Analysis of Variance (ANOVA)

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Analysis of Variance(ANOVA)
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When ANOVA is used..

• All the explanatory variables are categorical
  (factors)
• Each factor has two or more levels
• Example You have 60 DNA samples from 3 plants
  A, B, and C. And you measured DNA concentration
  once for each sample. Explanatory variable will
  be plants.

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One-way ANOVA

• A single factor with three or more levels.
• Example see previous example

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Multi-way ANOVA

• Two or more factors.
• Example You measured concentration of DNA
  samples from 3 plants (A, B, and C) that were
  grown in four different soils (K, M, N, P) -
  two-way ANOVA

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Multi-way ANOVA

• Null hypotheses The results of a two-way anova
  include tests of two null hypotheses that the
  means of observations grouped by one factor are
  the same that the means of observations grouped
  by the other factor are the same
• Modelconcentrationplantsoils

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Factorial ANOVA

• When there is replication at each combination of
  levels in a multi-way ANOVA.
• Example You measured 3 times concentration of
  DNA samples from 3 plants (A, B, and C) that were
  grown in four different soils (K, M, N, P) -
  two-way ANOVA with replication

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Factorial ANOVA (cont.)

• Null hypotheses The results of a two-way anova
  with replication include tests of three null
  hypotheses that the means of observations
  grouped by one factor are the same that the
  means of observations grouped by the other factor
  are the same and that there is no interaction
  between the two factors. The interaction test
  tells you whether the effects of one factor
  depend on the other factor.
• Modelconcentrationplantsoilsplantsoils

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ANOVA

• ANOVA compares the mean values by comparing
  variances.
• It calculates the total variation (SSY) and
  partitioning it into two components explained
  variation (SSA) and unexplained variation (SSE)
• SSY

SSA
SSE
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Total variation

• SSY- the total sum of squares is the sum of
  squares of the differences between the data
  points and the overall mean, n is number of
  samples per treatment, k is the number of
  treatments

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Unexplained variation

• SSE- error sum of squares is the sum of the
  squares of the differences between the data
  points and their individual treatments mean

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Explained variation

• SSA- treatment sum of squares is the sum of the
  squares of the differences between the individual
  treatment means and the overall mean
• The amount of the variation explained by
  differences between the treatment means

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Explained variation (cont.)

• SSASSY-SSE
• The larger difference between total variation and
  unexplained variation (SSY-SSE) the larger
  explained variation (SSA)
• the greater the deference between treatment means

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Before starting ANOVA
1. Check for constancy of variance Is the
variances differ by more than factor of 2?

• 2. Test homogeneity of variance
• Is Fligner-Killeen test showing significant
  p-value?

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Analysis of sample Assumptions

• Independence of samples elements
• Normality
• Homogeneity
• Sufficient sample sizes, equal sample sizes is
  the best