Background
Terminology
Correlation is not necessarily causation.
Correlation tests yield two
measurements:
r value (measure of correlation)
r ≈ 1
(strong, positive correlation)
r ≈ 0 (no correlation)
r ≈ −1 (strong, negative
correlation)
p value (measure of statistical
significance)
→ Get a feeling for it here http://guessthecorrelation.com/
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Analyses Contingency Coefficient
Purpose And Assumptions
Contingency Coefficent
ContCoef() in the DescTools package
Purpose:
To test whether variables are associated.
Assumptions: Variables must be categorical.
This test does not yield a significance assessment and only makes a statement
of whether variables are associated (|c| ≈ 1) or not (|c| ≈ 0).
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Analyses Contingency Coefficient
Minimal Working Example
Let’s see if the ContCoef() function will identify the association between two
categorical variables whose records are identical.
Samples <- c("Yes", "No")
set.seed(42)
counts <- sample(Samples, size = 1000, replace = TRUE)
table(counts, counts)
## counts
## counts No Yes
## No 501 0
## Yes 0 499
ContCoef(table(counts, counts))
## [1] 0.71
The association has been identified.
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Analyses Kendall’s Tau (Rank Correlation)
Purpose And Assumptions
Kendall’s Tau
cor.test() in base R using the method = "kendall" specification
Purpose:
To test whether the ranks of values of two variables are
correlated.
H
0
Ranks of variable values are not correlated.
Assumptions:
Variable value ranks are recorded as ‘numeric‘.
Variable values are ordinal, interval or ratio scaled.
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Analyses Kendall’s Tau (Rank Correlation)
Minimal Working Example
Let’s have a look at what happens when we supply the cor.test(...,
method = "kendall") with two correlated data sets:
cor.test(x = 1:1000, y = 1:1000, method = "kendall")
##
## Kendall's rank correlation tau
##
## data: 1:1000 and 1:1000
## z = 47, p-value <2e-16
## alternative hypothesis: true tau is not equal to 0
## sample estimates:
## tau
## 1
As expected, a strong correlation can be found.
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Analyses Spearman’s Rank Correlation
Purpose And Assumptions
Spearman Correlation
cor.test() in base R using the method = "spearman" specification
Purpose:
To test whether the values of two variables are correlated in a
non-parametric way.
H
0
Values of variables are not correlated.
Assumptions:
Variable values are recorded as ‘numeric‘.
Variable values are ordinal, interval or ratio scaled.
Pairs of variable values are monotonically related.
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Analyses Spearman’s Rank Correlation
Minimal Working Example
Let’s have a look at what happens when we supply the cor.test(...,
method = "spearman") with two non-correlated data sets:
set.seed(42)
cor.test(x = c(1:1000), y = sample(c(1:1000), size = 1000, replace = FALSE),
method = "spearman")
##
## Spearman's rank correlation rho
##
## data: c(1:1000) and sample(c(1:1000), size = 1000, replace = FALSE)
## S = 2e+08, p-value = 0.7
## alternative hypothesis: true rho is not equal to 0
## sample estimates:
## rho
## 0.01
As expected, no correlation can be found.
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Analyses Pearson Correlation
Purpose And Assumptions
Pearson Correlation
cor.test() in base R (method = "pearson" is the default)
Purpose:
To test whether the values of two variables are correlated in a
parametric way.
H
0
Values of variables are not correlated.
Assumptions:
Values of each variable follow a normal distribution.
Variable values are recorded as ‘numeric‘.
Variable values are ordinal, interval or ratio scaled.
Pairs of variable values are monotonically related.
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Analyses Pearson Correlation
Minimal Working Example
Let’s have a look at what happens when we supply the cor.test() with two
identical (hence correlated) data sets:
set.seed(42)
Data <- rnorm(n = 1000, mean = 500, sd = 50)
cor.test(x = Data, y = Data)
##
## Pearson's product-moment correlation
##
## data: Data and Data
## t = Inf, df = 998, p-value <2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 1 1
## sample estimates:
## cor
## 1
As expected, a strong correlation can be found.
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Our Data Choice Of Variables
Variables We Can Use
Which variables in our Passer domesticus data set are usable?
In general, correlation analyses can make use of all kinds of data as long as
they are recorded as numeric and we are able to convert categorical values
into numerical records if need be.
What about Pearson correlation and normal distributed data?
We need to check whether the assumption of normal distributed data records
holds true for each variable before applying Pearson correlation. If that is not
the case, we may wish to use Spearman correlation.
When dealing with climate types, remember to reduce confounding factors by
only using data belonging to the stations SI, UK, RE, and AU.
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Our Data Methods
What If There Are No Ranks?
We order our data!
set.seed(42)
Data <- sort(round(rnorm(n=10, mean=50, sd=2), 0))
Data # the data does not need to be sorted for ranking!
## [1] 49 50 50 50 51 51 51 53 53 54
The ties.method argument in the ranks() function controls what to do
when multiple values are the same.
rank(Data, ties.method = "average")
## [1] 1.0 3.0 3.0 3.0 6.0 6.0 6.0 8.5 8.5 10.0
rank(Data, ties.method = "min")
## [1] 1 2 2 2 5 5 5 8 8 10
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Our Data Research Questions
Research Questions And Hypotheses
So which of our major research questions (seminar 6) can we answer?
Contingency Coefficient
Predation: Are colour/nesting site and
predators associated?
Sexual Dimorphism: Are climate records
and sex ratios associated?
Kendall’s Tau
Climate Warming/Extremes: Do heavier
sparrows have heavier/less eggs? You
need to rank female sparrow weight and
egg weight for this.
Spearman
Climate Warming/Extremes: Do sparrow
weight/height and wing chord correlate with
latitude?
Climate Warming/Extremes: Do egg
weight/number of eggs correlate with
latitude?
Pearson
Fitness Constraints: Dies psarrow weight
and height correlate? Can you even run this
analysis?
Use the 1 - Sparrow_Data_READY.rds data set for these analyses.
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