# On the Relationships among Nurse Staffing, Inpatient Care Quality, and Hospital Competition under the Global Budget Payment Scheme of Taiwan’s National Health Insurance System: Mixed Frequency VAR Analyses

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data and Variables

#### 2.2. VAR Models

_{t}represents the quarterly PNR calculated as the average of monthly PNR (i.e., ${L}_{t}=({L}_{1t}+{L}_{2t}+{L}_{3t})/3$). Other notations used in Equation (4) share the same definitions as those used in Equations (2) and (3). Analogous to Equations (2) and (3), one of the possible linkages among PNR, ICQ, and real ICE per admission can be written as the 2nd low of Equation (4) as follows:

#### 2.3. Granger Causality Tests

#### 2.4. Impulse-Response and Variance Decomposition

## 3. Results

#### 3.1. Descriptive Statistics

#### 3.2. Unit Root Tests

#### 3.3. Granger Causality Tests

#### 3.4. Impulse-Response Analyses

#### 3.5. Variance Decomposition

**PNR**= ΣPNR

_{i}) in the MF-VAR model for medical centers are 1.61~4.58 (=21.70/13.46~37.20/8.13), 1.55~3.81(=45.87/29.53~44.06/11.56), and 1.62~3.77 (=46.45/27.73~46.80/12.40) times higher than those attributed to an aggregation of PNR (i.e., PNR

^{A}) in the LF-VAR model in the short-run (h = 2), medium-run (h = 7), and long-run (h = 12), respectively, based on whether the 3-day EDV rate or 14-day readmission rate was chosen to measure ICQ. In addition, the proportions of forecast error variance of the real ICE per admission attributed to the PNR within a 3-month cycle of a quarter timespan (i.e.,

**PNR**= ΣPNR

_{i}) in the MF-VAR model for medical centers are 1.24~1.36 (=74.07/59.63~73.87/54.34), 1.50~1.65 (=70.93/47.18~71.88/43.48), and 1.60~1.69 (=73.91/46.16~74.71/44.12) times higher than those attributed to the aggregation of PNR (i.e., PNR

^{A}) in the LF-VAR model in the short-run (h = 2), medium-run (h = 7), and long-run (h = 12), respectively, based on whether the 3-day re-EDV rate or 14-day readmission rate was chosen to measure ICQ. Similar results, wherein the MF-VAR model generated a higher explanatory power than the LF-VAR model, could also be found for the relationships among PNR, ICQ, and real ICE per admission for regional hospitals and district hospitals. Therefore, the findings of the forecast error variance decompositions shown in Table 4 indicate that the MF-VAR model has a greater explanatory power than the LF-VAR model in the investigation of interdependences between PNR, ICQ, and real ICE per admission for the three different types of hospitals.

## 4. Discussion

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Notes

1 | Inpatient care expenditure per admission was calculated using total inpatient care expenditure divided by total admissions in a specific type of hospital, and it was measured using the 2016 price level (constant 2016 USD). The patient-to-nurse ratio was defined as the mean of number of patients divided by the nurse staffing number within three shifts per day in a specific type of hospital. The quarterly and monthly sample periods start from 2015: Q1 to 2021: Q4 and 2015: M1 to 2021: M12, resulting in a total of 28 and 84 quarterly and monthly observations, respectively. The IQR and JB statistics represent the interquartile range and Jarque-Bera statistics, respectively. ”**”, ”*” denote 1% and 5% significance levels for the rejection of null hypothesis of the normality of time series, respectively. |

2 | All variables are defined in the same way as for Table 1. The lag length is selected based on Bayesian Information Criterion (BIC) with the maximal lag as eight. ”**” and “*” represent 1% and 5% significance levels, respectively. $\sum _{t=1}^{3}\mathrm{C}\_\mathrm{ln}\left({\mathrm{PNR}}_{i}^{k}\right),$, k = MC, RH, and DH define cyclic components of aggregate monthly PNR. |

3 | Quarterly data on cyclical components of quality of care indicators (such as the 3-day EDV rate and 14-day readmission rate), inpatient care expenditure per admission, and monthly data on cyclical components of the patient-to-nurse ratio were used to estimate the MF-VAR model. The monthly data on cyclical components of the patient-to-nurse ratio were aggregated into quarterly data (PNR ^{A}) when the LF-VAR model was estimated. The lag length is selected based on Newey and West’s automatic lag selection with the maximal lag as 3 [52]. “PNR^{A} $\ne >$ EDV”, for example, represents the null hypothesis of non-causality from PNR^{A} to RER. The bold font of PNR denotes the vector of cyclical components of PNR symbolized by [C_lnPNR1, C_lnPNR2, C_lnPNR3]^{’}. “PNR $\ne >$ EDV”, for example, represents the null hypothesis of joint non-causality from the vector of cyclical components of PNR to the cyclical component of EDV. The p values were calculated using the heteroscedasticity-robust parametric bootstrap of Gonçalves and Kilian [53] with 10,000 replications. “***”,”**”,”*” represent 1%, 5%, and 10% significance levels, respectively. |

