R Programming

11.

Statistical Analysis in R

11.1.

Descriptive Statistics

11.1.1.

Measures of Central Tendency

11.1.1.1.

Mean Calculation

11.1.1.1.1.

`mean()` Function

11.1.1.1.2.

11.1.1.1.3.

11.1.1.2.

Median Calculation

11.1.1.2.1.

`median()` Function

11.1.1.2.2.

Robust Central Tendency

11.1.1.3.

Mode Calculation

11.1.1.3.1.

Custom Mode Functions

11.1.1.3.2.

Modal Class for Continuous Data

11.1.2.

Measures of Variability

11.1.2.1.

Variance and Standard Deviation

11.1.2.1.1.

`var()` and `sd()` Functions

11.1.2.1.2.

Population vs. Sample Statistics

11.1.2.2.

Range and Interquartile Range

11.1.2.2.1.

`range()` Function

11.1.2.2.2.

`IQR()` Function

11.1.2.2.3.

`quantile()` Function

11.1.2.3.

Coefficient of Variation

11.1.2.4.

Mean Absolute Deviation

11.1.3.

Distribution Shape

11.1.3.1.

Skewness Measures

11.1.3.2.

Kurtosis Measures

11.1.3.3.

Distribution Visualization

11.1.4.

Summary Statistics

11.1.4.1.

`summary()` Function

11.1.4.2.

`describe()` from psych Package

11.1.4.3.

Custom Summary Functions

11.1.5.

Frequency Analysis

11.1.5.1.

`table()` Function

11.1.5.2.

Cross-tabulation

11.1.5.3.

Proportions and Percentages

11.2.

Probability Distributions

11.2.1.

Distribution Function Naming Convention

11.2.1.1.

Density Functions (`d*`)

11.2.1.2.

Cumulative Distribution Functions (`p*`)

11.2.1.3.

Quantile Functions (`q*`)

11.2.1.4.

Random Number Generation (`r*`)

11.2.2.

Continuous Distributions

11.2.2.1.

Normal Distribution

11.2.2.1.1.

`dnorm()`, `pnorm()`, `qnorm()`, `rnorm()`

11.2.2.1.2.

Standard Normal Distribution

11.2.2.1.3.

Normal Approximations

11.2.2.2.

Student's t-Distribution

11.2.2.2.1.

`dt()`, `pt()`, `qt()`, `rt()`

11.2.2.2.2.

Degrees of Freedom

11.2.2.3.

Chi-squared Distribution

11.2.2.3.1.

`dchisq()`, `pchisq()`, `qchisq()`, `rchisq()`

11.2.2.3.2.

Applications in Testing

11.2.2.4.

11.2.2.4.1.

`df()`, `pf()`, `qf()`, `rf()`

11.2.2.4.2.

ANOVA Applications

11.2.2.5.

Uniform Distribution

11.2.2.5.1.

`dunif()`, `punif()`, `qunif()`, `runif()`

11.2.2.6.

Exponential Distribution

11.2.2.6.1.

`dexp()`, `pexp()`, `qexp()`, `rexp()`

11.2.2.6.2.

11.2.3.

Discrete Distributions

11.2.3.1.

Binomial Distribution

11.2.3.1.1.

`dbinom()`, `pbinom()`, `qbinom()`, `rbinom()`

11.2.3.1.2.

Success Probability

11.2.3.1.3.

Number of Trials

11.2.3.2.

Poisson Distribution

11.2.3.2.1.

`dpois()`, `ppois()`, `qpois()`, `rpois()`

11.2.3.2.2.

11.2.3.2.3.

Count Data Applications

11.2.3.3.

Geometric Distribution

11.2.3.3.1.

`dgeom()`, `pgeom()`, `qgeom()`, `rgeom()`

11.2.3.3.2.

First Success Probability

11.2.3.4.

Negative Binomial Distribution

11.2.3.4.1.

`dnbinom()`, `pnbinom()`, `qnbinom()`, `rnbinom()`

11.2.3.4.2.

Overdispersion Modeling

11.2.4.

Distribution Fitting

11.2.4.1.

Method of Moments

11.2.4.2.

Maximum Likelihood Estimation

11.2.4.3.

Goodness of Fit Tests

11.3.

Hypothesis Testing

11.3.1.

Hypothesis Testing Framework

11.3.1.1.

Null and Alternative Hypotheses

11.3.1.2.

Type I and Type II Errors

11.3.1.3.

Significance Levels

11.3.1.4.

P-values and Interpretation

11.3.1.5.

Confidence Intervals

11.3.2.

One-sample Tests

11.3.2.1.

One-sample t-test

11.3.2.1.1.

`t.test()` Function

11.3.2.1.2.

Assumptions and Diagnostics

11.3.2.2.

One-sample Proportion Test

11.3.2.2.1.

`prop.test()` Function

11.3.2.3.

Wilcoxon Signed-rank Test

11.3.2.3.1.

`wilcox.test()` Function

11.3.2.3.2.

Non-parametric Alternative

11.3.3.

Two-sample Tests

11.3.3.1.

Independent Two-sample t-test

11.3.3.1.1.

Equal and Unequal Variances

11.3.3.1.2.

11.3.3.2.

11.3.3.2.1.

