• Test

• Impact of the factor on quantitative output

  • Impact of 1 factor – one way anova

  • Impact of 2 factors – two way anova

  • …

• Etc.

• Analysis of Variance - a set of methods used to test the impact of one or more factors on population mean

  • an inferential method that is used to test the equality of three or more population means

One-way ANOVA - Decomposition of total variance on sub-variances

Sub-variances contain information which influence x_ij values in observed distribution

s² → variance due to model s₁² + residual variance s_r²

One-way ANOVA model

One-way ANOVA model

• balanced model (equal n)

• unbalanced model (different n)

• model presumptions

  • Normal distribution of all samples

  • Independency of samples

  • Homogeneity of variances

• Null hypothesis statement

  • H₀: m₁ = m₂ = … = m_m

  • H₀: a₁ = a₂ = … = a_m = 0

• Alternative hypothesis H₁

  • at least one of the population means is different from the others

Multiple comparison

• In case of h1

  • Scheffé´s method

  • Tuckey´s method

  • Duncan´s method

  • Kramer´s method

• What is null hypothesis of ANOVA?

• What are assumptions of ANOVA?

• Why we make detailed pairwise comparison after we reject null hypothesis? What methods can we use?

• What represents residual variance?

• What is difference between balanced and non-balanced ANOVA model?

• What is difference between one-way and two-way ANOVA?

• Parametric tests

  • Parameters testing

  • Assume normal distribution (F-test, t-test)

• Nonparametric tests

  • Distribution is not required

  • Quantitative and qualitative variables

  • Simple calculation (especially for small samples)

  • Lower power

Two sample Wilcoxon test

• Nonparametric equivalent of the two sample t-test

• Test of two independent samples

• To test the hypothesis that two independent samples X=(x₁, x₂,…,x_m) and Y=(y₁, y₂,…,y_n) are from the same population

• procedure

  • all values are ranked in ascending order (pooled sample) with ties assigned the average of the next available ranks

Wilcoxon test

• Nonparametric equivalent of the paired t-test

• We assess whether two dependent samples are from the same population

• Procedure

  • For each pair of dependent observations (x_i, y_i) compute difference d_i=x_i-y_i

  • all values are ranked in ascending order (pooled sample) with ties assigned the average of the next available ranks

  • Absolute values of differences are ranked in ascending order (zero differences are not considered)

  • Summarize separately order of positive and negative differences

• W⁺ = sum of order for positive differences

• W^- = sum of order for negative differences

• Test criterion is to lower value from W⁺ a W^-

  • W=(W⁺ ,W^- )

• H₀ is rejected, if W≤W_α,n, where

  • W_α,n critical table value

  • n…number of nonzero differences

Sign test

• For two dependent samples

• Procedure

  • The test criterion Z represents the lower number from positive and negative differences

  • Z=min (Z⁺, Z^-)

  • if Z < Za; n, H0 is rejected

    • n - number of nonzero differences

    • a - significance level

Kruskall – Wallis test

• Nonparametric equivalent of one-way ANOVA

• Test of the hypothesis that m independent samples with sizes n₁, n₂,…n_m are from the same distribution

• Procedure

  • all values are ranked in ascending order (pooled sample) with ties assigned the average of the next available ranks

Dixon test of extrem deviations