One-Way ANOVA App

Introduction

This application performs One-Way ANOVA (Analysis of Variance) to compare means across multiple groups. One-Way ANOVA is a statistical method used to determine if there are statistically significant differences between the means of three or more independent groups.

Model of One-Way ANOVA

The One-Way ANOVA model can be expressed as:

$$ Y_{ij} = \mu + \alpha_i + \epsilon_{ij} $$

• Where \( Y_{ij} \) is the j-th observation in the i-th group.
• \( \mu \) is the overall mean of all observations.
• \( \alpha_i \) is the effect of the i-th group.
• \( \epsilon_{ij} \) is the random error component.

Assumptions of One-Way ANOVA

• Independence: The observations are independent of each other.
• Normality: The data in each group should be approximately normally distributed.
• Homogeneity of Variance: The variances among the groups should be approximately equal.

Features of the App

• Input Data: You can specify the number of groups and input your data values for each group.
• Data Frame: After entering your data, you can view it in a structured table format, making it easier to visualize the input data.
• Summary Statistics: This section provides key statistics for each group, including sample sizes, means, standard deviations, and pooled standard deviation.
• Boxplots: You can visualize your data using boxplots, which show the distribution and potential outliers of each group's data.
• ANOVA Results: This tab displays the ANOVA table, including sums of squares, degrees of freedom, mean squares, F-statistic, and p-value, along with detailed calculations.
• Fisher's LSD: Perform pairwise comparisons of means using Fisher's Least Significant Difference to identify which specific group means are significantly different, along with significance levels.
• Explanation of Calculations: This section explains the steps involved in the One-Way ANOVA process, including the formulas used for calculations.

By using this app, you can easily conduct a One-Way ANOVA analysis and visualize your data, making it a valuable tool for statistical analysis.

Input Data

Number of Groups: Process Data

Data Frame

Summary Statistics

Boxplots

ANOVA Results

Fisher's LSD

significance Level (%): Calculate LSD

Explanation of Calculations

Here are the steps to perform One-Way ANOVA:

1. Calculate the overall mean (Grand Mean): $$ \bar{Y} = \frac{\sum_{i=1}^{k} n_i \bar{Y}_i}{N} $$

2. Calculate the Sum of Squares Between Groups (SSB): $$ SSB = \sum_{i=1}^{k} n_i (\bar{Y}_i - \bar{Y})^2 $$

3. Calculate the Sum of Squares Within Groups (SSW): $$ SSW = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (Y_{ij} - \bar{Y}_i)^2 $$

4. Calculate the Total Sum of Squares (SST): $$ SST = SSB + SSW $$

5. Calculate the degrees of freedom:

• Between Groups: \( df_{B} = k - 1 \)
• Within Groups: \( df_{W} = N - k \)
• Total: \( df_{T} = N - 1 \)

6. Calculate Mean Squares:

• Mean Square Between (MSB): $$ MSB = \frac{SSB}{df_{B}} $$
• Mean Square Within (MSW): $$ MSW = \frac{SSW}{df_{W}} $$

7. Calculate the F-statistic: $$ F = \frac{MSB}{MSW} $$

8. Calculate the p-value using the F-distribution.

9. Fisher's Least Significant Difference (LSD): Fisher's LSD is a post-hoc test used after a significant ANOVA result to determine which specific group means are significantly different. The steps to calculate Fisher's LSD are as follows:

• Calculate the pooled standard deviation:

  $$ s_p = \sqrt{MSW} $$

• Determine the critical t-value based on the desired confidence level and degrees of freedom.
• Calculate the LSD using the formula:

  $$ LSD = t_{\alpha/2} \cdot s_p \cdot \sqrt{\frac{1}{n_i}+\frac{1}{n_j}} $$

• Compare the mean difference between groups \(i\) and \(j\) to the LSD value to assess significance. Repeat for each pair of two groups.

R Code Examples

ANOVA Table Generation

# Example R code for One-Way ANOVA
data <- data.frame(
    values = c(1.2, 2.3, 1.8, 3.1, 2.5, 2.9, 3.3, 1.5, 5.7, 6.8, 8.9, 10.4),
    group = factor(rep(c("A", "B", "C"), c(3, 4, 5)))
)

anova_results <- aov(values ~ group, data = data)
summary(anova_results)

Calculating Fisher's LSD

# Example R code for calculating Fisher's LSD
t_value <- qt((1 + CL) / 2, df) # CL is the Confidence level such as 0.95, df is the number of degrees of freedom of MSW
LSD <- t_value * SD * sqrt(1/n1 + 1/n2) # SD is the pooled standard deviation, n1 and n2 are the sample sizes of groups being compared