Dependent Samples

11.3.3.2.2.

Before-after Comparisons

11.3.3.3.

Two-sample Proportion Test

11.3.3.4.

Mann-Whitney U Test

11.3.3.4.1.

Non-parametric Two-sample Test

11.3.4.

Tests for Categorical Data

11.3.4.1.

Chi-squared Goodness of Fit

11.3.4.1.1.

`chisq.test()` Function

11.3.4.1.2.

Expected vs. Observed Frequencies

11.3.4.2.

Chi-squared Test of Independence

11.3.4.2.1.

Contingency Tables

11.3.4.2.2.

Association Testing

11.3.4.3.

Fisher's Exact Test

11.3.4.3.1.

`fisher.test()` Function

11.3.4.3.2.

Small Sample Sizes

11.3.5.

Analysis of Variance (ANOVA)

11.3.5.1.

11.3.5.1.1.

`aov()` Function

11.3.5.1.2.

Between-group Comparisons

11.3.5.1.3.

11.3.5.2.

11.3.5.2.1.

Main Effects and Interactions

11.3.5.2.2.

Factorial Designs

11.3.5.3.

Repeated Measures ANOVA

11.3.5.4.

ANOVA Assumptions

11.3.5.4.1.

Normality Testing

11.3.5.4.2.

Homogeneity of Variance

11.3.5.4.3.

11.3.6.

Non-parametric Tests

11.3.6.1.

Kruskal-Wallis Test

11.3.6.1.1.

Non-parametric ANOVA Alternative

11.3.6.2.

11.3.6.2.1.

Non-parametric Repeated Measures

11.4.

Correlation and Association

11.4.1.

Correlation Analysis

11.4.1.1.

Pearson Correlation

11.4.1.1.1.

`cor()` Function

11.4.1.1.2.

Linear Relationships

11.4.1.1.3.

11.4.1.2.

Spearman Rank Correlation

11.4.1.2.1.

Monotonic Relationships

11.4.1.2.2.

Non-parametric Correlation

11.4.1.3.

11.4.1.3.1.

Alternative Rank Correlation

11.4.1.4.

Partial Correlation

11.4.1.4.1.

Controlling for Other Variables

11.4.2.

Correlation Testing

11.4.2.1.

`cor.test()` Function

11.4.2.2.

Significance Testing

11.4.2.3.

Confidence Intervals

11.4.3.

Covariance Analysis

11.4.3.1.

`cov()` Function

11.4.3.2.

Covariance Matrices

11.4.3.3.

Relationship to Correlation

11.4.4.

Association Measures

11.4.4.1.

Contingency Table Analysis

11.4.4.2.

11.4.4.3.

Phi Coefficient

11.5.

Regression Analysis

11.5.1.

Simple Linear Regression

11.5.1.1.

`lm()` Function

11.5.1.2.

Model Formula Syntax

11.5.1.3.

Least Squares Estimation

11.5.1.4.

Model Interpretation

11.5.2.

Multiple Linear Regression

11.5.2.1.

Multiple Predictors

11.5.2.2.

Model Building Strategies

11.5.2.3.

Variable Selection

11.5.2.4.

Multicollinearity

11.5.3.

Model Diagnostics

11.5.3.1.

Residual Analysis

11.5.3.1.1.

11.5.3.1.2.

Normality of Residuals

11.5.3.1.3.

Homoscedasticity

11.5.3.2.

Influential Observations

11.5.3.2.1.

Cook's Distance

11.5.3.2.2.

11.5.3.2.3.

Outlier Detection

11.5.3.3.

Model Assumptions

11.5.3.3.1.

11.5.3.3.2.

11.5.3.3.3.

11.5.3.3.4.

Constant Variance

11.5.4.

Model Selection

11.5.4.1.

Forward Selection

11.5.4.2.

Backward Elimination

11.5.4.3.

Stepwise Selection

11.5.4.4.

Information Criteria (AIC, BIC)

11.5.5.

Generalized Linear Models

11.5.5.1.

`glm()` Function

11.5.5.2.

11.5.5.3.

Exponential Family Distributions

11.5.5.4.

Logistic Regression

11.5.5.4.1.

Binary Outcomes

11.5.5.4.2.

11.5.5.4.3.

Model Interpretation

11.5.5.5.

Poisson Regression

11.5.5.5.1.

11.5.5.5.2.

11.5.5.5.3.

11.5.6.

Model Validation

11.5.6.1.

Cross-validation

11.5.6.2.

Train-test Splits

11.5.6.3.

Performance Metrics

11.6.

Time Series Analysis

11.6.1.

Time Series Objects

11.6.1.1.

`ts()` Function

11.6.1.2.

Time Series Properties

11.6.1.3.

Frequency and Seasonality

11.6.2.

Time Series Decomposition

11.6.2.1.

Trend, Seasonal, and Irregular Components

11.6.2.2.

`decompose()` Function

11.6.2.3.

STL Decomposition

11.6.3.

11.6.3.1.

Stationarity Tests

11.6.3.2.

11.6.3.3.

Transformation Methods

11.6.4.

11.6.4.1.

Autoregressive Models

11.6.4.2.

Moving Average Models

11.6.4.3.

ARIMA Model Fitting

11.6.4.4.

Model Selection and Diagnostics

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