Variables What Are Variables?
Variables And Their Subsets
What is a variable?
A variable presents more or less valuable information about a potential
multitude of characteristics of a study system.
What’s the fuss?
Sampling effort is limited. We only ever sample a subset of globally present
values of any given variable.
So?
Every variable is subject to a certain distribution of its values and each
measurement is expected to follow the global distribution to a certain degree.
Variables are, more or less, raw data.
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Variables Types of Variables
Types of Variables
Variables can be classed into a multitude of types. The most common
classification system knows:
Categorical Variables
also known as Qualitative
Variables
Scales can be either:
Nominal
Ordinal
Continuous Variables
also known as Quantitative
Variables
Scales can be either:
Discrete
Continuous
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Variables Types of Variables
Categorical Variables
Categorical variables are those variables which
establish and fall into
distinct groups and classes.
Categorical variables:
can take on a finite number of values
assign each unit of the population to one of a finite number of groups
can sometimes be ordered
In R, categorical variables usually come up as object type factor or
character.
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Variables Types of Variables
Categorical Variables (Examples)
Examples of categorical variables:
Biome Classifications (e.g. "Boreal Forest", "Tundra", etc.)
Sex (e.g. "Male", "Female")
Hierarchy Position (e.g. "α-Individual", "β-Individual", etc.)
Soil Type (e.g. "Sandy", "Mud", "Permafrost", etc.)
Leaf Type (e.g. "Compound", "Single Blade", etc.)
Sexual Reproductive Stage (e.g. "Juvenile", "Mature", etc.)
Species Membership
Family Group Membership
...
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Variables Types of Variables
Continuous Variables
Continuous variables are those variables which establish a range of
possible data values.
Continuous variables:
can take on an infinite number of values
can take on a new value for each unit in the set-up
can always be ordered
In R, continuous variables usually come up as object type numeric.
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Variables Types of Variables
Converting Variable Types
Continuous variables can be converted into categorical variables via a method
called binning:
Given a variable range, one can establish however many “bins” as one wants.
For example:
Given a temperature range of
271K − 291K
, there may be 4 bins of equal
size:
Bin A: 271K ≤ X ≤ 276K
Bin B: 276K < X ≤ 281K
Bin C: 281K < X ≤ 286K
Bin D: 286K < X ≤ 291K
Whilst a continuous variable can be both continuous and categorical,
a categorical variable can only ever be categorical!
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Variables Variables And Scales
Variables On Scales
Another way of classifying variables are the scales they are represented on.
Different scales
of variables
require different statistical procedures
for analyses!
Variable scales include:
Nominal
Binary
Ordinal
Interval
Relation/Ratio
Some statistics books teach integer scales along the above mentioned scales.
Some people dispute this and claim these scales to be ratio scales.
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Variables Variables And Scales
Nominal And Binary
Nominal scales
of variables correspond to categorical variables which cannot
be put into a meaningful order.
Variables on nominal scales put units into distinct
categories
These variables may be numerical but offer no
mathematical interpretation
Examples:
Petal colour (red, green, blue, etc.)
Individual IDs
Binary scales are a special case of nominal scales taking only two possible
values: 0 and 1.
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Variables Variables And Scales
Ordinal
Ordinal scales of variables correspond to categorical variables which can be
put into meaningful order.
Variables on ordinal scales put units into distinct
categories
These variables may be numerical and
mathematical interpretation
Examples:
Size (small, medium, large, etc.)
Binned continuous variables
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Variables Variables And Scales
Interval/Discrete
Interval scales of variables correspond to a mix of continuous variables.
Variables on interval scales are measured on equal
intervals from a defined zero point/point of origin
The point of origin
does not imply an absence of
the measured characteristic
Examples:
Temperature [°C]
pH
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Variables Variables And Scales
Relation/Ratio
Relation/Ratio scales of variables correspond to continuous variables.
Variables on relation/ratio scales are measured on
equal intervals from a defined zero point/point of
origin
The point of origin
does imply an absence of the
measured characteristic
Examples:
Temperature [K]
Weight
Integer scales are a special case of ratio scales allowing only for integral
numbers.
