# The Effect of Threshold Values and Weighting Factors on the Association between Entropy Measures and Mortality after Myocardial Infarction in the Cardiac Arrhythmia Suppression Trial (CAST)

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Data

#### 2.2. Entropy Measures

- The approximate entropy $\mathrm{ApEn}$ as introduced by Pincus et al. [12] is calculated as$$\mathrm{ApEn}(m,r,N):=\frac{1}{(N-m+1)}\sum _{i=1}^{N-m+1}log\frac{{C}_{i}^{m}\left(r\right)}{{C}_{i}^{m+1}\left(r\right)},$$
- Since $\mathrm{ApEn}$ is biased by self-matches, the corrected approximate entropy $\mathrm{CApEn}$ was introduced by Porta et al. [16]. The correction is obtained by replacing the ratio ${C}_{i}^{m}\left(r\right)/{C}_{i}^{m+1}\left(r\right)$ in Equation (1) by $1/(N-m+1)$ when ${C}_{i}^{m}\left(r\right)=1$ or ${C}_{i}^{m+1}\left(r\right)=1$.
- Another modification of $\mathrm{ApEn}$ in order to correct its bias by self-matches is the sample entropy $\mathrm{SampEn}$ as introduced by Richman and Moorman [13]. It is defined as$$\mathrm{SampEn}(m,r,N):=log\left(\sum _{i=1}^{N-m}{C}_{i}^{m}\left(r\right)\right)-log\left(\sum _{i=1}^{N-m-1}{C}_{i}^{m+1}\left(r\right)\right);$$
- To soften the effects of a hard threshold r, Chen et al. [14] replaced it with the fuzzy membership function$$\mu (x,n,r):=exp(-0.69\xb7{(x/r)}^{n}).$$The factor of 0.69 was incorporated to get a value of 0.5 for $x/r=1$, which is important for comparisons between rectangular and fuzzy membership functions. Finally, with$${\varphi}^{m}(r,n,N):=\frac{1}{N-m}\sum _{i=1}^{N-m}\sum _{j=1,j\ne i}^{N-m}\frac{\mu (d({x}_{i}^{m},{x}_{j}^{m}),n,r)}{N-m-1},$$$$\mathrm{FuzzyEn}(m,r,n,N):=ln\left(\frac{{\varphi}^{m}(r,n,N)}{{\varphi}^{m+1}(r,n,N)}\right).$$
- The fuzzy measure entropy $\mathrm{FuzzyMEn}$, proposed by Liu et al. [15], introduces a distinction between local and global similarity based on $\mathrm{FuzzyEn}$:$$\mathrm{FuzzyMEn}(m,{r}_{L},{r}_{F},{n}_{L},{n}_{F},N):=ln\left(\frac{{\varphi}^{m}({r}_{L},{n}_{L},N)}{{\varphi}^{m+1}({r}_{L},{n}_{L},N)}\right)+ln\left(\frac{{\varphi}^{m}({r}_{F},{n}_{F},N)}{{\varphi}^{m+1}({r}_{F},{n}_{F},N)}\right).$$

#### 2.3. Application of Entropy Measures to CAST Data

#### 2.4. Statistical Analysis

## 3. Results

#### 3.1. Predictive Value with Standard Parameter Sets

#### 3.2. Parameter Selection Process

#### 3.2.1. Variation of the Threshold Value

#### 3.2.2. Variation of the Weighting Factor

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Kaplan–Meier survival curves according to risk groups based on $\mathrm{ApEn}$ for post-treatment data (see parameter set No. 1 in Table 2); for all patients (

**A**), for all patients w/o CABG (

**B**) and w/o CABG and DM (

**C**).

**Figure 2.**Significance of predictive values of entropy measures for different choices of r (multiples of ${r}_{\mathrm{Chon}}$); parameters: $m=2$, $N=1200$, $n={n}_{L}=2$, ${n}_{F}=1$; HRV data at baseline (

