Statistics: Сrash Course for Data Science. Part 2

Welcome to the second and final part of our article series on statistics for data science. In this installment, we will delve deeper into essential statistical concepts and techniques that are crucial for data analysis and modeling. We will explore the Student's T-test, analysis of variance (ANOVA), correlation, and regression. So, let's dive in!

Student's T-test

What is the Student's T-test?

The Student's T-test is a statistical test used to determine if there is a significant difference between the means of two groups. It is based on the T-distribution, which is a mathematical distribution that is similar to the normal distribution but has fatter tails. The T-test is commonly used when the sample size is small or the population standard deviation is unknown.

Why is knowledge of the T-test important?

Understanding the T-test is essential because it allows us to compare two groups and assess whether the observed difference between their means is statistically significant. This knowledge is crucial in various fields, such as medical research, social sciences, and business, where comparing group means is often necessary.

Where and when to apply knowledge of the T-test in practice?

The T-test finds applications in various scenarios, including A/B testing, clinical trials, market research, and quality control. Whenever you need to compare two groups or treatments and determine if there is a significant difference in their means, the T-test comes into play.

Example:

import numpy as np
from scipy.stats import ttest_ind

// Sample data for two groups
group1 = [10, 12, 15, 18, 20]
group2 = [8, 11, 14, 16, 19]

// Perform T-test
t_statistic, p_value = ttest_ind(group1, group2)

print("T-statistic:", t_statistic)
print("p-value:", p_value)

The T-test is named after its creator, William Sealy Gosset, who worked under the pseudonym "Student" while employed at the Guinness Brewery in Dublin, Ireland. Gosset developed the T-test as a statistical method to address the challenges of small sample sizes in quality control and brewing processes.

Due to the strict confidentiality policy at the Guinness Brewery, Gosset was not allowed to publish his work under his real name. Therefore, he used the pseudonym "Student" when publishing his findings in 1908

Analysis of Variance (ANOVA)

What is ANOVA?

Analysis of Variance (ANOVA) is a statistical method used to compare the means of three or more groups. It assesses whether there are any statistically significant differences between the group means and helps identify which groups differ from each other. ANOVA partitions the total variation in the data into two components: variation between groups and variation within groups.

Why is knowledge of ANOVA important?

ANOVA allows us to determine if there are significant differences among multiple groups, providing insights into the effects of different factors or treatments. It is widely used in experimental studies, social sciences, and industrial research to analyze the impact of various variables on a response variable.

Where and when to apply knowledge of ANOVA in practice?

ANOVA is applicable in scenarios where you need to compare the means of three or more groups. It is used in fields such as psychology, biology, marketing research, and manufacturing industries, where understanding the influence of different factors is crucial.

Example:

import numpy as np
from scipy.stats import f_oneway

// Sample data for multiple groups
group1 = [10, 12, 15, 18, 20]
group2 = [8, 11, 14, 16, 19]
group3 = [13, 14, 17, 21, 22]

// Perform ANOVA
f_statistic, p_value = f_oneway(group1, group2, group3)

print("F-statistic:", f_statistic)
print("p-value:", p_value)

Correlation

What is correlation?

Correlation is a statistical measure that quantifies the relationship between two variables. It assesses the strength and direction of the linear association between them. The correlation coefficient ranges from -1 to 1, with values close to -1 indicating a strong negative correlation, values close to 1 indicating a strong positive correlation, and values close to 0 indicating no or weak correlation.

Why is knowledge of correlation important?

Understanding correlation allows us to identify relationships between variables and assess their dependency. It helps in analyzing patterns, making predictions, and determining the strength of associations in datasets. Correlation analysis is widely used in fields such as finance, social sciences, and marketing to uncover meaningful insights.

Where and when to apply knowledge of correlation in practice?

Correlation analysis is valuable when studying the relationships between variables. It is used to identify factors that are strongly related, evaluate the impact of variables on an outcome, and guide decision-making processes. Correlation is commonly applied in fields like finance, economics, healthcare, and social sciences.

Example:

import numpy as np
import pandas as pd

// Sample data
data = pd.DataFrame({
    'X': [10, 15, 20, 25, 30],
    'Y': [20, 25, 30, 35, 40]
})

// Calculate correlation coefficient
correlation = data['X'].corr(data['Y'])

print("Correlation coefficient:", correlation)

Regression analysis

What is regression analysis?

Regression analysis is a statistical technique used to model the relationship between a dependent variable and one or more independent variables. It helps us understand how the dependent variable changes when the independent variables change. Regression analysis aims to find the best-fitting regression equation that predicts the dependent variable based on the independent variables.

Why is knowledge of regression important?

Regression analysis is essential because it allows us to make predictions, estimate relationships between variables, and understand the impact of independent variables on the dependent variable. It is widely used in various fields, including finance, economics, social sciences, and machine learning.

Where and when to apply knowledge of regression in practice?

Regression analysis is applied in scenarios where we want to predict or estimate a continuous dependent variable based on one or more independent variables. It helps in understanding the relationship between variables, making forecasts, and identifying factors that influence the outcome of interest.

Different types of regression are used based on the nature of the problem and the type of data. Let's figure out that!

- Linear Regression

Linear regression is one of the most widely used regression techniques. It models the relationship between a dependent variable and one or more independent variables using a linear equation. The goal is to find the best-fit line that minimizes the sum of squared differences between the observed and predicted values.