4 | Figure 1 plots the impulse response functions (IRFs) for monthly horizons h = 0, 1, 2, …, 12 based on the MF-VAR model of quarterly data on cyclical components of quality of care indicators (such as the 3-day EDV rate and 14-day readmission rate), inpatient care expenditure per admission, and three individual monthly cyclical components of the patient-to-nurse ratio symbolized by C_lnPNR1, C_lnPNR2, and C_lnPNR3 in a quarter timespan. The Cholesky decomposition with order PNR1, PNR2, PNR3, EDV (or RAD), and ICE is selected. The sample period covers 2015:Q1–2021:Q4. The responses of variable Y (say, EDV) to 1σ shock in X (say, PNR1) at monthly horizon h is written as “PNR1=>RER”. MC, RH, and DH represent medical centers, regional hospitals, and district hospitals, respectively. Blue shaded areas denote 90% confidence intervals of IRFs based on the Monte Carlo simulation method with 10,000 replications. |

5 | Figure 2 plots the impulse response functions (IRFs) for monthly horizons h = 0, 1, 2, …, 12 based on the MF-VAR model of quarterly data on cyclical components of quality of care indicators (such as the 3-day EDV rate and 14-day readmission rate), inpatient care expenditure per admission, and three individual monthly cyclical components of the patient-to-nurse ratio symbolized by C_lnPNR1, C_lnPNR2, and C_lnPNR3 in a quarter timespan. The Cholesky decomposition with order PNR1, PNR2, PNR3, EDV (or RAD), and ICE is selected. The sample period covers 2015:Q1~2021:Q4. The responses of variable Y (say, EDV) to 1σ shock in X (say, ICE) at monthly horizon h is written as “ICE =>EDV”. Blue shaded areas denote 90% confidence intervals of IRFs based on the Monte Carlo simulation method with 10,000 replications. MC, RH, and DH denote medical centers, regional hospitals, and district hospitals, respectively. |

6 | Notations presented in this table are the same as those used in Table 3. The sum of variance decomposition may not equal 100 due to rounding. |

7 | The directions of arrows were drawn based on the Granger causality tests. The arrows with bold (dot) lines represent significant (insignificant) paths connecting two target variables based on 90% confidence intervals of the impulse-response effects accumulated across a 3-month cycle of a quarter timespan over a 12-month period. MC, RH, and DH denote medical centers, regional hospitals, and district hospitals, respectively. |

8 | EDV and RAD represent the 3-day EDV rate and 14-day readmission rate, respectively. ICE is real inpatient care expenditure per admission at the 2016 price level (USD). PNR symbolizes the patient-to-nurse ratio. MC, RH, and DH represent medical centers, regional hospitals, and district hospitals, respectively. Light blue and grey shaded areas show the post-acute care intervention period and COVID-19 strike waves, respectively. |

9 | All notations used in this figure are the same as for Figure 1. $\mathrm{ln}(\xb7)$ represents the natural logarithm transformation. |

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**Figure 2.**Impulse Response Functions for Inpatient Care Expenditure, Nurse Staffing, and Quality of Care Indicator

^{5}.

**Figure 3.**Mechanisms for activating the vicious cycle of hospital competition. (

**a1**) M1: the PNR origin mechanism for the three different levels of hospitals; (

**a2**) M2: the EDV origin mechanism for medical centers only; (

**a3**) M3: the EDV rebound mechanism for medical centers and regional hospitals; (

**a4**) M4: the readmission rebound mechanism for district hospitals

^{7}.

Panel A: Quarterly DataDescription | Mean | StandardDeviation | Median | IQR | Max | Min | JB Stat |

$\text{Re-emergency-department-visit}\mathrm{rate}\mathrm{in}\mathrm{the}\mathrm{same}\mathrm{hospital}\mathrm{within}3\mathrm{days}\mathrm{after}\mathrm{discharge}\mathrm{at}\mathrm{medical}\mathrm{centers}({\mathrm{EDV}}_{}^{\mathrm{MC}}:\%)$ | 2.489 | 0.139 | 2.483 | 0.149 | 2.807 | 2.208 | 0.077 |

$\text{Re-emergency-department-visit}\mathrm{rate}\mathrm{in}\mathrm{the}\mathrm{same}\mathrm{hospital}\mathrm{within}3\mathrm{days}\mathrm{after}\mathrm{discharge}\mathrm{at}\mathrm{regional}\mathrm{hospitals}({\mathrm{EDV}}_{}^{\mathrm{RH}}:\%)$ | 2.814 | 0.170 | 2.832 | 0.202 | 3.199 | 2.504 | 0.396 |

$\text{Re-emergency-department-visit}\mathrm{rate}\mathrm{in}\mathrm{the}\mathrm{same}\mathrm{hospital}\mathrm{within}3\mathrm{days}\mathrm{after}\mathrm{discharge}\mathrm{at}\mathrm{district}\mathrm{hospitals}({\mathrm{EDV}}_{}^{\mathrm{DH}}:\%)$ | 2.559 | 0.175 | 2.539 | 0.246 | 2.918 | 2.241 | 0.723 |

