TP8 - Multiple Corresponding Analysis (MCA)

Exploratory Data Analysis & Unsuperivsed Learning
Course: PHAUK Sokkey, PhD
TP: HAS Sothea, PhD

Objective: “In the previous TP, we studied Correspondence Analysis (CA), which can be used to analyze the association between two qualitative variables. In this TP, we extend this to the more general case of Multiple Correspondence Analysis (MCA), which is used to study the association of multiple variables in an indicator or a Burt matrix or survey.

The Jupyter Notebook for this TP can be downloaded here: TP8_MCA.ipynb.

1. Kaggle Stroke Dataset

A stroke occurs when the blood supply to part of the brain is interrupted or reduced, preventing brain tissue from getting oxygen and nutrients. Brain cells begin to die within minutes. Strokes can be classified into two main types: ischemic stroke, caused by a blockage in an artery, and hemorrhagic stroke, caused by a blood vessel leaking or bursting. Immediate medical attention is crucial to minimize brain damage and potential complications (see, Mayo Clinic and WebMD).

The Kaggle Stroke Prediction Dataset (available here) is designed to predict the likelihood of a stroke based on various health parameters. We will analyze this association using MCA.

The Stroke dataset dataset can be imported from kaggle as follow.

import kagglehub

# Download latest version
path = kagglehub.dataset_download("fedesoriano/stroke-prediction-dataset")
# Import data
import pandas as pd
data = pd.read_csv(path + "/healthcare-dataset-stroke-data.csv")
data.head()
id gender age hypertension heart_disease ever_married work_type Residence_type avg_glucose_level bmi smoking_status stroke
0 9046 Male 67.0 0 1 Yes Private Urban 228.69 36.6 formerly smoked 1
1 51676 Female 61.0 0 0 Yes Self-employed Rural 202.21 NaN never smoked 1
2 31112 Male 80.0 0 1 Yes Private Rural 105.92 32.5 never smoked 1
3 60182 Female 49.0 0 0 Yes Private Urban 171.23 34.4 smokes 1
4 1665 Female 79.0 1 0 Yes Self-employed Rural 174.12 24.0 never smoked 1 
# To do

A. Perform data preprocessing to ensure that the columns are in correct types, clean and complete for further analysis.

# To do

B. Recall that in CA (or even MCA), if \(X\in\mathbb{R}^{n\times m}\) is the design matrix, let \(Z=X/n\) be the normalized matrix. The row and column marginal relative frequencies are given by \(\textbf{r}=Z\mathbb{1}_{m}\) and \(\textbf{c}=\mathbb{1}_{n}Z\) where \(\mathbb{1}_d\) denotes a \(d\times d\) matrix with all elements equal to 1. Let the row and column weights \(D_r=\text{diag}(\textbf{r})\) and \(D_c=\text{diag}(\textbf{c})\), then the factor scores are obtained from the following singular value decomposition:

\[D_r^{-1/2}(Z-\textbf{r}\textbf{c}^T)D_c^{-1/2}=UD V^T,\]

where \(D\) is the diagonal matrix of sigular values with \(\Sigma=D^2\) are the eigenvalues. The row and column scores are given by: \[F=D_r^{-1/2}UD\quad\text{and}\quad G=D_c^{-1/2}VD,\] which are the scores represented on the principal axes with respect to \(\chi^2\)-distance.

With Stroke dataset,

  • Perform MCA on this stroke dataset using prince available here: https://github.com/MaxHalford/prince.

  • Compute the explained varianced of the first two axes.

  • Compute the row factor scores.

  • Compute the column factor scores.

  • Create symmetric Biplot: a balanced view of both variables and observations, with both sets in principal coordinates. Explain the graph.

# To do

2. Sleep Health and Lifestyle Dataset

MCA has been widely used in studies examining the associations between instances based on different categorical characteristics. In this section, we will use MCA to analyze the association within kaggle Sleep Health and Lifestyle Dataset available here: https://www.kaggle.com/datasets/siamaktahmasbi/insights-into-sleep-patterns-and-daily-habits.

A. Apply data preprocessing on the dataset:

  • Make sure that there is no missing values nor duplicated data, and it is ready for the analysis.

  • Convert some continuous variables into grouped data for our analysis such as blood pressure into [‘low’, ‘normal’, ‘high’]. You have to decide that own your own and based on your research.

import kagglehub
import pandas as pd
# Download latest version
path = kagglehub.dataset_download("siamaktahmasbi/insights-into-sleep-patterns-and-daily-habits")

data = pd.read_csv(path + '/sleep_health_lifestyle_dataset.csv')
data.head()
Person ID Gender Age Occupation Sleep Duration (hours) Quality of Sleep (scale: 1-10) Physical Activity Level (minutes/day) Stress Level (scale: 1-10) BMI Category Blood Pressure (systolic/diastolic) Heart Rate (bpm) Daily Steps Sleep Disorder
0 1 Male 29 Manual Labor 7.4 7.0 41 7 Obese 124/70 91 8539 NaN
1 2 Female 43 Retired 4.2 4.9 41 5 Obese 131/86 81 18754 NaN
2 3 Male 44 Retired 6.1 6.0 107 4 Underweight 122/70 81 2857 NaN
3 4 Male 29 Office Worker 8.3 10.0 20 10 Obese 124/72 55 6886 NaN
4 5 Male 67 Retired 9.1 9.5 19 4 Overweight 133/78 97 14945 Insomnia

B. Perform MCA on the dataset:

  • Compute variances explained by the first two dimensions.

  • Compute the row factor scores on the first two dimensions.

  • Compute the column factor scores on the first two dimensions.

  • Create a symmetric Biplot: a balanced view of variables and observations, with both sets in principal coordinates. Explain the graph.

Further Readings