# Hospital Readmissions Data Analysis and Recommendations for Reduction

### Background

In October 2012, the US government's Center for Medicare and Medicaid Services (CMS) began reducing Medicare payments for Inpatient Prospective Payment System hospitals with excess readmissions. Excess readmissions are measured by a ratio, by dividing a hospital’s number of “predicted” 30-day readmissions for heart attack, heart failure, and pneumonia by the number that would be “expected,” based on an average hospital with similar patients. A ratio greater than 1 indicates excess readmissions.

### Exercise Directions

In this exercise, you will:

• critique a preliminary analysis of readmissions data and recommendations (provided below) for reducing the readmissions rate
• construct a statistically sound analysis and make recommendations of your own

More instructions provided below. Include your work in this notebook and submit to your Github account.

### Resources



In :

%matplotlib inline

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import bokeh.plotting as bkp
from mpl_toolkits.axes_grid1 import make_axes_locatable




In :



## Preliminary Analysis



In :

# deal with missing and inconvenient portions of data




/anaconda/lib/python3.6/site-packages/pandas/core/indexing.py:537: SettingWithCopyWarning:
A value is trying to be set on a copy of a slice from a DataFrame.
Try using .loc[row_indexer,col_indexer] = value instead

See the caveats in the documentation: http://pandas.pydata.org/pandas-docs/stable/indexing.html#indexing-view-versus-copy
self.obj[item] = s




In :

# generate a scatterplot for number of discharges vs. excess rate of readmissions
# lists work better with matplotlib scatterplot function
x = [a for a in clean_hospital_read_df['Number of Discharges'][81:-3]]

fig, ax = plt.subplots(figsize=(8,5))
ax.scatter(x, y,alpha=0.2)

ax.fill_between([0,350], 1.15, 2, facecolor='red', alpha = .15, interpolate=True)
ax.fill_between([800,2500], .5, .95, facecolor='green', alpha = .15, interpolate=True)

ax.set_xlim([0, max(x)])
ax.set_xlabel('Number of discharges', fontsize=12)
ax.set_title('Scatterplot of number of discharges vs. excess rate of readmissions', fontsize=14)

ax.grid(True)
fig.tight_layout()






## Preliminary Report

A. Initial observations based on the plot above

• Overall, rate of readmissions is trending down with increasing number of discharges
• With lower number of discharges, there is a greater incidence of excess rate of readmissions (area shaded red)
• With higher number of discharges, there is a greater incidence of lower rates of readmissions (area shaded green)

B. Statistics

• In hospitals/facilities with number of discharges < 100, mean excess readmission rate is 1.023 and 63% have excess readmission rate greater than 1
• In hospitals/facilities with number of discharges > 1000, mean excess readmission rate is 0.978 and 44% have excess readmission rate greater than 1

C. Conclusions

• There is a significant correlation between hospital capacity (number of discharges) and readmission rates.
• Smaller hospitals/facilities may be lacking necessary resources to ensure quality care and prevent complications that lead to readmissions.

D. Regulatory policy recommendations

• Hospitals/facilties with small capacity (< 300) should be required to demonstrate upgraded resource allocation for quality care to continue operation.
• Directives and incentives should be provided for consolidation of hospitals and facilities to have a smaller number of them with higher capacity and number of discharges.

### Exercise

Include your work on the following in this notebook and submit to your Github account.

A. Do you agree with the above analysis and recommendations? Why or why not?

B. Provide support for your arguments and your own recommendations with a statistically sound analysis:

1. Setup an appropriate hypothesis test.
2. Compute and report the observed significance value (or p-value).
3. Report statistical significance for $\alpha$ = .01.
4. Discuss statistical significance and practical significance. Do they differ here? How does this change your recommendation to the client?
5. Look at the scatterplot above.
• What are the advantages and disadvantages of using this plot to convey information?
• Construct another plot that conveys the same information in a more direct manner.