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Variables Variables And Scales
Checklist For Your Variables
What variables am I using and:
Of what mode are they?
Is every record of the same variable based on the same unit of measure?
Should I convert some of them?
What scales apply and:
What do they imply?
Does my data fit these scales?
→ You should be able to answer these question before you begin data
collection!
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Distributions The Basics of Distributions
What Are Distributions?
A distribution of a statistical data
set (sample/population) shows
all the possible values/intervals
of the data in question and their
frequency.
→ Basically data patterns we are considering/looking for.
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Distributions The Basics of Distributions
Frequency Distributions
Frequency Distributions:
Theory
Simple representations of data
value frequencies
Can be established for every
variable
Practice in R
Visualisation via the ‘hist()‘
function
Frequency Distribution
rnorm(1e+05, 20, 2)
Frequency
10 15 20 25 30
0 5000 10000 15000
hist(rnorm(100000,20,2),
main = "Frequency Distribution")
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Distributions The Basics of Distributions
Probability Density Distributions I
Probability Density Distributions:
Theory
Representation of data value
probabilities
Can be established for
continuous variables
Practice in R
Visualisation via the ‘density()‘
function
15 20 25
0.00 0.05 0.10 0.15 0.20
Probability Density Distribution
N = 100000 Bandwidth = 0.1798
Density
plot(density(rnorm(100000,20,2)),
main = "Probability Density Distribution")
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Distributions The Basics of Distributions
Probability Density Distributions II
Probability Density Distributions hold the majority of importance in
statistics!
A few key points about these distributions:
Area under the curve (AUC) sums to 1
A probability for every given single value is 0
The AUC between two values on the X-axis equals the probability to
randomly sample a value between these two points
Find a masterful explanation of the single-value probability here.
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Distributions Normality
Univariate Standard Normal/Gaussian Distribution
One of the most important distributions in natural sciences.
Used to represent real-valued
random variables whose
distributions are not known
The
central limit theorem
applies
(draw a sufficient number of
samples and you end up with the
normal distribution)
These distributions are usually
known also as "bell curves"
(
Attention:
other distributions take
this shape too)
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Distributions Normality
Testing For Normality
Testing for normality of the data is crucial for certain statistical
procedures.
The Shapiro-Wilks Test In Theory
Base assumption: The data is
normally distributed
If p-value < chosen significance
level, the data is not normally
distributed
Very sensitive to sample size
The QQ Plot In Theory
Method for comparing two
probability distributions by plotting
their quantiles against each other
If the two distributions being
compared are similar, the plot will
show the line y = x.
Compare the data distribution to
the normal distribution
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Distributions Normality
The Shapiro-Wilks Test In R
Using the shapiro.test() function:
shapiro.test(rnorm(5000,
20, 2))
##
## Shapiro-Wilk normality test
##
## data: rnorm(5000, 20, 2)
## W = 1, p-value = 0.7
→ Clearly a normal distributed set of values
shapiro.test(seq(1, 500,
5))
##
## Shapiro-Wilk normality test
##
## data: seq(1, 500, 5)
## W = 1, p-value = 0.002
→ Clearly no normal distributed set of values
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Distributions Normality
The Q-Q Plot
Using the qqnorm() function:
qqnorm(rnorm(5000,20,2))
−4 −2 0 2 4
14 16 18 20 22 24 26
Normal Q−Q Plot
Theoretical Quantiles
Sample Quantiles
→ Clearly a normal distributed set of values
qqnorm(seq(1,500,5))
−2 −1 0 1 2
0 100 200 300 400 500
Normal Q−Q Plot
Theoretical Quantiles
Sample Quantiles
→ Clearly no normal distributed set of values
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Distributions What Distributions To Consider
An Overview of Distributions
There is a plethora of distributions which variables could fall onto:
Bernoulli (probabilities of value 1 and 0 are interdependent)
Binomial (number of successes in a series)
Poisson (probability of a given number of events occurring in a fixed
interval of time or space)
Beta (family of two-parameter distributions with one mode)
Kent (three-dimensional sphere distribution)
Univariate Standard Normal/Gaussian
Multivariate Normal
Log-Normal
. . . and yet the hat goes deeper
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Distributions What Distributions To Consider
Binomial Distribution
One of the more important distributions. It is applicable to:
Variables which can only take two
possible values (e.g. “states”)
All records of the variable have the
same probability p of being in one
of the two states
It is made up of three criteria:
p - the "success" probability
n - sample size (how often we
sample)
N - the "binomial total" (for how
many individuals we sample each
time)
Binomial Distribution
n = 10000, N = 50, p = 0.6
Frequency
15 20 25 30 35 40
0 500 1000 1500 2000
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Distributions What Distributions To Consider
Poisson Distribution
Another one of the more important distributions. It is applicable to:
Focal objects are placed randomly
in one or more dimensions
A random “counting window”
(usually one considering time) is
placed above the sampling
scheme
It is made up of two criteria:
λ - the mean (= expectation,
average count, intensity) as well as
the variance (i.e., variance =
mean)
n - sample size
Poisson Distribution
n = 10000, Lambda = 5
Frequency
0 2 4 6 8 10 12
0 50 100 150
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Distributions Important Measures Of Distributions
How to Measure Distributions
Not all distributions are created equally.
