# Challenging Recently Published Parameter Sets for Entropy Measures in Risk Prediction for End-Stage Renal Disease Patients

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data

#### 2.2. HRV Analysis

#### 2.3. Entropy Measures

- As introduced by Pincus et al. [12], the $\mathrm{ApEn}$ is calculated as$$\mathrm{ApEn}(m,r,N):=\left(\right)open="("\; close=")">\frac{1}{N-m+1}\sum _{i=1}^{N-m+1}log{C}_{i}^{m}\left(r\right)$$
- Because of self-matches, $\mathrm{ApEn}$ is biased towards regularity. Porta et al. [16] therefore introduced the corrected approximate entropy $\mathrm{CApEn}$, where $\mathrm{ApEn}$ was reformulated and adapted as$$\mathrm{CApEn}(m,r,N):=-\frac{1}{N-m+1}\sum _{i=1}^{N-m+1}log\left(\mathsf{\Theta}\right)$$$$\mathsf{\Theta}:=\left(\right)open="\{"\; close>\begin{array}{cc}{(N-m+1)}^{-1}\hfill & \mathrm{if}\phantom{\rule{1.em}{0ex}}{C}_{i}^{m}\left(r\right)=1\phantom{\rule{1.em}{0ex}}\mathrm{or}\phantom{\rule{1.em}{0ex}}{C}_{i}^{m-1}\left(r\right)=1\hfill \\ {C}_{i}^{m}\left(r\right)/{C}_{i}^{m-1}\left(r\right)\hfill & \mathrm{otherwise}\hfill \end{array}$$
- Richman and Moorman [13] introduced another approach to correct the bias of $\mathrm{ApEn}$, namely, $\mathrm{SampEn}$. This is calculated as$$\mathrm{SampEn}(m,r,N):=log\left(\right)open="("\; close=")">\sum _{i=1}^{N-m}{C}_{i}^{m}\left(r\right)$$
- ApEn and SampEn are sensitive to small variations of r, because ${C}_{i}^{m}\left(r\right)$ is a counting function and the condition resembles a Heaviside function. To counteract this problem, Chen et al. [14] introduced $\mathrm{FuzzyEn}$, replacing ${C}_{i}^{m}\left(r\right)$ with the fuzzy membership function:$$\mu (x,n,r):=exp(-0.69\xb7{(x/r)}^{n})$$$\mathrm{FuzzyEn}$ is then calculated as$$\begin{array}{ccc}\hfill \mathrm{FuzzyEn}(m,r,n,N)& :=& ln\left(\right)open="("\; close=")">\frac{{\varphi}^{m}(r,n,N)}{{\varphi}^{m+1}(r,n,N)},\mathrm{where}\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\varphi}^{m}(r,n,N)& :=& \frac{1}{N-m}\sum _{i=1}^{N-m}\sum _{j\ne i}\frac{\mu (d({x}_{i}^{m},{x}_{j}^{m}),n,r)}{N-m-1}\hfill \end{array}$$$$d(x,y):=\underset{i}{max}\left(\right)open="("\; close=")">|{x}_{i}-{y}_{i}|$$
- In order to distinguish between local (using ${r}_{L}$ and ${n}_{L}$) and global (using ${r}_{F}$ and ${n}_{F}$) similarity, Liu et al. [15] extended the $\mathrm{FuzzyEn}$ to the FuzzyMEn:$$\mathrm{FuzzyMEn}(m,{r}_{L},{r}_{F},{n}_{L},{n}_{F},N):=ln\left(\right)open="("\; close=")">\frac{{\varphi}^{m}({r}_{L},{n}_{L},N)}{{\varphi}^{m+1}({r}_{L},{n}_{L},N)}$$

#### 2.4. Application of Entropy Measures

#### 2.5. Statistical Analysis

## 3. Results

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Hazard ratios (HR) for (

**A**) all data; (

**B**) subgroup of patients without heart disease (HD); and (

**C**) patients with HD. HRs are plotted as diamonds with their corresponding 95% confidence interval.

**Table 1.**Parameter sets used for entropy estimation in this work for template length m, the weighting factor(s) n, ${n}_{L}$ and ${n}_{F}$, and the threshold parameter(s) r, ${r}_{L}$ and ${r}_{F}$.

m | $\mathit{n}={\mathit{n}}_{\mathit{L}}$ | ${\mathit{n}}_{\mathit{F}}$ | $\mathit{r}={\mathit{r}}_{\mathit{L}}={\mathit{r}}_{\mathit{F}}$ | |
---|---|---|---|---|

$entropy\left({r}_{\mathrm{Chon}}\right)$ | 2 | 2 | 1 | ${r}_{\mathrm{Chon}}$ |

$entropy\left({r}_{\sigma}\right)$ | 2 | 1 | 3 | $0.2\xb7\sigma $ |

**Table 2.**Baseline data of included patients. Data given as number (%) for categorical data, mean (SD) or median [IQR].

