## # A tibble: 6 × 6
## species sex n mean_mass sd_mass se_mass
## <fct> <fct> <int> <dbl> <dbl> <dbl>
## 1 Adelie female 73 3369. 269. 31.5
## 2 Adelie male 73 4043. 347. 40.6
## 3 Chinstrap female 34 3527. 285. 48.9
## 4 Chinstrap male 34 3939. 362. 62.1
## 5 Gentoo female 58 4680. 282. 37.0
## 6 Gentoo male 61 5485. 313. 40.1Lecture: Two-Way ANOVA
Factorial designs
Where We Left Off
Covered last time — one-way ANOVA:
- The ANOVA model, and fixed vs. random predictors
- Partitioning variance: \(SS_{total} = SS_{groups} + SS_{residual}\)
- The null hypothesis, and the F-ratio that tests it
- Assumptions and diagnostics on the residuals
- Post-hoc tests — Tukey and others
- Reporting the result
Note
✅ Key idea from last lecture
One-way ANOVA asks “do these groups differ?” for one factor. Today: what happens when two factors act together — separately, and possibly interacting?
Two-Factor Designs (2-Way ANOVA)
- Very common in ecology
- Can have more factors (e.g., 3-way ANOVA), but interpretation gets very challenging
- Most multifactor designs are factorial or nested
- We’ll cover a 2-factor ANOVA — say fertilizer and sunlight, for plant biomass:
- each can have an effect independently
- or they may interact: under high light, high fertilizer grows better than either effect alone would predict
Note
📖 Reference
Gotelli & Ellison, A Primer of Ecological Statistics, Ch. 10 — The Analysis of Variance, extends to factorial designs like the one covered today.
Factorial ANOVA: Design Structure
Consider two factors:
- Factorial/crossed: every level of B in every level of A
Factorial ANOVA: Effect Types
In factorial designs, we look at two types of factor effects:
- Main effect of each factor (pooling across the other factor)
- Interaction effects — is there a synergistic/antagonistic effect between factors?
Descriptive Statistics
## # A tibble: 6 × 6
## species sex n mean_mass sd_mass se_mass
## <fct> <fct> <int> <dbl> <dbl> <dbl>
## 1 Adelie female 34 3335. 260. 44.5
## 2 Adelie male 34 4049. 318. 54.5
## 3 Chinstrap female 34 3527. 285. 48.9
## 4 Chinstrap male 34 3939. 362. 62.1
## 5 Gentoo female 34 4681. 309. 53.0
## 6 Gentoo male 34 5525 274. 47.0
SSA: Factor A Effects
\[SS_A = qn\sum_{i=1}^{p}\left(\bar{Y}_{i\cdot} - \bar{Y}\right)^2\]
- \(\bar{Y}_{i\cdot}\) is the marginal mean of level i of A — the EMM for that species
- Square its distance from the grand mean, and multiply by \(qn\), the number of observations that marginal mean stands for (q levels of B × n per cell)
- The multiplier is not optional. Without it you have the SS of 3 numbers, not of 204 observations, and the df would not match.
SSB: Factor B Effects
\[SS_B = pn\sum_{j=1}^{q}\left(\bar{Y}_{\cdot j} - \bar{Y}\right)^2\]
- Same move, the other way round: each column marginal mean against the grand mean
- Multiplier is now \(pn\) — p levels of A × n per cell — because each sex EMM stands for that many birds
SSAB: Interaction Effects
\[SS_{AB} = n\sum_{i=1}^{p}\sum_{j=1}^{q}\left(\bar{Y}_{ij} - \bar{Y}_{i\cdot} - \bar{Y}_{\cdot j} + \bar{Y}\right)^2\]
- Take a cell mean, subtract what the two main effects alone predict for it, and you are left with the interaction
- \(\bar{Y}_{i\cdot} + \bar{Y}_{\cdot j} - \bar{Y}\) is the additive prediction; the grand mean is added back because it was subtracted twice
- Multiplier is just \(n\) — a cell mean stands for only the n birds in that cell
Note
✅ Key idea
\(SS_{AB}\) is the leftovers. If the additive model predicted every cell mean perfectly, every one of those brackets would be zero and \(SS_{AB} = 0\).
Example Study Details
Let’s try the example with the palmerpenguins package data:
- Effect of species and sex on
body_mass_g - 3 species (Factor A)
- 2 sexes (Factor B)
- 34 replicates in each cell
Checking Model Assumptions
The four standard diagnostic plots, built automatically with ggfortify::autoplot().
The Interaction Plot
Introduction to Unbalanced Designs
The challenge of unbalanced data: real-world data is often unbalanced — unequal sample sizes across groups, missing data patterns, natural variation in sampling.
Key issues: Type I, II, and III SS stop agreeing; order of terms matters for Type I SS; marginal means ≠ simple averages.
The model is unchanged — \(Y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} + \varepsilon_{ijk}\) — but estimation of its parameters is not.
## [1] "Unbalanced sample sizes:"
## # A tibble: 6 × 3
## species sex n
## <fct> <fct> <int>
## 1 Adelie female 73
## 2 Adelie male 73
## 3 Chinstrap female 34
## 4 Chinstrap male 34
## 5 Gentoo female 58
## 6 Gentoo male 61
## [1] "Total N = 333"
## # A tibble: 3 × 4
## species female male prop_female
## <fct> <int> <int> <dbl>
## 1 Adelie 73 73 0.5
## 2 Chinstrap 34 34 0.5
## 3 Gentoo 58 61 0.487
Diagnostic Plots — Unbalanced Design
## [1] "Shapiro-Wilk test:"
##
## Shapiro-Wilk normality test
##
## data: residuals(model_u)
## W = 0.99776, p-value = 0.9367
## [1] "Levene's test:"
## Levene's Test for Homogeneity of Variance (center = median)
## Df F value Pr(>F)
## group 5 1.3908 0.2272
## 327
## [1] "Residual SD by group:"
## # A tibble: 6 × 3
## species sex sd_resid
## <fct> <fct> <dbl>
## 1 Adelie female 269.
## 2 Adelie male 347.
## 3 Chinstrap female 285.
## 4 Chinstrap male 362.
## 5 Gentoo female 282.
## 6 Gentoo male 313.Side-by-Side Comparison
## Balanced vs Unbalanced Results:
## Effect Balanced_F Balanced_p Unbalanced_F Unbalanced_p
## 1 Species 477.20 0e+00 746.92 0e+00
## 2 Sex 239.22 0e+00 311.84 0e+00
## 3 Interaction 9.10 2e-04 8.76 2e-04
##
##
## Sample sizes:
## Balanced total N: 204
## Unbalanced total N: 333
## Data discarded: 129 observations