Distributions can be described via classic parameters of descriptive
statistics:
Arithmetic Mean
Mode
Median
Minimum, Maximum, Range
...
Variance
Standard Deviation
Quantile Range
Skewness
Kurtosis
...
→ Most of these are dealt with in the next seminar.
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Distributions Important Measures Of Distributions
Skewness I
Definition:
Describes the symmetry and relative tail length of distributions.
Positive skew: Right-hand tail is longer than the left-hand tail
Skew = 0: Symmetric distribution
Negative skew: Left-hand tail is longer than the right-hand tail
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Distributions Important Measures Of Distributions
Kurtosis I
Definition: Describes the evenness/"tailedness" of distributions.
Positive
kurtosis:
Short-tailed distribution aka. leptokurtic
Kurtosis = 0: Base representation of a given distribution aka. mesokurtic
Negative
kurtosis:
Long-tailed distribution aka. platykurtic
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Distributions Estimations
Point and Range Estimations
Point and Range estimations are parameters obtained from a sample data set
and meant to represent the population data set.
With a probability of 95.5%, the following is true
x − 2σ
x
≤ µ ≤ x + 2σ
x
With a probability of 68%, the following is true
x − σ
x
≤ µ ≤ x + σ
x
x Arithmetic mean of the sample
µ Arithmetic mean of the population
σ
x
Standard error of x
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Distributions Estimations
Confidence Intervals I
For multiple confidence intervals (CIs) on a level of
α
(i.e. 95%), a
proportion of
α
CIs for a given population will contain the arithmetic
mean of the population.
[x − t(α, df) ; x + t(α, df)]
x Arithmetic mean of the sample
σ
x
Standard error of x
α Confidence level (usually 95%)
df Degrees of freedom
t(α, df) t-value given α and df
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Distributions Estimations
Confidence Intervals II
The Basics of Confidence Intervals:
CIs get larger when:
Smaller sample sizes
Bigger spread of data values
Higher statistical certainty (α)
CIs get narrower when:
Bigger sample sizes
Smaller spread of data values
Lower statistical certainty (α)
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Exercise
R Environment Objects
R
environment objects (stored as
.RData
) are highly valuable objects to any
R
user because:
They let you save your entire working environment
You cannot alter them outside of R (aside from deleting them)
How to create them?
Use the function
save.image()
(you can specify the argument
file
for
a specific name of the file)
How to load them?
Use the function load(...) (“. . . ” specifies the exact path to the file on
your machine)
What to do?
Load the Primer.RData file into R
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Exercise
Variables
Answer the following questions (take a notes for each variable!) for the
variables contained within Primer.RData:
What variables am I using and:
Of what mode are they?
What scales apply and:
What do they imply?
Does my data fit these scales? Use the functions barplot() and table() when
applicable!
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Exercise
Distributions
Plot the distributions of the values for the following variables:
Length
Reproducing
IndividualsPassingBy
Depth
What distributions are these? Use QQPlots or the Shapiro Test to assess
normality. If you stumble upon a non-normal distribution, what else could it be?
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