Two-Way ANOVA Application

STAT 218 - Week 9, Lecture 2 Lab 9

March 12th, 2024

Two-Way ANOVA in R

Let’s Load Data Set

library(palmerpenguins)
library(tidyverse)
data(penguins)
glimpse(penguins)
Rows: 344
Columns: 8
$ species           <fct> Adelie, Adelie, Adelie, Adelie, Adelie, Adelie, Adel…
$ island            <fct> Torgersen, Torgersen, Torgersen, Torgersen, Torgerse…
$ bill_length_mm    <dbl> 39.1, 39.5, 40.3, NA, 36.7, 39.3, 38.9, 39.2, 34.1, …
$ bill_depth_mm     <dbl> 18.7, 17.4, 18.0, NA, 19.3, 20.6, 17.8, 19.6, 18.1, …
$ flipper_length_mm <int> 181, 186, 195, NA, 193, 190, 181, 195, 193, 190, 186…
$ body_mass_g       <int> 3750, 3800, 3250, NA, 3450, 3650, 3625, 4675, 3475, …
$ sex               <fct> male, female, female, NA, female, male, female, male…
$ year              <int> 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007…

A Note for Assumption Checking

na.omit(penguins) |> 
ggplot(aes(x = bill_length_mm, fill = factor(sex))) +
geom_histogram(binwidth = 1) +
facet_grid(sex ~ species, scales = "free") +
theme_minimal()

A Note for Assumption Checking

na.omit(penguins) |> 
  group_by(sex:species) |> 
summarize(mean_bill_length = mean(bill_length_mm), sd_bill_length = sd(bill_length_mm))
# A tibble: 6 × 3
  `sex:species`    mean_bill_length sd_bill_length
  <fct>                       <dbl>          <dbl>
1 female:Adelie                37.3           2.03
2 female:Chinstrap             46.6           3.11
3 female:Gentoo                45.6           2.05
4 male:Adelie                  40.4           2.28
5 male:Chinstrap               51.1           1.56
6 male:Gentoo                  49.5           2.72

A Note for Assumption Checking

na.omit(penguins) |> 
group_by(sex:species) |> 
summarize(sd_bill_length = sd(bill_length_mm)) |> 
summarize(ratio_max_min_sd = max(sd_bill_length) / min(sd_bill_length))
# A tibble: 1 × 1
  ratio_max_min_sd
             <dbl>
1             1.99

Two-Way ANOVA Function

two.way <- aov(bill_length_mm ~ sex + species + sex*species, data = penguins)
summary(two.way)
             Df Sum Sq Mean Sq F value Pr(>F)    
sex           1   1175    1175 219.228 <2e-16 ***
species       2   6976    3488 650.479 <2e-16 ***
sex:species   2     24      12   2.284  0.103    
Residuals   327   1753       5                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
11 observations deleted due to missingness

Main Effect 1

pairwise.t.test(penguins$bill_length_mm, penguins$species, p.adj = "bonf")

    Pairwise comparisons using t tests with pooled SD 

data:  penguins$bill_length_mm and penguins$species 

          Adelie Chinstrap
Chinstrap <2e-16 -        
Gentoo    <2e-16 0.0095   

P value adjustment method: bonferroni

Main Effect 2

pairwise.t.test(penguins$bill_length_mm, penguins$sex, p.adj = "bonf")

    Pairwise comparisons using t tests with pooled SD 

data:  penguins$bill_length_mm and penguins$sex 

     female 
male 1.1e-10

P value adjustment method: bonferroni

Interaction Effect

TukeyHSD(two.way, which = "sex:species")
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = bill_length_mm ~ sex + species + sex * species, data = penguins)

$`sex:species`
                                     diff       lwr        upr     p adj
male:Adelie-female:Adelie        3.132877  2.034137  4.2316165 0.0000000
female:Chinstrap-female:Adelie   9.315995  7.937732 10.6942588 0.0000000
male:Chinstrap-female:Adelie    13.836583 12.458320 15.2148470 0.0000000
female:Gentoo-female:Adelie      8.306259  7.138639  9.4738788 0.0000000
male:Gentoo-female:Adelie       12.216236 11.064727 13.3677453 0.0000000
female:Chinstrap-male:Adelie     6.183118  4.804855  7.5613821 0.0000000
male:Chinstrap-male:Adelie      10.703707  9.325443 12.0819703 0.0000000
female:Gentoo-male:Adelie        5.173382  4.005762  6.3410021 0.0000000
male:Gentoo-male:Adelie          9.083360  7.931851 10.2348686 0.0000000
male:Chinstrap-female:Chinstrap  4.520588  2.910622  6.1305547 0.0000000
female:Gentoo-female:Chinstrap  -1.009736 -2.443514  0.4240412 0.3338130
male:Gentoo-female:Chinstrap     2.900241  1.479553  4.3209291 0.0000002
female:Gentoo-male:Chinstrap    -5.530325 -6.964102 -4.0965471 0.0000000
male:Gentoo-male:Chinstrap      -1.620347 -3.041035 -0.1996591 0.0149963
male:Gentoo-female:Gentoo        3.909977  2.692570  5.1273846 0.0000000

An Important Note

  • When interactions are present, the main effects of factors don’t have their usual interpretations.
    • Because of this, we usually test for the presence of interactions first.
      • If interactions are present, then we often stop the analysis at this stage and examine interaction effects.
      • If no evidence for an interaction effect is found (i.e., if we do not reject \(H_0\)), then we proceed to testing the main effects of the individual factors.
  • So in this example, we ignored Main Effect 1 & Main Effect 2 and examine the Interaction Effect!