One-factor NOVA theory

Introduction to Module Two
- Focus: Basics of Analysis of Variance (ANOVA)
- Purpose: Derive ANOVA by hand using the original variance ratio technique discussed in Module One
- Note: Understanding ANOVA is essential for more complex analyses, but this derivation process is not examinable.

Generic Example Overview
- Using three independent groups: mean one, mean two, mean three
- Standard deviations corresponding to each mean

Concept: Estimate the population variance from sample data
Definition of Mean Square Error (MSE)
- Independent of treatment differences
- Represented by: $MSE = \frac{\sum (s^2)}{k}$
- Where (s^2) = individual sample variances
- (k) = number of means (in this case, three)
Key Point: MSE serves as an estimate independent of the null hypothesis (H0)
- H0 states that all group means are drawn from the same population (no average differences)

Second Method Explanation
- This formula assumes that the null hypothesis (H0) is true.
- The variance of sample means: $Var(X) = \frac{\sigma^2}{n}$
- Where (\sigma^2) = population variance, (n) = sample size
Rearranged Formula for Population Variance
- Calculate population variance, leading to Mean Square Between (MSB)
Comparison of MSE and MSB
- If values are similar, H0 may be true.

Pretend Data Example: Three groups
- Group 1 Mean: 6.3, Variance: 10
- Group 2 Mean: 4.3, Variance: 10
- Group 3 Mean: 5.3, Variance: 7.75
- Calculation of MSE
- Formula Input: Variances yield MSE = 9.25
MSB Calculation
- Using the computational formula from raw data
- Overall Mean Calculation required
- Example results:
- Mean for Group 1: 8.3
- Computed MSB Value: 9
Conclusion from Calculated Values
- Ratio of MSE to MSB: $\frac{9}{9.25} = 0.97$
- Suggests no significant difference among the three means (H0 likely true)

New Groups Means:
- Mean1: 8.3, Mean2: 3.3, Mean3: 5.3
Observations
- Differences among means suggest potential rejection of H0.
Re-calculating MSE = 9.25
MSB Calculation again yields a different value: 57.
Comparison:
- Ratio of MSB to MSE: $\frac{57}{9.25} = 6.16$
- Indicates significant differences among group means.

Introduction of Five Groups
- Each group with 10 participants: Total n = 50, These are unique participants in a between-subjects design
- Sum of Squares Calculations for ANOVA
Sum of Squares Total (SST)
- Formula: $SST = \sum x^{2} - \frac{(\sum x)^{2}}{n}$
- Defined as total variability regardless of treatment
- Derivation shows calculation for simplicity
Sum of Squares Between (SSB)
- Formula related to treatment variability
Sum of Squares Error (SSE)
- Variability within treatments, allowing calculation using SSB and SST relationship:
- $SST = SSB + SSE$
- Degrees of Freedom for SSE = Total n - Number of Groups

Summary Table Creation Overview
- Formats may vary, but essential columns remain: Source, Degrees of Freedom, Sum of Squares, Mean Square, F Ratio
Filling in Table Data:
- Degrees of freedom calculated for each term
- Sums of squares previously calculated
Mean Squares Calculation:
- MSB and MSE calculated by dividing SSB and SSE by their respective degrees of freedom
F Ratio Calculation:
- Final step: $F = \frac{MSB}{MSE}$
- Example yields F value of 9.1, indicating significant differences.

F Critical Value
- Sourced from statistical tables based on degrees of freedom
- Example table describes critical value thresholds
Comparison of F value against F critical value to ascertain significance
- Example with F value (9.1) vs F critical (2.61)
Result: if F value exceeds F critical, reject H0, indicating significant differences among group means.

Summarization of Analysis of Variance process and its importance
Emphasis on derivation concept for better understanding of ANOVA outputs in statistical software like SPSS
Look forward: Future module on one-way ANOVA analysis and its SPSS implementation, including practical applications of the theoretical concepts covered.