Example: Predicting house prices based on variables such as area, number of bedrooms, and location

import numpy as np
import pandas as pd
from sklearn.linear_model import LinearRegression

// Load the dataset
data = pd.read_csv('house_data.csv')

// Separate the independent variables (features) and the dependent variable (target)
X = data[['area', 'bedrooms', 'location']]
y = data['price']

// Create a linear regression model
model = LinearRegression()

// Fit the model to the data
model.fit(X, y)

// Predict house prices
new_data = pd.DataFrame([[2000, 3, 'suburb']], columns=['area', 'bedrooms', 'location'])
predicted_prices = model.predict(new_data)

print(predicted_prices)

- Polynomial Regression

Polynomial regression extends linear regression by introducing polynomial terms to model nonlinear relationships between the variables. It fits a polynomial curve to the data points, allowing for more flexible and curved relationships.

Example: Predicting the height of a plant based on the number of days since planting

import numpy as np
import matplotlib.pyplot as plt
from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression

// Generate sample data
days_since_planting = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
plant_height = np.array([2, 5, 9, 15, 20, 22, 24, 23, 19, 15])

// Transform the features to include polynomial terms
poly_features = PolynomialFeatures(degree=2)
X_poly = poly_features.fit_transform(days_since_planting.reshape(-1, 1))

// Create a polynomial regression model
model = LinearRegression()

// Fit the model to the data
model.fit(X_poly, plant_height)

// Predict plant height
new_days = np.array([[11]])
new_X_poly = poly_features.transform(new_days)
predicted_height = model.predict(new_X_poly)

print(predicted_height)

- Logistic Regression

Logistic regression is used for binary classification problems where the dependent variable is categorical with two outcomes. It models the probability of the outcome based on the independent variables using the logistic function.

Example: Predicting whether a customer will churn (yes/no) based on their demographics and usage data

import numpy as np
import pandas as pd
from sklearn.linear_model import LogisticRegression

// Load the dataset
data = pd.read_csv('churn_data.csv')

// Separate the independent variables (features) and the dependent variable (target)
X = data[['age', 'gender', 'usage']]
y = data['churn']

// Create a logistic regression model
model = LogisticRegression()

// Fit the model to the data
model.fit(X, y)

// Predict churn probability
new_data = pd.DataFrame([[35, 'Male', 150]], columns=['age', 'gender', 'usage'])
churn_probability = model.predict_proba(new_data)[:, 1]

print(churn_probability)

- Generalized Linear Models (GLM)

Generalized Linear Models extend the concept of linear regression to handle a broader range of response variables, including binary, count, and categorical data. GLMs incorporate different types of link functions and probability distributions to model the relationship between the predictors and the response variable.

Example: Predicting the likelihood of customer purchases based on demographic variables using a logistic regression

import numpy as np
import pandas as pd
import statsmodels.api as sm

// Load the dataset
data = pd.read_csv('customer_data.csv')

// Add an intercept term
data['intercept'] = 1

// Separate the independent variables (features) and the dependent variable (target)
X = data[['age', 'gender', 'income', 'intercept']]
y = data['purchased']

// Create a logistic regression model with a logit link function
model = sm.GLM(y, X, family=sm.families.Binomial(link=sm.families.links.logit()))

// Fit the model to the data
results = model.fit()

// Print the model summary
print(results.summary())

- Generalized Additive Models (GAM)

Generalized Additive Models extend the idea of GLMs by allowing for non-linear relationships between the predictors and the response variable. GAMs use smooth functions and spline techniques to model the non-linear effects, making them suitable for capturing complex patterns in the data.

Example: Predicting the impact of temperature and humidity on electricity consumption using a GAM

import numpy as np
import pandas as pd
import statsmodels.api as sm
from pygam import GAM, s

// Load the dataset
data = pd.read_csv('electricity_data.csv')

// Separate the independent variables (features) and the dependent variable (target)
X = data[['temperature', 'humidity']]
y = data['electricity_consumption']

// Create a GAM with smooth functions for each predictor
model = GAM(s(0) + s(1))

// Fit the model to the data
model.fit(X, y)

// Print the model summary
print(model.summary())

- Ridge Regression

Ridge Regression is a regularization technique that addresses multicollinearity (high correlation) among the predictors by adding a penalty term to the loss function. It helps prevent overfitting and stabilizes the regression coefficients.

Example: Predicting housing prices using ridge regression

import numpy as np
import pandas as pd
from sklearn.linear_model import Ridge

// Load the dataset
data = pd.read_csv('house_data.csv')

// Separate the independent variables (features) and the dependent variable (target)
X = data[['area', 'bedrooms', 'location']]
y = data['price']

// Create a ridge regression model
model = Ridge(alpha=0.5)

// Fit the model to the data
model.fit(X, y)

// Predict house prices
new_data = pd.DataFrame([[2000, 3, 'suburb']], columns=['area', 'bedrooms', 'location'])
predicted_prices = model.predict(new_data)

print(predicted_prices)

Feel free to experiment with these techniques and explore their capabilities in solving various regression problems.

See you next time

That concludes our crash course on statistics for data science. I hope you found this series insightful and valuable for building a solid foundation in statistical analysis. By understanding these fundamental concepts and techniques, you are equipped to make informed decisions, draw meaningful insights, and navigate the vast world of data science.

Stay curious, stay hungry, stay foolish, continue learning, and always question the underlying assumptions and implications. See you!

If you missed the first part, don't worry, here it is Statistics: Сrash Course for Data Science. Part I