You can compose in notebook cells using Markdown:



In :



### The dataset



In :




Out:

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Hospital Name
Provider Number
State
Measure Name
Number of Discharges
Footnote
Start Date
End Date

8126
NAPLES COMMUNITY HOSPITAL
100018
FL
2716
NaN
0.9804
5.2
5.3
141.0
07/01/2010
06/30/2013

6643
COMMUNITY MEDICAL CENTER
310041
NJ
2740
NaN
1.0003
22.7
22.7
623.0
07/01/2010
06/30/2013

1892
FLORIDA HOSPITAL
100007
FL
3570
NaN
1.0896
24.5
22.5
879.0
07/01/2010
06/30/2013

13615
NEW ENGLAND BAPTIST HOSPITAL
220088
MA
3980
NaN
0.7682
3.7
4.8
142.0
07/01/2010
06/30/2013

13666
HOSPITAL FOR SPECIAL SURGERY
330270
NY
6793
NaN
0.7379
3.9
5.3
258.0
07/01/2010
06/30/2013



#### We are interested in the hospitals with 'Excess Readmission Ratio' > 0



In :

ranges = [0, 1, 2]

print("+---------------------------------------------------|---------------+")
print("| Total hospitals with excess readmission ratio > 0 | %s        |" %(format(tot_hosp_valid, ',')))
print("|---------------------------------------------------|---------------|")
print("|                                               min | %.2f          |" % excess_read_ratio_min)
print("|                                              mean | %.2f          |" % excess_read_ratio_mean)
print("|---------------------------------------------------|---------------|")
print("| Hospitals with excess readmission ratio <= 1      | %s (%.2f%%)|" %(format(h1, ','), (100*(h1/(h1+h2)))))
print("| Hospitals with excess readmission ratio > 1 & <=2 | %s (%.2f%%)|" %(format(h2, ','), (100*(h2/(h1+h2)))))
print("+---------------------------------------------------|---------------+")




+---------------------------------------------------|---------------+
| Total hospitals with excess readmission ratio > 0 | 11,497        |
|---------------------------------------------------|---------------|
| Excess readimission ratio:                    max | 1.91          |
|                                               min | 0.55          |
|                                              mean | 1.01          |
|---------------------------------------------------|---------------|
| Hospitals with excess readmission ratio <= 1      | 5,558 (48.34%)|
| Hospitals with excess readmission ratio > 1 & <=2 | 5,939 (51.66%)|
+---------------------------------------------------|---------------+



### 'Excess Readmission Ratio' in hospitals with number of discharges < 100



In :

print("+---------------------------------------------------+--------------+")
print("| Hospitals with discharges < 100          Total    |   %s      |" %(format(tot_h100,',')))
print("|                                           Mean    |   %.3f      |" %med_h100)
print("|               Have excess readmission rate > 1    |   %.2f%%     |" %(100*(tot_h100_exc_gt_one/tot_h100)))
print("+---------------------------------------------------+--------------+")




+---------------------------------------------------+--------------+
| Hospitals with discharges < 100          Total    |   1,188      |
|                                           Mean    |   1.023      |
|               Have excess readmission rate > 1    |   63.22%     |
+---------------------------------------------------+--------------+



### 'Excess Readmission Ratio' in hospitals with number of discharges > 1000



In :

print("+---------------------------------------------------+--------------+")
print("| Hospitals with discharges > 1000          Total   |     %d      |" %tot_h1000)
print("|                                            Mean   |   %.3f      |" %med_h1000)
print("|                Have excess readmission rate > 1   |   %.2f%%     |" %(100*(tot_h1000_exc_gt_one/tot_h1000)))
print("+---------------------------------------------------+--------------+")




+---------------------------------------------------+--------------+
| Hospitals with discharges > 1000          Total   |     463      |
|                                            Mean   |   0.978      |
|                Have excess readmission rate > 1   |   44.49%     |
+---------------------------------------------------+--------------+



### How to find out if there is a correlation between hospital capacity (number of discharges) and readmission rates.



In :

from scipy.stats import pearsonr




In :




In :




Out:

