Course 2 · Week 2 — ANOVA and non-linear extensions

Cheatsheet — biostats_courses

Author

R. Heller

One-way ANOVA

aov(y ~ group, data = df) |> summary()

ANOVA is a linear model with a categorical predictor. The F-test compares between-group to within-group variance.

Contrasts with emmeans

library(emmeans)
emm <- emmeans(fit, ~ group)
pairs(emm, adjust = "tukey")   # all pairwise
contrast(emm, list(TrtVsCtrl = c(-1, 1, 1, 1) / 3))

Pre-specify contrasts before looking at the data; correct for multiplicity.

Two-way / factorial ANOVA

aov(y ~ A * B, data = df) |> summary()
emmip(fit, A ~ B)   # interaction plot

Interaction means “effect of A differs by level of B”. Report the interaction first; main effects are conditional.

Repeated measures / blocking

  • RCBD: aov(y ~ treatment + Error(block)).
  • Repeated measures: move to a mixed model.
library(lme4); library(lmerTest)
lmer(y ~ treatment + time + (1 | subject), data = df)

GAMs — smooth non-linear terms

library(mgcv); library(gratia)
fit <- gam(y ~ s(x, k = 10) + z, data = df)
summary(fit)      # edf tells you how "wiggly"
draw(fit)         # smooth + CI

edf ≈ 1 → nearly linear; > 4 → clearly non-linear.

Non-linear regression (nls)

# Michaelis-Menten: y = Vmax * x / (K + x)
fit <- nls(y ~ Vmax * x / (K + x),
           data = df, start = list(Vmax = 1, K = 1))

Start values matter. If it fails, plot first to guess reasonable starts.

Decision rule for Week 2

  • Categorical predictor, > 2 levels → ANOVA + contrasts.
  • Factorial design → include interaction, report it first.
  • Effect obviously curved → GAM with spline; else try nls.
  • Repeated measures → mixed model, not repeated-measures ANOVA.

Common pitfalls

  • Tukey HSD without pre-specified contrasts of interest.
  • ANOVA p < 0.05 reported alone — without naming which groups differ.
  • Forcing a GAM onto monotonic data that nls fits cleanly.
  • Ignoring the random effect in clustered designs (pseudoreplication).

Further reading