### 2.3. Computing p with Follow-Up Time

**Theorem**

**1.**

#### 2.4. Marginal Survival Distributions

- 1.
- The exponential distribution:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {S}_{j}\left(t\right)=exp(-{\lambda}_{j}t),\phantom{\rule{1.em}{0ex}}{\lambda}_{j}>0,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {S}_{i}\left({S}_{j}^{-1}\left(v\right)\right)={v}^{\frac{{\lambda}_{i}}{{\lambda}_{j}}},\hfill \end{array}$$
- 2.
- The Weibull distribution:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {S}_{j}\left(t\right)=exp(-{\lambda}_{j}{t}^{{k}_{j}}),\phantom{\rule{1.em}{0ex}}{\lambda}_{j}>0,{k}_{j}>0,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {S}_{i}\left({S}_{j}^{-1}\left(v\right)\right)=exp\left(-{\lambda}_{i}{\left(-\frac{logv}{{\lambda}_{j}}\right)}^{\frac{{k}_{i}}{{k}_{j}}}\right),\hfill \end{array}$$
- 3.
- The gamma distribution:$${S}_{j}\left(t\right)=1-\frac{\gamma ({k}_{j},{\lambda}_{j}t)}{\mathsf{\Gamma}\left({k}_{j}\right)},\phantom{\rule{1.em}{0ex}}{\lambda}_{j}>0,{k}_{j}>0,$$
- 4.
- The log-normal distribution:$${S}_{j}\left(t\right)=\frac{1}{\sqrt{2\pi {\sigma}_{j}^{2}}}{\int}_{t}^{\infty}\frac{1}{y}exp\left\{-\frac{1}{2{\sigma}_{j}^{2}}{(logy-{\mu}_{j})}^{2}\right\}dy,$$
- 5.
- The Burr III distribution:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {S}_{j}\left(t\right)=1-{(1+{t}^{-{c}_{j}})}^{-{k}_{j}},\phantom{\rule{1.em}{0ex}}{c}_{j}>0,{k}_{j}>0,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {S}_{i}\left({S}_{j}^{-1}\left(v\right)\right)=1-{\left(1+{\left({(1-v)}^{-\frac{1}{{k}_{j}}}-1\right)}^{\frac{{c}_{i}}{{c}_{j}}}\right)}^{-{k}_{i}},\hfill \end{array}$$

**Example**

**1.**

**Example**

**2.**

#### 2.5. Sensitivity Analysis by Copulas

## 3. Software and Web App

#### 3.1. Input

#### 3.2. Output

#### 3.3. Example of Using the App

## 4. Simulation Studies

## 5. Data Analysis

#### 5.1. Tongue Cancer Data

#### 5.2. Prostate Cancer Data

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Kendall’s τ

- 1.
- The independence copula:$$\mathrm{Kendall}\u2019\mathrm{s}\phantom{\rule{4pt}{0ex}}\tau =0.$$
- 2.
- The Clayton copula:$$\mathrm{Kendall}\u2019\mathrm{s}\phantom{\rule{4pt}{0ex}}\tau =\frac{\theta}{\theta +2},\phantom{\rule{1.em}{0ex}}\theta \in [-1,\infty )\setminus \left\{0\right\}.$$
- 3.
- The Gumbel copula:$$\mathrm{Kendall}\u2019\mathrm{s}\phantom{\rule{4pt}{0ex}}\tau =\frac{\theta}{\theta +1},\phantom{\rule{1.em}{0ex}}\theta \in [0,\infty ).$$
- 4.
- The Frank copula:$$\mathrm{Kendall}\u2019\mathrm{s}\phantom{\rule{4pt}{0ex}}\tau =1-\frac{4}{\theta}+\frac{4{D}_{\theta}}{\theta},\phantom{\rule{1.em}{0ex}}\mathrm{where}\phantom{\rule{4pt}{0ex}}{D}_{\theta}=\frac{1}{\theta}{\int}_{0}^{\theta}\frac{x}{exp\left(x\right)-1}dx,\phantom{\rule{1.em}{0ex}}\theta \in (-\infty ,\infty )\setminus \left\{0\right\}.$$
- 5.
- The FGM copula:$$\mathrm{Kendall}\u2019\mathrm{s}\phantom{\rule{4pt}{0ex}}\tau =\frac{2}{9}\theta ,\phantom{\rule{1.em}{0ex}}\theta \in [-1,1].$$
- 6.
- The GB copula:$$\mathrm{Kendall}\u2019\mathrm{s}\phantom{\rule{4pt}{0ex}}\tau =1-\frac{4}{\theta}{\int}_{0}^{1}t(1-\theta logt)log(1-\theta logt)dt,\phantom{\rule{1.em}{0ex}}\theta \in [0,1].$$

