UNIVERSITY

09-06-2026 >eR-BioStat

Data Science Training

** Analysis of Variance**

Edna Chumoh based on ** chapter 19** in the Book of R* (https://www.openintro.org/book/biostat/)

## load/install libraries
.libPaths(c("./Rpackages",.libPaths()))
library(knitr)
library(tidyverse)
library(deSolve)
library(minpack.lm)
library(ggpubr)
library(readxl)
library(gamlss)
library(data.table)
library(grid)
library(png)
library(nlme)
library(gridExtra)
library(mvtnorm)
library(e1071)
library(lattice)
library(ggplot2)
library(dslabs)
library(NHANES)
library(plyr)
library(dplyr)
library(nasaweather)
library(ggplot2)
library(gganimate)
library(av)
library(gifski)
library(foreach)
library("DAAG")
library(DT)
library(TeachingDemos)
library(gridExtra)

One-Way ANOVA

One-way ANOVA is used when we want to compare the mean of a continuous outcome across more than two groups.

Hypotheses

For a one-way ANOVA with \(k\) groups:

\(H_0\): all group means are equal
\(H_A\): at least one group mean differs

General workflow

Visualize the outcome by group.
Check assumptions:
- independence
- approximate normality within groups / residuals
- roughly equal variances
Fit the ANOVA model.
Interpret the F-test and p-value.

Example 1: Iris data

## 'data.frame':    150 obs. of  5 variables:
##  $ Sepal.Length: num  5.1 4.9 4.7 4.6 5 5.4 4.6 5 4.4 4.9 ...
##  $ Sepal.Width : num  3.5 3 3.2 3.1 3.6 3.9 3.4 3.4 2.9 3.1 ...
##  $ Petal.Length: num  1.4 1.4 1.3 1.5 1.4 1.7 1.4 1.5 1.4 1.5 ...
##  $ Petal.Width : num  0.2 0.2 0.2 0.2 0.2 0.4 0.3 0.2 0.2 0.1 ...
##  $ Species     : Factor w/ 3 levels "setosa","versicolor",..: 1 1 1 1 1 1 1 1 1 1 ...

## 
##     setosa versicolor  virginica 
##         50         50         50

Side-by-side boxplots

Group means

##      Species Sepal.Length
## 1     setosa        5.006
## 2 versicolor        5.936
## 3  virginica        6.588

Fit one-way ANOVA

##              Df Sum Sq Mean Sq F value Pr(>F)    
## Species       2  63.21  31.606   119.3 <2e-16 ***
## Residuals   147  38.96   0.265                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Diagnostic checks

Variability by group

##     setosa versicolor  virginica 
##  0.3524897  0.5161711  0.6358796

Two-Way ANOVA

Two-way ANOVA is used when a continuous outcome is explained by two categorical predictors.
We usually examine:

the main effect of factor 1
the main effect of factor 2
the interaction effect between the two factors

Hypotheses

For factors \(G_1\) and \(G_2\):

\(H_0\): \(G_1\) has no main effect
\(H_0\): \(G_2\) has no main effect
\(H_0\): there is no interaction between \(G_1\) and \(G_2\)

Example: `warpbreaks`

## 'data.frame':    54 obs. of  3 variables:
##  $ breaks : num  26 30 54 25 70 52 51 26 67 18 ...
##  $ wool   : Factor w/ 2 levels "A","B": 1 1 1 1 1 1 1 1 1 1 ...
##  $ tension: Factor w/ 3 levels "L","M","H": 1 1 1 1 1 1 1 1 1 2 ...

##    
##     L M H
##   A 9 9 9
##   B 9 9 9

Means by wool and tension

##   wool tension        x
## 1    A       L 44.55556
## 2    B       L 28.22222
## 3    A       M 24.00000
## 4    B       M 28.77778
## 5    A       H 24.55556
## 6    B       H 18.77778

One-way ANOVA on each factor separately

##             Df Sum Sq Mean Sq F value Pr(>F)
## wool         1    451   450.7   2.668  0.108
## Residuals   52   8782   168.9

##             Df Sum Sq Mean Sq F value  Pr(>F)   
## tension      2   2034  1017.1   7.206 0.00175 **
## Residuals   51   7199   141.1                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Full two-way ANOVA with interaction

##              Df Sum Sq Mean Sq F value   Pr(>F)    
## wool          1    451   450.7   3.765 0.058213 .  
## tension       2   2034  1017.1   8.498 0.000693 ***
## wool:tension  2   1003   501.4   4.189 0.021044 *  
## Residuals    48   5745   119.7                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Interaction plot

Kruskal-Wallis Test

The Kruskal-Wallis test is a nonparametric alternative to one-way ANOVA.
It is useful when normality is doubtful and you want to compare several groups.

