Homework 8 — Factorial ANOVA

Factorial ANOVA

Two-factor / factorial ANOVA.

Author

Bill Perry

📚 Where these data come from

Original study: Sargent, L.W. & Lodge, D.M. (2014). Evolution of invasive traits in nonindigenous species: increased survival and faster growth in invasive populations of rusty crayfish (Orconectes rusticus). Evolutionary Applications 7: 949–961.

A common-garden experiment: young-of-year crayfish from native (Ohio) and invasive (Wisconsin) populations, grown in enclosures in three northern Wisconsin lakes in summer 2011. Not a textbook dataset — there is no chapter to look up, but the paper is worth reading.

You will see these data again in Homework 9 (Linear Mixed Models), where lake stops being a fixed factor and becomes a random effect.

library(skimr)
library(patchwork)
library(janitor)


Attaching package: 'janitor'

The following objects are masked from 'package:stats':

    chisq.test, fisher.test

library(readxl)
library(car)

Loading required package: carData

library(broom)
library(emmeans)

Welcome to emmeans.
Caution: You lose important information if you filter this package's results.
See '? untidy'

library(effectsize)
library(tidyverse)

── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.2.1     ✔ readr     2.2.0
✔ forcats   1.0.1     ✔ stringr   1.6.0
✔ ggplot2   4.0.3     ✔ tibble    3.3.1
✔ lubridate 1.9.5     ✔ tidyr     1.3.2
✔ purrr     1.2.2

── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
✖ dplyr::recode() masks car::recode()
✖ purrr::some()   masks car::some()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

theme_set(theme_light())

Assignment Overview

This homework assignment analyzes crayfish growth data from Sargent and Lodge (2014) to examine differences in growth rates between native and invasive populations of rusty crayfish (Orconectes rusticus) across different lake environments using two-way ANOVA.

Learning Objectives

By completing this assignment, you will be able to:

Understand two-way ANOVA concepts and applications
Perform exploratory data analysis for factorial designs
Conduct two-way ANOVA analysis
Test statistical assumptions for ANOVA
Interpret main effects and interactions
Create publication-quality figures
Write scientific methods and results sections

Data Description

The dataset contains growth measurements from a common garden experiment where young-of-year (YOY) rusty crayfish from native (Ohio) and invasive (Wisconsin) populations were grown in enclosures in three northern Wisconsin lakes during summer 2011.

Key variables:

- range: Population origin (Native vs Invasive)
- lake: Lake location (Big, High, Papoose)
- growth_per_day: Daily growth rate (mm/day) - response variable
- initial_length: Starting length (mm)
- final_length: Ending length (mm)
- days: Duration of experiment

Part 1: Data Loading and Preparation

1.1 Load and Clean the Data

Part 2: Statistical Analysis Setup

2.1 Analysis Type and Model

Type of Analysis: Two-way factorial ANOVA

Model Equation: Growth Rate = μ + Range + Lake + (Range × Lake) + ε

Where:

- Growth Rate = daily growth rate (mm/day)
- Range = population origin (Native vs Invasive)
- Lake = lake environment (Big vs High vs Papoose)
- Range × Lake = interaction between range and lake
- ε = random error

Hypotheses:

Main Effect
- - Range:
  - - H₀: μ_Native = μ_Invasive (no difference in growth between ranges)
  - - H₁: μ_Native ≠ μ_Invasive (difference exists between ranges)
- - Lake:
  - - H₀: μ_Big = μ_High = μ_Papoose (no difference among lakes)
  - - H₁: At least one lake mean differs from others
Interaction Effect:
- - H₀: No interaction between Range and Lake
- - H₁: Interaction exists between Range and Lake

Variables:

- Response: growth_per_day (continuous, mm/day)
- Factor 1: range (categorical, 2 levels: Native, Invasive)
- Factor 2: lake (categorical, 3 levels: Big, High, Papoose)

Part 3: Exploratory Data Analysis

3.1 Summary Statistics

# cray_df %>% group_by(range) %>% 
# skim()

3.2 Exploratory Visualizations

Part 4: Two-Way ANOVA Analysis

4.1 Fit the ANOVA Model

Car ANOVA for unbalanced

# Anova(????, type = 3)

4.2 Effect Sizes

The effectsize package does this in one line each — no sums of squares by hand, and it reports a confidence interval with every estimate. It is already loaded in the setup chunk at the top.

