R Programming - Unit 4

1. Write a note on statistical testing and modelling.

• Statistical model & steps to create statistical model
• Sampling distribution and type of sampling distribution
• Explain is sampling distribution of mean
• Explain sampling distribution of proportion
• Explain T distribution
• Testing mean
• Testing proportion
• Testing categorical variable & common statistical test used to analyse categorical variable

R Programming Overview

R is a programming language and environment primarily used for statistical computing and data analysis. It provides a wide variety of statistical and graphical techniques, and is highly extensible.

Key Features:

• Data Analysis: R is designed for data manipulation, calculation, and graphical display.
• Statistical Modeling: Supports linear and nonlinear modeling, time-series analysis, and clustering.
• Visualization: Excellent for creating high-quality plots and charts.

Simple R Code Example

# Simple linear regression example
data(mtcars)  # Load example dataset
model <- lm(mpg ~ wt, data = mtcars)  # Fit linear model
summary(model)  # Display model summary

Statistical Testing and Modeling

Statistical testing involves using data to determine if there is enough evidence to support a specific hypothesis. Common tests include t-tests, chi-square tests, and ANOVA.

Modeling, on the other hand, involves creating a mathematical representation of data. It can be used for prediction or to understand relationships between variables.

Mermaid Flowchart for Statistical Testing

Time Complexity of Common Tests

• t-test: $O(n)$
• ANOVA: $O(nk)$ where $k$ is the number of groups

Space Complexity

• All Tests: $O(1)$ for constant space, except for storing input data.

This concise overview covers R programming, statistical testing, and modeling, providing a foundational understanding for further exploration.

_{This was AI generated from github copilot on 2025-12-23}

2. Hypothesis testing & components of hypothesis testing

Hypothesis Testing in R

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0) and the alternative hypothesis (H1).

Components of Hypothesis Testing:

1. Null Hypothesis (H0): A statement that there is no effect or no difference.
2. Alternative Hypothesis (H1): A statement that contradicts the null hypothesis.
3. Significance Level (α): The probability of rejecting the null hypothesis when it is true, commonly set at 0.05.
4. Test Statistic: A standardized value used to determine whether to reject H0.
5. P-value: The probability of obtaining test results at least as extreme as the observed results, given that H0 is true.
6. Decision Rule: A method to decide whether to reject or fail to reject H0 based on the p-value and α.

Simple R Code Example:

Here’s a basic example of a t-test in R:

# Sample data
data1 <- c(5, 6, 7, 8, 9)
data2 <- c(6, 7, 8, 9, 10)

# Perform t-test
result <- t.test(data1, data2)

# Display result
print(result)

Visual Representation:

Complexity:

• Time Complexity: O(n) for the calculation of the test statistic.
• Space Complexity: O(1) since only a fixed amount of additional space is used.

This summarizes the key aspects of hypothesis testing in R.

_{This was AI generated from github copilot on 2025-12-23}

3. Chi square test of independence & implementation of chi squared test of independence in R

Chi-Square Test of Independence

The Chi-Square Test of Independence is a statistical method used to determine whether there is a significant association between two categorical variables. It compares the observed frequencies in each category of a contingency table with the expected frequencies if the variables were independent.

Steps to Perform Chi-Square Test

1. State Hypotheses:

   • Null Hypothesis ($H_0$): Variables are independent.
   • Alternative Hypothesis ($H_a$): Variables are not independent.

2. Calculate Chi-Square Statistic:

   $${\chi }^2=\sum \frac{(O_i-E_i{)}^2}{E_i}$$

   where $O_i$ is the observed frequency and $E_i$ is the expected frequency.

3. Determine Degrees of Freedom:

   $$df=(r-1)(c-1)$$

   where $r$ is the number of rows and $c$ is the number of columns.

4. Compare with Critical Value: Using the Chi-Square distribution table, compare the calculated statistic with the critical value at desired significance level (e.g., 0.05).

Implementation in R

Here’s a simple R code snippet to perform the Chi-Square Test of Independence:

# Create a contingency table
data <- matrix(c(10, 20, 30, 40), nrow = 2)

# Perform Chi-Square Test
result <- chisq.test(data)

# Print the result
print(result)

Mermaid Diagram

Below is a flowchart illustrating the steps involved in conducting the Chi-Square Test of Independence:

This diagram outlines the sequential steps from hypothesis formulation to conclusion.

_{This was AI generated from github copilot on 2025-12-23}

4. What is statistical power? What are the key factors affecting statistical power

What is Statistical Power?

Statistical power is the probability that a statistical test will correctly reject a false null hypothesis (i.e., detect an effect when there is one). It is a crucial concept in hypothesis testing as it helps to determine the likelihood of avoiding a Type II error (failing to detect an effect).

The power of a test can be expressed as:

$$\text{Power}=1-\beta$$

where $\beta$ is the probability of a Type II error.

Key Factors Affecting Statistical Power

1. Sample Size (n): Larger sample sizes generally increase power because they provide more information about the population.

2. Effect Size: The larger the true effect (the difference between groups), the more power the test has to detect it.

3. Significance Level (α): A higher alpha level (e.g., 0.05 vs. 0.01) increases power, as it makes it easier to reject the null hypothesis.

4. Variability: Lower variability or noise in the data increases power because it makes true effects easier to detect.

5. Test Type: One-tailed tests typically have more power than two-tailed tests when the effect is in the predicted direction.

Visualization of Factors Affecting Power

This diagram illustrates the primary factors that influence the statistical power of a test. Understanding these factors can help researchers design more effective studies.

_{This was AI generated from github copilot on 2025-12-23}

5. Explain ANOVA

• Types of ANOVA
• R implementation of
  • one way ANOVA
  • two way ANOVA

ANOVA Overview

ANOVA (Analysis of Variance) is a statistical method used to compare means between two or more groups to determine if at least one group mean is significantly different from the others.

Types of ANOVA

1. One-Way ANOVA: Tests differences between the means of three or more independent (unrelated) groups based on one factor.
2. Two-Way ANOVA: Tests differences between means based on two factors, which can include interaction effects between the factors.

R Implementation

# One-Way ANOVA
one_way_anova <- aov(response ~ factor, data = dataset)
summary(one_way_anova)

# Two-Way ANOVA
two_way_anova <- aov(response ~ factor1 * factor2, data = dataset)
summary(two_way_anova)

Mermaid Representation

Complexity

• One-Way ANOVA:

  • Time Complexity: $O(n)$
  • Space Complexity: $O(1)$

• Two-Way ANOVA:

  • Time Complexity: $O(n\cdot m)$ where $m$ is the number of levels of the second factor.
  • Space Complexity: $O(1)$

This concise explanation covers the essential aspects of ANOVA, its types, R implementations, and a simple flowchart representation.

_{This was AI generated from github copilot on 2025-12-23}