ANOVA

Function of ANOVA

• statistical test used to determin if a categorical independent variables
• with two or more groups
• has an effect on a continuous dependent variable

Types of ANOVA

1. One-way Indepdendent ANOVA - independet sample
2. Two-way Repeated mesure ANOVA

Notation to refer to an item in an ANOVA

Yg,i

g - group
i - individual

Why use an ANOVA

• only need to run a single test to test diffrences btween groups
• Need to run lest post hoc tests
• Less chance of finding diffrence by groups

Define sum of squares (SS)

• the sum of the diffrences from the mean of each point squares

Defne the F distribution

• The ratio between two variances (Model and residual)
• positive skew
  -Has 2 degrees of freedom
• if F is larger than Fcrit then can rejectnull hypothesis

How to report result of ANOVA

• Sentce saying interpretation
• F dfmodal df error = [f stat], P < 0.05

Assumptions of ANOVA

1. Independence of observation
2. Random sampling
3. Normality of data and errors
4. Homogenous variance
5. Responce variable must be continuous

Nonparametric alternatives to ANOVA

1. Kruskal-Wallis test - more than 2 groups
2. Mann-Whitney. U test - 2 groups

• Take rank sums
• no oneed normality

What does ANOVA not tell us

• Which groups differ from each other
• Direction of diffrence between groups
• Confidence intervals for the diffrence s
• Effect sizes between groups

Why Pos-hoc test after ANOVA

• Which specific groups differ from each other
• Provides detailed pairwise comparisons
• Helps understand practical significance
• Controls for multiple comparisons

Post hoc tests for after ANOVA

Bonferroni (concervative)

• divides signficance level by number of tests

Tuckey HSD (more leniant)

• balances error control and power

Name 3 types of ANOVA condtions

1. Repeated Measure ANOVA
2. Independent ANOVA
3. Mixed Factorial ANOVA

Define Repeated Mesure ANOVA

• For related groups (repated mesure experince)
• All factors vary within the individual
• so same individual must be used under diffrent conditons

Define Independent ANOVA

• For independent groups
• All factors vary between groups

Define Mised Factorial ANOVA

• Combination of both between group and between individual factors

Desfine Two-way ANOVA

-A statistical test used to determine if

• Two categorical independent variables
• Have an effect on a continuous dependent variable

Describe levels for a Two-way ANOVA

• different categories of a factor are referred as Levels
• The number of Levels can vary between factors
• level combinations of factors are called cell

Define Balanced design Two-way ANOVA

• Equal sample sizes per cell
• Standard ANOVA applies

Unbalanced design Two-way ANOVA

• unequal sample sizes per cells
• Special types of ANOVA

3. Hypothesis you can have with Two-Way ANOVA

1. Impact of factor A
   2, Impact of factor B
2. Intraction between factor A and B (The effects on one factors does not vary as a function of the other factor)

Describe a Euler Diagram for 2 way anova

• Statistical model
• partitions total variance
• data = (A + B + A:B) + error
• SStotal = SSa + SSb + SSa:SSb (interaction) + SSresifual

Assumptions of Two-Way ANOVA

Shared with one-way

1. Independent and random samping
2. Normality of data and residuals (outliars can distort)
3. homogeneity - equal varaince between groups

Two-way:

1. Two factors, each with multiple levels
2. Factioral independence (effects of two factors are independent to each other, no independece)
   3.Non-additicity - there is no indetraction between variables

Types of Model of SS in 2 way ANOVA

Type 1 - Sequential SS
Type 2 - Adjusted SS
Type 3 - Full Modal SS

Why there are multiple types of SS for 2 way ANOVA

• Most of the time, factors are not entirely indepenent
• There is some degree of overlap in the variance expain by each facotr
• Types of SS is how the shared varaince is split

SS Type 1

Sequential partitioning

• Each factor gets credit for whats keft after previous facots
• Order matters: first item given more credit
• Use when there is theoretical reason for order

SS Type 2

Adjusted SS

Partial partitioning

• Tests for each main effect after accounting for the other main effect
• What each factor independenly can explain

When to use:

• Insignificant interaction
• Factors are of equal intrest
• No reason for prioritising one factor over another

SS Type 3

Full Modal SS

Marginal partitioning

• tests each main effect after accounting for the other main effect

When to use

• Significant interaction is present
• No theoretical reason for order
• Unbalanced design

What is an omnibus test

• F statistic test
  -ANOVAs

Steps after conduting ANOVA

Omnibus (F-statistic test)

1. Examin interaction

• Plot data, look for non parallel line

1. Conduct post-hoc tests for interaction

• Test paimwise comparison
• Use multiple comparisons (e.g Turkey's HSD)

Present the results of Two-way ANOVA

"A two-way ANOVA was conducted to examine the effects of Age (young vs. old) and Education (degree vs. no degree) on fluid intelligence task performance.

• The results showed a significant significant main effect of Age (F(1, 276) = 118.8, p <0.001) and Education (F(1, 276) = 36.7, p < 0.001), with young participants and those with a degree performing better
• . A significant Age × Education interaction (F(1, 276) = 8.74, p = 0.004) indicated that older adults with no degree showed much lower performance compared to other groups."

Critical Steps before ANOVA

1. Design: ensure balanced design where possible
2. Check assumptions
3. Choose appropriate SS type