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Number of Discharges

1832
25
1.0914

1699
27
1.0961

1774
28
1.0934

1853
29
1.0908

1290
30
1.1123



### Pearson correlation

"The Pearson correlation coefficient measures the linear relationship between two datasets. ... Like other correlation coefficients, this one varies between -1 and +1 with 0 implying no correlation. Correlations of -1 or +1 imply an exact linear relationship. Positive correlations imply that as x increases, so does y. Negative correlations imply that as x increases, y decreases.".
Source: https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.stats.pearsonr.html

### A - All hospitals



In :

pearson, pvalue = pearsonr(df_temp[['Number of Discharges']], df_temp[['Excess Readmission Ratio']])
pvalue1 = pvalue
pearson1 = pearson
print("+----------------------------------------------------------+")
print("| 'Number of Discharges' versus 'Excess Readmission Ratio' |")
print("| for all hospitals:                                       |")
print("|----------------------------------------------------------|")
print("| Pearson Correlation = %.4f                            |" %pearson[])
print("| p-value = %.30f               |" %pvalue[])
print("+----------------------------------------------------------+")




+----------------------------------------------------------+
| 'Number of Discharges' versus 'Excess Readmission Ratio' |
| for all hospitals:                                       |
|----------------------------------------------------------|
| Pearson Correlation = -0.0974                            |
| p-value = 0.000000000000000000000000122255               |
+----------------------------------------------------------+




In :

sns.set(color_codes=True)
sns.regplot(x="Number of Discharges", y="Excess Readmission Ratio", data=df_temp)




Out:

<matplotlib.axes._subplots.AxesSubplot at 0x1175f4c50>



#### Calculating Pearson correlation and p-value without outlier hospitals. Let's verify only the hopitals with:

• Excess Readmission Ratio < 1.6
• Number of Discharges < 2,000


In :

df_temp_a = df_temp.loc[(df_temp['Number of Discharges'] < 2000) & (df_temp['Excess Readmission Ratio'] < 1.6)]
pearson, pvalue = pearsonr(df_temp_a[['Number of Discharges']], df_temp_a[['Excess Readmission Ratio']])
print("+----------------------------------------------------------+")
print("| 'Number of Discharges' versus 'Excess Readmission Ratio' |")
print("| for all hospitals with:                                  |")
print("|    a) Excess Readmission Ratio < 1.6                     |")
print("|    b) Number of Discharges < 2,000                       |")
print("|----------------------------------------------------------|")
print("| Pearson Correlation = %.4f                            |" %pearson[])
print("| p-value = %.30f               |" %pvalue[])
print("+----------------------------------------------------------+")




+----------------------------------------------------------+
| 'Number of Discharges' versus 'Excess Readmission Ratio' |
| for all hospitals with:                                  |
|    a) Excess Readmission Ratio < 1.6                     |
|    b) Number of Discharges < 2,000                       |
|----------------------------------------------------------|
| Pearson Correlation = -0.0989                            |
| p-value = 0.000000000000000000000000024326               |
+----------------------------------------------------------+




In :

sns.set(color_codes=True)
sns.regplot(x="Number of Discharges", y="Excess Readmission Ratio", data=df_temp_a)




Out:

<matplotlib.axes._subplots.AxesSubplot at 0x1175bfd68>



### B - Hospitals with discharges < 100



In :

df_temp_dischg100 = df_temp.loc[df_temp['Number of Discharges'] < 100]
pearson,pvalue = pearsonr(df_temp_dischg100[['Number of Discharges']], df_temp_dischg100[['Excess Readmission Ratio']])
pvalue2 = pvalue
pearson2 = pearson
print("+----------------------------------------------------------+")
print("| 'Number of Discharges' versus 'Excess Readmission Ratio' |")
print("| for hospitals with discharges < 100:                     |")
print("|----------------------------------------------------------|")
print("| Pearson Correlation = %.4f                            |" %pearson[])
print("| p-value = %.20f                         |" %pvalue[])
print("+----------------------------------------------------------+")




+----------------------------------------------------------+
| 'Number of Discharges' versus 'Excess Readmission Ratio' |
| for hospitals with discharges < 100:                     |
|----------------------------------------------------------|
| Pearson Correlation = -0.2446                            |
| p-value = 0.00000000000000001196                         |
+----------------------------------------------------------+