**A,B,C**) and after treatment (

**D,E,F**); for all patients (

**A,D**), for all patients w/o CABG (

**B,E**) and w/o CABG and DM (

**C,F**). p = 0.05 marks the threshold of statistical significance.

**Figure 3.**Significance of predictive values of entropy measures for different choices of r (multiples of σ); parameters: $m=2$, $N=1200$, $n={n}_{L}=1$, ${n}_{F}=3$; HRV data at baseline (

**A,B,C**) and after treatment (

**D,E,F**); for all patients (

**A,D**), for all patients w/o CABG (

**B,E**) and w/o CABG and DM (

**C,F**). p = 0.05 marks the threshold of statistical significance.

**Figure 4.**Significance of predictive values of $\mathrm{FuzzyMEn}$ for different choices of ${r}_{L}$ and ${r}_{F}$ (multiples of ${r}_{\mathrm{Chon}}$); parameters: $m=2$, $N=1200$, $n={n}_{L}=2$, ${n}_{F}=1$; HRV data at baseline (

**A,B,C**) and after treatment (

**D,E,F**); for all patients (

**A,D**), for all patients w/o CABG (

**B,E**) and w/o CABG and DM (

**C,F**).

**Figure 5.**Significance of predictive values of $\mathrm{FuzzyMEn}$ for different choices of ${r}_{L}$ and ${r}_{F}$ (multiples of σ); parameters: $m=2$, $N=1200$, $n={n}_{L}=1$, ${n}_{F}=3$; HRV data at baseline (

**A,B,C**) and after treatment (

**D,E,F**); for all patients (

**A,D**), for all patients w/o CABG (

**B,E**) and w/o CABG and DM (

**C,F**).

**Table 1.**Patient baseline data and number of records before and after treatment (data are stated as mean ± SD or median and $95\%$ confidence interval (CI)).

Before Treatment | After Treatment | |
---|---|---|

Sample size | 760 | 740 |

Age (years) | $60.8\pm 9.6$ | $60.7\pm 9.6$ |

Sex (m/f) | $624/136$ | $607/133$ |

Time after MI (days) | $38\phantom{\rule{0.166667em}{0ex}}\left(35\text{--}42\right)$ | $72\phantom{\rule{0.166667em}{0ex}}\left(68\text{--}76\right)$ |

CABG | 141 | 137 |

DM | 160 | 155 |

CABG & DM | 270 | 262 |

Follow-up time (days) | $333\phantom{\rule{0.166667em}{0ex}}\left(301\text{--}355\right)$ | $291.5\phantom{\rule{0.166667em}{0ex}}\left(268\text{--}329\right)$ |

**Table 2.**Parameter sets based on literature for template length m, data length N, threshold parameter r, the weighting factor(s) n, ${n}_{L}$ and ${n}_{F}$, and the threshold parameter(s) r, ${r}_{L}$ and ${r}_{F}$.

No. | Source | m | N | $n={n}_{L}$ | ${n}_{F}$ | $r={r}_{L}={r}_{F}$ | Entropy Type |
---|---|---|---|---|---|---|---|

1 | Mayer et al. [17] | 2 | 1200 | 2 | 1 | ${r}_{\mathrm{Chon}}$ | all |

2 | Mayer et al. [17] | 2 | 1200 | 1 | 3 | $0.2\xb7\sigma $ | all |

3 | Zhao et al. [18] | 2 | 1000 | - | - | 0.15 | $\mathrm{SampEn}$ |

4 | Zhao et al. [18] | 2 | 1000 | 3 | 2 | 0.15 | $\mathrm{FuzzyMEn}$ |

**Table 3.**Univariate entropy predictors of mortality in CAST at baseline with parameter sets 1 to 4 as defined in Table 2 (all data ($n=760,70$ deaths), w/o CABG ($n=619,63$ deaths) and w/o CABG, w/o DM ($n=490,39$ deaths).