$\mathrm{Unplanned}\text{re-admission}\mathrm{rate}\mathrm{within}14\mathrm{days}\mathrm{after}\mathrm{discharge}\mathrm{at}\mathrm{medical}\mathrm{centers}({\mathrm{RAD}}_{}^{\mathrm{MC}}:\%)$ | 6.428 | 0.241 | 6.467 | 0.411 | 6.871 | 6.103 | 1.860 |

$\mathrm{Unplanned}\text{re-admission}\mathrm{rate}\mathrm{within}14\mathrm{days}\mathrm{after}\mathrm{discharge}\mathrm{at}\mathrm{regional}\mathrm{hospitals}({\mathrm{RAD}}_{}^{\mathrm{RH}}:\%)$ | 7.259 | 0.228 | 7.239 | 0.413 | 7.675 | 6.947 | 2.198 |

$\mathrm{Unplanned}\text{re-admission}\mathrm{rate}\mathrm{within}14\mathrm{days}\mathrm{after}\mathrm{discharge}\mathrm{at}\mathrm{district}\mathrm{hospitals}({\mathrm{RAD}}_{}^{\mathrm{DH}}:\%)$ | 7.460 | 0.256 | 7.473 | 0.303 | 7.983 | 6.772 | 0.705 |

$\mathrm{Inpatient}\mathrm{care}\mathrm{expenditure}\mathrm{per}\mathrm{admission}\mathrm{at}\mathrm{medical}\mathrm{centers}({\mathrm{ICE}}_{}^{\mathrm{MC}}$:USD, Constant at 2016 price level, USD 1 = TWD 30 $)$ | 2629.971 | 179.855 | 2589.845 | 200.537 | 3078.093 | 2401.06 | 6.270 * |

$\mathrm{Inpatient}\mathrm{care}\mathrm{expenditure}\mathrm{per}\mathrm{admission}\mathrm{at}\mathrm{regional}\mathrm{hospitals}({\mathrm{ICE}}_{}^{\mathrm{RH}}$:USD, Constant at 2016 price level, USD 1 = TWD 30 $)$ | 1826.339 | 139.494 | 1795.955 | 182.638 | 2146.892 | 1649.037 | 4.762 |

$\mathrm{Inpatient}\mathrm{care}\mathrm{expenditure}\mathrm{per}\mathrm{admission}\mathrm{at}\mathrm{district}\mathrm{hospitals}({\mathrm{ICE}}_{}^{\mathrm{DH}}$:USD, Constant at 2016 price level, USD 1 = NTD 30 $)$ | 1679.079 | 97.236 | 1645.423 | 99.390 | 1943.214 | 1587.140 | 13.179 ** |

Panel B: Monthly DataDescription | Mean | StandardDeviation | Median | IQR | Max | Min | JB Stat |

$\text{Patient-to-nurse}\mathrm{ratio}\mathrm{at}\mathrm{acute}\mathrm{care}\mathrm{wards}\mathrm{of}\mathrm{medical}\mathrm{centers}({\mathrm{PNR}}_{}^{\mathrm{MC}}$$)$ | 7.436 | 0.256 | 7.434 | 0.388 | 7.880 | 6.706 | 0.809 |

$\text{Patient-to-nurse}\mathrm{ratio}\mathrm{at}\mathrm{acute}\mathrm{care}\mathrm{wards}\mathrm{of}\mathrm{medical}\mathrm{centers}\mathrm{in}\mathrm{the}1\mathrm{st}\mathrm{month}\mathrm{of}\mathrm{the}\mathrm{observed}\mathrm{quarter}({\mathrm{PNR}}_{1}^{\mathrm{MC}}$$)$ | 7.474 | 0.251 | 7.478 | 0.347 | 7.866 | 7.018 | 1.309 |

$\text{Patient-to-nurse}\mathrm{ratio}\mathrm{at}\mathrm{acute}\mathrm{care}\mathrm{wards}\mathrm{of}\mathrm{medical}\mathrm{centers}\mathrm{in}\mathrm{the}2\mathrm{nd}\mathrm{month}\mathrm{of}\mathrm{observed}\mathrm{quarter}({\mathrm{PNR}}_{2}^{\mathrm{MC}}$$)$ | 7.394 | 0.242 | 7.375 | 0.329 | 7.798 | 6.788 | 0.245 |

$\text{Patient-to-nurse}\mathrm{ratio}\mathrm{at}\mathrm{acute}\mathrm{care}\mathrm{wards}\mathrm{of}\mathrm{medical}\mathrm{centers}\mathrm{in}\mathrm{the}3\mathrm{rd}\mathrm{month}\mathrm{of}\mathrm{observed}\mathrm{quarter}({\mathrm{PNR}}_{3}^{\mathrm{MC}}$$)$ | 7.439 | 0.275 | 7.466 | 0.386 | 7.880 | 6.706 | 0.930 |