▶ Run this on your fitted model:

# eta_squared(crayfish_model, partial = FALSE)
#
# omega_squared(crayfish_model, partial = FALSE)

Each gives you one row per term — range, lake and range:lake — with the estimate and its 95% CI.

To get the small / medium / large label without looking it up in a table:

# eta_squared(crayfish_model, partial = FALSE) %>%
#   interpret_eta_squared()

What the two numbers mean

Eta-squared (η²) is the proportion of total variance explained by that term: η² = SS_term / SS_total. If η² = 0.10 for range, then 10% of the variance in growth rate is explained by whether the crayfish came from a native or invasive population.
Omega-squared (ω²) is the same idea corrected for bias. η² always overestimates a little, especially with small samples; ω² is the more honest estimate. They will be close.

Use partial = FALSE so the terms are fractions of the same total and can be compared to each other. (partial = TRUE, the default, divides each term by itself plus the residual, which is useful but does not add up to anything.)

Interpretation guidelines

Effect Size     η² / ω²       Interpretation
Small           0.01          1% of variance explained
Medium          0.06          6% of variance explained
Large           0.14          14% of variance explained

✏️ Your turn: Fill in from your own output, then answer the question underneath.

             eta2      omega2    label
range      ______     ______    ______
lake       ______     ______    ______
range:lake ______     ______    ______

A term can be highly significant and still explain very little. Which of your three terms has the smallest p-value, and which has the largest effect size? Are they the same term — and if not, what does that tell you about what p-values measure?

____________________________________________________________

____________________________________________________________

4.3 Post-hoc Tests

# Estimated marginal means for main effects

# Pairwise comparisons with Sidak correction

# Estimated marginal means for interaction effect

# Compact letter display for interaction effect

# Custom interaction plot using emmeans results
# emmeans_interaction_df <- as.data.frame(emmeans_interaction)
# 
# emmeans_interaction_plot <- emmeans_interaction_df %>%
#   ggplot(aes(x = lake, y = emmean, color = range, group = range)) +
#   geom_point(size = 3, position = position_dodge(width=0.2)) +
#   geom_line(size = 1, position = position_dodge(width=0.2)) +
#   geom_errorbar(aes(ymin = lower.CL, ymax = upper.CL), 
#                 width = 0.1, size = 1,
#                 , position = position_dodge(width=0.2)) +
#   labs(x = "Lake", 
#        y = "Estimated Marginal Mean Growth Rate (mm/day)",
#        color = "Range",
#        title = "Estimated Marginal Means with 95% Confidence Intervals") +
#   scale_color_manual(values = c("Native" = "coral", "Invasive" = "darkblue")) +
#   theme_classic()
# 
# emmeans_interaction_plot

Part 5: Assumption Testing

5.1 Check ANOVA Assumptions

# # Create diagnostic plots
# par(mfrow = c(2, 2))
# plot(crayfish_anova_model)
# par(mfrow = c(1, 1))

5.2 Formal Assumption Tests

Part 6: Publication Figure

6.1 Create Publication-Quality Figure

Submission Guidelines

What to turn in -

An R script (13_factorial_anova_homework.R) with all of your code, plus the data file if you modified the original. All of the code should run top to bottom with what you turn in. (2 points)
A PDF of your final figure from Part 6, saved with ggsave(). (2 points)
Comments in the script that say what each block of code is doing and what you concluded from the output - credit will be given even if the code does not work, as long as you detail what you were trying to do. As we move further into statistics you will be expected to interpret the results, not just produce them. (2 points)

No quarto file and no rendered html are required for this assignment - a commented .R script is enough.

Points

summary stats - 2 point
assumptions and hypotheses - 3 points
exploratory graphs - 2 point
interpretation - 4 points
Final figure - 1 point
Results - 2 points

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