In :

sns.regplot(x="Number of Discharges", y="Excess Readmission Ratio", data=df_dischg_lt100)




Out:

<matplotlib.axes._subplots.AxesSubplot at 0x117837b38>



#### Calculating Pearson correlation and p-value without outlier hospitals. Let's verify only the hopitals with:

• Excess Readmission Ratio < 1.2
• Number of Discharges > 40 (this dataset has already Number of Discharges < 100)


In :

df_temp_b = df_temp_dischg100.loc[(df_temp_dischg100['Number of Discharges'] > 40) & (df_temp_dischg100['Excess Readmission Ratio'] < 1.2)]
pearson, pvalue = pearsonr(df_temp_b[['Number of Discharges']], df_temp_b[['Excess Readmission Ratio']])
print("+----------------------------------------------------------+")
print("| 'Number of Discharges' versus 'Excess Readmission Ratio' |")
print("| for all hospitals with:                                  |")
print("|    a) Excess Readmission Ratio < 1.2                     |")
print("|    b) Number of Discharges < 40                          |")
print("|----------------------------------------------------------|")
print("| Pearson Correlation = %.4f                            |" %pearson[])
print("| p-value = %.30f               |" %pvalue[])
print("+----------------------------------------------------------+")




+----------------------------------------------------------+
| 'Number of Discharges' versus 'Excess Readmission Ratio' |
| for all hospitals with:                                  |
|    a) Excess Readmission Ratio < 1.2                     |
|    b) Number of Discharges < 40                          |
|----------------------------------------------------------|
| Pearson Correlation = -0.2933                            |
| p-value = 0.000000000000000000000005975351               |
+----------------------------------------------------------+




In :

sns.set(color_codes=True)
sns.regplot(x="Number of Discharges", y="Excess Readmission Ratio", data=df_temp_b)




Out:

<matplotlib.axes._subplots.AxesSubplot at 0x11762a278>



### C - Hospitals with discharges > 1,000



In :

df_temp_dischg1000 = df_temp.loc[df_temp['Number of Discharges'] > 1000]
pearson, pvalue = pearsonr(df_temp_dischg1000[['Number of Discharges']], df_temp_dischg1000[['Excess Readmission Ratio']])
pvalue3 = pvalue
pearson3 = pearson
print("+----------------------------------------------------------+")
print("| 'Number of Discharges' versus 'Excess Readmission Ratio' |")
print("| for hospitals with discharges > 1000:                    |")
print("|----------------------------------------------------------|")
print("| Pearson Correlation = %.4f                            |" %pearson[])
print("| p-value = %.20f                         |" %pvalue[])
print("+----------------------------------------------------------+")




+----------------------------------------------------------+
| 'Number of Discharges' versus 'Excess Readmission Ratio' |
| for hospitals with discharges > 1000:                    |
|----------------------------------------------------------|
| Pearson Correlation = -0.0793                            |
| p-value = 0.08839944177056585639                         |
+----------------------------------------------------------+




In :

sns.regplot(x="Number of Discharges", y="Excess Readmission Ratio", data=df_dischg_gt1000)




Out:

<matplotlib.axes._subplots.AxesSubplot at 0x11c09abe0>



#### Calculating Pearson correlation and p-value without outlier hospitals. Let's verify only the hopitals with:

• Excess Readmission Ratio < 1.2
• Number of Discharges < 2,000 (this dataset has already Number of Discharges > 1,000)


In :

df_temp_c = df_dischg_gt1000.loc[(df_dischg_gt1000['Number of Discharges'] < 2000) & (df_dischg_gt1000['Excess Readmission Ratio'] < 1.2)]
pearson, pvalue = pearsonr(df_temp_c[['Number of Discharges']], df_temp_c[['Excess Readmission Ratio']])
print("+----------------------------------------------------------+")
print("| 'Number of Discharges' versus 'Excess Readmission Ratio' |")
print("| for all hospitals with:                                  |")
print("|    a) Excess Readmission Ratio < 1.2                     |")
print("|    b) Number of Discharges < 2,000                       |")
print("|----------------------------------------------------------|")
print("| Pearson Correlation = %.4f                            |" %pearson[])
print("| p-value = %.30f               |" %pvalue[])
print("+----------------------------------------------------------+")




