Continuous Variables for Two or More Groups

CONTINUOUS VARIABLES FOR TWO OR MORE GROUPS

What does a one-way ANOVA test evaluate? (aov(), then summary())

Full name: analysis of variance

ANOVA tests whether three or more group means are significantly different.

• H₀: all population means are equal

  • the population means are the same for all groups

• H₁: at least one population mean is different

  • the population means are not the same for all groups

What is the ANOVA test statistic and how are degrees of freedom calculated?

Test statistic: F

• Higher F values = lower probability of H₀ being true

2 degrees of freedom:

• Between groups: G − 1

  • G = number of groups

• Within groups: N − G

  • N = total number of observations

*exact rejection regions depend on degrees of freedom

What do you do after a significant ANOVA?

Use post-hoc tests to determine which groups are significantly different from each other:

• Pairwise comparisons: TurkeyHSD() or posthocPairwiseT()

• Planned comparisons: for contrasts of a prior interest

  • Specify comparisons of interest

*Adjust p-values for multiple comparisons using Bonferroni correction: p′ = p*m  

• m = total number of comparisons

What is eta squared (η²) and how is it interpreted? - Effect size

Eta squared measures the proportion of variance explained by the group differences:

• 0.01: small

• 0.06: medium

• 0.14: large

*larger values correspond to more unequal means

• (Alternate scale: 0.02 / 0.13 / 0.26 from lecture slides) (?? sooo which scale)

What assumptions does ANOVA make?

• Normality of residuals: the residuals are normally distributed

  • Check with: Shapiro-Wilk test (shapiro.test()), histogram, QQ plot

  • If violated: use Kruskal-Wallis sum test (kruskal.test())

• Homogeneity of variance: the variance is the same in both groups

  • Check with: leveneTest())

  • If violated: use Welch’s one-way test (oneway.test())

• Independence of residuals: The residuals are independent