## Appendix B. Examples of p and p_{τ} with Different Copulas

- 1.
- The Clayton copula:$$p={\int}_{0}^{1}{v}^{-\theta -1}{\left({\left\{{S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)\right\}}^{-\theta}+{v}^{-\theta}-1\right)}^{-\frac{1}{\theta}-1}dv.$$
- 2.
- The Gumbel copula:$$\begin{array}{cc}\hfill p=& {\int}_{0}^{1}\left\{exp\left\{-{[{(-log{S}_{1}\left({S}_{2}^{-1}\left(v\right)\right))}^{\theta +1}+{(-logv)}^{\theta +1}]}^{\frac{1}{\theta +1}}\right\}\right.\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times \left.{\left[{\left(-log\left({S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)\right)\right)}^{\theta +1}+{\left(-log\left(v\right)\right)}^{\theta +1}\right]}^{-\frac{\theta}{\theta +1}}\frac{{(-log\left(v\right))}^{\theta}}{v}\right\}dv.\hfill \end{array}$$
- 3.
- The Frank copula:$$p={\int}_{0}^{1}\frac{{e}^{-\theta v}\left({e}^{-\theta {S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)}-1\right)}{\left({e}^{-\theta}-1\right)+\left({e}^{-\theta {S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)}-1\right)\left({e}^{-\theta v}-1\right)}dv.$$
- 4.
- The FGM copula:$$p={\int}_{0}^{1}\left\{{S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)+\theta {S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)\left(1-{S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)\right)(1-2v)\right\}dv.$$
- 5.
- The GB copula:$$p={\int}_{0}^{1}\left\{{S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)\left(1-\theta log{S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)\right){v}^{-\theta log{S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)}\right\}dv.$$

- 1.
- The Clayton copula:$${p}_{\tau}={\int}_{{S}_{2}\left(\tau \right)}^{1}\left\{{v}^{-\theta -1}{\left({\left\{{S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)\right\}}^{-\theta}+{v}^{-\theta}-1\right)}^{-\frac{1}{\theta}-1}\right\}dv+\frac{1}{2}{({S}_{1}{\left(\tau \right)}^{-\theta}+{S}_{2}{\left(\tau \right)}^{-\theta}-1)}^{-1/\theta}.$$
- 2.
- The Gumbel copula:$$\begin{array}{cc}\hfill {p}_{\tau}=& {\int}_{{S}_{2}\left(\tau \right)}^{1}\left\{exp\left\{-{[{(-log{S}_{1}\left({S}_{2}^{-1}\left(v\right)\right))}^{\theta +1}+{(-logv)}^{\theta +1}]}^{\frac{1}{\theta +1}}\right\}\right.\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \left.\times {\left[{\left(-log\left({S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)\right)\right)}^{\theta +1}+{\left(-log\left(v\right)\right)}^{\theta +1}\right]}^{-\frac{\theta}{\theta +1}}\frac{{(-log\left(v\right))}^{\theta}}{v}\right\}dv\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& +\frac{1}{2}exp\left\{-{[{(-log{S}_{1}\left(\tau \right))}^{\theta +1}+{(-log{S}_{2}\left(\tau \right))}^{\theta +1}]}^{\frac{1}{\theta +1}}\right\}.\hfill \end{array}$$
- 3.
- The Frank copula:$$\begin{array}{cc}\hfill {p}_{\tau}=& {\int}_{{S}_{2}\left(\tau \right)}^{1}\frac{{e}^{-\theta v}\left({e}^{-\theta {S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)}-1\right)}{\left({e}^{-\theta}-1\right)+\left({e}^{-\theta {S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)}-1\right)\left({e}^{-\theta v}-1\right)}dv\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& +\frac{1}{2}\left(-\frac{1}{\theta}log\left(1+\frac{({e}^{-\theta {S}_{1}\left(\tau \right)}-1)({e}^{-\theta {S}_{2}\left(\tau \right)}-1)}{{e}^{-\theta}-1}\right)\right).\hfill \end{array}$$
- 4.
- The FGM copula:$$\begin{array}{cc}\hfill {p}_{\tau}=& {\int}_{{S}_{2}\left(\tau \right)}^{1}\left\{{S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)+\theta {S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)\left(1-{S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)\right)(1-2v)\right\}dv\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& +\frac{1}{2}\left({S}_{1}\left(\tau \right){S}_{2}\left(\tau \right)+\theta {S}_{1}\left(\tau \right){S}_{2}\left(\tau \right)(1-{S}_{1}\left(\tau \right))(1-{S}_{2}\left(\tau \right))\right).\hfill \end{array}$$
- 5.
- The GB copula:$$\begin{array}{cc}\hfill {p}_{\tau}=& {\int}_{{S}_{2}\left(\tau \right)}^{1}\left\{{S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)\left(1-\theta log{S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)\right){v}^{-\theta log{S}_{1}\left({S}_{2}^{-1}\left(v\right)\right)}\right\}dv\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& +\frac{1}{2}\left({S}_{1}\left(\tau \right){S}_{2}\left(\tau \right)exp(-\theta log{S}_{1}\left(\tau \right)log{S}_{2}\left(\tau \right))\right).\hfill \end{array}$$