+----------------------------------------------------------+
| 'Number of Discharges' versus 'Excess Readmission Ratio' |
| for all hospitals with:                                  |
|    a) Excess Readmission Ratio < 1.2                     |
|    b) Number of Discharges < 2,000                       |
|----------------------------------------------------------|
| Pearson Correlation = -0.1197                            |
| p-value = 0.013894160454049312922175651863               |
+----------------------------------------------------------+




In :

sns.regplot(x="Number of Discharges", y="Excess Readmission Ratio", data=df_temp_c)




Out:

<matplotlib.axes._subplots.AxesSubplot at 0x11c035828>



#### D - What about hospitals with discharges between 100 and 1,000 (inclusive)?



In :

df_temp2 = df_temp.loc[(df_temp['Number of Discharges'] >= 100) & (df_temp['Number of Discharges'] <= 1000)]
pearson, pvalue = pearsonr(df_temp2[['Number of Discharges']], df_temp2[['Excess Readmission Ratio']])
pvalue4 = pvalue
print("+----------------------------------------------------------+")
print("| 'Number of Discharges' versus 'Excess Readmission Ratio' |")
print("| for hospitals with discharges >= 100 and <= 1,000:       |")
print("|----------------------------------------------------------|")
print("| Pearson Correlation = %.4f                            |" %pearson[])
print("| p-value = %.20f                         |" %pvalue[])
print("+----------------------------------------------------------+")




+----------------------------------------------------------+
| 'Number of Discharges' versus 'Excess Readmission Ratio' |
| for hospitals with discharges >= 100 and <= 1,000:       |
|----------------------------------------------------------|
| Pearson Correlation = -0.0559                            |
| p-value = 0.00000002797303043947                         |
+----------------------------------------------------------+




In :

sns.regplot(x="Number of Discharges", y="Excess Readmission Ratio", data=df_dischg_medium)




Out:

<matplotlib.axes._subplots.AxesSubplot at 0x11c1f8208>



## Preliminary Report

A. Initial observations based on the plot above

• Overall, rate of readmissions is trending down with increasing number of discharges
• With lower number of discharges, there is a greater incidence of excess rate of readmissions (area shaded red)
• With higher number of discharges, there is a greater incidence of lower rates of readmissions (area shaded green)

B. Statistics

• In hospitals/facilities with number of discharges < 100, mean excess readmission rate is 1.023 and 63% have excess readmission rate greater than 1
• In hospitals/facilities with number of discharges > 1000, mean excess readmission rate is 0.978 and 44% have excess readmission rate greater than 1

C. Conclusions

• There is a significant correlation between hospital capacity (number of discharges) and readmission rates.
• Smaller hospitals/facilities may be lacking necessary resources to ensure quality care and prevent complications that lead to readmissions.

D. Regulatory policy recommendations

• Hospitals/facilties with small capacity (< 300) should be required to demonstrate upgraded resource allocation for quality care to continue operation.
• Directives and incentives should be provided for consolidation of hospitals and facilities to have a smaller number of them with higher capacity and number of discharges.

### A. Do you agree with the above analysis and recommendations? Why or why not?

No.