$\text{Patient-to-nurse}\mathrm{ratio}\mathrm{at}\mathrm{acute}\mathrm{care}\mathrm{wards}\mathrm{of}\mathrm{regional}\mathrm{hospitals}({\mathrm{PNR}}_{}^{\mathrm{RH}}$$)$ | 9.261 | 0.365 | 9.365 | 0.393 | 10.007 | 7.649 | 75.585 ** |

$\text{Patient-to-nurse}\mathrm{ratio}\mathrm{at}\mathrm{acute}\mathrm{care}\mathrm{wards}\mathrm{of}\mathrm{regional}\mathrm{hospitals}\mathrm{in}\mathrm{the}1\mathrm{st}\mathrm{month}\mathrm{of}\mathrm{observed}\mathrm{quarter}({\mathrm{PNR}}_{1}^{\mathrm{RH}}$$)$ | 9.292 | 0.349 | 9.420 | 0.387 | 9.906 | 8.201 | 10.223 ** |

$\text{Patient-to-nurse}\mathrm{ratio}\mathrm{at}\mathrm{acute}\mathrm{care}\mathrm{wards}\mathrm{of}\mathrm{regional}\mathrm{hospitals}\mathrm{in}\mathrm{the}2\mathrm{nd}\mathrm{month}\mathrm{of}\mathrm{observed}\mathrm{quarter}({\mathrm{PNR}}_{2}^{\mathrm{RH}}$$)$ | 9.205 | 0.325 | 9.202 | 0.507 | 9.928 | 8.568 | 0.288 |

$\text{Patient-to-nurse}\mathrm{ratio}\mathrm{at}\mathrm{acute}\mathrm{care}\mathrm{wards}\mathrm{of}\mathrm{regional}\mathrm{hospitals}\mathrm{in}\mathrm{the}3\mathrm{rd}\mathrm{month}\mathrm{of}\mathrm{observed}\mathrm{quarter}({\mathrm{PNR}}_{3}^{\mathrm{RH}}$$)$ | 9.286 | 0.421 | 9.372 | 0.315 | 10.007 | 7.649 | 68.002 ** |

$\text{Patient-to-nurse}\mathrm{ratio}\mathrm{at}\mathrm{acute}\mathrm{care}\mathrm{wards}\mathrm{of}\mathrm{district}\mathrm{hospitals}({\mathrm{PNR}}_{}^{\mathrm{DH}}$$)$ | 7.573 | 0.399 | 7.602 | 0.314 | 8.314 | 5.943 | 51.089 ** |

$\text{Patient-to-nurse}\mathrm{ratio}\mathrm{at}\mathrm{acute}\mathrm{care}\mathrm{wards}\mathrm{of}\mathrm{district}\mathrm{hospitals}\mathrm{in}\mathrm{the}1\mathrm{st}\mathrm{month}\mathrm{of}\mathrm{observed}\mathrm{quarter}({\mathrm{PNR}}_{1}^{\mathrm{DH}}$$)$ | 7.605 | 0.392 | 7.614 | 0.337 | 8.314 | 6.333 | 13.500 ** |

$\text{Patient-to-nurse}\mathrm{ratio}\mathrm{at}\mathrm{acute}\mathrm{care}\mathrm{wards}\mathrm{of}\mathrm{district}\mathrm{hospitals}\mathrm{in}\mathrm{the}2\mathrm{nd}\mathrm{month}\mathrm{of}\mathrm{observed}\mathrm{quarter}({\mathrm{PNR}}_{2}^{\mathrm{DH}}$$)$ | 7.524 | 0.356 | 7.541 | 0.410 | 8.309 | 6.733 | 0.259 |

$\text{Patient-to-nurse}\mathrm{ratio}\mathrm{at}\mathrm{acute}\mathrm{care}\mathrm{wards}\mathrm{of}\mathrm{district}\mathrm{hospitals}\mathrm{in}\mathrm{the}3\mathrm{rd}\mathrm{month}\mathrm{of}\mathrm{observed}\mathrm{quarter}({\mathrm{PNR}}_{3}^{\mathrm{DH}}$$)$ | 7.589 | 0.453 | 7.669 | 0.314 | 8.295 | 5.943 | 38.238 ** |

Panel A: Quarterly Data | |||||||

Levels | Cyclical Components | ||||||

Mean | StandardDeviation | Constant(C) | Constant+Trend (T) | Mean | StandardDeviation | WithoutC+T | |

$\mathrm{ln}\left({\mathrm{EDV}}_{}^{\mathrm{MC}}\right)$ | 0.911 | 0.056 | −3.473 * | −3.405 | 0.000 | 0.052 | −3.952 ** |

$\mathrm{ln}\left({\mathrm{EDV}}_{}^{\mathrm{RH}}\right)$ | 1.033 | 0.060 | −2.329 | −3.330 | 0.000 | 0.050 | −3.874 ** |