Variable | All Data (n = 265) |
---|---|

Age (years) | 62 (15.1 SD) |

Sex—male, n (%) | 175 (66%) |

Body weight (kg) | 77.1 (19 SD) |

Height (m) | 1.71 (0.0842 SD) |

Body mass index (kg/m${}^{2}$) | 25 [22.5, 28.7] |

Presence of diabetes, n (%) | 92 (35%) |

Presence of hypertension, n (%) | 250 (94%) |

Current smokers, n (%) | 72 (27 %) |

Adapted CCI (-) | 2 [1, 5] |

Dialysis vintage (mo) | 47.3 [23.8, 79.4] |

Dialysis duration per session (h) | 4.23 [4.02, 4.55] |

UFV (mL) | 2196 (1157 SD) |

Kt/V (-) | 1.47 (0.398 SD) |

Serum albumin (g/dL) | 4.03 (0.395 SD) |

hsCRP (mg/dL) | 0.368 [0.159, 0.881] |

Total cholesterol (mg/dL) | 181 (44.8 SD) |

HDL cholesterol (mg/dL) | 42 [36, 52] |

LDL cholesterol (mg/dL) | 112 (36 SD) |

Calcium × phosphate (mmol${}^{2}$/L${}^{2}$) | 3.94 (1.13 SD) |

Antihypertensive drugs, n (%) | 237 (89%) |

Statins, n (%) | 97 (37%) |

Anticoagulant, n (%) | 29 (11%) |

**Table 3.**Heart rate variability (HRV) parameters of the time and frequency domain and entropy measures at baseline. Data given as mean (SD) or median [IQR].

Variable | All Data (n = 265) |
---|---|

AVNN (ms) | 887 (131 SD) |

SDNN (ms) | 33.9 (17.5 SD) |

RMSSD (ms) | 13.4 [9.42, 20.2] |

pNN50 (%) | 0.539 [0.0838, 2.48] |

HRVTI (-) | 30.3 (9.84 SD) |

Total P (ms${}^{2}$) | 1090 [495, 2198] |

LF (ms${}^{2}$) | 189 [72, 461] |

HF (ms${}^{2}$) | 59.9 [25.6, 158] |

LF/HF (-) | 3.21 [1.37, 6.41] |

ApEn$\left({r}_{\mathrm{Chon}}\right)$ (-) | 0.336 [0.184, 0.526] |

ApEn$\left({r}_{\sigma}\right)$ (-) | 1.05 (0.294 SD) |

SampEn$\left({r}_{\mathrm{Chon}}\right)$ (-) | 3.36 (0.597 SD) |

SampEn$\left({r}_{\sigma}\right)$ (-) | 0.968 (0.345 SD) |

FuzzyEn$\left({r}_{\mathrm{Chon}}\right)$ (-) | 3.58 (0.581 SD) |

FuzzyEn$\left({r}_{\sigma}\right)$ (-) | 0.688 (0.238 SD) |

FuzzyMEn$\left({r}_{\mathrm{Chon}}\right)$ (-) | 6.95 [6.29, 7.74] |

FuzzyMEn$\left({r}_{\sigma}\right)$ (-) | 1.61 (0.552 SD) |

CApEn$\left({r}_{\mathrm{Chon}}\right)$ (-) | 7.4 [7.23, 7.53] |

CApEn$\left({r}_{\sigma}\right)$ (-) | 1.36 (0.488 SD) |

Predictor | Unit | HR (95% CI) | p |
---|---|---|---|

Age | 1 year | 1.05 (1.03, 1.07) | <0.001 |

Height | 1 cm | 0.97 (0.94, 1.00) | 0.04 |

Adapted CCI | 1 | 1.24 (1.16, 1.33) | <0.001 |

Dialysis duration per session | 1 h | 0.52 (0.30, 0.89) | 0.02 |

Serum albumin | 1 g/dL | 0.24 (0.13, 0.45) | <0.001 |

hsCRP | log(1 mg/dL) | 1.26 (1.02, 1.56) | 0.04 |

Anticoagulant | No/Yes | 0.31 (0.17, 0.55) | <0.001 |

**Table 5.**Unadjusted and adjusted hazard ratios of heart rate variability (HRV) parameters ($n=265$).

Variable | Unit | Unadjusted | Model A | Model B | |||
---|---|---|---|---|---|---|---|

