This document outlines the steps taken to perform a one-way/two-way ANOVA analysis using R. The goal is to compare the effects of three different treatments on a continuous outcome variable.

**Objective**: Determine if there are statistically significant differences among the means of three or more independent groups.

```
anova_result <- aov(value ~ treatment, data = data_anova)
summary(anova_result)
```

```
# Set seed for reproducibility
set.seed(123)
# Simulate data for a one-way ANOVA with three treatments
group_a <- rnorm(20, mean = 30, sd = 10) # Treatment A
group_b <- rnorm(20, mean = 75, sd = 10) # Treatment B
group_c <- rnorm(20, mean = 50, sd = 10) # Treatment C
# Combine the data into a data frame
data_anova <- data.frame(
value = c(group_a, group_b, group_c),
treatment = factor(rep(c("A", "B", "C"), each = 20))
)
```

```
# Perform one-way ANOVA
anova_result <- aov(value ~ treatment, data = data_anova)
summary(anova_result)
```

```
Df Sum Sq Mean Sq F value Pr(>F)
treatment 2 18599 9299 109.3 <2e-16 ***
Residuals 57 4848 85
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
```

```
# QQ plot of residuals from one-way ANOVA
qqnorm(residuals(anova_result))
qqline(residuals(anova_result), col = "steelblue", lwd = 2)
```

- Objective: Assess the normality of the residuals, an assumption of ANOVA.

```
# Residual plot from one-way ANOVA
plot(residuals(anova_result), main = "Residuals Plot", xlab = "Index", ylab = "Residuals")
abline(h = 0, col = "red", lwd = 2)
```

- Objective: Evaluate the homogeneity of variances and check for any patterns in the residuals.

```
# Perform linear model
lm_result <- lm(value ~ treatment, data = data_anova)
summary(lm_result)
```

```
Call:
lm(formula = value ~ treatment, data = data_anova)
Residuals:
Min 1Q Median 3Q Max
-21.0824 -5.9407 -0.5103 6.2225 20.6247
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 31.416 2.062 15.235 < 2e-16 ***
treatmentB 43.071 2.916 14.769 < 2e-16 ***
treatmentC 19.649 2.916 6.738 8.7e-09 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 9.222 on 57 degrees of freedom
Multiple R-squared: 0.7932, Adjusted R-squared: 0.786
F-statistic: 109.3 on 2 and 57 DF, p-value: < 2.2e-16
```

- Objective: Provide an alternative method to analyze the data, offering detailed statistics and insights into the relationship between the treatment groups and the outcome variable.

```
# Plotting using ggplot2
library(ggplot2)
ggplot(data_anova, aes(x = treatment, y = value, fill = treatment)) +
geom_boxplot() +
theme_minimal() +
labs(title = "One-way ANOVA", y = "Value", x = "Treatment")
```

**Objective**: The objective of a two-way ANOVA is to evaluate the main effects of two independent variables on a dependent variable and to investigate if there’s a significant interaction between them that affects the outcome.

```
set.seed(123)
# Simulate data for a two-way ANOVA with three treatments and two blocks
group_a_day1 <- rnorm(10, mean = 30, sd = 10) # Treatment A, Day 1
group_b_day1 <- rnorm(10, mean = 75, sd = 10) # Treatment B, Day 1
group_c_day1 <- rnorm(10, mean = 50, sd = 10) # Treatment C, Day 1
group_a_day2 <- rnorm(10, mean = 30, sd = 10) # Treatment A, Day 2
group_b_day2 <- rnorm(10, mean = 75, sd = 10) # Treatment B, Day 2
group_c_day2 <- rnorm(10, mean = 50, sd = 10) # Treatment C, Day 2
# Combine the data into a data frame
data_anova_block <- data.frame(
value = c(group_a_day1, group_b_day1, group_c_day1, group_a_day2, group_b_day2, group_c_day2),
treatment = factor(rep(c("A", "B", "C", "A", "B", "C"), each = 10)),
day = factor(rep(c("Day1", "Day1", "Day1", "Day2", "Day2", "Day2"), each = 10))
)
```

```
# Perform two-way ANOVA with interaction between treatment and block
anova_block_result <- aov(value ~ treatment + day + treatment:day, data = data_anova_block)
summary(anova_block_result)
```

```
Df Sum Sq Mean Sq F value Pr(>F)
treatment 2 19708 9854 117.379 <2e-16 ***
day 1 76 76 0.908 0.345
treatment:day 2 187 93 1.113 0.336
Residuals 54 4533 84
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
```

```
# QQ plot of residuals from two-way ANOVA
qqnorm(residuals(anova_block_result))
qqline(residuals(anova_block_result), col = "steelblue", lwd = 2)
```

```
# Residual plot from two-way ANOVA
plot(residuals(anova_block_result), main = "Residuals Plot", xlab = "Index", ylab = "Residuals")
abline(h = 0, col = "red", lwd = 2)
```

```
# Perform linear model with interaction
lm_block_result <- lm(value ~ treatment * day, data = data_anova_block)
summary(lm_block_result)
```

```
Call:
lm(formula = value ~ treatment * day, data = data_anova_block)
Residuals:
Min 1Q Median 3Q Max
-21.7524 -6.1984 -0.8624 4.9443 21.7767
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 30.746 2.897 10.612 8.02e-15 ***
treatmentB 46.340 4.098 11.309 7.35e-16 ***
treatmentC 15.008 4.098 3.663 0.000569 ***
dayDay2 2.474 4.098 0.604 0.548495
treatmentB:dayDay2 -4.648 5.795 -0.802 0.426061
treatmentC:dayDay2 3.988 5.795 0.688 0.494249
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 9.163 on 54 degrees of freedom
Multiple R-squared: 0.815, Adjusted R-squared: 0.7979
F-statistic: 47.58 on 5 and 54 DF, p-value: < 2.2e-16
```

```
# Plotting using ggplot2, showing interaction
ggplot(data_anova_block, aes(x = treatment, y = value, fill = day)) +
geom_boxplot() +
theme_minimal() +
labs(title = "Two-way ANOVA with Blocking", y = "Value", x = "Treatment")
```

```
# Perform one-way ANOVA focusing only on treatment effects
anova_result <- aov(value ~ treatment, data = data_anova_block)
summary(anova_result)
ggplot(data_anova_block, aes(x = treatment, y = value, fill = treatment)) +
geom_boxplot() +
theme_minimal() +
labs(title = "One-way ANOVA", y = "Value", x = "Treatment")
```

```
Df Sum Sq Mean Sq F value Pr(>F)
treatment 2 19708 9854 117.1 <2e-16 ***
Residuals 57 4796 84
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
```

**Focus**: Examines the effect of a single factor (treatment) on a dependent variable.**Objective**: Determine if there are significant differences between the means of three or more groups.**Usage**: Ideal when interested in the impact of one independent variable.

**Focus**: Assesses the impact of two independent variables (treatments and blocks) on a dependent variable.**Objective**: Evaluate the main effects of each independent variable and their interaction effect on the dependent variable.**Usage**: Suitable for exploring the combined effect of two variables and their interaction.