Hypotheses

\(H_0\): group distributions / medians are equal
\(H_A\): at least one group differs

Example: `survey` data from MASS

## 'data.frame':    237 obs. of  12 variables:
##  $ Sex   : Factor w/ 2 levels "Female","Male": 1 2 2 2 2 1 2 1 2 2 ...
##  $ Wr.Hnd: num  18.5 19.5 18 18.8 20 18 17.7 17 20 18.5 ...
##  $ NW.Hnd: num  18 20.5 13.3 18.9 20 17.7 17.7 17.3 19.5 18.5 ...
##  $ W.Hnd : Factor w/ 2 levels "Left","Right": 2 1 2 2 2 2 2 2 2 2 ...
##  $ Fold  : Factor w/ 3 levels "L on R","Neither",..: 3 3 1 3 2 1 1 3 3 3 ...
##  $ Pulse : int  92 104 87 NA 35 64 83 74 72 90 ...
##  $ Clap  : Factor w/ 3 levels "Left","Neither",..: 1 1 2 2 3 3 3 3 3 3 ...
##  $ Exer  : Factor w/ 3 levels "Freq","None",..: 3 2 2 2 3 3 1 1 3 3 ...
##  $ Smoke : Factor w/ 4 levels "Heavy","Never",..: 2 4 3 2 2 2 2 2 2 2 ...
##  $ Height: num  173 178 NA 160 165 ...
##  $ M.I   : Factor w/ 2 levels "Imperial","Metric": 2 1 NA 2 2 1 1 2 2 2 ...
##  $ Age   : num  18.2 17.6 16.9 20.3 23.7 ...

## 
## Heavy Never Occas Regul  <NA> 
##    11   189    19    17     1

Kruskal-Wallis test

## 
##  Kruskal-Wallis rank sum test
## 
## data:  Age by Smoke
## Kruskal-Wallis chi-squared = 3.9262, df = 3, p-value = 0.2695

Exercise 19.1

Archaeological site data

(a) Boxplots with sample means

(b) Diagnostic checks & (c) One-way ANOVA

##             Df Sum Sq Mean Sq F value Pr(>F)
## site         3    780   260.1   0.473  0.703
## Residuals   42  23082   549.6

Exercise 19.2

Quakes data

## 'data.frame':    1000 obs. of  5 variables:
##  $ lat     : num  -20.4 -20.6 -26 -18 -20.4 ...
##  $ long    : num  182 181 184 182 182 ...
##  $ depth   : int  562 650 42 626 649 195 82 194 211 622 ...
##  $ mag     : num  4.8 4.2 5.4 4.1 4 4 4.8 4.4 4.7 4.3 ...
##  $ stations: int  41 15 43 19 11 12 43 15 35 19 ...

##       lat              long           depth            mag      
##  Min.   :-38.59   Min.   :165.7   Min.   : 40.0   Min.   :4.00  
##  1st Qu.:-23.47   1st Qu.:179.6   1st Qu.: 99.0   1st Qu.:4.30  
##  Median :-20.30   Median :181.4   Median :247.0   Median :4.60  
##  Mean   :-20.64   Mean   :179.5   Mean   :311.4   Mean   :4.62  
##  3rd Qu.:-17.64   3rd Qu.:183.2   3rd Qu.:543.0   3rd Qu.:4.90  
##  Max.   :-10.72   Max.   :188.1   Max.   :680.0   Max.   :6.40  
##     stations     
##  Min.   : 10.00  
##  1st Qu.: 18.00  
##  Median : 27.00  
##  Mean   : 33.42  
##  3rd Qu.: 42.00  
##  Max.   :132.00

(a) Create depth categories

## 
##   0-200 200-400 400-680 
##     418     185     397

(b) Decide between tests & (c) Run the chosen test

##              Df Sum Sq Mean Sq F value Pr(>F)  
## depth_cat     2   4364  2182.1   4.582 0.0104 *
## Residuals   997 474783   476.2                 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

## 
##  Kruskal-Wallis rank sum test
## 
## data:  stations by depth_cat
## Kruskal-Wallis chi-squared = 9.0943, df = 2, p-value = 0.0106

Cars93 data

Full two-way ANOVA

##                         Df Sum Sq Mean Sq F value   Pr(>F)    
## AirBags                  2   3752    1876  18.048 2.78e-07 ***
## Man.trans.avail          1   6388    6388  61.447 1.05e-11 ***
## AirBags:Man.trans.avail  2    433     217   2.084    0.131    
## Residuals               87   9044     104                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

One-Way ANOVA

Hypotheses

General workflow

Example 1: Iris data

Side-by-side boxplots

Group means

Fit one-way ANOVA

Diagnostic checks

Variability by group

Two-Way ANOVA

Hypotheses

Example: warpbreaks

Means by wool and tension

One-way ANOVA on each factor separately

Full two-way ANOVA with interaction

Interaction plot

Kruskal-Wallis Test

Hypotheses

Example: survey data from MASS

Kruskal-Wallis test

Exercise 19.1

Archaeological site data

(a) Boxplots with sample means

(b) Diagnostic checks & (c) One-way ANOVA

Exercise 19.2

Quakes data

(a) Create depth categories

(b) Decide between tests & (c) Run the chosen test

Cars93 data

Full two-way ANOVA

Example: `warpbreaks`

Example: `survey` data from MASS