No. | Variable | λ | All | w/o CABG | w/o CABG, w/o DM | |||
---|---|---|---|---|---|---|---|---|

$\mathit{HR}\phantom{\rule{0.166667em}{0ex}}(95\%\phantom{\rule{0.166667em}{0ex}}\mathit{CI})$ | p | $\mathit{HR}\phantom{\rule{0.166667em}{0ex}}(95\%\phantom{\rule{0.166667em}{0ex}}\mathit{CI})$ | p | $\mathit{HR}\phantom{\rule{0.166667em}{0ex}}(95\%\phantom{\rule{0.166667em}{0ex}}\mathit{CI})$ | p | |||

1 | ApEn | $-0.09$ | $0.990\phantom{\rule{0.166667em}{0ex}}(0.909\text{--}1.079)$ | $0.825$ | $0.976\phantom{\rule{0.166667em}{0ex}}(0.891\text{--}1.069)$ | $0.604$ | $0.969\phantom{\rule{0.166667em}{0ex}}(0.864\text{--}1.088)$ | $0.594$ |

CApEn | $-0.15$ | $0.989\phantom{\rule{0.166667em}{0ex}}(0.916\text{--}1.067)$ | $0.768$ | $0.977\phantom{\rule{0.166667em}{0ex}}(0.901\text{--}1.060)$ | $0.578$ | $0.974\phantom{\rule{0.166667em}{0ex}}(0.879\text{--}1.079)$ | $0.612$ | |

SampEn | $-0.12$ | $0.993\phantom{\rule{0.166667em}{0ex}}(0.920\text{--}1.071)$ | $0.854$ | $0.981\phantom{\rule{0.166667em}{0ex}}(0.904\text{--}1.063)$ | $0.635$ | $0.975\phantom{\rule{0.166667em}{0ex}}(0.880\text{--}1.081)$ | $0.632$ | |

FuzzyEn | $-0.02$ | $0.965\phantom{\rule{0.166667em}{0ex}}(0.781\text{--}1.192)$ | $0.738$ | $0.905\phantom{\rule{0.166667em}{0ex}}(0.718\text{--}1.140)$ | $0.396$ | $0.926\phantom{\rule{0.166667em}{0ex}}(0.686\text{--}1.249)$ | $0.615$ | |

FuzzyMEn | $0.21$ | $0.840\phantom{\rule{0.166667em}{0ex}}(0.469\text{--}1.504)$ | $0.557$ | $0.726\phantom{\rule{0.166667em}{0ex}}(0.390\text{--}1.351)$ | $0.313$ | $0.759\phantom{\rule{0.166667em}{0ex}}(0.342\text{--}1.685)$ | $0.498$ | |

2 | ApEn | $1.66$ | $0.962\phantom{\rule{0.166667em}{0ex}}(0.441\text{--}2.098)$ | $0.923$ | $0.889\phantom{\rule{0.166667em}{0ex}}(0.399\text{--}1.981)$ | $0.774$ | $0.638\phantom{\rule{0.166667em}{0ex}}(0.239\text{--}1.700)$ | $0.369$ |

CApEn | $0.61$ | $0.957\phantom{\rule{0.166667em}{0ex}}(0.711\text{--}1.288)$ | $0.770$ | $0.879\phantom{\rule{0.166667em}{0ex}}(0.650\text{--}1.188)$ | $0.402$ | $0.833\phantom{\rule{0.166667em}{0ex}}(0.569\text{--}1.221)$ | $0.349$ | |

SampEn | $0.88$ | $0.911\phantom{\rule{0.166667em}{0ex}}(0.566\text{--}1.464)$ | $0.699$ | $0.818\phantom{\rule{0.166667em}{0ex}}(0.506\text{--}1.323)$ | $0.413$ | $0.674\phantom{\rule{0.166667em}{0ex}}(0.368\text{--}1.235)$ | $0.202$ | |

