Bivariate and Multivariate Analysis: A Comprehensive Guide

Introduction

What is Bivariate and Multivariate Analysis?

• Bivariate Analysis: Examines the relationship between two variables.
• Multivariate Analysis: Analyzes the relationship between more than two variables simultaneously.

Both techniques help in understanding dependencies, correlations, patterns, and trends in data, essential for feature selection and predictive modeling.

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Step 1: Importing Required Libraries

import pandas as pd
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
from scipy.stats import pearsonr, spearmanr

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Step 2: Loading the Dataset

We’ll use the Titanic dataset for demonstration.

df = pd.read_csv("https://raw.githubusercontent.com/datasciencedojo/datasets/master/titanic.csv")
print(df.head())

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I. BIVARIATE ANALYSIS (Two-Variable Analysis)

Bivariate analysis is classified into three types based on the nature of the variables:

1. Numerical vs Numerical
2. Numerical vs Categorical
3. Categorical vs Categorical

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Case 1: Numerical vs Numerical Variables

1. Scatter Plot (Visualizing Relationships)

Used when both variables are continuous.

sns.scatterplot(x=df["Age"], y=df["Fare"])
plt.title("Scatter Plot: Age vs Fare")
plt.show()

✅ Interpretation:

• Positive Relationship: If Fare increases with Age.
• Negative Relationship: If Fare decreases with Age.
• No Relationship: If points are randomly scattered.

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2. Correlation Analysis (Measuring Relationships)

The Pearson correlation coefficient measures the strength and direction of a linear relationship between two numerical variables.

corr, _ = pearsonr(df["Age"].dropna(), df["Fare"].dropna())
print(f"Pearson Correlation: {corr:.3f}")

✅ Interpretation:

• +1 → Strong positive correlation
• 0 → No correlation
• -1 → Strong negative correlation

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3. Heatmap (Overall Correlation Matrix)

Displays correlation between multiple numerical variables.

plt.figure(figsize=(8,5))
sns.heatmap(df.corr(), annot=True, cmap="coolwarm", linewidths=0.5)
plt.title("Correlation Heatmap")
plt.show()

✅ Helps identify relationships across all numerical variables.

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Case 2: Numerical vs Categorical Variables

1. Box Plot (Comparing Distributions)

Used to compare a numerical variable across different categories.

sns.boxplot(x=df["Survived"], y=df["Age"])
plt.title("Box Plot: Age vs Survived")
plt.show()

✅ Interpretation:

• Shows median, quartiles, and outliers for each category.
• Helps understand if Age differs significantly between survived and non-survived groups.

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2. Violin Plot (Comparing Distributions with Density)

sns.violinplot(x=df["Pclass"], y=df["Fare"])
plt.title("Violin Plot: Pclass vs Fare")
plt.show()

✅ Shows distribution shape and data density along with box plot statistics.

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3. Bar Plot (Mean of Numerical Feature per Category)

sns.barplot(x=df["Sex"], y=df["Fare"])
plt.title("Bar Plot: Average Fare Paid by Gender")
plt.show()

✅ Helps compare group-wise averages.

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Case 3: Categorical vs Categorical Variables

1. Cross Tabulation (Frequency Table)

cross_tab = pd.crosstab(df["Sex"], df["Survived"])
print(cross_tab)

✅ Shows how survival varies by gender.

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2. Grouped Bar Chart

sns.countplot(x="Sex", hue="Survived", data=df)
plt.title("Survival Count by Gender")
plt.show()

✅ Stacked comparison of categories across another categorical feature.

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3. Chi-Square Test (Statistical Relationship)

To check if two categorical variables are independent.

from scipy.stats import chi2_contingency
chi2, p, _, _ = chi2_contingency(cross_tab)
print(f"Chi-square Test p-value: {p:.3f}")

✅ If p < 0.05, the relationship is significant.

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II. MULTIVARIATE ANALYSIS (More Than Two Variables)

Multivariate analysis helps in understanding how multiple factors interact.

1. Pair Plot (Visualizing Pairwise Relationships)

sns.pairplot(df[["Age", "Fare", "Pclass", "Survived"]], hue="Survived")
plt.show()

✅ Plots scatter plots for numerical variables and distributions for categorical variables.

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2. Multivariate Correlation Heatmap

plt.figure(figsize=(8,5))
sns.heatmap(df.corr(), annot=True, cmap="coolwarm", linewidths=0.5)
plt.title("Multivariate Correlation Heatmap")
plt.show()

✅ Highlights relationships between multiple numerical features.

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3. Multivariate Regression Analysis

Examines how multiple independent variables impact a dependent variable.

import statsmodels.api as sm

df.dropna(inplace=True)  # Removing missing values
X = df[["Pclass", "Age", "Fare"]]
y = df["Survived"]

X = sm.add_constant(X)  # Adding intercept
model = sm.Logit(y, X).fit()
print(model.summary())

✅ Used to predict survival probability based on multiple features.

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4. Principal Component Analysis (PCA)

Used to reduce dimensionality while preserving variance.

from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler

features = ["Age", "Fare", "Pclass"]
X = StandardScaler().fit_transform(df[features])

pca = PCA(n_components=2)
X_pca = pca.fit_transform(X)

plt.scatter(X_pca[:,0], X_pca[:,1], c=df["Survived"], cmap="coolwarm")
plt.xlabel("Principal Component 1")
plt.ylabel("Principal Component 2")
plt.title("PCA - Titanic Data")
plt.show()

✅ Reduces high-dimensional data to two main components for visualization.

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Key Insights from Bivariate & Multivariate Analysis

✔ Bivariate Analysis:

• Scatter plots, correlation, and regression for numerical vs numerical relationships.
• Box plots, violin plots, and bar charts for numerical vs categorical comparisons.
• Cross-tabulation, count plots, and chi-square tests for categorical vs categorical relationships.

✔ Multivariate Analysis:

• Pair plots & Heatmaps highlight relationships between multiple variables.
• Regression & PCA help in feature selection & dimensionality reduction.

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