I made up this example myself: Three geoscientists from China, USA and Europe were interested in exploring the mineralogical and textural characteristics and organic carbon composition of carbonate concretions. A concretion is a hard, compact mass of matter formed by the precipitation of mineral cement within the spaces between particles and is found in sedimentary rock or soil.
They conducted Rock pyrolysis analysis of the concretions and their host shale. Their primary goal was to compare the population mean hydrogen indices (HI measured in mg/g) for the three independent regions using an Analysis of Variance (ANOVA) procedure.
They make the usual assumptions of their samples being drawn from normal populations with equal population variances. The geoscientists reached out to a Statistics Professor to help them with the analyses using a software tool. The data is stored in an Excel file (Example1.xlsx)
H0: mu1 = mu2 = mu3
Ha: At least two means differ
The critical value (CV) from the F-table for α = 0.05 is 3.403.
# install.packages("readxl") library(readxl) data <- read_excel("Example1.xlsx") data data$Country <- factor(data$Country) boxplot(HI ~ Country, data = data, xlab="Country", ylab = "HI") fit <- aov(HI ~ Country, data = data) summary(fit)
At α = 0.05 since the test statistic falls in the rejection region we reject H0 and conclude at least two populations means differ. Post-hoc tests are warranted.
So I ran the Tukey's pairwise comparison using
TukeyHSD() function from the
stats package and the
glht() function from the
multcomp R package.
# Tukey's HSD TukeyHSD(fit) pairwise <- TukeyHSD(fit) plotdata <- as.data.frame(pairwise[]) plotdata$conditions <- row.names(plotdata) library(ggplot2) ggplot(data=plotdata, aes(x=conditions, y=diff)) + geom_errorbar(aes(ymin=lwr, ymax=upr, width=.2)) + geom_hline(yintercept=0, color="red", linetype="dashed") + geom_point(size=3, color="red") + theme_bw() + labs(y="Difference in mean levels", x="", title="95% family-wise confidence level") + coord_flip() # Multiple comparisons the multcomp package library(multcomp) tuk <- glht(fit, linfct = mcp(Country = "Tukey")) summary(tuk)
In both cases none of the pairwise test is significant. What is happening here? The fact that the main ANOVA test rejects the Ho shouldn't there be at least one significant test in the pairwise comparisons?
In other words, all group means are statistically indistinguishable from each other and can be considered to be equivalent. This result indicates that there is no evidence of significant differences in the population means of the three groups with respect to the variable being studied. Doesn't the absence of significant differences necessarily mean that the means are exactly equal in the population?