## Appendix C. Examples of Using the Shiny Web App

**Figure A1.**The web app showing the results for computing p and ${p}_{\tau}$ under the Burr III distributions.

**Example**

**A1.**

**Figure A2.**The web app showing the results for computing p and ${p}_{\tau}$ under the log-normal distribution.

**Example**

**A2.**

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**Figure 1.**Scatter plots of 3000 data points generated from the copula distribution with parameter $\theta $.

**Figure 3.**The web app showing the results for computing p and ${p}_{\tau}$ under the exponential distribution.

**Figure 4.**KM estimators for the DNA-aneuploid tumor and the DNA-diploid tumor group and the exponential survival curves with the MLEs of exponential hazard rate (darker blue and red lines), ${\widehat{\lambda}}_{1}=0.00736,\text{}{\widehat{\lambda}}_{2}=0.0130$. The vertical line signifies the follow-up time $\tau =167$.

**Figure 5.**Example for the tongue cancer dataset on the app. This setting is marginal distribution: “Exponential”, ${\lambda}_{1}=0.00736,\text{}{\lambda}_{2}=0.0130,\tau =167$, copula: “Clayton”, copula parameter: $\theta =1$, and language: “English”.

**Figure 6.**KM estimators for the moderately differentiated and the poorly differentiated group and exponential survival curves with the MLEs of exponential hazard rate (darker blue and red lines), ${\widehat{\lambda}}_{1}=0.000817,\text{}{\widehat{\lambda}}_{2}=0.00374$. The vertical line signifies the follow-up time $\tau =108$.

**Figure 7.**Example for the tongue cancer dataset on the app. This setting is marginal distribution: “Exponential”, ${\lambda}_{1}=0.000817,\text{}{\lambda}_{2}=0.00374,\tau =108$, copula: “Gumbel”, copula parameter: $\theta =4$, and language: “English”.

Copula | $\mathit{\theta}$ | Kendall’s $\mathit{\tau}$ |
---|---|---|

Clayton | 1.0 | 0.33 |

5.0 | 0.71 | |

10.0 | 0.83 | |

Gumbel | 0.0 | 0.00 |

4.0 | 0.80 | |

Frank | −20.0 | −0.82 |

−5.0 | −0.46 | |

1.0 | 0.11 | |

5.0 | 0.46 | |

FGM | −1.0 | −0.22 |

0.0 | 0.00 | |

1.0 | 0.22 | |

GB | 0.5 | −0.21 |

1.0 | −0.36 |

**Table 2.**Comparison of the theoretical value and the simulation value (SE < 0.002) for calculating ${p}_{\tau}$ defined in Theorem 1. (The exponential, Weibull, gamma, log-normal, and Burr III distributions).

