ANOVA: one way repeated measures
A fellow grad student asked if I had ever done a repeated measures ANOVA. I went back to my stats labs and found some code in a .R script. Here I re-use and update that code within this markdown report to provide some commentary.
Scenario
The weights of moose calves from different regions are are compared over three time periods. 1. Using a repeated measures ANOVA we need to determine if the weights at each time interval are significantly different from one another.
Is there a change in the weight of moose calves over the three time periods?
Null hypothesis
The weights of the calves remain constant over time.
\[H_0 : \mu_1 = \mu_2 = \mu_3\]
Analysis
This analysis examined the change on weight of the moose calves over three time periods. However, this does not consider the influence of the eco regions. The independent variable is time with 3 time periods and the dependent variables are the weight.
Load libraries
library(lsr)
library(car) #for levene test
library(onewaytests) #for 'Brown-Forsyth' and 'Welch Test'
library(reshape2) #to rearrange the date (i.e. melt)
library(tidyr) #to rearrange the data (i.e. gather)
library(moments) #for Skewness and Kurtosis
library(nortest) #For testing normality
library(reshape2)
#Libraries for Graphs
library(qqplotr) # library extention to ggplot2 gives Q-Q ability
library(ggplot2) # Utilizes ggplot 2 graphics library
require(gridExtra) # used to plot 2 graphs side by side (using ggplot)
library(tidyverse)Import data
The data is imported and rearranged for analysis
# myData <- read.csv("./rawData/AnovaDatabase_Labels.csv", header = TRUE)
myData <- read.csv("e:/workspace/NRES776/Chisholm_LabProject2/Lab06_ANOVA/rawData/AnovaDatabase_Labels.csv")
weights <- myData %>% select(Weight1, Weight2, Weight3) %>%
pivot_longer(Weight1: Weight3, names_to = "Period", values_to = "Weights") %>%
mutate(Period = as.factor(Period)) ## need period as a factor for lm() analysis # activity2 <- myData[,c("Weight1", "Weight2", "Weight3")] ## just examine the weights
knitr::kable(head(weights, 10),
caption = "First few weights in the data") %>%
kableExtra::kable_styling()| Period | Weights |
|---|---|
| Weight1 | 50 |
| Weight2 | 48 |
| Weight3 | 45 |
| Weight1 | 47 |
| Weight2 | 45 |
| Weight3 | 44 |
| Weight1 | 44 |
| Weight2 | 39 |
| Weight3 | 36 |
| Weight1 | 47 |
Complete ANOVA
a <- lm(weights$Weights ~ weights$Period)
a.aov <- anova(a)
print(a.aov)## Analysis of Variance Table
##
## Response: weights$Weights
## Df Sum Sq Mean Sq F value Pr(>F)
## weights$Period 2 365.87 182.933 6.1424 0.003197 **
## Residuals 87 2591.03 29.782
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Complete Post-hoc test if the ANOVA is significant
if (a.aov[1,"Pr(>F)"] < 0.05) {
print("Post Hoc Analysis:")
pairwise.t.test(weights$Weights, weights$Period, p.adj = "bonferroni")
} else {
print("H0 affirmed, no post hoc required")
}## [1] "Post Hoc Analysis:"##
## Pairwise comparisons using t tests with pooled SD
##
## data: weights$Weights and weights$Period
##
## Weight1 Weight2
## Weight2 0.1852 -
## Weight3 0.0022 0.3340
##
## P value adjustment method: bonferroniEta-squared – power test
etaSquared(a, type = 2, anova = FALSE)## eta.sq eta.sq.part
## weights$Period 0.1237332 0.1237332Results
A repeated measure ANOVA was showed that moose calf weights changed significantly over the three time periods (\(F−value = 6.1424 [df: 2; 87 ]\), \(p−value : 0.0032\)). Effect size calculated as medium to large (\(\eta^2 = 0.1237332\)). A Post hoc analysis using the Bonferroni test2showed that the only significant difference in weights was between period 1 and period 3 (\(p−value = 0.0022\)).
