Bonferroni and Holm Corrections

• Introduces an adjustment to the p-value to oversee a few variables within the test, known as a correction for multiple comparisons or simultaneous inference.
• Mentions that multiple ways exist to perform this adjustment with discussions found throughout Chapter 16.8.

Bonferroni Corrections

• Adjustment method that involves multiplying all raw p-values by the number of separate tests (m).
• The corrected p-value will be p' = m × p.
• Using the Bonferroni correction: reject the null hypothesis if p' < α.
• The logic ensures that the total Type I error rate across m tests is no more than α.
• Method is simple and general.
• In R, the Bonferroni correction can be implemented using pairwise.t.test() with p.adjust.method = "bonferroni" or posthocPairwiseT() in the lsr package.

Holm Corrections

• Method used instead of the Bonferroni correction.
• Employs a sequential testing approach by adjusting p-values based on their rank.
• The j-th largest p-value adjustment is pj = j * pj, or pj = pj+1, whichever is larger.
• Sort all p-values in order from smallest to largest.
• Multiply the smallest p-value by m.
• The other values enter a two-stage process: Multiply the secondary p-value by m - 1.
• Use the greater value for the adjusted p.
• Holm correction: more powerful due to a lower Type II error rate, while maintaining the same Type I error rate.
• Holm correction: the default setting in pairwise.t.test() and posthocPairwiseT().

Assumptions of One-Way ANOVA

• Statistical test that relies on assumptions about data, including normality, homogeneity of variance, and independence.
• A statistical model underpinning ANOVA is expressed as Yᵢₖ = μ + ϵᵢₖ
• Alternatively, Yᵢₖ = μₖ + ϵᵢₖ.
• Testing procedures rely on specific assumption about residuals (ϵᵢₖ ~ Normal(0, σ²)).
• Normality: residuals are normally distributed, assessed via QQ plots or Shapiro-Wilk tests.
• Homogeneity of Variance: population standard deviation is the same across all groups.
• Independence: knowing one residual provides no information about any other residual.

Checking Homogeneity of Variance

• Levene's Test: Commonly used test in literature, closely related to Brown-Forsythe test.

• The population means of Z which are identical for all groups are tested with the null hypothesis.

• The leveneTest() function: found in the car package, used for performing Levene's test.

• Use leveneTest(my.anova) by inputting original aov object as a function.

• A Brown-Forsythe test differs: constructs the transformed variable Z using deviations from group medians.

• Apply Zik = |Yik - mediank(Y)|.

• Although R (the most up-to-date version being utilized) will report test as F-value, it could also be W.

Removing Homogeneity of Variance

• Welch One-Way Test: saves ANOVA when homogeneity is violated - implemented using oneway.test() function.
• Important arguments includes:
  • Formula: outcome ~ group.
  • Data: specifies dateframe containing variables
• var.equal: If FALSE, the Welch test runs. If TRUE, a regular ANOVA runs

Checking the Normality Assumption

• Residual extraction performed by R function residuals().
• Residual extraction can be done through the hist,qqnorm and shapiro.test to generate histogram and QQ plots.
• Results of the Shapiro-Wilk test supports the hypothesis that normality is not violated if the p value is >0.05

Removing the Normality Assumption

• Easiest solution is switching to a non parametric test such as the Kruskal-Wallis which doesn't relate to a particular assumption of normality
• The test consists of ranking all Yik values while further analyzing ranked data.
• With Rik refers to ranking, average rank (Rk): (1/Nk)Σ Rik. The grand mean rank R = ΣΣ Rik.
• Kruskal-Wallis Statistic: K = (N-1)RSSb/RSStot.
  • With new calculation: K = [12/N(N+1)] Σk (SRk²/Nk)-3(N+1) for when there are correct matching variables.
• K sampling: distribution ≈ chi-square with (G–1) degrees of freedom, one-sided test (reject H₀ when K sufficiently large).
• Adjust equation for K: K = [12/N(N+1)] Σk (SRk²/Nk)-3(N+1). Corrects for when there are matching variables.

How to Run the Kruskal-Wallis Test

• Easily performed in R since R function kruskal.test.
• Specify data by variable mood.gain and tree.
• Specify outcome and grouping variable as separate input arguments, X + Y.
• Bundle all three together as list:
  • mood.gain<- list(placebo, joyzepam, anxifree
  • kruskal.test<-( mood.gain)

Relationship Between ANOVA and The Students Test:

• ANOVA with two groups is nearly identical to the Student t test with a few exceptions.
  • If the residual df from ANOVA and t test align themselves up to t(16) =-1.13068
  • P-Values are almost identical to when we value p=0.21 but we were able to achieve the result.
  • The new equation for F-statistic being F(0) = 1.71

Overview of Bonferroni and Holm corrections. These adjustments to p-values control the error rate across multiple tests. The Bonferroni method multiplies raw p-values by the number of tests, while the Holm method uses a sequential testing approach.