$\mathit{\tau}=0.5$ | $\mathit{\tau}=2$ | $\mathit{\tau}=5$ | $\mathit{\tau}=\mathit{\infty}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Distribution | Copula | $\mathit{\theta}$ | $\mathbf{Kendall}\u2019\mathbf{s}\phantom{\rule{4pt}{0ex}}\mathit{\tau}$ | ${\mathit{p}}_{\mathit{\tau},\mathbf{theory}}$ | ${\mathit{p}}_{\mathit{\tau},\mathbf{sim}}$ | ${\mathit{p}}_{\mathit{\tau},\mathbf{theory}}$ | ${\mathit{p}}_{\mathit{\tau},\mathbf{sim}}$ | ${\mathit{p}}_{\mathit{\tau},\mathbf{theory}}$ | ${\mathit{p}}_{\mathit{\tau},\mathbf{sim}}$ | ${\mathit{p}}_{\mathit{\tau},\mathrm{theory}}$ | ${\mathit{p}}_{\mathit{\tau},\mathrm{sim}}$ |

Exponential | Clayton | 1.0 | 0.33 | 0.645 | 0.643 | 0.737 | 0.738 | 0.744 | 0.746 | 0.744 | 0.746 |

5.0 | 0.71 | 0.704 | 0.706 | 0.872 | 0.872 | 0.881 | 0.883 | 0.881 | 0.883 | ||

10.0 | 0.83 | 0.746 | 0.745 | 0.920 | 0.921 | 0.930 | 0.930 | 0.930 | 0.929 | ||

Gumbel | 0.0 | 0.00 | 0.629 | 0.631 | 0.666 | 0.666 | 0.666 | 0.665 | 0.667 | 0.666 | |

4.0 | 0.80 | 0.798 | 0.799 | 0.961 | 0.961 | 0.970 | 0.969 | 0.970 | 0.968 | ||

Frank | −5.0 | −0.46 | 0.615 | 0.615 | 0.622 | 0.622 | 0.622 | 0.622 | 0.622 | 0.620 | |

1.0 | 0.11 | 0.636 | 0.636 | 0.684 | 0.684 | 0.685 | 0.685 | 0.685 | 0.685 | ||

5.0 | 0.46 | 0.674 | 0.674 | 0.771 | 0.768 | 0.773 | 0.772 | 0.773 | 0.775 | ||

FGM | −1.0 | −0.22 | 0.617 | 0.617 | 0.633 | 0.634 | 0.633 | 0.631 | 0.633 | 0.633 | |

0.0 | 0.00 | 0.629 | 0.629 | 0.666 | 0.666 | 0.666 | 0.666 | 0.666 | 0.664 | ||

1.0 | 0.22 | 0.642 | 0.641 | 0.699 | 0.697 | 0.700 | 0.702 | 0.700 | 0.698 | ||

GB | 0.5 | −0.21 | 0.623 | 0.624 | 0.642 | 0.643 | 0.642 | 0.641 | 0.642 | 0.640 | |

1.0 | −0.36 | 0.617 | 0.616 | 0.629 | 0.628 | 0.629 | 0.632 | 0.629 | 0.630 | ||

Weibull | Clayton | 1.0 | 0.33 | 0.497 | 0.496 | 0.594 | 0.589 | 0.603 | 0.602 | 0.603 | 0.602 |

5.0 | 0.71 | 0.482 | 0.480 | 0.644 | 0.645 | 0.653 | 0.654 | 0.653 | 0.650 | ||

10.0 | 0.83 | 0.472 | 0.472 | 0.645 | 0.645 | 0.654 | 0.653 | 0.654 | 0.652 | ||

Gumbel | 0.0 | 0.00 | 0.511 | 0.509 | 0.560 | 0.562 | 0.562 | 0.562 | 0.562 | 0.563 | |

4.0 | 0.80 | 0.425 | 0.424 | 0.584 | 0.585 | 0.593 | 0.594 | 0.593 | 0.592 | ||

Frank | −5.0 | −0.46 | 0.528 | 0.530 | 0.542 | 0.541 | 0.542 | 0.543 | 0.542 | 0.541 | |

