Introduction to Hypothesis Testing

What is Hypothesis Testing?

Hypothesis testing is a statistical method used to determine whether there is enough evidence in a sample of data to support a particular hypothesis about a population. The process involves:

1. Formulating Hypotheses:

   • Null Hypothesis (H₀): The hypothesis that there is no effect or difference.
   • Alternative Hypothesis (H₁): The hypothesis that there is an effect or difference.

2. Choosing a Significance Level (α): Typically set at 0.05, this is the probability of rejecting the null hypothesis when it is actually true.

3. Performing the Test: Calculating a test statistic and comparing it to a critical value to decide whether to reject the null hypothesis.

4. Interpreting the Results: Based on the test statistic and p-value, determine whether to reject the null hypothesis.

T-Tests, Chi-Square Tests, and ANOVA in R

T-Tests

T-tests are used to compare the means of two groups to determine if they are significantly different from each other.

One-Sample T-Test

Used to compare the mean of a single sample to a known value.

Example:

# One-sample t-test
data <- c(10, 12, 9, 11, 13)
t.test(data, mu = 10)  # Test if mean is different from 10

Two-Sample T-Test

Used to compare the means of two independent samples.

Example:

# Two-sample t-test
group1 <- c(20, 22, 23, 19, 21)
group2 <- c(30, 31, 32, 29, 28)
t.test(group1, group2)  # Test if means are different

Paired T-Test

Used to compare means from the same group at different times.

Example:

# Paired t-test
before <- c(5, 6, 7, 8, 9)
after <- c(6, 7, 8, 9, 10)
t.test(before, after, paired = TRUE)  # Test if means are different

Chi-Square Tests

Chi-square tests are used to examine the association between categorical variables.

Chi-Square Test of Independence

Used to determine if two categorical variables are independent.

Example:

# Chi-square test of independence
data <- matrix(c(20, 30, 25, 25), nrow = 2)
chisq.test(data)  # Test if variables are independent

Chi-Square Goodness of Fit Test

Used to determine if a sample distribution matches a theoretical distribution.

Example:

# Chi-square goodness of fit test
observed <- c(20, 30, 25, 25)
expected <- c(25, 25, 25, 25)
chisq.test(observed, p = expected / sum(expected))  # Test if observed fits expected distribution

ANOVA (Analysis of Variance)

ANOVA is used to compare the means of three or more groups to see if at least one group mean is significantly different from the others.

One-Way ANOVA

Used to compare means of different groups based on one factor.

Example:

# One-way ANOVA
data <- data.frame(
  value = c(23, 25, 30, 28, 27, 31, 29, 34, 30),
  group = rep(c("A", "B", "C"), each = 3)
)
anova_result <- aov(value ~ group, data = data)
summary(anova_result)  # Test if group means are different

Two-Way ANOVA

Used to examine the effect of two factors on a response variable.

Example:

# Two-way ANOVA
data <- data.frame(
  value = c(20, 22, 25, 30, 28, 29, 35, 34, 33),
  factor1 = rep(c("A", "B", "C"), each = 3),
  factor2 = rep(c("X", "Y"), times = 3)
)
anova_result <- aov(value ~ factor1 * factor2, data = data)
summary(anova_result)  # Test if factors affect the response variable

Building Linear and Logistic Regression Models

Linear Regression

Linear regression models the relationship between a dependent variable and one or more independent variables.

Simple Linear Regression

Used when there is one predictor variable.

Example:

# Simple linear regression
data <- data.frame(
  x = c(1, 2, 3, 4, 5),
  y = c(2, 4, 6, 8, 10)
)
lm_model <- lm(y ~ x, data = data)
summary(lm_model)  # Model summary and coefficients

Multiple Linear Regression

Used when there are multiple predictor variables.

Example:

# Multiple linear regression
data <- data.frame(
  x1 = c(1, 2, 3, 4, 5),
  x2 = c(2, 3, 4, 5, 6),
  y = c(3, 5, 7, 9, 11)
)
lm_model <- lm(y ~ x1 + x2, data = data)
summary(lm_model)  # Model summary and coefficients

Logistic Regression

Logistic regression models the probability of a binary outcome based on predictor variables.

Example:

# Logistic regression
data <- data.frame(
  x1 = c(1, 2, 3, 4, 5),
  y = factor(c(0, 0, 1, 1, 1))
)
logit_model <- glm(y ~ x1, data = data, family = binomial)
summary(logit_model)  # Model summary and coefficients

Model Interpretation and Diagnostics

Interpreting Model Output

• Coefficients: Represent the effect of predictor variables on the response variable.
• R-squared: Indicates the proportion of variance explained by the model.
• p-values: Assess the significance of each predictor variable.

Diagnostics

Model diagnostics help assess the validity of the model and identify potential issues.

• Residual Plots: Check for patterns in residuals to validate assumptions.
• Normality of Residuals: Use Q-Q plots to assess if residuals follow a normal distribution.
• Multicollinearity: Check for correlations among predictor variables.

Example:

# Residual plot
plot(lm_model, which = 1)  # Residuals vs. Fitted values

# Q-Q plot
qqnorm(residuals(lm_model))
qqline(residuals(lm_model))

# Check for multicollinearity
library(car)
vif(lm_model)  # Variance Inflation Factor

Practical Examples Using Real-World Datasets

Example 1: Hypothesis Testing with a Real Dataset

Using the mtcars dataset, test if the mean miles per gallon (mpg) of cars with automatic and manual transmissions is different.

Example:

# Load dataset
data(mtcars)

# T-test comparing mpg between automatic and manual cars
auto_mpg <- mtcars[mtcars$am == 0, "mpg"]
manual_mpg <- mtcars[mtcars$am == 1, "mpg"]
t.test(auto_mpg, manual_mpg)

Example 2: Building a Linear Regression Model

Predict the miles per gallon (mpg) based on weight (wt) and horsepower (hp) using the mtcars dataset.

Example:

# Load dataset
data(mtcars)

# Linear regression model
lm_model <- lm(mpg ~ wt + hp, data = mtcars)
summary(lm_model)

Example 3: Logistic Regression with a Real Dataset

Using the iris dataset, classify species based on sepal length and sepal width.

Example:

# Load dataset
data(iris)

# Logistic regression model (species is a factor)
logit_model <- glm(Species ~ Sepal.Length + Sepal.Width, data = iris, family = binomial)
summary(logit_model)

Conclusion

Statistical analysis in R provides powerful tools for hypothesis testing and building regression models. Understanding and applying methods such as t-tests, chi-square tests, and ANOVA are fundamental for drawing inferences from data. Building and interpreting linear and logistic regression models help to understand relationships between variables and make predictions. By utilizing R’s statistical functions and packages, you can effectively analyze data, test hypotheses, and develop models to support data-driven decisions. As you gain experience, exploring more advanced statistical methods and diagnostics will further enhance your analytical capabilities.