FuzzyEn | $0.85$ | $0.979\phantom{\rule{0.166667em}{0ex}}(0.375\text{--}2.558)$ | $0.965$ | $1.047\phantom{\rule{0.166667em}{0ex}}(0.359\text{--}3.057)$ | $0.932$ | $0.557\phantom{\rule{0.166667em}{0ex}}(0.146\text{--}2.127)$ | $0.392$ | |

FuzzyMEn | $0.93$ | $0.962\phantom{\rule{0.166667em}{0ex}}(0.622\text{--}1.490)$ | $0.863$ | $0.988\phantom{\rule{0.166667em}{0ex}}(0.617\text{--}1.582)$ | $0.959$ | $0.726\phantom{\rule{0.166667em}{0ex}}(0.404\text{--}1.307)$ | $0.286$ | |

3 | SampEn | $0.09$ | $0.951\phantom{\rule{0.166667em}{0ex}}(0.773\text{--}1.169)$ | $0.633$ | $0.890\phantom{\rule{0.166667em}{0ex}}(0.710\text{--}1.114)$ | $0.308$ | $0.876\phantom{\rule{0.166667em}{0ex}}(0.653\text{--}1.176)$ | $0.378$ |

4 | FuzzyMEn | $0.08$ | $0.965\phantom{\rule{0.166667em}{0ex}}(0.740\text{--}1.259)$ | $0.795$ | $0.889\phantom{\rule{0.166667em}{0ex}}(0.669\text{--}1.180)$ | $0.415$ | $0.887\phantom{\rule{0.166667em}{0ex}}(0.615\text{--}1.281)$ | $0.523$ |

**Table 4.**Univariate entropy predictors of mortality in CAST after treatment with parameter sets 1 to 4 as defined in Table 2 (all data ($n=740,69$ deaths), w/o CABG ($n=603,62$ deaths) and w/o CABG, w/o DM ($n=478,39$ deaths).

No. | Variable | λ | All | w/o CABG | w/o CABG, w/o DM | |||
---|---|---|---|---|---|---|---|---|

$\mathit{HR}\phantom{\rule{0.166667em}{0ex}}(95\%\phantom{\rule{0.166667em}{0ex}}\mathit{CI})$ | p | $\mathit{HR}\phantom{\rule{0.166667em}{0ex}}(95\%\phantom{\rule{0.166667em}{0ex}}\mathit{CI})$ | p | $\mathit{HR}\phantom{\rule{0.166667em}{0ex}}(95\%\phantom{\rule{0.166667em}{0ex}}\mathit{CI})$ | p | |||

1 | ApEn | $-0.19$ | $0.927\phantom{\rule{0.166667em}{0ex}}(0.875\text{--}0.981)$ | $\mathit{0}.\mathit{009}$ | $0.910\phantom{\rule{0.166667em}{0ex}}(0.857\text{--}0.966)$ | $\mathit{0}.\mathit{002}$ | $0.930\phantom{\rule{0.166667em}{0ex}}(0.866\text{--}0.999)$ | $\mathit{0}.\mathit{048}$ |

CApEn | $-0.28$ | $0.939\phantom{\rule{0.166667em}{0ex}}(0.895\text{--}0.985)$ | $\mathit{0}.\mathit{010}$ | $0.926\phantom{\rule{0.166667em}{0ex}}(0.881\text{--}0.974)$ | $\mathit{0}.\mathit{003}$ | $0.943\phantom{\rule{0.166667em}{0ex}}(0.888\text{--}1.001)$ | $0.053$ | |

SampEn | $-0.23$ | $0.937\phantom{\rule{0.166667em}{0ex}}(0.892\text{--}0.985)$ | $\mathit{0}.\mathit{010}$ | $0.922\phantom{\rule{0.166667em}{0ex}}(0.876\text{--}0.972)$ | $\mathit{0}.\mathit{002}$ | $0.940\phantom{\rule{0.166667em}{0ex}}(0.884\text{--}1.000)$ | $\mathit{0}.\mathit{049}$ | |