1.0 | 0.11 | 0.505 | 0.504 | 0.566 | 0.565 | 0.569 | 0.572 | 0.569 | 0.568 | ||

5.0 | 0.46 | 0.486 | 0.486 | 0.592 | 0.595 | 0.597 | 0.597 | 0.597 | 0.594 | ||

FGM | −1.0 | −0.22 | 0.521 | 0.523 | 0.548 | 0.549 | 0.549 | 0.551 | 0.549 | 0.549 | |

0.0 | 0.00 | 0.511 | 0.509 | 0.560 | 0.563 | 0.562 | 0.562 | 0.562 | 0.564 | ||

1.0 | 0.22 | 0.501 | 0.503 | 0.572 | 0.574 | 0.575 | 0.577 | 0.575 | 0.574 | ||

GB | 0.5 | −0.21 | 0.519 | 0.520 | 0.549 | 0.546 | 0.549 | 0.548 | 0.549 | 0.553 | |

1.0 | −0.36 | 0.526 | 0.526 | 0.545 | 0.545 | 0.545 | 0.545 | 0.545 | 0.546 | ||

Gamma | Clayton | 1.0 | 0.33 | 0.529 | 0.529 | 0.651 | 0.649 | 0.679 | 0.678 | 0.679 | 0.680 |

5.0 | 0.71 | 0.530 | 0.528 | 0.763 | 0.763 | 0.809 | 0.810 | 0.809 | 0.810 | ||

10.0 | 0.83 | 0.534 | 0.533 | 0.817 | 0.816 | 0.862 | 0.863 | 0.863 | 0.863 | ||

Gumbel | 0.0 | 0.00 | 0.530 | 0.530 | 0.611 | 0.612 | 0.615 | 0.614 | 0.615 | 0.616 | |

4.0 | 0.80 | 0.545 | 0.546 | 0.813 | 0.813 | 0.853 | 0.853 | 0.854 | 0.853 | ||

Frank | −5.0 | −0.46 | 0.532 | 0.530 | 0.584 | 0.584 | 0.584 | 0.583 | 0.584 | 0.584 | |

1.0 | 0.11 | 0.530 | 0.529 | 0.622 | 0.622 | 0.628 | 0.628 | 0.628 | 0.628 | ||

5.0 | 0.46 | 0.530 | 0.530 | 0.679 | 0.679 | 0.694 | 0.692 | 0.694 | 0.697 | ||

FGM | −1.0 | −0.22 | 0.531 | 0.533 | 0.591 | 0.591 | 0.592 | 0.591 | 0.592 | 0.592 | |

0.0 | 0.00 | 0.530 | 0.529 | 0.611 | 0.610 | 0.615 | 0.614 | 0.615 | 0.617 | ||

1.0 | 0.22 | 0.529 | 0.529 | 0.631 | 0.631 | 0.639 | 0.640 | 0.639 | 0.641 | ||

GB | 0.5 | −0.21 | 0.531 | 0.532 | 0.597 | 0.598 | 0.598 | 0.598 | 0.598 | 0.597 | |

1.0 | −0.36 | 0.532 | 0.532 | 0.589 | 0.592 | 0.590 | 0.588 | 0.589 | 0.588 | ||

Log-normal | Clayton | 1.0 | 0.33 | 0.580 | 0.582 | 0.596 | 0.594 | 0.593 | 0.591 | 0.573 | 0.574 |

5.0 | 0.71 | 0.599 | 0.599 | 0.658 | 0.658 | 0.675 | 0.677 | 0.619 | 0.619 | ||

10.0 | 0.83 | 0.618 | 0.617 | 0.715 | 0.715 | 0.756 | 0.755 | 0.684 | 0.684 | ||

Gumbel | 0.0 | 0.00 | 0.574 | 0.575 | 0.576 | 0.578 | 0.569 | 0.570 | 0.564 | 0.562 | |

4.0 | 0.80 | 0.652 | 0.652 | 0.745 | 0.746 | 0.760 | 0.759 | 0.728 | 0.726 | ||

Frank | −5.0 | −0.46 | 0.567 | 0.567 | 0.553 | 0.554 | 0.547 | 0.547 | 0.547 | 0.544 | |