HR (95% CI) | p | HR (95% CI) | p | HR (95% CI) | p | ||

AVNN | 10 ms | 1.01 (0.99, 1.03) | 0.3 | - | - | - | - |

SDNN | $log\left(\mathrm{ms}\right)$ | 0.66 (0.43, 1.01) | 0.05 | 0.76 (0.47, 1.21) | 0.25 | 0.67 (0.41, 1.10) | 0.12 |

RMSSD | $log\left(\mathrm{ms}\right)$ | 0.94 (0.63, 1.40) | 0.75 | - | - | - | - |

pnn50 | $log(\%)$ | 1.02 (0.97, 1.07) | 0.51 | - | - | - | - |

HRVTI | 1 | 0.97 (0.94, 0.99) | 0.01 | 0.98 (0.95, 1.00) | 0.07 | 0.98 (0.95, 1.01) | 0.12 |

Total P | $log\left({\mathrm{ms}}^{2}\right)$ | 0.81 (0.65, 0.99) | 0.04 | 0.87 (0.69, 1.10) | 0.25 | 0.83 (0.65, 1.05) | 0.12 |

LF | $log\left({\mathrm{ms}}^{2}\right)$ | 0.82 (0.69, 0.96) | 0.01 | 0.84 (0.70, 1.00) | 0.05 | 0.82 (0.68, 0.99) | 0.04 |

HF | $log\left({\mathrm{ms}}^{2}\right)$ | 1.02 (0.85, 1.22) | 0.84 | - | - | - | - |

LF/HF | 1 | 0.74 (0.60, 0.90) | 0.003 | 0.82 (0.66, 1.01) | 0.06 | 0.74 (0.59, 0.92) | 0.007 |

ApEn$\left({r}_{\mathrm{Chon}}\right)$ | $log\left(1\right)$ | 1.30 (0.95, 1.77) | 0.1 | - | - | - | - |

ApEn$\left({r}_{\sigma}\right)$ | 1 | 0.55 (0.25, 1.23) | 0.15 | - | - | - | - |

SampEn$\left({r}_{\mathrm{Chon}}\right)$ | 1 | 0.68 (0.45, 1.03) | 0.07 | - | - | - | - |

SampEn$\left({r}_{\sigma}\right)$ | 1 | 0.55 (0.27, 1.10) | 0.09 | - | - | - | - |

FuzzyEn$\left({r}_{\mathrm{Chon}}\right)$ | 1 | 0.59 (0.39, 0.90) | 0.01 | 0.58 (0.37, 0.92) | 0.02 | 0.65 (0.41, 1.04) | 0.07 |

FuzzyEn$\left({r}_{\sigma}\right)$ | $log\left(1\right)$ | 0.65 (0.33, 1.27) | 0.21 | - | - | - | - |

FuzzyMEn$\left({r}_{\mathrm{Chon}}\right)$ | $log\left(1\right)$ | 0.32 (0.09, 1.13) | 0.08 | - | - | - | - |

FuzzyMEn$\left({r}_{\sigma}\right)$ | 1 | 0.74 (0.48, 1.15) | 0.18 | - | - | - | - |

CApEn$\left({r}_{\mathrm{Chon}}\right)$ | ⋄ | 0.91 (0.84, 0.98) | 0.01 | 0.92 (0.84, 1.00) | 0.04 | 0.92 (0.84, 1.01) | 0.08 |

CApEn$\left({r}_{\sigma}\right)$ | 1 | 0.90 (0.55, 1.47) | 0.67 | - | - | - | - |

**Table 6.**Unadjusted and adjusted HRs of entropies for the subgroup patients without heart disease ($n=166$).

No Heart Disease | |||||||
---|---|---|---|---|---|---|---|

Variable | Unit | Unadjusted | Model A | Model B | |||

HR (95% CI) | p | HR (95% CI) | p | HR (95% CI) | p | ||

ApEn$\left({r}_{\mathrm{Chon}}\right)$ | $log\left(1\right)$ | 1.28 (0.85, 1.90) | 0.23 | - | - | - | - |

ApEn$\left({r}_{\sigma}\right)$ | 1 | 1.21 (0.39, 3.77) | 0.74 | - | - | - | - |

SampEn$\left({r}_{\mathrm{Chon}}\right)$ | 1 | 0.78 (0.44, 1.40) | 0.41 | - | - | - | - |

SampEn$\left({r}_{\sigma}\right)$ | 1 | 1.04 (0.40, 2.73) | 0.93 | - | - | - | - |

FuzzyEn$\left({r}_{\mathrm{Chon}}\right)$ | 1 | 0.66 (0.38, 1.14) | 0.14 | - | - | - | - |