• I could not find a significant correlation - Pearson correlation close to ABS(1) - between hospital capacity (number of discharges) and readmission rates;
• Considering:
i) all hospitals
ii) hospitals/facilities with number of discharges < 100
iii) hospitals/facilities with number of discharges > 1000


In :

print("+-----------------------------------------------------------+-------------+")
print("| Hospital/facilities                                       | Correlation*|")
print("|-----------------------------------------------------------|-------------|")
print("| i) All hospitals                                          | %.4f     |" %pearson1)
print("| ii) Hospitals/facilities with number of discharges < 100  | %.4f     |" %pearson2)
print("| ii) Hospitals/facilities with number of discharges > 1,000| %.4f     |" %pearson3)
print("+-----------------------------------------------------------+-------------+")
print("* Pearson Correlation for: Hospital capacity(number of discharges) / readmission rates")




+-----------------------------------------------------------+-------------+
| Hospital/facilities                                       | Correlation*|
|-----------------------------------------------------------|-------------|
| i) All hospitals                                          | -0.0974     |
| ii) Hospitals/facilities with number of discharges < 100  | -0.2446     |
| ii) Hospitals/facilities with number of discharges > 1,000| -0.0793     |
+-----------------------------------------------------------+-------------+
* Pearson Correlation for: Hospital capacity(number of discharges) / readmission rates


• For all three groups above there is a negative correlation,
• The Pearson Correlation values is "far" from a strong correlation (close to 1) and, finally,
• Even without the outliers, we found similar values for the the Pearson Correlation (three groups above).

### B. Provide support for your arguments and your own recommendations with a statistically sound analysis:

#### 1. Setup an appropriate hypothesis test



In :

print("+--------------------------------------------------------------+")
print("| Null Hypothesis:                                             |")
print("|      Ho: There is *not* a significant correlation between    |")
print("|          hospital capacity (discharges) and readmission rates|")
print("|--------------------------------------------------------------|")
print("| Alternative Hypothesis:                                      |")
print("|      Ha: There is a significant correlation between          |")
print("|          hospital capacity (discharges) and readmission rates|")
print("+--------------------------------------------------------------+")




+--------------------------------------------------------------+
| Null Hypothesis:                                             |
|      Ho: There is *not* a significant correlation between    |
|          hospital capacity (discharges) and readmission rates|
|--------------------------------------------------------------|
| Alternative Hypothesis:                                      |
|      Ha: There is a significant correlation between          |
|          hospital capacity (discharges) and readmission rates|
+--------------------------------------------------------------+



#### 2. Compute and report the observed significance value (or p-value).



In :

print("+--------------------------------------------------------------+")
print("| Scenario 1:                                                  |")
print("| All Hospitals:  P-Value = %.30f   |" %pvalue1)
print("|--------------------------------------------------------------|")
print("| Scenario 2:                                                  |")
print("| Hospitals discharges < 100: P-Value = %.20f |" %pvalue2)
print("|--------------------------------------------------------------|")
print("| Scenario 3:                                                  |")
print("| Hospitals with discharges > 1,000: P-Value = %.4f          |" %pvalue3)
print("|--------------------------------------------------------------|")
print("| Scenario 4:                                                  |")
print("| Hospitals discharges>100 and <1,000: P-Value = %.12f|" %pvalue4)
print("+--------------------------------------------------------------+")




+--------------------------------------------------------------+
| Scenario 1:                                                  |
| All Hospitals:  P-Value = 0.000000000000000000000000122255   |
|--------------------------------------------------------------|
| Scenario 2:                                                  |
| Hospitals discharges < 100: P-Value = 0.00000000000000001196 |
|--------------------------------------------------------------|
| Scenario 3:                                                  |
| Hospitals with discharges > 1,000: P-Value = 0.0884          |
|--------------------------------------------------------------|
| Scenario 4:                                                  |
| Hospitals discharges>100 and <1,000: P-Value = 0.000000027973|
+--------------------------------------------------------------+



#### 3. Report statistical significance for α = .01



In [ ]:



4. Discuss statistical significance and practical significance. Do they differ here? How does this change your recommendation to the client?



In [ ]:



#### 5. Look at the scatterplot above

• What are the advantages and disadvantages of using this plot to convey information?

a) It shows how much one variable affect the other or, in other words. The relationship between two variables is called their correlation;
b) Outliers: the maximum and minimum value, usually, can be easily determined;
c) It is possible to show many variables in a single plot.

d) Discretization: it is difficult to see the values (x,y) if we have many of them that are very close;

• Construct another plot that conveys the same information in a more direct manner


In [ ]: