Data Preprocessing : Data Quality Assessment, Preprocessing and Exploration for a Classification Modelling Problem¶

John Pauline Pineda

November 21, 2023

  • 1. Table of Contents
    • 1.1 Data Background
    • 1.2 Data Description
    • 1.3 Data Quality Assessment
    • 1.4 Data Preprocessing
      • 1.4.1 Data Cleaning
      • 1.4.2 Missing Data Imputation
      • 1.4.3 Outlier Treatment
      • 1.4.4 Collinearity
      • 1.4.5 Shape Transformation
      • 1.4.6 Centering and Scaling
      • 1.4.7 Data Encoding
      • 1.4.8 Preprocessed Data Description
    • 1.5 Data Exploration
      • 1.5.1 Exploratory Data Analysis
      • 1.5.2 Hypothesis Testing
  • 2. Summary
  • 3. References

1. Table of Contents ¶

This project explores the various methods in assessing Data Quality, implementing Data Preprocessing and conducting Data Exploration for prediction problems with categorical responses using various helpful packages in Python. A non-exhaustive list of methods to detect missing data, extreme outlying points, near-zero variance, multicollinearity, and skewed distributions were evaluated. Remedial procedures on addressing data quality issues including missing data imputation, centering and scaling transformation, shape transformation and outlier treatment were similarly considered, as applicable. All results were consolidated in a Summary presented at the end of the document.

Data quality assessment involves profiling and assessing the data to understand its suitability for machine learning tasks. The quality of training data has a huge impact on the efficiency, accuracy and complexity of machine learning tasks. Data remains susceptible to errors or irregularities that may be introduced during collection, aggregation or annotation stage. Issues such as incorrect labels, synonymous categories in a categorical variable or heterogeneity in columns, among others, which might go undetected by standard pre-processing modules in these frameworks can lead to sub-optimal model performance, inaccurate analysis and unreliable decisions.

Data preprocessing involves changing the raw feature vectors into a representation that is more suitable for the downstream modelling and estimation processes, including data cleaning, integration, reduction and transformation. Data cleaning aims to identify and correct errors in the dataset that may negatively impact a predictive model such as removing outliers, replacing missing values, smoothing noisy data, and correcting inconsistent data. Data integration addresses potential issues with redundant and inconsistent data obtained from multiple sources through approaches such as detection of tuple duplication and data conflict. The purpose of data reduction is to have a condensed representation of the data set that is smaller in volume, while maintaining the integrity of the original data set. Data transformation converts the data into the most appropriate form for data modeling.

Data exploration involves analyzing and investigating data sets to summarize their main characteristics, often employing data visualization methods. It helps determine how best to manipulate data sources to discover patterns, spot anomalies, test a hypothesis, or check assumptions. This process is primarily used to see what data can reveal beyond the formal modeling or hypothesis testing task and provides a better understanding of data set variables and the relationships between them.

1.1. Data Background ¶

Datasets used for the analysis were separately gathered and consolidated from various sources including:

  1. Cancer Rates from World Population Review
  2. Social Protection and Labor Indicator from World Bank
  3. Education Indicator from World Bank
  4. Economy and Growth Indicator from World Bank
  5. Environment Indicator from World Bank
  6. Climate Change Indicator from World Bank
  7. Agricultural and Rural Development Indicator from World Bank
  8. Social Development Indicator from World Bank
  9. Health Indicator from World Bank
  10. Science and Technology Indicator from World Bank
  11. Urban Development Indicator from World Bank
  12. Human Development Indices from Human Development Reports
  13. Environmental Performance Indices from Yale Center for Environmental Law and Policy

This study hypothesized that various global development indicators and indices influence cancer rates across countries.

The target variable for the study is:

  • CANRAT - Dichotomized category based on age-standardized cancer rates, per 100K population (2022)

The predictor variables for the study are:

  • GDPPER - GDP per person employed, current US Dollars (2020)
  • URBPOP - Urban population, % of total population (2020)
  • PATRES - Patent applications by residents, total count (2020)
  • RNDGDP - Research and development expenditure, % of GDP (2020)
  • POPGRO - Population growth, annual % (2020)
  • LIFEXP - Life expectancy at birth, total in years (2020)
  • TUBINC - Incidence of tuberculosis, per 100K population (2020)
  • DTHCMD - Cause of death by communicable diseases and maternal, prenatal and nutrition conditions, % of total (2019)
  • AGRLND - Agricultural land, % of land area (2020)
  • GHGEMI - Total greenhouse gas emissions, kt of CO2 equivalent (2020)
  • RELOUT - Renewable electricity output, % of total electricity output (2015)
  • METEMI - Methane emissions, kt of CO2 equivalent (2020)
  • FORARE - Forest area, % of land area (2020)
  • CO2EMI - CO2 emissions, metric tons per capita (2020)
  • PM2EXP - PM2.5 air pollution, population exposed to levels exceeding WHO guideline value, % of total (2017)
  • POPDEN - Population density, people per sq. km of land area (2020)
  • GDPCAP - GDP per capita, current US Dollars (2020)
  • ENRTER - Tertiary school enrollment, % gross (2020)
  • HDICAT - Human development index, ordered category (2020)
  • EPISCO - Environment performance index , score (2022)

1.2. Data Description ¶

  1. The dataset is comprised of:
    • 177 rows (observations)
    • 22 columns (variables)
      • 1/22 metadata (object)
        • COUNTRY
      • 1/22 target (categorical)
        • CANRAT
      • 19/22 predictor (numeric)
        • GDPPER
        • URBPOP
        • PATRES
        • RNDGDP
        • POPGRO
        • LIFEXP
        • TUBINC
        • DTHCMD
        • AGRLND
        • GHGEMI
        • RELOUT
        • METEMI
        • FORARE
        • CO2EMI
        • PM2EXP
        • POPDEN
        • GDPCAP
        • ENRTER
        • EPISCO
      • 1/22 predictor (categorical)
        • HDICAT
In [1]:
##################################
# Loading Python Libraries
##################################
import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
import itertools
import os
%matplotlib inline

from operator import add,mul,truediv
from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PowerTransformer
from sklearn.preprocessing import StandardScaler
from scipy import stats
In [2]:
##################################
# Defining file paths
##################################
DATASETS_ORIGINAL_PATH = r"datasets\original"
In [3]:
##################################
# Loading the dataset
# from the DATASETS_ORIGINAL_PATH
##################################
cancer_rate = pd.read_csv(os.path.join("..", DATASETS_ORIGINAL_PATH, 'CategoricalCancerRates.csv'))
In [4]:
##################################
# Performing a general exploration of the dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate.shape)

Dataset Dimensions: 

(177, 22)
In [5]:
##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate.dtypes)

Column Names and Data Types:

COUNTRY     object
CANRAT      object
GDPPER     float64
URBPOP     float64
PATRES     float64
RNDGDP     float64
POPGRO     float64
LIFEXP     float64
TUBINC     float64
DTHCMD     float64
AGRLND     float64
GHGEMI     float64
RELOUT     float64
METEMI     float64
FORARE     float64
CO2EMI     float64
PM2EXP     float64
POPDEN     float64
ENRTER     float64
GDPCAP     float64
HDICAT      object
EPISCO     float64
dtype: object
In [6]:
##################################
# Taking a snapshot of the dataset
##################################
cancer_rate.head()
Out[6]:
COUNTRY CANRAT GDPPER URBPOP PATRES RNDGDP POPGRO LIFEXP TUBINC DTHCMD ... RELOUT METEMI FORARE CO2EMI PM2EXP POPDEN ENRTER GDPCAP HDICAT EPISCO
0 Australia High 98380.63601 86.241 2368.0 NaN 1.235701 83.200000 7.2 4.941054 ... 13.637841 131484.763200 17.421315 14.772658 24.893584 3.335312 110.139221 51722.06900 VH 60.1
1 New Zealand High 77541.76438 86.699 348.0 NaN 2.204789 82.256098 7.2 4.354730 ... 80.081439 32241.937000 37.570126 6.160799 NaN 19.331586 75.734833 41760.59478 VH 56.7
2 Ireland High 198405.87500 63.653 75.0 1.23244 1.029111 82.556098 5.3 5.684596 ... 27.965408 15252.824630 11.351720 6.768228 0.274092 72.367281 74.680313 85420.19086 VH 57.4
3 United States High 130941.63690 82.664 269586.0 3.42287 0.964348 76.980488 2.3 5.302060 ... 13.228593 748241.402900 33.866926 13.032828 3.343170 36.240985 87.567657 63528.63430 VH 51.1
4 Denmark High 113300.60110 88.116 1261.0 2.96873 0.291641 81.602439 4.1 6.826140 ... 65.505925 7778.773921 15.711000 4.691237 56.914456 145.785100 82.664330 60915.42440 VH 77.9

5 rows × 22 columns

In [7]:
##################################
# Setting the levels of the categorical variables
##################################
cancer_rate['CANRAT'] = cancer_rate['CANRAT'].astype('category')
cancer_rate['CANRAT'] = cancer_rate['CANRAT'].cat.set_categories(['Low', 'High'], ordered=True)
cancer_rate['HDICAT'] = cancer_rate['HDICAT'].astype('category')
cancer_rate['HDICAT'] = cancer_rate['HDICAT'].cat.set_categories(['L', 'M', 'H', 'VH'], ordered=True)
In [8]:
##################################
# Performing a general exploration of the numeric variables
##################################
print('Numeric Variable Summary:')
display(cancer_rate.describe(include='number').transpose())