$\mathrm{ln}\left({\mathrm{EDV}}_{}^{\mathrm{DH}}\right)$ | 0.937 | 0.068 | −3.247 * | −3.341 | 0.000 | 0.060 | −3.917 ** |

$\mathrm{ln}\left({\mathrm{RAD}}_{}^{\mathrm{MC}}\right)$ | 1.860 | 0.038 | −2.518 | −2.455 | 0.000 | 0.033 | −2.904 ** |

$\mathrm{ln}\left({\mathrm{RAD}}_{}^{\mathrm{RH}}\right)$ | 1.982 | 0.031 | −2.698 | −3.462 | 0.000 | 0.025 | −3.719 ** |

$\mathrm{ln}\left({\mathrm{RAD}}_{}^{\mathrm{DH}}\right)$ | 2.009 | 0.035 | −3.370 * | −4.306 * | 0.000 | 0.031 | −4.250 ** |

$\mathrm{ln}\left({\mathrm{ICE}}_{}^{\mathrm{MC}}\right)$ | 7.873 | 0.066 | 0.036 | −2.412 | 0.000 | 0.025 | −4.347 ** |

$\mathrm{ln}\left({\mathrm{ICE}}_{}^{\mathrm{RH}}\right)$ | 7.507 | 0.074 | 1.087 | −1.824 | 0.000 | 0.025 | −3.164 ** |

$\mathrm{ln}\left({\mathrm{ICE}}_{}^{\mathrm{DH}}\right)$ | 7.424 | 0.056 | 0.676 | −1.303 | 0.000 | 0.024 | −2.221 * |

Panel B: Monthly Data | |||||||

Levels | Cyclical Components | ||||||

Mean | StandardDeviation | Constant(C) | Constant+Trend (T) | Mean | StandardDeviation | WithoutC+T | |

$\mathrm{ln}\left({\mathrm{PNR}}_{}^{\mathrm{MC}}\right)$ | 2.006 | 0.035 | −4.253 ** | −4.611 ** | 0.000 | 0.027 | −5.740 ** |

$\mathrm{ln}\left({\mathrm{PNR}}_{}^{\mathrm{RH}}\right)$ | 2.225 | 0.041 | −4.434 ** | −4.596 ** | 0.000 | 0.035 | −5.397 ** |

$\mathrm{ln}\left({\mathrm{PNR}}_{}^{\mathrm{DH}}\right)$ | 2.023 | 0.055 | −3.434 * | −5.079 ** | 0.000 | 0.037 | −5.515 ** |

Panel C: Aggregate Monthly Data | |||||||

Levels | Cyclical Components | ||||||

Mean | StandardDeviation | Constant(C) | Constant+Trend (T) | Mean | StandardDeviation | WithoutC+T | |

${\sum}_{t=1}^{3}{\mathrm{C}\_\mathrm{ln}(\mathrm{PNR}}_{t}^{\mathrm{MC}}})/3$ | -------- | -------- | -------- | -------- | 0.000 | 0.057 | −5.383 ** |

${\sum}_{t=1}^{3}{\mathrm{C}\_\mathrm{ln}(\mathrm{PNR}}_{t}^{\mathrm{RH}}})/3$ | -------- | -------- | -------- | -------- | 0.000 | 0.071 | −6.482 ** |

${\sum}_{t=1}^{3}{\mathrm{C}\_\mathrm{ln}(\mathrm{PNR}}_{t}^{\mathrm{DH}}})/3$ | -------- | -------- | -------- | -------- | 0.000 | 0.073 | −2.231 * |

Panel A: | Re-Emergency-Department-Visit Rate in the Same Hospital within 3 Days after Discharge as the Quality of Care Indicator | |||||||||

Types of | MF-VAR Model | LF-VAR Model | ||||||||

Hospitals | Null Hypothesis | χ2 | p Value | Null Hypothesis | χ2 | p Value | ||||

EDV | $\ne >$ | PNR | 10.935 | 0.090 * | EDV | $\ne >$ | PNR^{A} | 1.977 | 0.372 | |

ICE | $\ne >$ | PNR | 3.213 | 0.782 | ICE | $\ne >$ | PNR^{A} | 2.346 | 0.309 | |

Medical | PNR | $\ne >$ | EDV | 16.029 | 0.014 ** | PNR^{A} | $\ne >$ | EDV | 3.075 | 0.215 |

Centers | ICE | $\ne >$ | EDV | 3.254 | 0.776 | ICE | $\ne >$ | EDV | 2.841 | 0.242 |

PNR | $\ne >$ | ICE | 14.095 | 0.029 ** | PNR^{A} | $\ne >$ | ICE | 1.291 | 0.524 | |

EDV | $\ne >$ | ICE | 14.635 | 0.023 ** | EDV | $\ne >$ | ICE | 7.013 | 0.030 ** | |

