Lecture Notes: Analysis of Variance (ANOVA) - Part 3

Introduction

• This is the third and final video on ANOVA.
• Important to watch the first two videos; otherwise, this lecture may not be clear.

Rejecting the Null Hypothesis

• When the null hypothesis is rejected in ANOVA, it indicates that the means are not equal.
• Multiple comparisons can determine which means differ from one another.
• Key Point: Null hypothesis must be rejected before doing multiple comparisons.

Example: Fish in Different Bays

• ANOVA showed significant p-value; hence, null hypothesis is rejected.
• Statistical difference in the size of fish among bays indicated.
• Goal: Identify which bays have differing fish sizes.

Least Significant Difference (LSD)

• Used for pairwise comparisons between groups.
• Criteria: LSD is the minimum difference indicating statistical difference.
• If the difference between two group means is greater than or equal to LSD, they are statistically different.

Calculating LSD

1. Formula: LSD = Q * sqrt(Mean Sum of Squares Within Groups / Sample Size)
2. Components:
   • Mean Sum of Squares from ANOVA table.
   • Sample size from data.
   • Q Value: Derived from a table using number of groups and error degrees of freedom.*

Using the Q Table

• Number of Groups: E.g., 3 different bays.
• Error Degrees of Freedom: Equivalent to degrees of freedom within groups.
• Q Example: Q = 3.77 in this scenario.

Applying LSD to Data

• Calculate means and differences between group means.
• Compare differences to LSD.
• Only differences exceeding LSD are statistically significant.

Example Results

• Galvis Bay vs. Corpus Christi Bay had a mean difference of 13.88 cm, exceeding the LSD of 11.69 cm.
• Statistically, fish in Galvis Bay differ from those in Corpus Christi Bay.
• Matagorda Bay is not statistically different from either.

Visual Representation

• Box and whisker plots can depict group differences.
• Label groups based on statistical differences (e.g., 'A' and 'B').

Key Takeaways

• When to Use ANOVA: Instead of multiple T-tests to prevent Type I error.
• Calculations: Total sum of squares, among group sum of squares, within group sum of squares, degrees of freedom, and F value.
• F Value: Determines if the null hypothesis should be rejected.
• LSD Method: Identify which means are significantly different using LSD if ANOVA indicates significant differences.

Next Steps

• Moving to correlation analysis.
• Transitioning from categorical data to continuous numerical variables.