# Dynamic Risk Prediction via a Joint Frailty-Copula Model and IPD Meta-Analysis: Building Web Applications

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Review on Dynamic Prediction

#### 2.2. Review on a Joint Frailty-Copula Model

#### 2.3. Dynamic Prediction under the Joint Frailty-Copula Model

#### 2.4. Online Web Applications

## 3. Validation Methods

#### 3.1. Calibration Plot

#### 3.2. Brier Score

#### 3.3. The C-Index for Discrimination Performance

## 4. Tutorial: Building Web Applications

**Step 1**:- Fit a training dataset to a joint frailty-copula model using the R package joint.Cox;
**Step 2**:- Validate the fitted model;
**Step 3**:- Use the “app.R” file to build a web application using the R package Shiny.

**Step 1: Fit a training dataset to a model**.

- Estrogen receptor status (
**ER**= 1 for positive; = 0 for negative); - Tumor size (
**Size**= 1 for > 2 cm; = 0 for ≤ 2 cm); - Lymph nodal status (
**Node**= 1 for present; =0 for absent); - Age at diagnosis (
**Age**= 1 for age ≤40; =2 for 40 < age ≤ 50; = 3 for age>50); - The 70-gene signature developed by [1] (
**MammaPrint**= 1 for high; = −1 for low); - The gene expression grade index (GGI) defined by [3] (
**GGI**= 1 for high; = −1 for low).

**Step 2: Validate the prediction formula.**

- (i)
- Check the confidence interval (CI) for the prediction formula;
- (ii)
- Check the calibration plot, Brier score, and c-index.

**Step 3: Make a web application using Shiny.**

## 5. Results

#### 5.1. Model Fitting

#### 5.2. Developing and Validating a Predictor

**Patient 1:**

- Age at diagnosis: 45 years
- Estrogen receptor: positive
- Tumor size: >2cm
- Lymph nodal status: present
- MammaPrint: high
- GGI: high

#### 5.3. Upload a Web Application

**Step 1**:- Open the “app_breast” file in R studio;
**Step 2**:- Run the code in the file (as with the usual R code), and then the application is generated in a window;
**Step 3**:- Check if the application works properly;
**Step 4**:- Publish the application (click the “Publish” icon).

## 6. Conclusions and Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A risk prediction scheme in the framework of dynamic prediction. The measure of prediction is the conditional probability of death between t and t + w given the observed status of a patient at time t. The expressions for F are defined in Section 2.3.

**Figure 2.**The web application for a clinical prediction tool made by applying the proposed methods to breast cancer data. The interactive version is available at https://takeshi.shinyapps.io/Breast-2022-0218/ (accessed on 1 April 2022).

**Figure 3.**The predicted probability of death when the prediction time is set at t = 1000 days. The 95% CIs are indicated by the dotted lines (……), and their widths are shown by the vertical lines (and the number below the lines) at three time points.

**Figure 4.**The calibration plots comparing the observed and predicted survival rates, consisting of $\left\{\mathrm{Pred}\left({w}_{k}\right),\mathrm{Obs}\left({w}_{k}\right)\right\}$ with equally spaced prediction horizons, $0<{w}_{1}<\dots <{w}_{50}=8108$ (days). If the plots are placed on the diagonal line, the ideal performance of the prediction formula is achieved.

**Left panel**: the joint frailty-copula model;

**right panel**: the dynamic KM estimator.

**Figure 5.**

**(Left panel**): prediction errors (Brier score) based on the breast cancer data at the prediction time at 1000 days. (

**Right panel**): the c-index for discrimination ability with the 95% CI based on the same setting.

**Figure 6.**The web application for a clinical prediction tool made by applying the proposed methods to ovarian cancer data. The interactive version is available at https://takeshi.shinyapps.io/Ovarian-2022-0218/ (accessed on 1 April 2022).

**Table 1.**The breast cancer dataset of Haibe-Kains et al. [4].

Maximum Follow-Up Days | Dataset ^{a} | N | The Number of Observed Events (Event Rates) | ||
---|---|---|---|---|---|

Metastasis | Death | Censoring | |||

5165 | CAL | 109 | 24 (22%) | 75 (69%) | 34 (31%) |

6694 | NKI | 295 | 101 (34%) | 79 (27%) | 216 (73%) |

9108 | TRANSBIG | 196 | 62 (32%) | 56 (29%) | 140 (71%) |

8267 | UCSF | 120 | 19 (16%) | 39 (32%) | 81 (68%) |

9108 | Total | 720 | 206 (29%) | 249(35%) | 471 (65%) |

**Notes:**The R code for obtaining the data is available in the Supplementary Materials. The data are a subset from the file “jnci-JNCI-11-0924-s02.csv” available in the Supplementary Data of Haibe-Kains et al. [4]; the file is available on the journal webpage.

^{a}Datasets are signified as acronyms: CAL = dataset of breast cancer patients from the University of California, San Francisco, and the California Pacific Medical Center (United States); NKI = National Kanker Institute (the Netherlands); TRANSBIG = dataset collected by the TransBIG consortium (Europe); UCSF = University of California, San Francisco (United States). The extracted data are the subset having complete values of “t.dmfs: time for distant metastasis-free survival (days)”, “e.dmfs: event for distant metastasis-free survival”, “t.os: time for overall survival (days)”, and “e.os: event for overall survival”, as well as covariates (ER, Size, Node, Age, MammaPrint, and GGI). The median follow-up time was calculated from the Kaplan–Meier estimator for the time to censoring for each study. The event rates were calculated separately for each study.

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**MDPI and ACS Style**

Emura, T.; Michimae, H.; Matsui, S.
Dynamic Risk Prediction via a Joint Frailty-Copula Model and IPD Meta-Analysis: Building Web Applications. *Entropy* **2022**, *24*, 589.
https://doi.org/10.3390/e24050589

**AMA Style**

Emura T, Michimae H, Matsui S.
Dynamic Risk Prediction via a Joint Frailty-Copula Model and IPD Meta-Analysis: Building Web Applications. *Entropy*. 2022; 24(5):589.
https://doi.org/10.3390/e24050589

**Chicago/Turabian Style**

Emura, Takeshi, Hirofumi Michimae, and Shigeyuki Matsui.
2022. "Dynamic Risk Prediction via a Joint Frailty-Copula Model and IPD Meta-Analysis: Building Web Applications" *Entropy* 24, no. 5: 589.
https://doi.org/10.3390/e24050589