Numeric Variable Summary:
count mean std min 25% 50% 75% max
GDPPER 165.0 45284.424283 3.941794e+04 1718.804896 13545.254510 34024.900890 66778.416050 2.346469e+05
URBPOP 174.0 59.788121 2.280640e+01 13.345000 42.432750 61.701500 79.186500 1.000000e+02
PATRES 108.0 20607.388889 1.340683e+05 1.000000 35.250000 244.500000 1297.750000 1.344817e+06
RNDGDP 74.0 1.197474 1.189956e+00 0.039770 0.256372 0.873660 1.608842 5.354510e+00
POPGRO 174.0 1.127028 1.197718e+00 -2.079337 0.236900 1.179959 2.031154 3.727101e+00
LIFEXP 174.0 71.746113 7.606209e+00 52.777000 65.907500 72.464610 77.523500 8.456000e+01
TUBINC 174.0 105.005862 1.367229e+02 0.770000 12.000000 44.500000 147.750000 5.920000e+02
DTHCMD 170.0 21.260521 1.927333e+01 1.283611 6.078009 12.456279 36.980457 6.520789e+01
AGRLND 174.0 38.793456 2.171551e+01 0.512821 20.130276 40.386649 54.013754 8.084112e+01
GHGEMI 170.0 259582.709895 1.118550e+06 179.725150 12527.487367 41009.275980 116482.578575 1.294287e+07
RELOUT 153.0 39.760036 3.191492e+01 0.000296 10.582691 32.381668 63.011450 1.000000e+02
METEMI 170.0 47876.133575 1.346611e+05 11.596147 3662.884908 11118.976025 32368.909040 1.186285e+06
FORARE 173.0 32.218177 2.312001e+01 0.008078 11.604388 31.509048 49.071780 9.741212e+01
CO2EMI 170.0 3.751097 4.606479e+00 0.032585 0.631924 2.298368 4.823496 3.172684e+01
PM2EXP 167.0 91.940595 2.206003e+01 0.274092 99.627134 100.000000 100.000000 1.000000e+02
POPDEN 174.0 200.886765 6.453834e+02 2.115134 27.454539 77.983133 153.993650 7.918951e+03
ENRTER 116.0 49.994997 2.970619e+01 2.432581 22.107195 53.392460 71.057467 1.433107e+02
GDPCAP 170.0 13992.095610 1.957954e+04 216.827417 1870.503029 5348.192875 17421.116227 1.173705e+05
EPISCO 165.0 42.946667 1.249086e+01 18.900000 33.000000 40.900000 50.500000 7.790000e+01
In [9]:
##################################
# Performing a general exploration of the object variable
##################################
print('Object Variable Summary:')
display(cancer_rate.describe(include='object').transpose())

Object Variable Summary:
count unique top freq
COUNTRY 177 177 Australia 1
In [10]:
##################################
# Performing a general exploration of the categorical variables
##################################
print('Categorical Variable Summary:')
display(cancer_rate.describe(include='category').transpose())

Categorical Variable Summary:
count unique top freq
CANRAT 177 2 Low 132
HDICAT 167 4 VH 59

1.3. Data Quality Assessment ¶

Data quality findings based on assessment are as follows:

  1. No duplicated rows observed.
  2. Missing data noted for 20 variables with Null.Count>0 and Fill.Rate<1.0.
    • RNDGDP: Null.Count = 103, Fill.Rate = 0.418
    • PATRES: Null.Count = 69, Fill.Rate = 0.610
    • ENRTER: Null.Count = 61, Fill.Rate = 0.655
    • RELOUT: Null.Count = 24, Fill.Rate = 0.864
    • GDPPER: Null.Count = 12, Fill.Rate = 0.932
    • EPISCO: Null.Count = 12, Fill.Rate = 0.932
    • HDICAT: Null.Count = 10, Fill.Rate = 0.943
    • PM2EXP: Null.Count = 10, Fill.Rate = 0.943
    • DTHCMD: Null.Count = 7, Fill.Rate = 0.960
    • METEMI: Null.Count = 7, Fill.Rate = 0.960
    • CO2EMI: Null.Count = 7, Fill.Rate = 0.960
    • GDPCAP: Null.Count = 7, Fill.Rate = 0.960
    • GHGEMI: Null.Count = 7, Fill.Rate = 0.960
    • FORARE: Null.Count = 4, Fill.Rate = 0.977
    • TUBINC: Null.Count = 3, Fill.Rate = 0.983
    • AGRLND: Null.Count = 3, Fill.Rate = 0.983
    • POPGRO: Null.Count = 3, Fill.Rate = 0.983
    • POPDEN: Null.Count = 3, Fill.Rate = 0.983
    • URBPOP: Null.Count = 3, Fill.Rate = 0.983
    • LIFEXP: Null.Count = 3, Fill.Rate = 0.983
  3. 120 observations noted with at least 1 missing data. From this number, 14 observations reported high Missing.Rate>0.2.
    • COUNTRY=Guadeloupe: Missing.Rate= 0.909
    • COUNTRY=Martinique: Missing.Rate= 0.909
    • COUNTRY=French Guiana: Missing.Rate= 0.909
    • COUNTRY=New Caledonia: Missing.Rate= 0.500
    • COUNTRY=French Polynesia: Missing.Rate= 0.500
    • COUNTRY=Guam: Missing.Rate= 0.500
    • COUNTRY=Puerto Rico: Missing.Rate= 0.409
    • COUNTRY=North Korea: Missing.Rate= 0.227
    • COUNTRY=Somalia: Missing.Rate= 0.227
    • COUNTRY=South Sudan: Missing.Rate= 0.227
    • COUNTRY=Venezuela: Missing.Rate= 0.227
    • COUNTRY=Libya: Missing.Rate= 0.227
    • COUNTRY=Eritrea: Missing.Rate= 0.227
    • COUNTRY=Yemen: Missing.Rate= 0.227
  4. Low variance observed for 1 variable with First.Second.Mode.Ratio>5.
    • PM2EXP: First.Second.Mode.Ratio = 53.000
  5. No low variance observed for any variable with Unique.Count.Ratio>10.
  6. High skewness observed for 5 variables with Skewness>3 or Skewness<(-3).
    • POPDEN: Skewness = +10.267
    • GHGEMI: Skewness = +9.496
    • PATRES: Skewness = +9.284
    • METEMI: Skewness = +5.801
    • PM2EXP: Skewness = -3.141
In [11]:
##################################
# Counting the number of duplicated rows
##################################
cancer_rate.duplicated().sum()
Out[11]:
np.int64(0)
In [12]:
##################################
# Gathering the data types for each column
##################################
data_type_list = list(cancer_rate.dtypes)
In [13]:
##################################
# Gathering the variable names for each column
##################################
variable_name_list = list(cancer_rate.columns)
In [14]:
##################################
# Gathering the number of observations for each column
##################################
row_count_list = list([len(cancer_rate)] * len(cancer_rate.columns))
In [15]:
##################################
# Gathering the number of missing data for each column
##################################
null_count_list = list(cancer_rate.isna().sum(axis=0))
In [16]:
##################################
# Gathering the number of non-missing data for each column
##################################
non_null_count_list = list(cancer_rate.count())
In [17]:
##################################
# Gathering the missing data percentage for each column
##################################
fill_rate_list = map(truediv, non_null_count_list, row_count_list)
In [18]:
##################################
# Formulating the summary
# for all columns
##################################
all_column_quality_summary = pd.DataFrame(zip(variable_name_list,
                                              data_type_list,
                                              row_count_list,
                                              non_null_count_list,
                                              null_count_list,
                                              fill_rate_list), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count',                                                 
                                                 'Fill.Rate'])
display(all_column_quality_summary)
Column.Name Column.Type Row.Count Non.Null.Count Null.Count Fill.Rate
0 COUNTRY object 177 177 0 1.000000
1 CANRAT category 177 177 0 1.000000
2 GDPPER float64 177 165 12 0.932203
3 URBPOP float64 177 174 3 0.983051
4 PATRES float64 177 108 69 0.610169
5 RNDGDP float64 177 74 103 0.418079
6 POPGRO float64 177 174 3 0.983051
7 LIFEXP float64 177 174 3 0.983051
8 TUBINC float64 177 174 3 0.983051
9 DTHCMD float64 177 170 7 0.960452
10 AGRLND float64 177 174 3 0.983051
11 GHGEMI float64 177 170 7 0.960452
12 RELOUT float64 177 153 24 0.864407
13 METEMI float64 177 170 7 0.960452
14 FORARE float64 177 173 4 0.977401
15 CO2EMI float64 177 170 7 0.960452
16 PM2EXP float64 177 167 10 0.943503
17 POPDEN float64 177 174 3 0.983051
18 ENRTER float64 177 116 61 0.655367
19 GDPCAP float64 177 170 7 0.960452
20 HDICAT category 177 167 10 0.943503
21 EPISCO float64 177 165 12 0.932203
In [19]:
##################################
# Counting the number of columns
# with Fill.Rate < 1.00
##################################
len(all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<1)])
Out[19]:
20
In [20]:
##################################
# Identifying the columns
# with Fill.Rate < 1.00
##################################
display(all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<1)].sort_values(by=['Fill.Rate'], ascending=True))
Column.Name Column.Type Row.Count Non.Null.Count Null.Count Fill.Rate
5 RNDGDP float64 177 74 103 0.418079
4 PATRES float64 177 108 69 0.610169
18 ENRTER float64 177 116 61 0.655367
12 RELOUT float64 177 153 24 0.864407
21 EPISCO float64 177 165 12 0.932203
2 GDPPER float64 177 165 12 0.932203
16 PM2EXP float64 177 167 10 0.943503
20 HDICAT category 177 167 10 0.943503
15 CO2EMI float64 177 170 7 0.960452
13 METEMI float64 177 170 7 0.960452
11 GHGEMI float64 177 170 7 0.960452
9 DTHCMD float64 177 170 7 0.960452
19 GDPCAP float64 177 170 7 0.960452
14 FORARE float64 177 173 4 0.977401
6 POPGRO float64 177 174 3 0.983051
3 URBPOP float64 177 174 3 0.983051
17 POPDEN float64 177 174 3 0.983051
10 AGRLND float64 177 174 3 0.983051
7 LIFEXP float64 177 174 3 0.983051
8 TUBINC float64 177 174 3 0.983051
In [21]:
##################################
# Identifying the rows
# with Fill.Rate < 0.90
##################################
column_low_fill_rate = all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<0.90)]
In [22]:
##################################
# Gathering the metadata labels for each observation
##################################
row_metadata_list = cancer_rate["COUNTRY"].values.tolist()
In [23]:
##################################
# Gathering the number of columns for each observation
##################################
column_count_list = list([len(cancer_rate.columns)] * len(cancer_rate))
In [24]:
##################################
# Gathering the number of missing data for each row
##################################
null_row_list = list(cancer_rate.isna().sum(axis=1))
In [25]:
##################################
# Gathering the missing data percentage for each column
##################################
missing_rate_list = map(truediv, null_row_list, column_count_list)
In [26]:
##################################
# Identifying the rows
# with missing data
##################################
all_row_quality_summary = pd.DataFrame(zip(row_metadata_list,
                                           column_count_list,
                                           null_row_list,
                                           missing_rate_list), 
                                        columns=['Row.Name',
                                                 'Column.Count',
                                                 'Null.Count',                                                 
                                                 'Missing.Rate'])
display(all_row_quality_summary)
Row.Name Column.Count Null.Count Missing.Rate
0 Australia 22 1 0.045455
1 New Zealand 22 2 0.090909
2 Ireland 22 0 0.000000
3 United States 22 0 0.000000
4 Denmark 22 0 0.000000
... ... ... ... ...
172 Congo Republic 22 3 0.136364
173 Bhutan 22 2 0.090909
174 Nepal 22 2 0.090909
175 Gambia 22 4 0.181818
176 Niger 22 2 0.090909

