METRIC TESTS (MORE-THAN-TWO-SAMPLE
SITUATIONS)
Erik Kusch
erik.kusch@i-solution.de
Section for Ecoinformatics & Biodiversity
Center for Biodiversity and Dynamics in a Changing World (BIOCHANGE)
Aarhus University
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1 Background
2 Analyses
Kruskal-Wallis Test
Friedman Test
3 Our Data
Choice Of Variables
Methods
Research Questions
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Background
Introduction
Metric tests are used to compare parameters of metric/ordinal variable values
among groups/individuals.
Prominent metric tests for more-than-two-sample situations include:
Kruskal-Wallis Test
Friedman Test
t Test (dealt with in Seminar 12)
...
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Background
Terminology
Remember:
Got two samples?
Independent. → Mann-Whitney U Test
Dependent. → Wilcoxon Signed Rank Test
Got more than two samples?
Independent. → Kruskal-Wallis Test
Dependent. → Friedman Test
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Analyses Kruskal-Wallis Test
Purpose And Assumptions
Kruskal-Wallis Test
kruskal.test() in base R
Purpose:
To identify whether groups of variable values are different from
one another.
H
0
There is no difference in characteristics of the response
variable values in dependence of the classes of the predictor
variable.
Assumptions:
Predictor variable is categorical (not binary!)
Response variable is ordinal or metric
Variable values are independent (not paired)
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Analyses Kruskal-Wallis Test
Minimal Working Example
Let’s use the kruskal.test() function to test whether the medians of an
unnamed variable of three unconnected populations (a, b and c) with 10
individuals each are truly different:
set.seed(42)
a <- rnorm(n = 10, mean = 15, sd = 3)
b <- rnorm(n = 10, mean = 10, sd = 3)
c <- rnorm(n = 10, mean = 5, sd = 3)
groups <- as.factor(rep(c("a", "b", "c"), each = 10))
kruskal.test(x = c(a, b, c), g = groups)
##
## Kruskal-Wallis rank sum test
##
## data: c(a, b, c) and groups
## Kruskal-Wallis chi-squared = 19, df = 2, p-value =
## 7e-05
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Analyses Friedman Test
Purpose And Assumptions
Friedman Test
friedman.test() in base R
Purpose:
To identify whether groups of variable values are different from
one another.
H
0
There is no difference in characteristics of the response
variable values in dependence of the classes of the predictor
variable.
Assumptions:
Predictor variable is categorical (not binary!)
Response variable is ordinal or metric
Variable values are dependent (paired)
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Analyses Friedman Test
Minimal Working Example
Let’s use the friedman.test() function to test whether the medians of an
unnamed variable of three connected samples (a, b and c) with 10 individuals
each (i.e. one re-sampled population) are truly different:
set.seed(42)
a <- rnorm(n = 10, mean = 15, sd = 3)
b <- rnorm(n = 10, mean = 10, sd = 3)
c <- rnorm(n = 10, mean = 5, sd = 3)
TestData <- matrix(c(a, b, c), nrow = 10, byrow = FALSE,
dimnames = list(1:10, c("a", "b", "c")))
friedman.test(y = TestData)
##
## Friedman rank sum test
##
## data: TestData
## Friedman chi-squared = 15, df = 2, p-value = 7e-04
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Our Data Choice Of Variables
Variables We Can Use
Response variables (metric/ordinal)
Weight
Height
Wing Chord
Nesting Height
Number of Eggs
Egg Weight
Home Range
Predictor variables (categorical but
not binary)
Home Range (3 levels - Small,
Medium, Large)
Site Index (11 levels)
Predator Presence/Type (3 levels -
Avian vs. Non-Avian vs. None)
Climate (3 levels - Continental,
Semi-Coastal, Coastal)
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Our Data Methods
Writng A Function In R I
Establishing user-defined functions is at the heart of ‘R‘!
A function requires:
+ A name
+ Arguments
+ to be called
Fun <- function(argument) {
print(argument)
}
Fun(argument = "Test")
## [1] "Test"
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Our Data Methods
Writng A Function In R II
A function that adds 1 to whatever the input is:
AddFun <- function(input) {
output <- input + 1
# this gives the result back to the non-function
# environment
return(output)
}
AddFun(input = 1)
## [1] 2
AddFun(input = 2)
## [1] 3
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Our Data Research Questions
Research Questions And Hypotheses
So which of our major research questions (seminar 6) can we answer?
Kruskal-Wallis Test
Climate Warming/Extremes: Does
sparrow morphology depend on
climate?
Predation: Does nesting height
depend on predator characteristics?
Competition: Does home range
depend on climate?
Use the
1 - Sparrow_Data_READY.rds
data
set for these analyses.
Friedman Test (suppose a resettling
program)
Climate Warming/Extremes: Does
sparrow morphology change depend
on climate?
Predation: Does nesting height
depend on predator characteristics?
Competition: Does home range
depend on climate?
Use the 2a -
Sparrow_ResettledSIMA_READY.rds and
2b -
Sparrow_ResettledSIUK_READY.rds data
sets for these analyses.
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