FuzzyEn | $-0.09$ | $0.962\phantom{\rule{0.166667em}{0ex}}(0.815\text{--}1.136)$ | $0.649$ | $0.882\phantom{\rule{0.166667em}{0ex}}(0.742\text{--}1.050)$ | $0.158$ | $0.897\phantom{\rule{0.166667em}{0ex}}(0.715\text{--}1.124)$ | $0.343$ | |

FuzzyMEn | $0.11$ | $0.844\phantom{\rule{0.166667em}{0ex}}(0.529\text{--}1.348)$ | $0.478$ | $0.689\phantom{\rule{0.166667em}{0ex}}(0.430\text{--}1.104)$ | $0.121$ | $0.727\phantom{\rule{0.166667em}{0ex}}(0.394\text{--}1.340)$ | $0.307$ | |

2 | ApEn | $1.80$ | $2.252\phantom{\rule{0.166667em}{0ex}}(0.994\text{--}5.102)$ | $0.052$ | $2.349\phantom{\rule{0.166667em}{0ex}}(1.006\text{--}5.489)$ | $\mathit{0}.\mathit{049}$ | $1.690\phantom{\rule{0.166667em}{0ex}}(0.610\text{--}4.686)$ | $0.313$ |

CApEn | $0.58$ | $1.516\phantom{\rule{0.166667em}{0ex}}(1.100\text{--}2.089)$ | $0.011$ | $1.441\phantom{\rule{0.166667em}{0ex}}(1.040\text{--}1.996)$ | $\mathit{0}.\mathit{028}$ | $1.334\phantom{\rule{0.166667em}{0ex}}(0.890\text{--}1.998)$ | $0.162$ | |

SampEn | $0.92$ | $1.713\phantom{\rule{0.166667em}{0ex}}(1.023\text{--}2.869)$ | $\mathit{0}.\mathit{041}$ | $1.705\phantom{\rule{0.166667em}{0ex}}(1.008\text{--}2.884)$ | $\mathit{0}.\mathit{046}$ | $1.367\phantom{\rule{0.166667em}{0ex}}(0.720\text{--}2.597)$ | $0.340$ | |

FuzzyEn | $0.77$ | $3.375\phantom{\rule{0.166667em}{0ex}}(1.381\text{--}8.247)$ | $\mathit{0}.\mathit{008}$ | $3.828\phantom{\rule{0.166667em}{0ex}}(1.454\text{--}10.075)$ | $\mathit{0}.\mathit{007}$ | $1.825\phantom{\rule{0.166667em}{0ex}}(0.535\text{--}6.229)$ | $0.337$ | |

FuzzyMEn | $0.86$ | $1.634\phantom{\rule{0.166667em}{0ex}}(1.054\text{--}2.535)$ | $\mathit{0}.\mathit{028}$ | $1.767\phantom{\rule{0.166667em}{0ex}}(1.104\text{--}2.826)$ | $\mathit{0}.\mathit{018}$ | $1.200\phantom{\rule{0.166667em}{0ex}}(0.668\text{--}2.157)$ | $0.542$ | |

3 | SampEn | $0.06$ | $0.953\phantom{\rule{0.166667em}{0ex}}(0.788\text{--}1.153)$ | $0.622$ | $0.857\phantom{\rule{0.166667em}{0ex}}(0.699\text{--}1.050)$ | $0.136$ | $0.845\phantom{\rule{0.166667em}{0ex}}(0.650\text{--}1.098)$ | $0.207$ |

4 | FuzzyMEn | $0.02$ | $1.124\phantom{\rule{0.166667em}{0ex}}(0.881\text{--}1.435)$ | $0.347$ | $0.980\phantom{\rule{0.166667em}{0ex}}(0.758\text{--}1.269)$ | $0.880$ | $0.967\phantom{\rule{0.166667em}{0ex}}(0.690\text{--}1.356)$ | $0.846$ |

