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Intro to Analysis of Variance (ANOVA), a independent or quasi-independent…

- - - - SSₜₒₜₐₗ= ∑X² - (G²/N)
      - SSwᵢₜₕᵢₙ= Sum of SS for each treatment condition (i.e.) SS₁ +SS₂ + SS₃*
        
        SSbₑₜwₑₑₙ= Sum of Squares for Sample Means = n(SSₘₑₐₙₛ)
        
        ONLY used when samples are exact same size n
        
        alternative formula:
        SSbₑₜwₑₑₙ = ∑(T²/ n) - (G²/ N)
      - SSₜₒₜₐₗ = SSwᵢₜₕᵢₙ + SSbₑₜwₑₑₙ
  - - - OR
        dfₜₒₜₐₗ= dfbₑₜwₑₑₙ +dfwᵢₜₕᵢₙ*
    - - or the sum of degrees of freedom in each treatment
      - or dfwᵢₜₕᵢₙ = N - k
    - - number of treatments - 1
- - - - similar to t-distribution, but only positive and not a symmetrical curve.
      - smaller df creates more spread out distribution
      - larger df = F-values more highly clustered around 1.0
  - - - = SSbₑₜwₑₑₙ/SSₜₒₜₐₗ
        
        use η² to describe, instead of r²
  - - - populations must have equal variance - Homogeneity of Variance
- - - - HSD = q √ (MSwᵢₜₕᵢₙ/n)
        
        determines the mean difference necessary for any two samples to be significantly different
      - q= Studentized Range Statistic
        
        locate value in table, using k and error term or dfwᵢₜₕᵢₙ
    - - one of the safest post hoc tests, due to smallest risk of Type I error
      - uses an F-ratio to evaluate significance of difference between two treatments
        
        same formula for F, but SSbₑₜwₑₑₙ just for two treatments, df is k-1
        
        F evaluated using same critical region and error term as overall ANOVA
- - - - Experimentwise Alpha Level
        
        ANOVA has significant advantage over t-test here
    - - F-Ratio
        
        F= variance differences between sample means / variance expected with no treatment effect
        
        OR
        F= systematic treatment effects + random differences/ random differences
        
        OR
        F= Between-Treatments Variance / Within-Treatments Variance