EDV | $\ne >$ | PNR | 12.035 | 0.061 * | EDV | $\ne >$ | PNR^{A} | 0.520 | 0.771 | |

ICE | $\ne >$ | PNR | 4.706 | 0.582 | ICE | $\ne >$ | PNR^{A} | 5.716 | 0.057 * | |

Regional | PNR | $\ne >$ | EDV | 13.365 | 0.038 ** | PNR^{A} | $\ne >$ | EDV | 3.121 | 0.210 |

Hospitals | ICE | $\ne >$ | EDV | 6.021 | 0.421 | ICE | $\ne >$ | EDV | 3.564 | 0.168 |

PNR | $\ne >$ | ICE | 18.311 | 0.005 *** | PNR^{A} | $\ne >$ | ICE | 1.871 | 0.392 | |

EDV | $\ne >$ | ICE | 4.700 | 0.583 | EDV | $\ne >$ | ICE | 0.366 | 0.833 | |

RER | $\ne >$ | PNR | 8.592 | 0.198 | EDV | $\ne >$ | PNR^{A} | 0.557 | 0.757 | |

ICE | $\ne >$ | PNR | 4.797 | 0.570 | ICE | $\ne >$ | PNR^{A} | 4.162 | 0.125 | |

District | PNR | $\ne >$ | EDV | 12.614 | 0.049 ** | PNR^{A} | $\ne >$ | EDV | 5.049 | 0.080 * |

Hospitals | ICE | $\ne >$ | EDV | 2.749 | 0.840 | ICE | $\ne >$ | EDV | 0.540 | 0.763 |

PNR | $\ne >$ | ICE | 13.519 | 0.035 ** | PNR^{A} | $\ne >$ | ICE | 1.438 | 0.487 | |

RER | $\ne >$ | ICE | 7.127 | 0.309 | EDV | $\ne >$ | ICE | 5.761 | 0.056 * | |

Panel B: | Unplanned Re-Admission Rate within 14 Days after Discharge as the Quality of Care Indicator | |||||||||

Types of | MF-VAR Model | LF-VAR Model | ||||||||

Hospitals | Null Hypothesis | χ2 | p Value | Null Hypothesis | χ2 | p Value | ||||

RAD | $\ne >$ | PNR | 8.933 | 0.177 | RAD | $\ne >$ | PNR^{A} | 5.054 | 0.080 * | |

ICE | $\ne >$ | PNR | 6.004 | 0.423 | ICE | $\ne >$ | PNR^{A} | 4.763 | 0.092 * | |

Medical | PNR | $\ne >$ | RAD | 7.822 | 0.251 | PNR^{A} | $\ne >$ | RAD | 6.941 | 0.031 ** |

Centers | ICE | $\ne >$ | RAD | 8.972 | 0.175 | ICE | $\ne >$ | RAD | 7.418 | 0.024 ** |

PNR | $\ne >$ | ICE | 13.079 | 0.042 ** | PNR^{A} | $\ne >$ | ICE | 2.848 | 0.241 | |

RAD | $\ne >$ | ICE | 4.386 | 0.625 | RAD | $\ne >$ | ICE | 1.159 | 0.560 | |

RAD | $\ne >$ | PNR | 9.952 | 0.127 | RER | $\ne >$ | PNR^{A} | 0.281 | 0.869 | |

ICE | $\ne >$ | PNR | 5.990 | 0.424 | ICE | $\ne >$ | PNR^{A} | 3.839 | 0.147 | |

Regional | PNR | $\ne >$ | RAD | 8.200 | 0.224 | PNR^{A} | $\ne >$ | RAD | 6.799 | 0.033 ** |

Hospitals | ICE | $\ne >$ | RAD | 8.635 | 0.195 | ICE | $\ne >$ | RAD | 4.140 | 0.126 |

PNR | $\ne >$ | ICE | 13.674 | 0.033 ** | PNR^{A} | $\ne >$ | ICE | 3.684 | 0.158 | |

RAD | $\ne >$ | ICE | 3.943 | 0.684 | RER | $\ne >$ | ICE | 2.161 | 0.339 | |

RAD | $\ne >$ | PNR | 13.511 | 0.036 ** | RER | $\ne >$ | PNR^{A} | 1.753 | 0.416 | |

ICE | $\ne >$ | PNR | 8.614 | 0.196 | ICE | $\ne >$ | PNR^{A} | 4.366 | 0.113 | |

District | PNR | $\ne >$ | RAD | 4.441 | 0.617 | PNR^{A} | $\ne >$ | RAD | 2.971 | 0.226 |

Hospitals | ICE | $\ne >$ | RAD | 5.582 | 0.472 | ICE | $\ne >$ | RAD | 1.988 | 0.370 |

PNR | $\ne >$ | ICE | 12.169 | 0.058 * | PNR^{A} | $\ne >$ | ICE | 1.653 | 0.438 | |

