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Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA). When ANOVA is used. All the explanatory variables are categorical (factors) Each factor has two or more levels

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Analysis of Variance (ANOVA)

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  1. Analysis of Variance(ANOVA)

  2. 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.

  3. One-way ANOVA • A single factor with three or more levels. • Example: see previous example

  4. 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

  5. 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; • Model=concentration~plant+soils

  6. 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

  7. 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. • Model=concentration~plant+soils+plant:soils

  8. 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

  9. 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

  10. 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

  11. 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

  12. Explained variation (cont.) • SSA=SSY-SSE • The larger difference between total variation and unexplained variation (SSY-SSE) the larger explained variation (SSA) • the greater the deference between treatment means

  13. Before starting ANOVA 2. Test homogeneity of variance Is Fligner-Killeen test showing significant p-value? 1. Check for constancy of variance Is the variances differ by more than factor of 2?

  14. Analysis of sample Assumptions • Independence of samples elements • Normality • Homogeneity • Sufficient sample sizes, equal sample sizes is the best

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