FuzzyEn$\left({r}_{\sigma}\right)$ | $log\left(1\right)$ | 1.23 (0.46, 3.26) | 0.67 | - | - | - | - |

FuzzyMEn$\left({r}_{\mathrm{Chon}}\right)$ | $log\left(1\right)$ | 0.63 (0.12, 3.45) | 0.60 | - | - | - | - |

FuzzyMEn$\left({r}_{\sigma}\right)$ | 1 | 1.12 (0.62, 2.03) | 0.71 | - | - | - | - |

CApEn$\left({r}_{\mathrm{Chon}}\right)$ | ⋄ | 0.99 (0.98, 1.00) | 0.10 | - | - | - | - |

CApEn$\left({r}_{\sigma}\right)$ | 1 | 1.43 (0.74, 2.77) | 0.29 | - | - | - | - |

**Table 7.**Unadjusted and adjusted HRs of entropies for the subgroup patients with heart disease ($n=99$).

Heart Disease (HD) | |||||||
---|---|---|---|---|---|---|---|

Variable | Unit | Unadjusted | Model A | Model B | |||

HR (95% CI) | p | HR (95% CI) | p | HR (95% CI) | p | ||

ApEn$\left({r}_{\mathrm{Chon}}\right)$ | $log\left(1\right)$ | 1.27 (0.76, 2.11) | 0.37 | - | - | - | - |

ApEn$\left({r}_{\sigma}\right)$ | 1 | 0.29 (0.09, 0.91) | 0.03 | 0.13 (0.04, 0.44) | 0.001 | 0.20 (0.05, 0.82) | 0.02 |

SampEn$\left({r}_{\mathrm{Chon}}\right)$ | 1 | 0.64 (0.36, 1.12) | 0.11 | - | - | - | - |

SampEn$\left({r}_{\sigma}\right)$ | 1 | 0.31 (0.11, 0.90) | 0.03 | 0.15 (0.05, 0.46) | <0.001 | 0.22 (0.06, 0.81) | 0.02 |

FuzzyEn$\left({r}_{\mathrm{Chon}}\right)$ | 1 | 0.56 (0.30, 1.06) | 0.08 | - | - | - | - |

FuzzyEn$\left({r}_{\sigma}\right)$ | $log\left(1\right)$ | 0.39 (0.15, 1.00) | 0.05 | 0.21 (0.07, 0.58) | 0.003 | 0.32 (0.10, 0.99) | 0.05 |

FuzzyMEn$\left({r}_{\mathrm{Chon}}\right)$ | $log\left(1\right)$ | 0.19 (0.03, 1.18) | 0.08 | - | - | - | - |

FuzzyMEn$\left({r}_{\sigma}\right)$ | 1 | 0.50 (0.25, 0.99) | 0.05 | 0.32 (0.16, 0.67) | 0.002 | 0.43 (0.19, 0.96) | 0.04 |

CApEn$\left({r}_{\mathrm{Chon}}\right)$ | ⋄ | 0.99 (0.98, 1.00) | 0.06 | - | - | - | - |

CApEn$\left({r}_{\sigma}\right)$ | 1 | 0.57 (0.27, 1.20) | 0.14 | - | - | - | - |

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Hagmair, S.; Bachler, M.; Braunisch, M.C.; Lorenz, G.; Schmaderer, C.; Hasenau, A.-L.; Stülpnagel, L.V.; Bauer, A.; Rizas, K.D.; Wassertheurer, S.;
et al. Challenging Recently Published Parameter Sets for Entropy Measures in Risk Prediction for End-Stage Renal Disease Patients. *Entropy* **2017**, *19*, 582.
https://doi.org/10.3390/e19110582

**AMA Style**

Hagmair S, Bachler M, Braunisch MC, Lorenz G, Schmaderer C, Hasenau A-L, Stülpnagel LV, Bauer A, Rizas KD, Wassertheurer S,
et al. Challenging Recently Published Parameter Sets for Entropy Measures in Risk Prediction for End-Stage Renal Disease Patients. *Entropy*. 2017; 19(11):582.
https://doi.org/10.3390/e19110582

**Chicago/Turabian Style**

Hagmair, Stefan, Martin Bachler, Matthias C. Braunisch, Georg Lorenz, Christoph Schmaderer, Anna-Lena Hasenau, Lukas Von Stülpnagel, Axel Bauer, Kostantinos D. Rizas, Siegfried Wassertheurer,
and et al. 2017. "Challenging Recently Published Parameter Sets for Entropy Measures in Risk Prediction for End-Stage Renal Disease Patients" *Entropy* 19, no. 11: 582.
https://doi.org/10.3390/e19110582