177 rows × 4 columns

In [27]:
##################################
# Counting the number of rows
# with Missing.Rate > 0.00
##################################
len(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.00)])
Out[27]:
120
In [28]:
##################################
# Counting the number of rows
# with Missing.Rate > 0.20
##################################
len(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)])
Out[28]:
14
In [29]:
##################################
# Identifying the rows
# with Missing.Rate > 0.20
##################################
row_high_missing_rate = all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)]
In [30]:
##################################
# Identifying the rows
# with Missing.Rate > 0.20
##################################
display(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)].sort_values(by=['Missing.Rate'], ascending=False))
Row.Name Column.Count Null.Count Missing.Rate
35 Guadeloupe 22 20 0.909091
39 Martinique 22 20 0.909091
56 French Guiana 22 20 0.909091
13 New Caledonia 22 11 0.500000
44 French Polynesia 22 11 0.500000
75 Guam 22 11 0.500000
53 Puerto Rico 22 9 0.409091
85 North Korea 22 6 0.272727
168 South Sudan 22 6 0.272727
132 Somalia 22 6 0.272727
117 Libya 22 5 0.227273
73 Venezuela 22 5 0.227273
161 Eritrea 22 5 0.227273
164 Yemen 22 5 0.227273
In [31]:
##################################
# Formulating the dataset
# with numeric columns only
##################################
cancer_rate_numeric = cancer_rate.select_dtypes(include='number')
In [32]:
##################################
# Gathering the variable names for each numeric column
##################################
numeric_variable_name_list = cancer_rate_numeric.columns
In [33]:
##################################
# Gathering the minimum value for each numeric column
##################################
numeric_minimum_list = cancer_rate_numeric.min()
In [34]:
##################################
# Gathering the mean value for each numeric column
##################################
numeric_mean_list = cancer_rate_numeric.mean()
In [35]:
##################################
# Gathering the median value for each numeric column
##################################
numeric_median_list = cancer_rate_numeric.median()
In [36]:
##################################
# Gathering the maximum value for each numeric column
##################################
numeric_maximum_list = cancer_rate_numeric.max()
In [37]:
##################################
# Gathering the first mode values for each numeric column
##################################
numeric_first_mode_list = [cancer_rate[x].value_counts(dropna=True).index.tolist()[0] for x in cancer_rate_numeric]
In [38]:
##################################
# Gathering the second mode values for each numeric column
##################################
numeric_second_mode_list = [cancer_rate[x].value_counts(dropna=True).index.tolist()[1] for x in cancer_rate_numeric]
In [39]:
##################################
# Gathering the count of first mode values for each numeric column
##################################
numeric_first_mode_count_list = [cancer_rate_numeric[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_numeric]
In [40]:
##################################
# Gathering the count of second mode values for each numeric column
##################################
numeric_second_mode_count_list = [cancer_rate_numeric[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_numeric]
In [41]:
##################################
# Gathering the first mode to second mode ratio for each numeric column
##################################
numeric_first_second_mode_ratio_list = map(truediv, numeric_first_mode_count_list, numeric_second_mode_count_list)
In [42]:
##################################
# Gathering the count of unique values for each numeric column
##################################
numeric_unique_count_list = cancer_rate_numeric.nunique(dropna=True)
In [43]:
##################################
# Gathering the number of observations for each numeric column
##################################
numeric_row_count_list = list([len(cancer_rate_numeric)] * len(cancer_rate_numeric.columns))
In [44]:
##################################
# Gathering the unique to count ratio for each numeric column
##################################
numeric_unique_count_ratio_list = map(truediv, numeric_unique_count_list, numeric_row_count_list)
In [45]:
##################################
# Gathering the skewness value for each numeric column
##################################
numeric_skewness_list = cancer_rate_numeric.skew()
In [46]:
##################################
# Gathering the kurtosis value for each numeric column
##################################
numeric_kurtosis_list = cancer_rate_numeric.kurtosis()
In [47]:
numeric_column_quality_summary = pd.DataFrame(zip(numeric_variable_name_list,
                                                numeric_minimum_list,
                                                numeric_mean_list,
                                                numeric_median_list,
                                                numeric_maximum_list,
                                                numeric_first_mode_list,
                                                numeric_second_mode_list,
                                                numeric_first_mode_count_list,
                                                numeric_second_mode_count_list,
                                                numeric_first_second_mode_ratio_list,
                                                numeric_unique_count_list,
                                                numeric_row_count_list,
                                                numeric_unique_count_ratio_list,
                                                numeric_skewness_list,
                                                numeric_kurtosis_list), 
                                        columns=['Numeric.Column.Name',
                                                 'Minimum',
                                                 'Mean',
                                                 'Median',
                                                 'Maximum',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio',
                                                 'Skewness',
                                                 'Kurtosis'])
display(numeric_column_quality_summary)
Numeric.Column.Name Minimum Mean Median Maximum First.Mode Second.Mode First.Mode.Count Second.Mode.Count First.Second.Mode.Ratio Unique.Count Row.Count Unique.Count.Ratio Skewness Kurtosis
0 GDPPER 1718.804896 45284.424283 34024.900890 2.346469e+05 98380.636010 77541.764380 1 1 1.000000 165 177 0.932203 1.517574 3.471992
1 URBPOP 13.345000 59.788121 61.701500 1.000000e+02 100.000000 86.699000 2 1 2.000000 173 177 0.977401 -0.210702 -0.962847
2 PATRES 1.000000 20607.388889 244.500000 1.344817e+06 6.000000 2.000000 4 3 1.333333 97 177 0.548023 9.284436 91.187178
3 RNDGDP 0.039770 1.197474 0.873660 5.354510e+00 1.232440 3.422870 1 1 1.000000 74 177 0.418079 1.396742 1.695957
4 POPGRO -2.079337 1.127028 1.179959 3.727101e+00 1.235701 2.204789 1 1 1.000000 174 177 0.983051 -0.195161 -0.423580
5 LIFEXP 52.777000 71.746113 72.464610 8.456000e+01 83.200000 82.256098 1 1 1.000000 174 177 0.983051 -0.357965 -0.649601
6 TUBINC 0.770000 105.005862 44.500000 5.920000e+02 12.000000 4.100000 4 3 1.333333 131 177 0.740113 1.746333 2.429368
7 DTHCMD 1.283611 21.260521 12.456279 6.520789e+01 4.941054 4.354730 1 1 1.000000 170 177 0.960452 0.900509 -0.691541
8 AGRLND 0.512821 38.793456 40.386649 8.084112e+01 46.252480 38.562911 1 1 1.000000 174 177 0.983051 0.074000 -0.926249
9 GHGEMI 179.725150 259582.709895 41009.275980 1.294287e+07 571903.119900 80158.025830 1 1 1.000000 170 177 0.960452 9.496120 101.637308
10 RELOUT 0.000296 39.760036 32.381668 1.000000e+02 100.000000 80.081439 3 1 3.000000 151 177 0.853107 0.501088 -0.981774
11 METEMI 11.596147 47876.133575 11118.976025 1.186285e+06 131484.763200 32241.937000 1 1 1.000000 170 177 0.960452 5.801014 38.661386
12 FORARE 0.008078 32.218177 31.509048 9.741212e+01 17.421315 37.570126 1 1 1.000000 173 177 0.977401 0.519277 -0.322589
13 CO2EMI 0.032585 3.751097 2.298368 3.172684e+01 14.772658 6.160799 1 1 1.000000 170 177 0.960452 2.721552 10.311574
14 PM2EXP 0.274092 91.940595 100.000000 1.000000e+02 100.000000 100.000000 106 2 53.000000 61 177 0.344633 -3.141557 9.032386
15 POPDEN 2.115134 200.886765 77.983133 7.918951e+03 3.335312 19.331586 1 1 1.000000 174 177 0.983051 10.267750 119.995256
16 ENRTER 2.432581 49.994997 53.392460 1.433107e+02 110.139221 75.734833 1 1 1.000000 116 177 0.655367 0.275863 -0.392895
17 GDPCAP 216.827417 13992.095610 5348.192875 1.173705e+05 51722.069000 41760.594780 1 1 1.000000 170 177 0.960452 2.258568 5.938690
18 EPISCO 18.900000 42.946667 40.900000 7.790000e+01 29.600000 43.600000 3 3 1.000000 137 177 0.774011 0.641799 0.035208
In [48]:
##################################
# Counting the number of numeric columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['First.Second.Mode.Ratio']>5)])
Out[48]:
1
In [49]:
##################################
# Identifying the numeric columns
# with First.Second.Mode.Ratio > 5.00
##################################
display(numeric_column_quality_summary[(numeric_column_quality_summary['First.Second.Mode.Ratio']>5)].sort_values(by=['First.Second.Mode.Ratio'], ascending=False))
Numeric.Column.Name Minimum Mean Median Maximum First.Mode Second.Mode First.Mode.Count Second.Mode.Count First.Second.Mode.Ratio Unique.Count Row.Count Unique.Count.Ratio Skewness Kurtosis
14 PM2EXP 0.274092 91.940595 100.0 100.0 100.0 100.0 106 2 53.0 61 177 0.344633 -3.141557 9.032386
In [50]:
##################################
# Counting the number of numeric columns
# with Unique.Count.Ratio > 10.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['Unique.Count.Ratio']>10)])
Out[50]:
0
In [51]:
##################################
# Counting the number of numeric columns
# with Skewness > 3.00 or Skewness < -3.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['Skewness']>3) | (numeric_column_quality_summary['Skewness']<(-3))])
Out[51]:
5
In [52]:
##################################
# Identifying the numeric columns
# with Skewness > 3.00 or Skewness < -3.00
##################################
display(numeric_column_quality_summary[(numeric_column_quality_summary['Skewness']>3) | (numeric_column_quality_summary['Skewness']<(-3))].sort_values(by=['Skewness'], ascending=False))
Numeric.Column.Name Minimum Mean Median Maximum First.Mode Second.Mode First.Mode.Count Second.Mode.Count First.Second.Mode.Ratio Unique.Count Row.Count Unique.Count.Ratio Skewness Kurtosis
15 POPDEN 2.115134 200.886765 77.983133 7.918951e+03 3.335312 19.331586 1 1 1.000000 174 177 0.983051 10.267750 119.995256
9 GHGEMI 179.725150 259582.709895 41009.275980 1.294287e+07 571903.119900 80158.025830 1 1 1.000000 170 177 0.960452 9.496120 101.637308
2 PATRES 1.000000 20607.388889 244.500000 1.344817e+06 6.000000 2.000000 4 3 1.333333 97 177 0.548023 9.284436 91.187178
11 METEMI 11.596147 47876.133575 11118.976025 1.186285e+06 131484.763200 32241.937000 1 1 1.000000 170 177 0.960452 5.801014 38.661386
14 PM2EXP 0.274092 91.940595 100.000000 1.000000e+02 100.000000 100.000000 106 2 53.000000 61 177 0.344633 -3.141557 9.032386
In [53]:
##################################
# Formulating the dataset
# with object column only
##################################
cancer_rate_object = cancer_rate.select_dtypes(include='object')
In [54]:
##################################
# Gathering the variable names for the object column
##################################
object_variable_name_list = cancer_rate_object.columns
In [55]:
##################################
# Gathering the first mode values for the object column
##################################
object_first_mode_list = [cancer_rate[x].value_counts().index.tolist()[0] for x in cancer_rate_object]
In [56]:
##################################
# Gathering the second mode values for each object column
##################################
object_second_mode_list = [cancer_rate[x].value_counts().index.tolist()[1] for x in cancer_rate_object]
In [57]:
##################################
# Gathering the count of first mode values for each object column
##################################
object_first_mode_count_list = [cancer_rate_object[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_object]
In [58]:
##################################
# Gathering the count of second mode values for each object column
##################################
object_second_mode_count_list = [cancer_rate_object[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_object]
In [59]:
##################################
# Gathering the first mode to second mode ratio for each object column
##################################
object_first_second_mode_ratio_list = map(truediv, object_first_mode_count_list, object_second_mode_count_list)
In [60]:
##################################
# Gathering the count of unique values for each object column
##################################
object_unique_count_list = cancer_rate_object.nunique(dropna=True)
In [61]:
##################################
# Gathering the number of observations for each object column
##################################
object_row_count_list = list([len(cancer_rate_object)] * len(cancer_rate_object.columns))
In [62]:
##################################
# Gathering the unique to count ratio for each object column
##################################
object_unique_count_ratio_list = map(truediv, object_unique_count_list, object_row_count_list)
In [63]:
object_column_quality_summary = pd.DataFrame(zip(object_variable_name_list,
                                                 object_first_mode_list,
                                                 object_second_mode_list,
                                                 object_first_mode_count_list,
                                                 object_second_mode_count_list,
                                                 object_first_second_mode_ratio_list,
                                                 object_unique_count_list,
                                                 object_row_count_list,
                                                 object_unique_count_ratio_list), 
                                        columns=['Object.Column.Name',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio'])
display(object_column_quality_summary)
Object.Column.Name First.Mode Second.Mode First.Mode.Count Second.Mode.Count First.Second.Mode.Ratio Unique.Count Row.Count Unique.Count.Ratio
0 COUNTRY Australia New Zealand 1 1 1.0 177 177 1.0
In [64]:
##################################
# Counting the number of object columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(object_column_quality_summary[(object_column_quality_summary['First.Second.Mode.Ratio']>5)])
Out[64]:
0
In [65]:
##################################
# Counting the number of object columns
# with Unique.Count.Ratio > 10.00
##################################
len(object_column_quality_summary[(object_column_quality_summary['Unique.Count.Ratio']>10)])
Out[65]:
0
In [66]:
##################################
# Formulating the dataset
# with categorical columns only
##################################
cancer_rate_categorical = cancer_rate.select_dtypes(include='category')
In [67]:
##################################
# Gathering the variable names for the categorical column
##################################
categorical_variable_name_list = cancer_rate_categorical.columns
In [68]:
##################################
# Gathering the first mode values for each categorical column
##################################
categorical_first_mode_list = [cancer_rate[x].value_counts().index.tolist()[0] for x in cancer_rate_categorical]
In [69]:
##################################
# Gathering the second mode values for each categorical column
##################################
categorical_second_mode_list = [cancer_rate[x].value_counts().index.tolist()[1] for x in cancer_rate_categorical]
In [70]:
##################################
# Gathering the count of first mode values for each categorical column
##################################
categorical_first_mode_count_list = [cancer_rate_categorical[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_categorical]
In [71]:
##################################
# Gathering the count of second mode values for each categorical column
##################################
categorical_second_mode_count_list = [cancer_rate_categorical[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_categorical]
In [72]:
##################################
# Gathering the first mode to second mode ratio for each categorical column
##################################
categorical_first_second_mode_ratio_list = map(truediv, categorical_first_mode_count_list, categorical_second_mode_count_list)
In [73]:
##################################
# Gathering the count of unique values for each categorical column
##################################
categorical_unique_count_list = cancer_rate_categorical.nunique(dropna=True)
In [74]:
##################################
# Gathering the number of observations for each categorical column
##################################
categorical_row_count_list = list([len(cancer_rate_categorical)] * len(cancer_rate_categorical.columns))
In [75]:
##################################
# Gathering the unique to count ratio for each categorical column
##################################
categorical_unique_count_ratio_list = map(truediv, categorical_unique_count_list, categorical_row_count_list)
In [76]:
categorical_column_quality_summary = pd.DataFrame(zip(categorical_variable_name_list,
                                                    categorical_first_mode_list,
                                                    categorical_second_mode_list,
                                                    categorical_first_mode_count_list,
                                                    categorical_second_mode_count_list,
                                                    categorical_first_second_mode_ratio_list,
                                                    categorical_unique_count_list,
                                                    categorical_row_count_list,
                                                    categorical_unique_count_ratio_list), 
                                        columns=['Categorical.Column.Name',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio'])
display(categorical_column_quality_summary)
Categorical.Column.Name First.Mode Second.Mode First.Mode.Count Second.Mode.Count First.Second.Mode.Ratio Unique.Count Row.Count Unique.Count.Ratio
0 CANRAT Low High 132 45 2.933333 2 177 0.011299
1 HDICAT VH H 59 39 1.512821 4 177 0.022599
In [77]:
##################################
# Counting the number of categorical columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(categorical_column_quality_summary[(categorical_column_quality_summary['First.Second.Mode.Ratio']>5)])
Out[77]:
0
In [78]:
##################################
# Counting the number of categorical columns
# with Unique.Count.Ratio > 10.00
##################################
len(categorical_column_quality_summary[(categorical_column_quality_summary['Unique.Count.Ratio']>10)])
Out[78]:
0

1.4. Data Preprocessing ¶

1.4.1 Data Cleaning ¶

  1. Subsets of rows and columns with high rates of missing data were removed from the dataset:
    • 4 variables with Fill.Rate<0.9 were excluded for subsequent analysis.
      • RNDGDP: Null.Count = 103, Fill.Rate = 0.418
      • PATRES: Null.Count = 69, Fill.Rate = 0.610
      • ENRTER: Null.Count = 61, Fill.Rate = 0.655
      • RELOUT: Null.Count = 24, Fill.Rate = 0.864
    • 14 rows with Missing.Rate>0.2 were exluded for subsequent analysis.
      • COUNTRY=Guadeloupe: Missing.Rate= 0.909
      • COUNTRY=Martinique: Missing.Rate= 0.909
      • COUNTRY=French Guiana: Missing.Rate= 0.909
      • COUNTRY=New Caledonia: Missing.Rate= 0.500
      • COUNTRY=French Polynesia: Missing.Rate= 0.500
      • COUNTRY=Guam: Missing.Rate= 0.500
      • COUNTRY=Puerto Rico: Missing.Rate= 0.409
      • COUNTRY=North Korea: Missing.Rate= 0.227
      • COUNTRY=Somalia: Missing.Rate= 0.227
      • COUNTRY=South Sudan: Missing.Rate= 0.227
      • COUNTRY=Venezuela: Missing.Rate= 0.227
      • COUNTRY=Libya: Missing.Rate= 0.227
      • COUNTRY=Eritrea: Missing.Rate= 0.227
      • COUNTRY=Yemen: Missing.Rate= 0.227
  2. No variables were removed due to zero or near-zero variance.
  3. The cleaned dataset is comprised of:
    • 163 rows (observations)
    • 18 columns (variables)
      • 1/18 metadata (object)
        • COUNTRY
      • 1/18 target (categorical)
        • CANRAT
      • 15/18 predictor (numeric)
        • GDPPER
        • URBPOP
        • POPGRO
        • LIFEXP
        • TUBINC
        • DTHCMD
        • AGRLND
        • GHGEMI
        • METEMI
        • FORARE
        • CO2EMI
        • PM2EXP
        • POPDEN
        • GDPCAP
        • EPISCO
      • 1/18 predictor (categorical)
        • HDICAT
In [79]:
##################################
# Performing a general exploration of the original dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate.shape)

Dataset Dimensions: 

(177, 22)
In [80]:
##################################
# Filtering out the rows with
# with Missing.Rate > 0.20
##################################
cancer_rate_filtered_row = cancer_rate.drop(cancer_rate[cancer_rate.COUNTRY.isin(row_high_missing_rate['Row.Name'].values.tolist())].index)
In [81]:
##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_filtered_row.shape)

Dataset Dimensions: 

(163, 22)
In [82]:
##################################
# Filtering out the columns with
# with Fill.Rate < 0.90
##################################
cancer_rate_filtered_row_column = cancer_rate_filtered_row.drop(column_low_fill_rate['Column.Name'].values.tolist(), axis=1)
In [83]:
##################################
# Formulating a new dataset object
# for the cleaned data
##################################
cancer_rate_cleaned = cancer_rate_filtered_row_column
In [84]:
##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_cleaned.shape)

Dataset Dimensions: 

(163, 18)

1.4.2 Missing Data Imputation ¶

Iterative Imputer is based on the Multivariate Imputation by Chained Equations (MICE) algorithm - an imputation method based on fully conditional specification, where each incomplete variable is imputed by a separate model. As a sequential regression imputation technique, the algorithm imputes an incomplete column (target column) by generating plausible synthetic values given other columns in the data. Each incomplete column must act as a target column, and has its own specific set of predictors. For predictors that are incomplete themselves, the most recently generated imputations are used to complete the predictors prior to prior to imputation of the target columns.

Linear Regression explores the linear relationship between a scalar response and one or more covariates by having the conditional mean of the dependent variable be an affine function of the independent variables. The relationship is modeled through a disturbance term which represents an unobserved random variable that adds noise. The algorithm is typically formulated from the data using the least squares method which seeks to estimate the coefficients by minimizing the squared residual function. The linear equation assigns one scale factor represented by a coefficient to each covariate and an additional coefficient called the intercept or the bias coefficient which gives the line an additional degree of freedom allowing to move up and down a two-dimensional plot.

  1. Missing data for numeric variables were imputed using the iterative imputer algorithm with a linear regression estimator.
    • GDPPER: Null.Count = 1
    • FORARE: Null.Count = 1
    • PM2EXP: Null.Count = 5
  2. Missing data for categorical variables were imputed using the most frequent value.
    • HDICAP: Null.Count = 1
In [85]:
##################################
# Formulating the summary
# for all cleaned columns
##################################
cleaned_column_quality_summary = pd.DataFrame(zip(list(cancer_rate_cleaned.columns),
                                                  list(cancer_rate_cleaned.dtypes),
                                                  list([len(cancer_rate_cleaned)] * len(cancer_rate_cleaned.columns)),
                                                  list(cancer_rate_cleaned.count()),
                                                  list(cancer_rate_cleaned.isna().sum(axis=0))), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count'])
display(cleaned_column_quality_summary)
Column.Name Column.Type Row.Count Non.Null.Count Null.Count
0 COUNTRY object 163 163 0
1 CANRAT category 163 163 0
2 GDPPER float64 163 162 1
3 URBPOP float64 163 163 0
4 POPGRO float64 163 163 0
5 LIFEXP float64 163 163 0
6 TUBINC float64 163 163 0
7 DTHCMD float64 163 163 0
8 AGRLND float64 163 163 0
9 GHGEMI float64 163 163 0
10 METEMI float64 163 163 0
11 FORARE float64 163 162 1
12 CO2EMI float64 163 163 0
13 PM2EXP float64 163 158 5
14 POPDEN float64 163 163 0
15 GDPCAP float64 163 163 0
16 HDICAT category 163 162 1
17 EPISCO float64 163 163 0
In [86]:
##################################
# Formulating the cleaned dataset
# with categorical columns only
##################################
cancer_rate_cleaned_categorical = cancer_rate_cleaned.select_dtypes(include='object')
In [87]:
##################################
# Formulating the cleaned dataset
# with numeric columns only
##################################
cancer_rate_cleaned_numeric = cancer_rate_cleaned.select_dtypes(include='number')
In [88]:
##################################
# Taking a snapshot of the cleaned dataset
##################################
cancer_rate_cleaned_numeric.head()
Out[88]:
GDPPER URBPOP POPGRO LIFEXP TUBINC DTHCMD AGRLND GHGEMI METEMI FORARE CO2EMI PM2EXP POPDEN GDPCAP EPISCO
0 98380.63601 86.241 1.235701 83.200000 7.2 4.941054 46.252480 5.719031e+05 131484.763200 17.421315 14.772658 24.893584 3.335312 51722.06900 60.1
1 77541.76438 86.699 2.204789 82.256098 7.2 4.354730 38.562911 8.015803e+04 32241.937000 37.570126 6.160799 NaN 19.331586 41760.59478 56.7
2 198405.87500 63.653 1.029111 82.556098 5.3 5.684596 65.495718 5.949773e+04 15252.824630 11.351720 6.768228 0.274092 72.367281 85420.19086 57.4
3 130941.63690 82.664 0.964348 76.980488 2.3 5.302060 44.363367 5.505181e+06 748241.402900 33.866926 13.032828 3.343170 36.240985 63528.63430 51.1
4 113300.60110 88.116 0.291641 81.602439 4.1 6.826140 65.499675 4.113555e+04 7778.773921 15.711000 4.691237 56.914456 145.785100 60915.42440 77.9
In [89]:
##################################
# Defining the estimator to be used
# at each step of the round-robin imputation
##################################
lr = LinearRegression()
In [90]:
##################################
# Defining the parameter of the
# iterative imputer which will estimate 
# the columns with missing values
# as a function of the other columns
# in a round-robin fashion
##################################
iterative_imputer = IterativeImputer(
    estimator = lr,
    max_iter = 10,
    tol = 1e-10,
    imputation_order = 'ascending',
    random_state=88888888
)
In [91]:
##################################
# Implementing the iterative imputer 
##################################
cancer_rate_imputed_numeric_array = iterative_imputer.fit_transform(cancer_rate_cleaned_numeric)
In [92]:
##################################
# Transforming the imputed data
# from an array to a dataframe
##################################
cancer_rate_imputed_numeric = pd.DataFrame(cancer_rate_imputed_numeric_array, 
                                           columns = cancer_rate_cleaned_numeric.columns)
In [93]:
##################################
# Taking a snapshot of the imputed dataset
##################################
cancer_rate_imputed_numeric.head()
Out[93]:
GDPPER URBPOP POPGRO LIFEXP TUBINC DTHCMD AGRLND GHGEMI METEMI FORARE CO2EMI PM2EXP POPDEN GDPCAP EPISCO
0 98380.63601 86.241 1.235701 83.200000 7.2 4.941054 46.252480 5.719031e+05 131484.763200 17.421315 14.772658 24.893584 3.335312 51722.06900 60.1
1 77541.76438 86.699 2.204789 82.256098 7.2 4.354730 38.562911 8.015803e+04 32241.937000 37.570126 6.160799 65.867296 19.331586 41760.59478 56.7
2 198405.87500 63.653 1.029111 82.556098 5.3 5.684596 65.495718 5.949773e+04 15252.824630 11.351720 6.768228 0.274092 72.367281 85420.19086 57.4
3 130941.63690 82.664 0.964348 76.980488 2.3 5.302060 44.363367 5.505181e+06 748241.402900 33.866926 13.032828 3.343170 36.240985 63528.63430 51.1
4 113300.60110 88.116 0.291641 81.602439 4.1 6.826140 65.499675 4.113555e+04 7778.773921 15.711000 4.691237 56.914456 145.785100 60915.42440 77.9
In [94]:
##################################
# Formulating the cleaned dataset
# with categorical columns only
##################################
cancer_rate_cleaned_categorical = cancer_rate_cleaned.select_dtypes(include='category')
In [95]:
##################################
# Imputing the missing data
# for categorical columns with
# the most frequent category
##################################
cancer_rate_cleaned_categorical['HDICAT'] = cancer_rate_cleaned_categorical['HDICAT'].fillna(cancer_rate_cleaned_categorical['HDICAT'].mode()[0])
cancer_rate_imputed_categorical = cancer_rate_cleaned_categorical.reset_index(drop=True)
In [96]:
##################################
# Formulating the imputed dataset
##################################
cancer_rate_imputed = pd.concat([cancer_rate_imputed_numeric,cancer_rate_imputed_categorical], axis=1, join='inner')
In [97]:
##################################
# Gathering the data types for each column
##################################
data_type_list = list(cancer_rate_imputed.dtypes)
In [98]:
##################################
# Gathering the variable names for each column
##################################
variable_name_list = list(cancer_rate_imputed.columns)
In [99]:
##################################
# Gathering the number of observations for each column
##################################
row_count_list = list([len(cancer_rate_imputed)] * len(cancer_rate_imputed.columns))
In [100]:
##################################
# Gathering the number of missing data for each column
##################################
null_count_list = list(cancer_rate_imputed.isna().sum(axis=0))
In [101]:
##################################
# Gathering the number of non-missing data for each column
##################################
non_null_count_list = list(cancer_rate_imputed.count())
In [102]:
##################################
# Gathering the missing data percentage for each column
##################################
fill_rate_list = map(truediv, non_null_count_list, row_count_list)
In [103]:
##################################
# Formulating the summary
# for all imputed columns
##################################
imputed_column_quality_summary = pd.DataFrame(zip(variable_name_list,
                                                  data_type_list,
                                                  row_count_list,
                                                  non_null_count_list,
                                                  null_count_list,
                                                  fill_rate_list), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count',                                                 
                                                 'Fill.Rate'])
display(imputed_column_quality_summary)
Column.Name Column.Type Row.Count Non.Null.Count Null.Count Fill.Rate
0 GDPPER float64 163 163 0 1.0
1 URBPOP float64 163 163 0 1.0
2 POPGRO float64 163 163 0 1.0
3 LIFEXP float64 163 163 0 1.0
4 TUBINC float64 163 163 0 1.0
5 DTHCMD float64 163 163 0 1.0
6 AGRLND float64 163 163 0 1.0
7 GHGEMI float64 163 163 0 1.0
8 METEMI float64 163 163 0 1.0
9 FORARE float64 163 163 0 1.0
10 CO2EMI float64 163 163 0 1.0
11 PM2EXP float64 163 163 0 1.0
12 POPDEN float64 163 163 0 1.0
13 GDPCAP float64 163 163 0 1.0
14 EPISCO float64 163 163 0 1.0
15 CANRAT category 163 163 0 1.0
16 HDICAT category 163 163 0 1.0

1.4.3 Outlier Detection ¶

  1. High number of outliers observed for 5 numeric variables with Outlier.Ratio>0.10 and marginal to high Skewness.
    • PM2EXP: Outlier.Count = 37, Outlier.Ratio = 0.226, Skewness=-3.061
    • GHGEMI: Outlier.Count = 27, Outlier.Ratio = 0.165, Skewness=+9.299
    • GDPCAP: Outlier.Count = 22, Outlier.Ratio = 0.134, Skewness=+2.311
    • POPDEN: Outlier.Count = 20, Outlier.Ratio = 0.122, Skewness=+9.972
    • METEMI: Outlier.Count = 20, Outlier.Ratio = 0.122, Skewness=+5.688
  2. Minimal number of outliers observed for 5 numeric variables with Outlier.Ratio<0.10 and normal Skewness.
    • TUBINC: Outlier.Count = 12, Outlier.Ratio = 0.073, Skewness=+1.747
    • CO2EMI: Outlier.Count = 11, Outlier.Ratio = 0.067, Skewness=+2.693
    • GDPPER: Outlier.Count = 3, Outlier.Ratio = 0.018, Skewness=+1.554
    • EPISCO: Outlier.Count = 3, Outlier.Ratio = 0.018, Skewness=+0.635
    • CANRAT: Outlier.Count = 2, Outlier.Ratio = 0.012, Skewness=+0.910
In [104]:
##################################
# Formulating the imputed dataset
# with numeric columns only
##################################
cancer_rate_imputed_numeric = cancer_rate_imputed.select_dtypes(include='number')
In [105]:
##################################
# Gathering the variable names for each numeric column
##################################
numeric_variable_name_list = list(cancer_rate_imputed_numeric.columns)
In [106]:
##################################
# Gathering the skewness value for each numeric column
##################################
numeric_skewness_list = cancer_rate_imputed_numeric.skew()
In [107]:
##################################
# Computing the interquartile range
# for all columns
##################################
cancer_rate_imputed_numeric_q1 = cancer_rate_imputed_numeric.quantile(0.25)
cancer_rate_imputed_numeric_q3 = cancer_rate_imputed_numeric.quantile(0.75)
cancer_rate_imputed_numeric_iqr = cancer_rate_imputed_numeric_q3 - cancer_rate_imputed_numeric_q1
In [108]:
##################################
# Gathering the outlier count for each numeric column
# based on the interquartile range criterion
##################################
numeric_outlier_count_list = ((cancer_rate_imputed_numeric < (cancer_rate_imputed_numeric_q1 - 1.5 * cancer_rate_imputed_numeric_iqr)) | (cancer_rate_imputed_numeric > (cancer_rate_imputed_numeric_q3 + 1.5 * cancer_rate_imputed_numeric_iqr))).sum()
In [109]:
##################################
# Gathering the number of observations for each column
##################################
numeric_row_count_list = list([len(cancer_rate_imputed_numeric)] * len(cancer_rate_imputed_numeric.columns))
In [110]:
##################################
# Gathering the unique to count ratio for each categorical column
##################################
numeric_outlier_ratio_list = map(truediv, numeric_outlier_count_list, numeric_row_count_list)
In [111]:
##################################
# Formulating the outlier summary
# for all numeric columns
##################################
numeric_column_outlier_summary = pd.DataFrame(zip(numeric_variable_name_list,
                                                  numeric_skewness_list,
                                                  numeric_outlier_count_list,
                                                  numeric_row_count_list,
                                                  numeric_outlier_ratio_list), 
                                        columns=['Numeric.Column.Name',
                                                 'Skewness',
                                                 'Outlier.Count',
                                                 'Row.Count',
                                                 'Outlier.Ratio'])
display(numeric_column_outlier_summary)
Numeric.Column.Name Skewness Outlier.Count Row.Count Outlier.Ratio
0 GDPPER 1.554457 3 163 0.018405
1 URBPOP -0.212327 0 163 0.000000
2 POPGRO -0.181666 0 163 0.000000
3 LIFEXP -0.329704 0 163 0.000000
4 TUBINC 1.747962 12 163 0.073620
5 DTHCMD 0.930709 0 163 0.000000
6 AGRLND 0.035315 0 163 0.000000
7 GHGEMI 9.299960 27 163 0.165644
8 METEMI 5.688689 20 163 0.122699
9 FORARE 0.563015 0 163 0.000000
10 CO2EMI 2.693585 11 163 0.067485
11 PM2EXP -3.088403 37 163 0.226994
12 POPDEN 9.972806 20 163 0.122699
13 GDPCAP 2.311079 22 163 0.134969
14 EPISCO 0.635994 3 163 0.018405
In [112]:
##################################
# Formulating the individual boxplots
# for all numeric columns
##################################
for column in cancer_rate_imputed_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_imputed_numeric, x=column)
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1.4.4 Collinearity ¶

Pearson’s Correlation Coefficient is a parametric measure of the linear correlation for a pair of features by calculating the ratio between their covariance and the product of their standard deviations. The presence of high absolute correlation values indicate the univariate association between the numeric predictors and the numeric response.

  1. Majority of the numeric variables reported moderate to high correlation which were statistically significant.
  2. Among pairwise combinations of numeric variables, high Pearson.Correlation.Coefficient values were noted for:
    • GDPPER and GDPCAP: Pearson.Correlation.Coefficient = +0.921
    • GHGEMI and METEMI: Pearson.Correlation.Coefficient = +0.905
  3. Among the highly correlated pairs, variables with the lowest correlation against the target variable were removed.
    • GDPPER: Pearson.Correlation.Coefficient = +0.690
    • METEMI: Pearson.Correlation.Coefficient = +0.062
  4. The cleaned dataset is comprised of:
    • 163 rows (observations)
    • 16 columns (variables)
      • 1/16 metadata (object)
        • COUNTRY
      • 1/16 target (categorical)
        • CANRAT
      • 13/16 predictor (numeric)
        • URBPOP
        • POPGRO
        • LIFEXP
        • TUBINC
        • DTHCMD
        • AGRLND
        • GHGEMI
        • FORARE
        • CO2EMI
        • PM2EXP
        • POPDEN
        • GDPCAP
        • EPISCO
      • 1/16 predictor (categorical)
        • HDICAT
In [113]:
##################################
# Formulating a function 
# to plot the correlation matrix
# for all pairwise combinations
# of numeric columns
##################################
def plot_correlation_matrix(corr, mask=None):
    f, ax = plt.subplots(figsize=(11, 9))
    sns.heatmap(corr, 
                ax=ax,
                mask=mask,
                annot=True, 
                vmin=-1, 
                vmax=1, 
                center=0,
                cmap='coolwarm', 
                linewidths=1, 
                linecolor='gray', 
                cbar_kws={'orientation': 'horizontal'})
In [114]:
##################################
# Computing the correlation coefficients
# and correlation p-values
# among pairs of numeric columns
##################################
cancer_rate_imputed_numeric_correlation_pairs = {}
cancer_rate_imputed_numeric_columns = cancer_rate_imputed_numeric.columns.tolist()
for numeric_column_a, numeric_column_b in itertools.combinations(cancer_rate_imputed_numeric_columns, 2):
    cancer_rate_imputed_numeric_correlation_pairs[numeric_column_a + '_' + numeric_column_b] = stats.pearsonr(
        cancer_rate_imputed_numeric.loc[:, numeric_column_a], 
        cancer_rate_imputed_numeric.loc[:, numeric_column_b])
In [115]:
##################################
# Formulating the pairwise correlation summary
# for all numeric columns
##################################
cancer_rate_imputed_numeric_summary = cancer_rate_imputed_numeric.from_dict(cancer_rate_imputed_numeric_correlation_pairs, orient='index')
cancer_rate_imputed_numeric_summary.columns = ['Pearson.Correlation.Coefficient', 'Correlation.PValue']
display(cancer_rate_imputed_numeric_summary.sort_values(by=['Pearson.Correlation.Coefficient'], ascending=False).head(20))
Pearson.Correlation.Coefficient Correlation.PValue
GDPPER_GDPCAP 0.921010 8.158179e-68
GHGEMI_METEMI 0.905121 1.087643e-61
POPGRO_DTHCMD 0.759470 7.124695e-32
GDPPER_LIFEXP 0.755787 2.055178e-31
GDPCAP_EPISCO 0.696707 5.312642e-25
LIFEXP_GDPCAP 0.683834 8.321371e-24
GDPPER_EPISCO 0.680812 1.555304e-23
GDPPER_URBPOP 0.666394 2.781623e-22
GDPPER_CO2EMI 0.654958 2.450029e-21
TUBINC_DTHCMD 0.643615 1.936081e-20
URBPOP_LIFEXP 0.623997 5.669778e-19
LIFEXP_EPISCO 0.620271 1.048393e-18
URBPOP_GDPCAP 0.559181 8.624533e-15
CO2EMI_GDPCAP 0.550221 2.782997e-14
URBPOP_CO2EMI 0.550046 2.846393e-14
LIFEXP_CO2EMI 0.531305 2.951829e-13
URBPOP_EPISCO 0.510131 3.507463e-12
POPGRO_TUBINC 0.442339 3.384403e-09
DTHCMD_PM2EXP 0.283199 2.491837e-04
CO2EMI_EPISCO 0.282734 2.553620e-04
In [116]:
##################################
# Plotting the correlation matrix
# for all pairwise combinations
# of numeric columns
##################################
cancer_rate_imputed_numeric_correlation = cancer_rate_imputed_numeric.corr()
mask = np.triu(cancer_rate_imputed_numeric_correlation)
plot_correlation_matrix(cancer_rate_imputed_numeric_correlation,mask)
plt.show()
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In [117]:
##################################
# Formulating a function 
# to plot the correlation matrix
# for all pairwise combinations
# of numeric columns
# with significant p-values only
##################################
def correlation_significance(df=None):
    p_matrix = np.zeros(shape=(df.shape[1],df.shape[1]))
    for col in df.columns:
        for col2 in df.drop(col,axis=1).columns:
            _ , p = stats.pearsonr(df[col],df[col2])
            p_matrix[df.columns.to_list().index(col),df.columns.to_list().index(col2)] = p
    return p_matrix
In [118]:
##################################
# Plotting the correlation matrix
# for all pairwise combinations
# of numeric columns
# with significant p-values only
##################################
cancer_rate_imputed_numeric_correlation_p_values = correlation_significance(cancer_rate_imputed_numeric)                     
mask = np.invert(np.tril(cancer_rate_imputed_numeric_correlation_p_values<0.05)) 
plot_correlation_matrix(cancer_rate_imputed_numeric_correlation,mask)
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In [119]:
##################################
# Filtering out one among the 
# highly correlated variable pairs with
# lesser Pearson.Correlation.Coefficient
# when compared to the target variable
##################################
cancer_rate_imputed_numeric.drop(['GDPPER','METEMI'], inplace=True, axis=1)
In [120]:
##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_imputed_numeric.shape)

Dataset Dimensions: 

(163, 13)

1.4.5 Shape Transformation ¶

Yeo-Johnson Transformation applies a new family of distributions that can be used without restrictions, extending many of the good properties of the Box-Cox power family. Similar to the Box-Cox transformation, the method also estimates the optimal value of lambda but has the ability to transform both positive and negative values by inflating low variance data and deflating high variance data to create a more uniform data set. While there are no restrictions in terms of the applicable values, the interpretability of the transformed values is more diminished as compared to the other methods.

  1. A Yeo-Johnson transformation was applied to all numeric variables to improve distributional shape.
  2. Most variables achieved symmetrical distributions with minimal outliers after transformation.
  3. One variable which remained skewed even after applying shape transformation was removed.
    • PM2EXP
  4. The transformed dataset is comprised of:
    • 163 rows (observations)
    • 15 columns (variables)
      • 1/15 metadata (object)
        • COUNTRY
      • 1/15 target (categorical)
        • CANRAT
      • 12/15 predictor (numeric)
        • URBPOP
        • POPGRO
        • LIFEXP
        • TUBINC
        • DTHCMD
        • AGRLND
        • GHGEMI
        • FORARE
        • CO2EMI
        • POPDEN
        • GDPCAP
        • EPISCO
      • 1/15 predictor (categorical)
        • HDICAT
In [121]:
##################################
# Conducting a Yeo-Johnson Transformation
# to address the distributional
# shape of the variables
##################################
yeo_johnson_transformer = PowerTransformer(method='yeo-johnson',
                                          standardize=False)
cancer_rate_imputed_numeric_array = yeo_johnson_transformer.fit_transform(cancer_rate_imputed_numeric)
In [122]:
##################################
# Formulating a new dataset object
# for the transformed data
##################################
cancer_rate_transformed_numeric = pd.DataFrame(cancer_rate_imputed_numeric_array,
                                               columns=cancer_rate_imputed_numeric.columns)
In [123]:
##################################
# Formulating the individual boxplots
# for all transformed numeric columns
##################################
for column in cancer_rate_transformed_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_transformed_numeric, x=column)
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In [124]:
##################################
# Filtering out the column
# which remained skewed even
# after applying shape transformation
##################################
cancer_rate_transformed_numeric.drop(['PM2EXP'], inplace=True, axis=1)
In [125]:
##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_transformed_numeric.shape)

Dataset Dimensions: 

(163, 12)

1.4.6 Centering and Scaling ¶

  1. All numeric variables were transformed using the standardization method to achieve a comparable scale between values.
  2. The scaled dataset is comprised of:
    • 163 rows (observations)
    • 15 columns (variables)
      • 1/15 metadata (object)
        • COUNTRY
      • 1/15 target (categorical)
        • CANRAT
      • 12/15 predictor (numeric)
        • URBPOP
        • POPGRO
        • LIFEXP
        • TUBINC
        • DTHCMD
        • AGRLND
        • GHGEMI
        • FORARE
        • CO2EMI
        • POPDEN
        • GDPCAP
        • EPISCO
      • 1/15 predictor (categorical)
        • HDICAT
In [126]:
##################################
# Conducting standardization
# to transform the values of the 
# variables into comparable scale
##################################
standardization_scaler = StandardScaler()
cancer_rate_transformed_numeric_array = standardization_scaler.fit_transform(cancer_rate_transformed_numeric)
In [127]:
##################################
# Formulating a new dataset object
# for the scaled data
##################################
cancer_rate_scaled_numeric = pd.DataFrame(cancer_rate_transformed_numeric_array,
                                          columns=cancer_rate_transformed_numeric.columns)
In [128]:
##################################
# Formulating the individual boxplots
# for all transformed numeric columns
##################################
for column in cancer_rate_scaled_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_scaled_numeric, x=column)
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1.4.7 Data Encoding ¶

  1. One-hot encoding was applied to the HDICAP_VH variable resulting to 4 additional columns in the dataset:
    • HDICAP_L
    • HDICAP_M
    • HDICAP_H
    • HDICAP_VH
In [129]:
##################################
# Formulating the categorical column
# for encoding transformation
##################################
cancer_rate_categorical_encoded = pd.DataFrame(cancer_rate_cleaned_categorical.loc[:, 'HDICAT'].to_list(),
                                               columns=['HDICAT'])
In [130]:
##################################
# Applying a one-hot encoding transformation
# for the categorical column
##################################
cancer_rate_categorical_encoded = pd.get_dummies(cancer_rate_categorical_encoded, columns=['HDICAT'])

1.4.8 Preprocessed Data Description ¶

  1. The preprocessed dataset is comprised of:
    • 163 rows (observations)
    • 18 columns (variables)
      • 1/18 metadata (object)
        • COUNTRY
      • 1/18 target (categorical)
        • CANRAT
      • 12/18 predictor (numeric)
        • URBPOP
        • POPGRO
        • LIFEXP
        • TUBINC
        • DTHCMD
        • AGRLND
        • GHGEMI
        • FORARE
        • CO2EMI
        • POPDEN
        • GDPCAP
        • EPISCO
      • 4/18 predictor (categorical)
        • HDICAT_L
        • HDICAT_M
        • HDICAT_H
        • HDICAT_VH
In [131]:
##################################
# Consolidating both numeric columns
# and encoded categorical columns
##################################
cancer_rate_preprocessed = pd.concat([cancer_rate_scaled_numeric,cancer_rate_categorical_encoded], axis=1, join='inner')
In [132]:
##################################
# Performing a general exploration of the consolidated dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_preprocessed.shape)

Dataset Dimensions: 

(163, 16)

1.5. Data Exploration ¶

1.5.1 Exploratory Data Analysis ¶

  1. Bivariate analysis identified individual predictors with generally positive association to the target variable based on visual inspection.
  2. Higher values or higher proportions for the following predictors are associated with the CANRAT HIGH category:
    • URBPOP
    • LIFEXP
    • CO2EMI
    • GDPCAP
    • EPISCO
    • HDICAP_VH=1
  3. Decreasing values or smaller proportions for the following predictors are associated with the CANRAT LOW category:
    • POPGRO
    • TUBINC
    • DTHCMD
    • HDICAP_L=0
    • HDICAP_M=0
    • HDICAP_H=0
  4. Values for the following predictors are not associated with the CANRAT HIGH or LOW categories:
    • AGRLND
    • GHGEMI
    • FORARE
    • POPDEN
In [133]:
##################################
# Segregating the target
# and predictor variable lists
##################################
cancer_rate_preprocessed_target = cancer_rate_filtered_row['CANRAT'].to_frame()
cancer_rate_preprocessed_target.reset_index(inplace=True, drop=True)
cancer_rate_preprocessed_categorical = cancer_rate_preprocessed[cancer_rate_categorical_encoded.columns]
cancer_rate_preprocessed_categorical_combined = cancer_rate_preprocessed_categorical.join(cancer_rate_preprocessed_target)
cancer_rate_preprocessed = cancer_rate_preprocessed.drop(cancer_rate_categorical_encoded.columns, axis=1) 
cancer_rate_preprocessed_predictors = cancer_rate_preprocessed.columns
cancer_rate_preprocessed_combined = cancer_rate_preprocessed.join(cancer_rate_preprocessed_target)
In [134]:
##################################
# Segregating the target
# and predictor variable names
##################################
y_variable = 'CANRAT'
x_variables = cancer_rate_preprocessed_predictors
In [135]:
##################################
# Defining the number of 
# rows and columns for the subplots
##################################
num_rows = 6
num_cols = 2
In [136]:
##################################
# Formulating the subplot structure
##################################
fig, axes = plt.subplots(num_rows, num_cols, figsize=(15, 30))

##################################
# Flattening the multi-row and
# multi-column axes
##################################
axes = axes.ravel()

##################################
# Formulating the individual boxplots
# for all scaled numeric columns
##################################
for i, x_variable in enumerate(x_variables):
    ax = axes[i]
    ax.boxplot([group[x_variable] for name, group in cancer_rate_preprocessed_combined.groupby(y_variable, observed=True)])
    ax.set_title(f'{y_variable} Versus {x_variable}')
    ax.set_xlabel(y_variable)
    ax.set_ylabel(x_variable)
    ax.set_xticks(range(1, len(cancer_rate_preprocessed_combined[y_variable].unique()) + 1), ['Low', 'High'])

##################################
# Adjusting the subplot layout
##################################
plt.tight_layout()

##################################
# Presenting the subplots
##################################
plt.show()
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In [137]:
##################################
# Segregating the target
# and predictor variable names
##################################
y_variables = cancer_rate_preprocessed_categorical.columns
x_variable = 'CANRAT'

##################################
# Defining the number of 
# rows and columns for the subplots
##################################
num_rows = 2
num_cols = 2

##################################
# Formulating the subplot structure
##################################
fig, axes = plt.subplots(num_rows, num_cols, figsize=(15, 10))

##################################
# Flattening the multi-row and
# multi-column axes
##################################
axes = axes.ravel()

##################################
# Formulating the individual stacked column plots
# for all categorical columns
##################################
for i, y_variable in enumerate(y_variables):
    ax = axes[i]
    category_counts = cancer_rate_preprocessed_categorical_combined.groupby([x_variable, y_variable], observed=True).size().unstack(fill_value=0)
    category_proportions = category_counts.div(category_counts.sum(axis=1), axis=0)
    category_proportions.plot(kind='bar', stacked=True, ax=ax)
    ax.set_title(f'{x_variable} Versus {y_variable}')
    ax.set_xlabel(x_variable)
    ax.set_ylabel('Proportions')

##################################
# Adjusting the subplot layout
##################################
plt.tight_layout()

##################################
# Presenting the subplots
##################################
plt.show()
No description has been provided for this image

