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Completely Randomized Design in R

Completely Randomized Design (CRD) is one of the simplest forms of experimental design, typically used when the experimental units are homogeneous. In CRD, treatments are assigned randomly to the experimental units, and each unit receives exactly one treatment.

Here's a tutorial on how to analyze data from a Completely Randomized Design in R:

1. Sample Data

Let's say you conducted an experiment to measure the growth of plants under three different fertilizers, and you have a completely randomized design.

# Sample data
data <- data.frame(
  Growth = c(10, 11, 12, 14, 11, 9, 14, 13, 15, 13, 14, 15),
  Fertilizer = factor(rep(c("A", "B", "C"), each = 4))
)

2. Analyzing CRD using ANOVA

To determine if there's a significant difference in plant growth between the three fertilizers, you can use Analysis of Variance (ANOVA):

model <- aov(Growth ~ Fertilizer, data=data)
summary(model)

The output will give you the F-statistic and the associated p-value for the Fertilizer factor. A low p-value (typically < 0.05) would indicate a significant difference in growth between at least two of the fertilizers.

3. Post-hoc Tests

If the ANOVA indicates a significant difference, you might want to know which specific pairs of fertilizers are different. The Tukey Honest Significant Differences (HSD) test is a common post-hoc test for this:

library(agricolae)
result <- HSD.test(model, "Fertilizer", group=TRUE)
print(result)

4. Visualizing Results

A boxplot can help visualize the growth distribution for each fertilizer.

boxplot(Growth ~ Fertilizer, data=data, main="Plant Growth by Fertilizer", 
        xlab="Fertilizer", ylab="Growth")

5. Assumptions and Checks

For ANOVA to be valid, several assumptions need to be met:

Homogeneity of variances
Normally distributed residuals
Independent observations

To check these assumptions in R:

a. Homogeneity of Variances: Use Bartlett's test or Levene's test.

bartlett.test(Growth ~ Fertilizer, data=data)

b. Normally Distributed Residuals: A Q-Q plot can help visualize this:

qqnorm(resid(model))
qqline(resid(model))

c. Independence: This is usually ensured by design, but you can also inspect the residuals vs fitted values plot.

plot(model, which=1)

Summary:

Analyzing data from a Completely Randomized Design in R is relatively straightforward, primarily using ANOVA. Ensure to always check the assumptions of ANOVA and use post-hoc tests for pairwise comparisons if necessary.

R Completely Randomized Design Example:

CRD is a design where experimental units are randomly assigned to treatments.

# Example of CRD
set.seed(123)
treatments <- c("A", "B", "C")
data <- data.frame(Treatment = sample(treatments, 30, replace = TRUE),
                   Response = rnorm(30))

How to Implement Completely Randomized Design in R:
Implementing CRD involves randomly assigning treatments to experimental units.
```
# Implement CRD
data$Treatment <- sample(treatments, nrow(data), replace = TRUE)
```

Random Assignment of Treatments in R:

Randomly assign treatments to experimental units.

# Random assignment of treatments
data$Treatment <- sample(treatments, nrow(data), replace = TRUE)

ANOVA for Completely Randomized Design in R:

Perform ANOVA for CRD.

# ANOVA for CRD
anova_model <- aov(Response ~ Treatment, data = data)
summary(anova_model)

Analysis of Variance with CRD in R:

Analyze variance using the aov() function.

# Analysis of variance with CRD
anova_model <- aov(Response ~ Treatment, data = data)

Comparing Treatment Means in Completely Randomized Design in R:
Compare treatment means using post-hoc tests.
```
# Comparing treatment means
posthoc_test <- TukeyHSD(anova_model)
summary(posthoc_test)
```

Randomization Tests in R for CRD:

Conduct randomization tests for CRD.

# Randomization tests for CRD
library(randomize)
random_test <- randtest(Response ~ Treatment, data = data)
summary(random_test)

Post-hoc Tests for CRD in R:

Perform post-hoc tests after ANOVA.

# Post-hoc tests for CRD
posthoc_test <- TukeyHSD(anova_model)
summary(posthoc_test)

Visualizing Data in Completely Randomized Design in R:
Visualize data using boxplots or other suitable plots.
```
# Visualizing data in CRD
boxplot(Response ~ Treatment, data = data, col = "lightblue")
```
Assumptions of Completely Randomized Design in R:
Assumptions include independence, normality, and homogeneity of variances.
```
# Check assumptions
plot(anova_model, 1)
```