1.0 | 0.11 | 0.577 | 0.576 | 0.584 | 0.586 | 0.578 | 0.580 | 0.571 | 0.570 | ||

5.0 | 0.46 | 0.593 | 0.593 | 0.624 | 0.625 | 0.623 | 0.621 | 0.609 | 0.607 | ||

FGM | −1.0 | −0.22 | 0.568 | 0.567 | 0.560 | 0.561 | 0.553 | 0.553 | 0.550 | 0.551 | |

0.0 | 0.00 | 0.574 | 0.574 | 0.576 | 0.576 | 0.569 | 0.569 | 0.564 | 0.560 | ||

1.0 | 0.22 | 0.580 | 0.580 | 0.591 | 0.593 | 0.585 | 0.586 | 0.577 | 0.579 | ||

GB | 0.5 | −0.21 | 0.571 | 0.572 | 0.563 | 0.566 | 0.558 | 0.555 | 0.558 | 0.557 | |

1.0 | −0.36 | 0.568 | 0.567 | 0.557 | 0.557 | 0.550 | 0.554 | 0.549 | 0.550 | ||

Burr III | Clayton | 1.0 | 0.33 | 0.661 | 0.662 | 0.753 | 0.753 | 0.760 | 0.761 | 0.747 | 0.748 |

5.0 | 0.71 | 0.665 | 0.665 | 0.819 | 0.820 | 0.883 | 0.883 | 0.863 | 0.865 | ||

10.0 | 0.83 | 0.666 | 0.666 | 0.832 | 0.832 | 0.913 | 0.914 | 0.896 | 0.897 | ||

Gumbel | 0.0 | 0.00 | 0.660 | 0.661 | 0.714 | 0.715 | 0.701 | 0.701 | 0.697 | 0.696 | |

4.0 | 0.80 | 0.667 | 0.665 | 0.832 | 0.832 | 0.885 | 0.883 | 0.871 | 0.871 | ||

Frank | −5.0 | −0.46 | 0.658 | 0.658 | 0.663 | 0.662 | 0.650 | 0.649 | 0.650 | 0.653 | |

1.0 | 0.11 | 0.661 | 0.660 | 0.730 | 0.731 | 0.720 | 0.719 | 0.715 | 0.716 | ||

5.0 | 0.46 | 0.664 | 0.663 | 0.788 | 0.789 | 0.798 | 0.797 | 0.789 | 0.788 | ||

FGM | −1.0 | −0.22 | 0.658 | 0.658 | 0.683 | 0.683 | 0.666 | 0.666 | 0.665 | 0.664 | |

0.0 | 0.00 | 0.660 | 0.660 | 0.714 | 0.711 | 0.701 | 0.701 | 0.697 | 0.699 | ||

1.0 | 0.22 | 0.661 | 0.662 | 0.744 | 0.747 | 0.737 | 0.736 | 0.730 | 0.729 | ||

GB | 0.5 | −0.21 | 0.659 | 0.659 | 0.692 | 0.693 | 0.676 | 0.677 | 0.675 | 0.676 | |

1.0 | −0.36 | 0.658 | 0.658 | 0.673 | 0.671 | 0.659 | 0.659 | 0.659 | 0.659 |

**Table 3.**Estimates ${\widehat{p}}_{\tau}(\tau =167)$ for fitting the KM estimator (independent) and with exponential marginal survival distributions (the independent, Clayton, Gumbel, Frank, FGM, GB copulas) for the tongue cancer dataset.

Copula | Marginal Distribution | $\mathit{\theta}$ | $\widehat{\mathit{p}}$ | SE | p-Value | ${\widehat{\mathit{p}}}_{\mathit{\tau}}(\mathit{\tau}=167)$ | SE | p-Value |
---|---|---|---|---|---|---|---|---|

