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Selection of mixed copula for association modeling with tied observations

Abstract

The link between Obesity and Hypertension is among the most popular topics which have been explored in medical research in recent decades. However, it is challenging to establish the relationship comprehensively and accurately because the distribution of BMI and blood pressure is usually fat tailed and severely tied. In this paper, we propose a data-driven copulas selection approach via penalized likelihood which can deal with tied data by interval censoring estimation. Minimax Concave Penalty is involved to perform the unbiased selection of mixed copula model for its convergence property to get un-penalized solution. Interval censoring and maximizing pseudo-likelihood, inspired from survival analysis, is introduced by considering ranks as intervals with upper and lower limits. This paper describes the model and corresponding iterative algorithm. Simulations to compare the proposed approach versus existing methods in different scenarios are presented. Additionally, the proposed method is also applied to the association modeling on the China Health and Nutrition Survey (CHNS) data. Both numerical studies and real data analysis reveal good performance of the proposed method.

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Acknowledgement

Dr. Y Li is supported by Platform of Public Health & Disease Control and Prevention, Major Innovation & Planning Interdisciplinary Platform for the “Double-First Class” Initiative, Renmin University of China and the National Scientific Foundation of China (71771211).

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Correspondence to Jiesheng Si.

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Appendices

Appendix

See Tables 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20.

1 Supported tables for simulation design

Table 5 True model of single copula model
Table 6 True model of mixed copula model consisted with two copulas
Table 7 List of simulated datasets in different scenarios

2 Line plots of correctness rates

Figures 3 and 4 are based on results from Table 2 and Table 3, respectively. In below figures, the proposed method is mentioned as ‘Proposed’ due to limited space.

Fig. 3
figure 3

Single copula model - change of correctness rate over various values of kendall’s tau. Note, % - rates of correctly selecting true copula over 100 replicates; ‘X - #’: ‘X’ indicates true copula and ‘#’ means one side tied or two sides tied data, e.g. N-1: Normal copula with one side tied data

Fig. 4
figure 4

Mixed copula model - change of correctness rate over various values of kendall’s tau. Note, % - rates of correctly selecting true copula over 100 replicates; ‘XX - #’: ‘XX’ indicates true mixed copula and ‘#’ means one side tied or two sides tied data, e.g. NC-2: Normal + Clayton mixed copula with two sides tied data

3 Estimated results (\(n=500\)) of MSE and Bias

Table 8 Single copula model (\(n=500\)): Bias and MSE of parameter estimation
Table 9 Mixed copula model (\(n=500\)): Bias and MSE of parameter estimation

4 Simulated results of sample size \(n=200\)

Table 10 Single copula model (n = 200): rate of correct selection (bolded values)/incorrect selection (values in parentheses)
Table 11 Mixed copula model (\(n=200\)): rate of correct selection (bolded values)/incorrect selection (values in parentheses)
Table 12 Single copula model (\(n=200\)): Bias and MSE of parameter estimation
Table 13 Mixed copula model (\(n=200\)): Bias and MSE of parameter estimation

5 Supportive simulations

Table 14 Single copula model (n=200) of proposed method with SCAD versus MCP: rate of correct selection (bolded values)/incorrect selection (values in parentheses)
Table 15 Single copula model (\(n=200\)) of proposed method with SCAD versus MCP: Bias and MSE of parameter estimation
Table 16 Proposed method with different levels of ties (n=200 and \(\tau =0.5\)): rate of correct selection (bolded values)/incorrect selection (values in parentheses)
Table 17 Proposed method with different levels of ties (n=200 and \(\tau =0.5\)): Bias and MSE of parameter estimation
Table 18 Proposed method of mixed model with different weights (n=200): rate of correct selection (bolded values)/incorrect selection (values in parentheses)
Table 19 Proposed method of mixed model with different weights (n=200): Bias and MSE of parameter estimation
Table 20 Additional estimations of the proposed method under the setting when true model is \(0.5C+0.5G\): rate of correct selection (bolded values)/incorrect selection (values in parentheses)

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Li, Y., Wang, F., Shen, Y. et al. Selection of mixed copula for association modeling with tied observations. Stat Methods Appl (2022). https://doi.org/10.1007/s10260-022-00628-3

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  • DOI: https://doi.org/10.1007/s10260-022-00628-3

Keywords

  • Copula selection
  • Interval censoring
  • MCP
  • Tied data