## 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
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?
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
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
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.
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
Full worked data table for the sea urchin example: for each grazing level (A1-A4, 0-100%) the density mean is shown boxed in blue (e.g., 39.2 for 0%), and for each patch nested within it the patch mean is shown boxed in green, with the overall mean mu = 20.26 marked on the right by a red bar spanning all values.
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
The full sea urchin data table repeated (density means per grazing level, patch means nested within each, overall mean y-bar = 20.26), shown here alongside the nested ANOVA table structure it feeds into.
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.
A First Look at the Data