RAD | $\ne >$ | ICE | 2.303 | 0.890 | RER | $\ne >$ | ICE | 0.765 | 0.682 |

Quality | Model | Medical Centers (%) | Regional Hospitals (%) | District Hospitals (%) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Variable | Horizon | h = 2 | h = 7 | h = 12 | Variable | Horizon | h = 2 | h = 7 | h = 12 | Variable | Horizon | h = 2 | h = 7 | h = 12 | ||

Panel A: RER as Quality Indicator | EDV | EDV ICE PNR ^{A} | 86.320.22 13.46 | 63.67 6.80 29.53 | 61.75 9.52 28.73 | EDV | EDV ICE PNR ^{A} | 82.96 7.04 10.0 | 64.87 16.37 18.76 | 58.74 15.33 25.93 | EDV | EDV ICE PNR ^{A} | 88.71 1.79 9.49 | 45.95 9.40 44.65 | 33.08 20.12 46.79 | |

LF-VAR | ICE | EDV ICE PNR ^{A} | 0.38 45.28 54.34 | 30.38 26.15 43.48 | 30.51 25.37 44.12 | ICE | EDV ICE PNR ^{A} | 0.81 57.91 41.28 | 4.79 55.43 39.78 | 7.17 53.07 39.75 | ICE | EDV ICE PNR ^{A} | 0.58 53.66 45.76 | 19.57 33.91 46.53 | 19.38 35.37 45.25 | |

PNR ^{A} | EDV ICE PNR ^{A} | 8.13 0.69 91.19 | 8.87 11.78 79.35 | 10.57 12.15 77.27 | PNR ^{A} | EDV ICE PNR ^{A} | 0.01 7.32 92.67 | 9.94 27.75 62.32 | 11.55 27.13 61.31 | PNR ^{A} | EDV ICE PNR ^{A} | 0.06 3.86 96.08 | 1.19 23.89 74.92 | 4.89 25.31 69.80 | ||

EDV | EDV ICE PNR=ΣPNR_{i} | 78.28 0.03 21.70 | 53.24 0.88 45.87 | 51.96 1.59 46.45 | EDV | EDV ICE PNR=ΣPNR_{i} | 91.76 1.80 6.45 | 62.08 4.82 33.11 | 45.48 8.05 46.47 | EDV | EDV ICE PNR=ΣPNR_{i} | 74.44 0.37 25.20 | 42.62 1.18 56.19 | 36.38 2.41 61.21 | ||

ICE | EDV ICE PNR=ΣPNR_{i} | 0.40 25.72 73.87 | 16.42 11.71 71.88 | 15.23 10.05 74.71 | ICE | EDV ICE PNR=ΣPNR_{i} | 2.22 47.92 49.88 | 6.22 40.84 52.94 | 7.69 31.27 61.03 | ICE | EDV ICE PNR=ΣPNR_{i} | 0.60 25.28 74.12 | 6.57 14.65 78.77 | 5.97 13.03 80.99 | ||

MF-VAR | PNR _{1} | EDV ICE PNR=ΣPNR_{i} | 0.88 0.29 98.84 | 1.44 1.02 97.53 | 2.23 1.39 96.38 | PNR _{1} | EDV ICE PNR=ΣPNR_{i} | 9.88 3.16 86.97 | 8.52 8.00 83.49 | 15.74 6.56 77.71 | PNR _{1} | EDV ICE PNR=ΣPNR_{i} | 4.76 0.72 94.52 | 3.91 2.68 93.40 | 3.60 3.64 92.77 | |

PNR _{2} | EDV ICE PNR=ΣPNR_{i} | 32.33 0.00 67.66 | 27.31 2.83 69.85 | 29.15 2.67 68.18 | PNR _{2} | EDV ICE PNR=ΣPNR_{i} | 11.67 0.30 88.03 | 11.67 18.31 70.01 | 9.69 14.84 75.47 | PNR _{2} | EDV ICE PNR=ΣPNR_{i} | 2.62 0.00 97.38 | 3.80 4.32 91.88 | 4.10 4.09 91.80 | ||

PNR _{3} | EDV ICE PNR=ΣPNR_{i} | 0.13 0.00 99.88 | 2.29 1.29 96.43 | 2.29 1.62 96.09 | PNR _{3} | EDV ICE PNR=ΣPNR_{i} | 5.23 0.70 94.08 | 13.00 7.94 79.07 | 16.19 4.86 78.95 | PNR _{3} | EDV ICE PNR=ΣPNR_{i} | 0.67 0.00 99.32 | 3.88 2.96 93.16 | 3.29 3.34 93.36 | ||

Quality | Model | Variable | Horizon | h = 2 | h = 7 | h = 12 | Variable | Horizon | h = 2 | h = 7 | h = 12 | Variable | Horizon | h = 2 | h = 7 | h = 12 |

Panel B: URR as Quality Indicator | RAD | RAD ICE PNR ^{A} | 83.73 8.14 8.13 | 73.30 15.14 11.56 | 71.73 15.87 12.40 | RAD | RAD ICE PNR ^{A} | 75.56 12.24 12.20 | 51.37 24.29 24.34 | 46.51 27.97 25.51 | RAD | RAD ICE PNR ^{A} | 89.73 9.58 0.69 | 56.37 20.43 23.21 | 43.57 27.87 28.56 | |

