ads2-anova-factorial

name	ads2-anova-factorial
description	Use this skill for ADS/ADS2 one-way ANOVA, two-way/factorial ANOVA, interaction effects, Tukey HSD post-hoc tests, ANOVA assumption checks, numeric response by categorical factors, tooth-growth vitamin C designs, and ANOVA power. Trigger on ANOVA, aov, Tukey, interaction, factorial, two-way, dose and treatment, supplement, or numeric outcome across groups.

ADS2 ANOVA And Factorial Designs

Use ANOVA when the response is numeric and predictors are categorical factors. ADS2 mock material includes a two-factor design, so do not assume ANOVA is only one-way.

One-Way ANOVA

dat$group <- factor(dat$group)
model <- aov(y ~ group, data = dat)
summary(model)

Hypotheses:

H0: all group means are equal.
H1: at least one group mean differs.

Two-Way / Factorial ANOVA

Example tooth-growth structure:

dat$supp <- factor(dat$supp)
dat$dose <- factor(dat$dose)
model <- aov(len ~ supp * dose, data = dat)
summary(model)

supp * dose includes:

• main effect of supp;
• main effect of dose;
• interaction supp:dose.

If interaction is significant, interpret the interaction before main effects:

The effect of supplement depends on dose, so the supplement effect should be compared within dose levels rather than only as an overall main effect.

Assumption Checks

Independence is from study design. Check residuals:

hist(resid(model))
shapiro.test(resid(model))
plot(model, 1)  # residuals vs fitted
plot(model, 2)  # normal Q-Q

Report:

The residual normality check did not show strong evidence against normality (Shapiro p = [p]). The residuals-vs-fitted plot did not show an obvious pattern, so the equal-variance assumption appears reasonable.

If assumptions fail badly, consider Kruskal-Wallis for one-factor designs or simulation/bootstrap; for complex two-factor designs, state the limitation clearly.

Post-Hoc Tests

After significant ANOVA:

TukeyHSD(model)

Report:

Tukey's HSD suggests the main pairwise difference is between [A] and [B] with adjusted p = [p].

Plot

boxplot(len ~ supp * dose, data = dat, xlab = "Group", ylab = "Length")
interaction.plot(dat$dose, dat$supp, dat$len)

ANOVA Power

a <- anova(model)
ss_factor <- a$"Sum Sq"[1]
ss_resid <- a$"Sum Sq"[nrow(a)]
eta_sq <- ss_factor / (ss_factor + ss_resid)
f <- sqrt(eta_sq / (1 - eta_sq))

library(pwr)
k <- length(unique(dat$group))
n_per_group <- min(table(dat$group))
pwr.anova.test(k = k, n = n_per_group, f = f, sig.level = 0.05)

Remember: n is per group.