TP8 - Multiple Corresponding Analysis (CA)

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 analysze this asssociation 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

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

  • Data types
data = data.drop(columns=['id'])
data.describe()
age hypertension heart_disease avg_glucose_level bmi stroke
count 5110.000000 5110.000000 5110.000000 5110.000000 4909.000000 5110.000000
mean 43.226614 0.097456 0.054012 106.147677 28.893237 0.048728
std 22.612647 0.296607 0.226063 45.283560 7.854067 0.215320
min 0.080000 0.000000 0.000000 55.120000 10.300000 0.000000
25% 25.000000 0.000000 0.000000 77.245000 23.500000 0.000000
50% 45.000000 0.000000 0.000000 91.885000 28.100000 0.000000
75% 61.000000 0.000000 0.000000 114.090000 33.100000 0.000000
max 82.000000 1.000000 1.000000 271.740000 97.600000 1.000000 
quan_col = ["age", "avg_glucose_level", "bmi"]

for va in data.columns:
    if va not in quan_col:
        data[va] = data[va].astype(object)
data.dtypes
gender                object
age                  float64
hypertension          object
heart_disease         object
ever_married          object
work_type             object
Residence_type        object
avg_glucose_level    float64
bmi                  float64
smoking_status        object
stroke                object
dtype: object
  • Missing values
data.isna().sum(), data.shape
(gender                 0
 age                    0
 hypertension           0
 heart_disease          0
 ever_married           0
 work_type              0
 Residence_type         0
 avg_glucose_level      0
 bmi                  201
 smoking_status         0
 stroke                 0
 dtype: int64,
 (5110, 11))
import seaborn as sns
sns.set(style="whitegrid")
import matplotlib.pyplot as plt

_, axs = plt.subplots(2, 3, figsize=(12, 6))

for i, va in enumerate(quan_col):
    sns.histplot(data, x=va, ax=axs[0,i])
    axs[0,i].set_title(f"Distribution of {va} before removing NA")

    sns.histplot(data.dropna(), x=va, ax=axs[1,i])
    axs[1,i].set_title(f"Distribution of {va} after removing NA")

plt.tight_layout()
plt.show()


qual_cols = data.select_dtypes(include='object').columns
_, axs = plt.subplots(2, len(qual_cols), figsize=(22, 6))
for i, va in enumerate(qual_cols):
    sns.countplot(data, x=va, ax=axs[0,i])
    axs[0,i].set_title(f"{va} before removing NA")

    sns.countplot(data.dropna(), x=va, ax=axs[1,i])
    axs[1,i].set_title(f"{va} after removing NA")
    if len(axs[0,i].containers[0])> 3:
        axs[0,i].tick_params(labelrotation=45)
        axs[1,i].tick_params(labelrotation=45)

plt.tight_layout()
plt.show()

According to the analysis, the missing values are “Completely At Random (MCAR)”, therefore we can drop them or impute them with some constant.

data = data.dropna()
data = data.drop_duplicates()

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.

import prince

mca = prince.MCA(n_components=22)
mca = mca.fit(data.select_dtypes(include='object'))
mca.percentage_of_variance_[:2].round()
array([15.,  9.])
import plotly.io as pio
pio.renderers.default = 'notebook'

import matplotlib.pyplot as plt
df = data.select_dtypes(include="object")      # select only categorical variables
plt.figure(figsize=(10,7))
mca.plot(
    df,
    x_component=0,
    y_component=1,
    show_column_markers=True,
    show_row_markers=True,
    show_column_labels=True,
    show_row_labels=False
)
<Figure size 1000x700 with 0 Axes>
  • The plot function from prince.MCA does not offer much flexibility for building the biplot of MCA. We can define our own biplot function as follows:
def MCABiplot(mca, data, w = 600, h = 500):
    import plotly.io as pio
    pio.renderers.default = 'notebook'
    row_coords = mca.row_coordinates(data)
    col_coords = mca.column_coordinates(data)

    import plotly.express as px
    import plotly.graph_objects as go
    row_coords.columns = ['dimension ' + str(i+1) for i in range(len(row_coords.columns))]
    col_coords.columns = ['dimension ' + str(i+1) for i in range(len(col_coords.columns))]
    fig0 = px.scatter(
        data_frame=col_coords, 
        x="dimension 1", 
        y="dimension 2",
        text=col_coords.index, 
        hover_name=col_coords.index, 
        opacity=0.8)
    fig0.update_traces(marker = dict(color="#e73232", size = 10))
    fig = px.scatter(
        data_frame=row_coords,
        x="dimension 1", 
        y="dimension 2",
        hover_name=row_coords.index)
    fig.add_trace(fig0.data[0])
    fig.update_xaxes(title = f"Dim 1 ({mca.percentage_of_variance_[0].round()}%)", range=[col_coords[['dimension 1']].min().values[0] * 1.4, col_coords[['dimension 1']].max().values[0] * 1.4])
    fig.update_yaxes(title = f"Dim 1 ({mca.percentage_of_variance_[1].round()}%)", range=[col_coords[['dimension 2']].min().values[0] * 1.4, col_coords[['dimension 2']].max().values[0] * 1.4])
    fig.update_layout(title = "Symmetric Biplot of MCA", width=w, height=h)
    fig.update_traces(textposition='top center')
    fig.show()
# 
MCABiplot(mca, data=df, w=750, h=600)

Interpretation:

  • Explained Variances: The first and second dimensions capture approximately 15% and 9% of the total inertia of the data points, respectively. Projecting the data onto these two axes may highlight certain aspects of the data associations but will not encompass all the information present in the dataset.

  • Individual Points: Each point on the plot represents an individual in the dataset. The position of an individual point indicates its profile based on the categorical variables. The proximity of individual points indicates the similarity between the data points. In other words, individuals clustered close together tend to belong to similar categories across the different variables.

  • Variable Points: The proximity of a variable point to an individual point indicates the strength of the association between the individual and the variable. Additionally, the proximity of variables indicates that they are strongly associated or correlated. This proximity suggests that the categories of these variables tend to occur together frequently in the dataset. For example, the biplot suggests that stroke is correlated with heart disease and hypertension. Moreover, people who have never worked, women, and children tend to have a lower chance of stroke.

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 dimension.

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

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

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

Further Readings