LF-VAR | ICE | RAD ICE PNR ^{A} | 2.55 37.82 59.63 | 19.93 32.89 47.18 | 19.43 34.41 46.16 | ICE | RAD ICE PNR ^{A} | 2.08 66.07 31.84 | 3.13 58.67 38.21 | 3.39 58.37 38.23 | ICE | RAD ICE PNR ^{A} | 0.50 59.39 40.10 | 4.00 53.57 42.43 | 4.08 53.93 41.99 | |

PNR ^{A} | RAD ICE PNR ^{A} | 14.55 0.17 85.28 | 14.89 29.15 55.96 | 14.69 29.28 56.04 | PNR ^{A} | RAD ICE PNR ^{A} | 1.62 3.33 95.05 | 1.55 28.30 70.15 | 2.18 32.03 65.78 | PNR ^{A} | RAD ICE PNR ^{A} | 0.21 7.04 92.75 | 2.76 32.36 64.87 | 3.72 35.16 61.12 | ||

RAD | RAD ICE PNR=ΣPNR_{i} | 60.54 2.27 37.20 | 50.90 5.03 44.06 | 47.97 5.24 46.80 | RAD | RAD ICE PNR=ΣPNR_{i} | 73.70 4.72 21.57 | 45.46 8.05 46.49 | 38.00 12.14 49.87 | RAD | RAD ICE PNR=ΣPNR_{i} | 81.99 6.11 11.89 | 56.39 15.33 28.29 | 48.59 20.14 31.27 | ||

ICE | RAD ICE PNR=ΣPNR_{i} | 2.99 22.94 74.07 | 16.21 12.87 70.93 | 14.00 12.08 73.91 | ICE | RAD ICE PNR=ΣPNR_{i} | 7.23 56.11 36.65 | 8.59 35.53 55.87 | 8.50 30.44 61.06 | ICE | RAD ICE PNR=ΣPNR_{i} | 0.67 49.48 49.86 | 7.19 32.90 59.90 | 10.06 30.98 58.95 | ||

MF-VAR | PNR _{1} | RAD ICE PNR=ΣPNR_{i} | 0.59 0.05 99.37 | 6.37 5.54 88.09 | 6.66 8.03 85.31 | PNR _{1} | RAD ICE PNR=ΣPNR_{i} | 16.34 1.25 82.42 | 11.66 9.88 78.46 | 9.22 9.43 81.35 | PNR _{1} | RAD ICE PNR=ΣPNR_{i} | 16.46 2.33 81.21 | 15.23 9.89 74.88 | 13.06 15.41 71.52 | |

PNR _{2} | RAD ICE PNR=ΣPNR_{i} | 16.47 0.65 82.87 | 15.16 14.33 70.51 | 12.61 9.87 77.53 | PNR _{2} | RAD ICE PNR=ΣPNR_{i} | 0.02 3.62 96.37 | 12.73 15.16 72.11 | 10.13 13.02 76.85 | PNR _{2} | RAD ICE PNR=ΣPNR_{i} | 2.50 2.23 95.27 | 22.17 11.29 66.54 | 21.45 10.76 67.79 | ||

PNR _{3} | RAD ICE PNR=ΣPNR_{i} | 0.17 0.56 99.27 | 6.98 4.28 88.74 | 7.79 4.85 87.35 | PNR _{3} | RAD ICE PNR=ΣPNR_{i} | 0.91 0.07 99.02 | 8.61 6.45 84.93 | 9.49 5.81 84.70 | PNR _{3} | RAD ICE PNR=ΣPNR_{i} | 0.48 0.26 99.25 | 16.06 5.07 78.87 | 15.53 7.37 77.09 |

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## Share and Cite

**MDPI and ACS Style**

Chen, W.-Y.
On the Relationships among Nurse Staffing, Inpatient Care Quality, and Hospital Competition under the Global Budget Payment Scheme of Taiwan’s National Health Insurance System: Mixed Frequency VAR Analyses. *Systems* **2022**, *10*, 187.
https://doi.org/10.3390/systems10050187

**AMA Style**

Chen W-Y.
On the Relationships among Nurse Staffing, Inpatient Care Quality, and Hospital Competition under the Global Budget Payment Scheme of Taiwan’s National Health Insurance System: Mixed Frequency VAR Analyses. *Systems*. 2022; 10(5):187.
https://doi.org/10.3390/systems10050187

**Chicago/Turabian Style**

Chen, Wen-Yi.
2022. "On the Relationships among Nurse Staffing, Inpatient Care Quality, and Hospital Competition under the Global Budget Payment Scheme of Taiwan’s National Health Insurance System: Mixed Frequency VAR Analyses" *Systems* 10, no. 5: 187.
https://doi.org/10.3390/systems10050187