1.5.2 Hypothesis Testing ¶

  1. The relationship between the numeric predictors to the CANRAT target variable was statistically evaluated using the following hypotheses:
    • Null: Difference in the means between groups LOW and HIGH is equal to zero
    • Alternative: Difference in the means between groups LOW and HIGH is not equal to zero
  2. There is sufficient evidence to conclude of a statistically significant difference between the means of the numeric measurements obtained from LOW and HIGH groups of the CANRAT target variable in 9 of the 12 numeric predictors given their high t-test statistic values with reported low p-values less than the significance level of 0.05.
    • GDPCAP: T.Test.Statistic=-11.937, Correlation.PValue=0.000
    • EPISCO: T.Test.Statistic=-11.789, Correlation.PValue=0.000
    • LIFEXP: T.Test.Statistic=-10.979, Correlation.PValue=0.000
    • TUBINC: T.Test.Statistic=+9.609, Correlation.PValue=0.000
    • DTHCMD: T.Test.Statistic=+8.376, Correlation.PValue=0.000
    • CO2EMI: T.Test.Statistic=-7.031, Correlation.PValue=0.000
    • URBPOP: T.Test.Statistic=-6.541, Correlation.PValue=0.000
    • POPGRO: T.Test.Statistic=+4.905, Correlation.PValue=0.000
    • GHGEMI: T.Test.Statistic=-2.243, Correlation.PValue=0.026
  3. The relationship between the categorical predictors to the CANRAT target variable was statistically evaluated using the following hypotheses:
    • Null: The categorical predictor is independent of the categorical target variable
    • Alternative: The categorical predictor is dependent of the categorical target variable
  4. There is sufficient evidence to conclude of a statistically significant relationship difference between the categories of the categorical predictors and the LOW and HIGH groups of the CANRAT target variable in all 4 categorical predictors given their high chisquare statistic values with reported low p-values less than the significance level of 0.05.
    • HDICAT_VH: ChiSquare.Test.Statistic=76.764, ChiSquare.Test.PValue=0.000
    • HDICAT_H: ChiSquare.Test.Statistic=13.860, ChiSquare.Test.PValue=0.000
    • HDICAT_M: ChiSquare.Test.Statistic=10.286, ChiSquare.Test.PValue=0.001
    • HDICAT_L: ChiSquare.Test.Statistic=9.081, ChiSquare.Test.PValue=0.002
In [138]:
##################################
# Computing the t-test 
# statistic and p-values
# between the target variable
# and numeric predictor columns
##################################
cancer_rate_preprocessed_numeric_ttest_target = {}
cancer_rate_preprocessed_numeric = cancer_rate_preprocessed_combined
cancer_rate_preprocessed_numeric_columns = cancer_rate_preprocessed_predictors
for numeric_column in cancer_rate_preprocessed_numeric_columns:
    group_0 = cancer_rate_preprocessed_numeric[cancer_rate_preprocessed_numeric.loc[:,'CANRAT']=='Low']
    group_1 = cancer_rate_preprocessed_numeric[cancer_rate_preprocessed_numeric.loc[:,'CANRAT']=='High']
    cancer_rate_preprocessed_numeric_ttest_target['CANRAT_' + numeric_column] = stats.ttest_ind(
        group_0[numeric_column], 
        group_1[numeric_column], 
        equal_var=True)
In [139]:
##################################
# Formulating the pairwise ttest summary
# between the target variable
# and numeric predictor columns
##################################
cancer_rate_preprocessed_numeric_summary = cancer_rate_preprocessed_numeric.from_dict(cancer_rate_preprocessed_numeric_ttest_target, orient='index')
cancer_rate_preprocessed_numeric_summary.columns = ['T.Test.Statistic', 'T.Test.PValue']
display(cancer_rate_preprocessed_numeric_summary.sort_values(by=['T.Test.PValue'], ascending=True).head(12))
T.Test.Statistic T.Test.PValue
CANRAT_GDPCAP -11.936988 6.247937e-24
CANRAT_EPISCO -11.788870 1.605980e-23
CANRAT_LIFEXP -10.979098 2.754214e-21
CANRAT_TUBINC 9.608760 1.463678e-17
CANRAT_DTHCMD 8.375558 2.552108e-14
CANRAT_CO2EMI -7.030702 5.537463e-11
CANRAT_URBPOP -6.541001 7.734940e-10
CANRAT_POPGRO 4.904817 2.269446e-06
CANRAT_GHGEMI -2.243089 2.625563e-02
CANRAT_FORARE -1.174143 2.420717e-01
CANRAT_POPDEN -0.495221 6.211191e-01
CANRAT_AGRLND -0.047628 9.620720e-01
In [140]:
##################################
# Computing the chisquare
# statistic and p-values
# between the target variable
# and categorical predictor columns
##################################
cancer_rate_preprocessed_categorical_chisquare_target = {}
cancer_rate_preprocessed_categorical = cancer_rate_preprocessed_categorical_combined
cancer_rate_preprocessed_categorical_columns = ['HDICAT_L','HDICAT_M','HDICAT_H','HDICAT_VH']
for categorical_column in cancer_rate_preprocessed_categorical_columns:
    contingency_table = pd.crosstab(cancer_rate_preprocessed_categorical[categorical_column], 
                                    cancer_rate_preprocessed_categorical['CANRAT'])
    cancer_rate_preprocessed_categorical_chisquare_target['CANRAT_' + categorical_column] = stats.chi2_contingency(
        contingency_table)[0:2]
In [141]:
##################################
# Formulating the pairwise chisquare summary
# between the target variable
# and categorical predictor columns
##################################
cancer_rate_preprocessed_categorical_summary = cancer_rate_preprocessed_categorical.from_dict(cancer_rate_preprocessed_categorical_chisquare_target, orient='index')
cancer_rate_preprocessed_categorical_summary.columns = ['ChiSquare.Test.Statistic', 'ChiSquare.Test.PValue']
display(cancer_rate_preprocessed_categorical_summary.sort_values(by=['ChiSquare.Test.PValue'], ascending=True).head(4))
ChiSquare.Test.Statistic ChiSquare.Test.PValue
CANRAT_HDICAT_VH 76.764134 1.926446e-18
CANRAT_HDICAT_M 13.860367 1.969074e-04
CANRAT_HDICAT_L 10.285575 1.340742e-03
CANRAT_HDICAT_H 9.080788 2.583087e-03

2. Summary ¶

Project41_Summary.png

3. References ¶

  • [Book] Data Preparation for Machine Learning: Data Cleaning, Feature Selection, and Data Transforms in Python by Jason Brownlee
  • [Book] Feature Engineering and Selection: A Practical Approach for Predictive Models by Max Kuhn and Kjell Johnson
  • [Book] Feature Engineering for Machine Learning by Alice Zheng and Amanda Casari
  • [Book] Applied Predictive Modeling by Max Kuhn and Kjell Johnson
  • [Book] Data Mining: Practical Machine Learning Tools and Techniques by Ian Witten, Eibe Frank, Mark Hall and Christopher Pal
  • [Book] Data Cleaning by Ihab Ilyas and Xu Chu
  • [Book] Data Wrangling with Python by Jacqueline Kazil and Katharine Jarmul
  • [Book] Regression Modeling Strategies by Frank Harrell
  • [Python Library API] NumPy by NumPy Team
  • [Python Library API] pandas by Pandas Team
  • [Python Library API] seaborn by Seaborn Team
  • [Python Library API] matplotlib.pyplot by MatPlotLib Team
  • [Python Library API] itertools by Python Team
  • [Python Library API] operator by Python Team
  • [Python Library API] sklearn.experimental by Scikit-Learn Team
  • [Python Library API] sklearn.impute by Scikit-Learn Team
  • [Python Library API] sklearn.linear_model by Scikit-Learn Team
  • [Python Library API] sklearn.preprocessing by Scikit-Learn Team
  • [Python Library API] scipy by SciPy Team
  • [Article] Exploratory Data Analysis in Python — A Step-by-Step Process by Andrea D'Agostino (Towards Data Science)
  • [Article] Exploratory Data Analysis with Python by Douglas Rocha (Medium)
  • [Article] 4 Ways to Automate Exploratory Data Analysis (EDA) in Python by Abdishakur Hassan (BuiltIn)
  • [Article] 10 Things To Do When Conducting Your Exploratory Data Analysis (EDA) by Alifia Harmadi (Medium)
  • [Article] How to Handle Missing Data with Python by Jason Brownlee (Machine Learning Mastery)
  • [Article] Statistical Imputation for Missing Values in Machine Learning by Jason Brownlee (Machine Learning Mastery)
  • [Article] Imputing Missing Data with Simple and Advanced Techniques by Idil Ismiguzel (Towards Data Science)
  • [Article] Missing Data Imputation Approaches | How to handle missing values in Python by Selva Prabhakaran (Machine Learning +)
  • [Article] Master The Skills Of Missing Data Imputation Techniques In Python(2022) And Be Successful by Mrinal Walia (Analytics Vidhya)
  • [Article] How to Preprocess Data in Python by Afroz Chakure (BuiltIn)
  • [Article] Easy Guide To Data Preprocessing In Python by Ahmad Anis (KDNuggets)
  • [Article] Data Preprocessing in Python by Tarun Gupta (Towards Data Science)
  • [Article] Data Preprocessing using Python by Suneet Jain (Medium)
  • [Article] Data Preprocessing in Python by Abonia Sojasingarayar (Medium)
  • [Article] Data Preprocessing in Python by Afroz Chakure (Medium)
  • [Article] Detecting and Treating Outliers | Treating the Odd One Out! by Harika Bonthu (Analytics Vidhya)
  • [Article] Outlier Treatment with Python by Sangita Yemulwar (Analytics Vidhya)
  • [Article] A Guide to Outlier Detection in Python by Sadrach Pierre (BuiltIn)
  • [Article] How To Find Outliers in Data Using Python (and How To Handle Them) by Eric Kleppen (Career Foundry)
  • [Article] Statistics in Python — Collinearity and Multicollinearity by Wei-Meng Lee (Towards Data Science)
  • [Article] Understanding Multicollinearity and How to Detect it in Python by Terence Shin (Towards Data Science)
  • [Article] A Python Library to Remove Collinearity by Gianluca Malato (Your Data Teacher)
  • [Article] 8 Best Data Transformation in Pandas by Tirendaz AI (Medium)
  • [Article] Data Transformation Techniques with Python: Elevate Your Data Game! by Siddharth Verma (Medium)
  • [Article] Data Scaling with Python by Benjamin Obi Tayo (KDNuggets)
  • [Article] How to Use StandardScaler and MinMaxScaler Transforms in Python by Jason Brownlee (Machine Learning Mastery)
  • [Article] Feature Engineering: Scaling, Normalization, and Standardization by Aniruddha Bhandari (Analytics Vidhya)
  • [Article] How to Normalize Data Using scikit-learn in Python by Jayant Verma (Digital Ocean)
  • [Article] What are Categorical Data Encoding Methods | Binary Encoding by Shipra Saxena (Analytics Vidhya)
  • [Article] Guide to Encoding Categorical Values in Python by Chris Moffitt (Practical Business Python)
  • [Article] Categorical Data Encoding Techniques in Python: A Complete Guide by Soumen Atta (Medium)
  • [Article] Categorical Feature Encoding Techniques by Tara Boyle (Medium)
  • [Article] Ordinal and One-Hot Encodings for Categorical Data by Jason Brownlee (Machine Learning Mastery)
  • [Article] Hypothesis Testing with Python: Step by Step Hands-On Tutorial with Practical Examples by Ece Işık Polat (Towards Data Science)
  • [Article] 17 Statistical Hypothesis Tests in Python (Cheat Sheet) by Jason Brownlee (Machine Learning Mastery)
  • [Article] A Step-by-Step Guide to Hypothesis Testing in Python using Scipy by Gabriel Rennó (Medium)
  • [Publication] Data Quality for Machine Learning Tasks by Nitin Gupta, Shashank Mujumdar, Hima Patel, Satoshi Masuda, Naveen Panwar, Sambaran Bandyopadhyay, Sameep Mehta, Shanmukha Guttula, Shazia Afzal, Ruhi Sharma Mittal and Vitobha Munigala (KDD ’21: Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining)
  • [Publication] Overview and Importance of Data Quality for Machine Learning Tasks by Abhinav Jain, Hima Patel, Lokesh Nagalapatti, Nitin Gupta, Sameep Mehta, Shanmukha Guttula, Shashank Mujumdar, Shazia Afzal, Ruhi Sharma Mittal and Vitobha Munigala (KDD ’20: Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining)
  • [Publication] Multiple Imputation of Discrete and Continuous Data by Fully Conditional Specification by Stef van Buuren (Statistical Methods in Medical Research)
  • [Publication] Mathematical Contributions to the Theory of Evolution: Regression, Heredity and Panmixia by Karl Pearson (Royal Society)
  • [Publication] A New Family of Power Transformations to Improve Normality or Symmetry by In-Kwon Yeo and Richard Johnson (Biometrika)
In [142]:
from IPython.display import display, HTML
display(HTML("<style>.rendered_html { font-size: 15px; font-family: 'Trebuchet MS'; }</style>"))