Independent | KM estimator | - | - | - | - | 0.624 | 0.071 | 0.079 |

Independent | exponential | - | 0.638 | 0.076 | 0.070 | 0.633 | 0.075 | 0.076 |

Clayton | exponential | 1.0 | 0.709 | 0.096 | 0.029 | 0.676 | 0.096 | 0.067 |

5.0 | 0.856 | 0.075 | <0.001 | 0.799 | 0.100 | 0.003 | ||

Gumbel | exponential | 4.0 | 0.944 | 0.084 | <0.001 | 0.895 | 0.095 | <0.001 |

Frank | exponential | −20.0 | 0.596 | 0.055 | 0.080 | 0.596 | 0.055 | 0.080 |

−5.0 | 0.600 | 0.057 | 0.081 | 0.600 | 0.057 | 0.081 | ||

5.0 | 0.733 | 0.111 | 0.036 | 0.714 | 0.108 | 0.046 | ||

FGM | exponential | −1.0 | 0.609 | 0.063 | 0.082 | 0.609 | 0.063 | 0.083 |

1.0 | 0.666 | 0.090 | 0.063 | 0.658 | 0.088 | 0.072 | ||

GB | exponential | 0.5 | 0.617 | 0.066 | 0.077 | 0.617 | 0.066 | 0.078 |

1.0 | 0.606 | 0.060 | 0.079 | 0.606 | 0.060 | 0.080 |

**Table 4.**Estimates ${p}_{\tau}(\tau =108)$ for fitting the KM estimator (independent) and ${p}_{\tau}(\tau =108)$ with the exponential marginal survival distributions (the independent, Clayton, Gumbel, Frank, FGM, GB copulas) for the prostate cancer dataset.

Copula | Marginal Distribution | $\mathit{\theta}$ | $\widehat{\mathit{p}}$ | SE | p-Value | ${\widehat{\mathit{p}}}_{\mathit{\tau}}(\mathit{\tau}=108)$ | SE | p-Value |
---|---|---|---|---|---|---|---|---|

Independent | KM estimator | - | - | - | - | 0.635 | 0.013 | <0.001 |

Independent | exponential | - | 0.821 | 0.010 | <0.001 | 0.625 | 0.007 | <0.001 |

Clayton | exponential | 1.0 | 0.889 | 0.008 | <0.001 | 0.626 | 0.007 | <0.001 |

5.0 | 0.958 | 0.003 | <0.001 | 0.635 | 0.007 | <0.001 | ||

Gumbel | exponential | 4.0 | 0.999 | <0.001 | <0.001 | 0.666 | 0.006 | <0.001 |

Frank | exponential | −20.0 | 0.741 | 0.009 | <0.001 | 0.624 | 0.007 | <0.001 |

−5.0 | 0.753 | 0.010 | <0.001 | 0.624 | 0.007 | <0.001 | ||

5.0 | 0.924 | 0.007 | <0.001 | 0.632 | 0.007 | <0.001 | ||

FGM | exponential | −1.0 | 0.777 | 0.011 | <0.001 | 0.623 | 0.007 | <0.001 |

1.0 | 0.865 | 0.010 | <0.001 | 0.626 | 0.007 | <0.001 | ||

GB | exponential | 0.5 | 0.786 | 0.010 | <0.001 | 0.624 | 0.007 | <0.001 |

1.0 | 0.764 | 0.010 | <0.001 | 0.623 | 0.007 | <0.001 |

Copula | Marginal Distribution | |
---|---|---|

Domma & Giordano [47] | FGM, Generalized FGM, Frank | Burr III, Dagum, Singh–Maddala |

Gao et al. [48] | Mixed (Clayton, Gumbel, Frank) | Empirical |

de Andrade et al. [49] | Clayton, Gumbel, Frank, Gauss, Plackett | Weibull, Gamma, Log-normal, Dagum |

Rathie et al. [50] | Frank | Dagum, Log-Dagum |

James et al. [51] | FGM | Rayleigh |

Shang & Yan [52] | Clayton | Weibull, Kumaraswamy |

Lima et al. [53] | Clayton, Frank, Gumbel–Houggard | Generalized extreme value, Weibull, gamma |

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**MDPI and ACS Style**

Nakazono, K.; Lin, Y.-C.; Liao, G.-Y.; Uozumi, R.; Emura, T.
Computation of the Mann–Whitney Effect under Parametric Survival Copula Models. *Mathematics* **2024**, *12*, 1453.
https://doi.org/10.3390/math12101453

**AMA Style**

Nakazono K, Lin Y-C, Liao G-Y, Uozumi R, Emura T.
Computation of the Mann–Whitney Effect under Parametric Survival Copula Models. *Mathematics*. 2024; 12(10):1453.
https://doi.org/10.3390/math12101453

**Chicago/Turabian Style**

Nakazono, Kosuke, Yu-Cheng Lin, Gen-Yih Liao, Ryuji Uozumi, and Takeshi Emura.
2024. "Computation of the Mann–Whitney Effect under Parametric Survival Copula Models" *Mathematics* 12, no. 10: 1453.
https://doi.org/10.3390/math12101453