## Data
## # A tibble: 6 × 4
## treat patch quad algae
## <fct> <fct> <dbl> <dbl>
## 1 high 1 1 0
## 2 high 1 2 0
## 3 high 1 3 0
## 4 high 1 4 6
## 5 high 1 5 2
## 6 high 2 1 0
##
##
## Treatment levels:
## [1] "none" "low" "medium" "high"Lecture: Nested ANOVA
Hierarchical designs
ANOVA
Factorial vs. nested designs, the nested-ANOVA linear model and variance partitioning, and a naive-vs-correct worked example (sea urchin grazing and algae cover) using the traditional aov() approach — the modern mixed-model (lmer) alternative continues in the Linear Mixed Models lecture.
Where We Left Off
Covered last time — two-factor ANOVA:
- Example, linear model, analysis of variance
- Null hypotheses
- Interactions and main effects
- Unequal sample size
- Assumptions
Note
✅ Key idea from last lecture
In a factorial design, every level of B appears under every level of A. Today: what happens when B is only ever nested inside one specific level of A?
Part 1 · Multifactor Designs
Two-Factor (2-Way) ANOVA, Recap
- We often consider more than 1 factor (independent categorical variables) to reduce unexplained variance and look at interactions
- 2-factor designs (2-way ANOVA) are very common in ecology
- Can have more factors (e.g., 3-way ANOVA) — interpretation gets tricky
- Most multifactor designs are: factorial, nested, or mixed
Factorial Versus Nested Designs
Consider two factors, A and B:
- Factorial/crossed: every level of B in every level of A
- Nested/hierarchical: levels of B occur only in 1 level of A
- both fixed — rare
- one fixed, one random
- both random — rare in ecology, common in genetics
Part 2 · Factorial Designs — Examples
Factorial Designs Overview
Factorial designs: both factors are typically fixed (but not always).
Factorial Design Example: Seedling Growth
Effects of light level on growth of seedlings of different size:
- 3 light levels (Factor A)
- 3 size classes (Factor B)
- 5 replicate seedlings in each cell
Factorial Design Example: Salamander Growth
Effects of food level and tadpole presence on larval salamander growth:
- 2 food levels (Factor A)
- Presence/absence of tadpoles (Factor B)
- 8 replicates in each cell
Factorial Design Example: Limpet Fecundity
Effect of season and density on limpet fecundity:
- 2 seasons (Factor A)
- 4 density treatments (Factor B)
- 3 replicates in each cell
Part 3 · Nested Designs — Examples
Nested Designs Overview
Nested designs cover: nested design examples, the linear model, analysis of variance, null hypotheses, unbalanced designs, assumptions, mixed-model ANOVA.
Nested designs: Factor A is usually fixed; Factor B is usually random.
Nested Design Example: Limpet Growth
Study on effects of enclosure size on limpet growth:
- 2 enclosure sizes (Factor A)
- 5 replicate enclosures (Factor B)
- 5 replicate limpets per enclosure
Nested Design Example: Reef Fish
Study on reef fish recruitment:
- 5 sites (Factor A)
- 6 transects at each site (Factor B)
- Replicate observations along each transect
Nested Design Example: Sea Urchin Grazing
Effects of sea urchin grazing on biomass of filamentous algae:
- 4 levels of urchin grazing: none, low, medium, high
- 4 patches of rocky bottom (3–4 m²) nested in each grazing level
- 5 replicate quadrats per patch
This is our worked example for the rest of the lecture.
Part 4 · The Nested Design Model
Nested Design: Linear Model Structure
Consider a nested design with:
- p levels of factor A (i = 1…p) — e.g., 4 grazing levels
- q levels of factor B (j = 1…q), nested within each level of A — e.g., 4 different patches per grazing level
- n replicates (k = 1…n) in each combination of A and B — e.g., 5 replicate quadrats in each patch in each grazing level
Calculating Means in a Nested Design
We can calculate several means:
- Overall mean (across all levels of A and B) = \(\bar{y}\)
- A mean for each level of A (across all levels of B in that A) = \(\bar{y}_i\)
- A mean for each level of B within each A = \(\bar{y}_{j(i)}\)
Nested Design Means, Visualized
The Nested Design Linear Model
\[y_{ijk} = \mu + \alpha_i + \beta_{j(i)} + \varepsilon_{ijk}\]
- \(y_{ijk}\) is the response variable — the value of the k-th replicate in the j-th level of B in the i-th level of A (e.g., algal biomass in the 3rd quadrat, in the 2nd patch, in the low-grazing treatment)
- \(\mu\) is the overall mean (overall average algal biomass)
Tip
🖐 Notice
This looks almost identical to the two-way ANOVA model — the difference is entirely in the subscript: \(\beta_{j(i)}\), not \(\beta_j\). B only exists inside a specific A.
Fixed and Random Effects in the Nested Model
\[y_{ijk} = \mu + \alpha_i + \beta_{j(i)} + \varepsilon_{ijk}\]
- \(\alpha_i\) is the fixed effect of factor A — the difference between the average biomass in all low-grazing-level quadrats and the overall mean
- \(\beta_{j(i)}\) is the random effect of factor B nested within factor A — usually a random variable measuring variance among all possible levels of B within each level of A (variance among all possible patches that could have been used in the low-grazing treatment)
Note
✅ Key idea
B being random is the whole point of nesting: you don’t care about these specific patches — you care about patch-to-patch variability in general.