**Table 5.**Univariated traditional HRV predictors as reported by Stein et al. [7] using the same dateset (i.e., CAST).

All | w/o CABG | w/o CABG, w/o DM | ||||
---|---|---|---|---|---|---|

Variable | $\mathit{HR}\phantom{\rule{0.166667em}{0ex}}(95\%\phantom{\rule{0.166667em}{0ex}}\mathit{CI})$ | p | $\mathit{HR}\phantom{\rule{0.166667em}{0ex}}(95\%\phantom{\rule{0.166667em}{0ex}}\mathit{CI})$ | p | $\mathit{HR}\phantom{\rule{0.166667em}{0ex}}(95\%\phantom{\rule{0.166667em}{0ex}}\mathit{CI})$ | p |

AVGNN | $1.001\phantom{\rule{0.166667em}{0ex}}(0.999\text{--}1.002)$ | $0.36$ | $1.000\phantom{\rule{0.166667em}{0ex}}(0.998\text{--}1.002)$ | $0.77$ | $1.000\phantom{\rule{0.166667em}{0ex}}(0.998\text{--}1.003)$ | $0.77$ |

SDNN | $0.997\phantom{\rule{0.166667em}{0ex}}(0.9991\text{--}1.003)$ | $0.36$ | $0.994\phantom{\rule{0.166667em}{0ex}}(0.987\text{--}1.001)$ | $0.09$ | $0.992\phantom{\rule{0.166667em}{0ex}}(0.982\text{--}1.001)$ | $0.07$ |

Ln SDANN | $0.800\phantom{\rule{0.166667em}{0ex}}(0.485\text{--}1.319)$ | $0.38$ | $0.576\phantom{\rule{0.166667em}{0ex}}(0.322\text{--}1.028)$ | $0.06$ | $0.400\phantom{\rule{0.166667em}{0ex}}(0.184\text{--}0.871)$ | $\mathit{0}.\mathit{02}$ |

Ln SDNNIDX | $0.919\phantom{\rule{0.166667em}{0ex}}(0.581\text{--}1.452)$ | $0.72$ | $0.740\phantom{\rule{0.166667em}{0ex}}(0.435\text{--}1.257)$ | $0.26$ | $0.748\phantom{\rule{0.166667em}{0ex}}(0.360\text{--}1.553)$ | $0.44$ |

LN rMSSD | $1.271\phantom{\rule{0.166667em}{0ex}}(0.803\text{--}2.010)$ | $0.31$ | $1.127\phantom{\rule{0.166667em}{0ex}}(0.683\text{--}1.860)$ | $0.64$ | $1.371\phantom{\rule{0.166667em}{0ex}}(0.735\text{--}2.557)$ | $0.32$ |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mayer, C.; Bachler, M.; Holzinger, A.; Stein, P.K.; Wassertheurer, S. The Effect of Threshold Values and Weighting Factors on the Association between Entropy Measures and Mortality after Myocardial Infarction in the Cardiac Arrhythmia Suppression Trial (CAST). *Entropy* **2016**, *18*, 129.
https://doi.org/10.3390/e18040129

**AMA Style**

Mayer C, Bachler M, Holzinger A, Stein PK, Wassertheurer S. The Effect of Threshold Values and Weighting Factors on the Association between Entropy Measures and Mortality after Myocardial Infarction in the Cardiac Arrhythmia Suppression Trial (CAST). *Entropy*. 2016; 18(4):129.
https://doi.org/10.3390/e18040129

**Chicago/Turabian Style**

Mayer, Christopher, Martin Bachler, Andreas Holzinger, Phyllis K. Stein, and Siegfried Wassertheurer. 2016. "The Effect of Threshold Values and Weighting Factors on the Association between Entropy Measures and Mortality after Myocardial Infarction in the Cardiac Arrhythmia Suppression Trial (CAST)" *Entropy* 18, no. 4: 129.
https://doi.org/10.3390/e18040129