The Error Term in the Nested Model
\[y_{ijk} = \mu + \alpha_i + \beta_{j(i)} + \varepsilon_{ijk}\]
- \(\varepsilon_{ijk}\) is the error term — the unexplained variance associated with the k-th replicate in the j-th level of B in the i-th level of A
- \(\alpha_i\) is the effect of the i-th level of A: \(\mu_i - \mu\)
- \(\varepsilon_{ijk}\): the difference between the observed algal biomass in the 3rd quadrat in the 2nd patch in the low-grazing treatment, and the predicted biomass (the average biomass in that 2nd patch in the low-grazing treatment)
Tip
🖐 Notice
The residual is measured against the cell mean (this specific patch), not the treatment mean — that’s what makes it “nested,” arithmetically.
Part 5 · Partitioning Variance
Analysis of Variance: Residual and Total
- \(SS_{resid}\) is the difference between each observation and the mean for its level of factor B, summed over all observations
- \(SS_{total} = SS_A + SS_B + SS_{resid}\)
- SS can be turned into MS by dividing by the appropriate df
Analysis of Variance: SSA
As before, we partition the variance in the response variable using SS. \(SS_A\) is the SS of differences between the mean in each level of A and the overall mean.
Analysis of Variance: SSB
\(SS_B\) is the SS of the difference between the mean in each level of B and the mean of the corresponding level of A, summed across all levels of A.
The Nested ANOVA Table
Part 6 · Hypotheses and Assumptions
Null Hypotheses: Factor A
Two hypotheses are tested on values of MS. 1. No effects of factor A — assuming A is fixed:
- \(H_0(A)\): \(\mu_1 = \mu_2 = ... = \mu_i = \mu\)
- Same as in 1-factor ANOVA, using means from B factors nested within each level of A
- (no difference in algal biomass across all grazing levels: \(\mu_{none} = \mu_{low} = \mu_{med} = \mu_{high}\))
Null Hypotheses: Factor B
2. No effects of factor B nested in A — assuming B is random:
- \(H_0(B)\): \(\sigma^2_\beta = 0\) (no variance added due to differences between all possible levels of B)
- (no variance added due to differences between patches)
Conclusions from the Analysis
Conclusion:
“Significant variation between replicate patches within each treatment, but no significant difference in the amount of filamentous algae between treatments.”
Unbalanced Nested Designs
Unequal sample sizes can arise from:
- An uneven number of B levels within each A
- An uneven number of replicates within each level of B
Not a problem unless you have unequal variance or a large deviation from normality.
Nested Design Assumptions
As usual, we assume: equal variance, normality, no outliers, independence of observations.
Equal variance + normality need to be assessed at both levels:
- Since means for each level of B within each A are used for the H-test about A, need to assess whether those means meet normality and equal variance
- Examine residuals for the H-test about B
- Transformations can be used
Warning
⚠️ Watch out!
This is the part people most often get wrong: checking assumptions on the raw data isn’t enough for a nested design — you need to check them at the level each hypothesis test actually operates on.
Part 7 · The Sea Urchin Example — Data
The Nested Design, the Hard Way
This analysis examines the effects of varying sea urchin densities on the percentage cover of filamentous algae. The experiment was designed with:
- Four random patches
- Four urchin density treatments (none = removed, low = control, medium = 33% of original density, high = 66% of original density)
- Five replicate quadrats measured within each treatment-patch combination
Note
📖 Reference
Quinn & Keough (2002), Experimental Design and Data Analysis for Biologists — this dataset and analysis follow their treatment of the sea urchin/algae example.
Data Overview
The data frame contains: patch (random patches 1–16 where treatments were applied), treat (urchin density treatment), quad (replicate quadrats within each treatment-patch combination), algae (percentage cover of filamentous algae — the response variable).
## Summary statistics by treatment:
## # A tibble: 4 × 7
## treat n mean sd se min max
## <fct> <int> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 none 20 39.2 28.7 6.41 0 83
## 2 low 20 21.6 25.1 5.62 0 79
## 3 medium 20 19 25.7 5.74 0 71
## 4 high 20 1.3 3.18 0.711 0 13A First Look at the Data
Part 8 · The Wrong Way — Naive Factorial Model
Manual Nested ANOVA — Setting Up
In this experimental design, patch is nested within treat because each patch received only one treatment level — a hierarchical design where the effect of patches must be considered within each treatment (Quinn & Keough 2002).
We’ll use a traditional nested ANOVA. You could make this easy by averaging algae per patch and running a one-way ANOVA — but your power is significantly reduced, and you lose estimates of variability among patches.
This first attempt is really a regular factorial ANOVA — not appropriate, since it’s pseudoreplicated. Also, the F-value for Treat is using \(MS_{resid}\) when it should use \(MS_{treat:patch}\).
u_fact_model <- aov(algae ~ treat + treat:patch, data = u_df)
cat("Factorial model summary:\n\n")
## Factorial model summary:
summary(u_fact_model)
## Df Sum Sq Mean Sq F value Pr(>F)
## treat 3 14429 4810 16.108 0.0000000658 ***
## treat:patch 12 21242 1770 5.928 0.0000008323 ***
## Residuals 64 19110 299
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Why the Factorial ANOVA Is Not Correct
- We can correct this by specifying the error term
- The MS needed for the F-value is really MS(patch nested in treatment), because those are the true replicates
- We can specify that — the problem is this approach can’t really handle unbalanced designs
u_nested_model <- aov(algae ~ treat + Error(treat:patch), data = u_df)
cat("Correctly specified nested ANOVA:\n\n")
## Correctly specified nested ANOVA:
summary(u_nested_model)
##
## Error: treat:patch
## Df Sum Sq Mean Sq F value Pr(>F)
## treat 3 14429 4810 2.717 0.0913 .
## Residuals 12 21242 1770
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Error: Within
## Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 64 19110 298.6