Statistical Inference

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Statistical Inference
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Parametric perspective on inference
Data
Inference
Scientific Model (Hypothesis test) Often with
linear models
Probability Model (Normal typically)
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Likelihood perspective on inference
Data
Inference
Probability Model
Scientific Model (hypothesis)
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An example...
The Data xi measurements of DBH on 50 trees yi
measurements of crown radius on those
trees The Scientific Model yi a b xi e
(linear relationship, with 2 parameters (a, b)
and an error term (e) (the residuals)) The
Probability Model e is normally distributed,
with Ee and variance estimated from the
observed variance of the residuals...
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The triangle of statistical inference Model

• Models clarify our understanding of nature.
• Help us understand the importance (or
  unimportance) of individuals processes and
  mechanisms.
• Since they are not hypotheses, they can never be
  correct.
• We dont reject models we assess their
  validity.
• Establish whats true by establishing which
  model the data support.

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The triangle of statistical inference Probability
distributions

• Data are never clean.
• Most models are deterministic, they describe the
  average behavior of a system but not the noise or
  variability. To compare models with data, we need
  a statistical model which describes the
  variability.
• We must understand the the processes giving rise
  to variability to select the correct probability
  density function (error structure) that gives
  rise to the variability or noise.

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An example Can we predict crown radius using
tree diameter?
The Data
xi measurements of DBH
on 50 trees yi
measurements of crown radius on those trees The
Scientific Model
yi a b DBHi e The
Probability Model
e is normally distributed.
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Why do we care about probability?

• Foundation of theory of statistics.
• Description of uncertainty (error).
• Measurement error
• Process error
• Needed to understand likelihood theory which is
  required for
• Estimating model parameters.
• Model selection (What hypothesis do data
  support?).

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Error (noise, variability) is your friend!

• Classical statistics are built around the
  assumption that the variability is normally
  distributed.
• Butnormality is in fact rare in ecology.
• Non-normality is an opportunity to
• Represent variability in a more realistic way.
• Gain insights into the process of interest.

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The likelihood framework
Ask biological question
Collect data
Ecological Model Model signal
Probability Model Model noise
Model selection
Estimate parameters
Estimate support regions
Answer questions
Bolker, Notes
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Probability Concepts

• An experiment is an operation with uncertain
  outcome.
• A sample space is a set of all possible outcomes
  of an experiment.
• An event is a particular outcome of an
  experiment, a subset of the sample space.

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Random Variables

• A random variable is a function that assigns a
  numeric value to every outcome of an experiment
  (event) or sample. For instance

Tree Growth f (DBH, light, soil)
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Functions and probability density functions
Function formula expressing a relationship
between two variables. All pdfs are
functions BUT NOT all functions are PDFs.
WE WILL TALK ABOUT THIS LATER
Functions Scientific Model
pdfs
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Probability Density Functions properties

• A function that assigns probabilities to ALL the
  possible values of a random variable (x).

Probability density f(x)
x
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Probability Density Functions Expectations

• The expectation of a random variable x is the
  weighted value of the possible values that x can
  take, each value weighted by the probability that
  x assumes it.
• Analogous to center of gravity. First moment.

-1 0 1
2 p(-1)0.10 p(0)0.25 p(1)0.3
p(2)0.35
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Probability Density Functions Variance

• The variance of a random variable reflects the
  spread of X values around the expected value.
• Second moment of a distribution.

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Probability Distributions

• A function that assigns probabilities to the
  possible values of a random variable (X).
• They come in two flavors
• DISCRETE outcomes are a set of discrete
  possibilities such as integers (e.g, counting).
• CONTINUOUS A probability distribution over a
  continuous range (real numbers or the
  non-negative real numbers).

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Probability Mass Functions
For a discrete random variable, X, the
probability that x takes on a value x is a
discrete density function, f(x) also known as
probability mass or distribution function.
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Probability Density Functions Continuous
variables
A probability density function (f(x)) gives the
probability that a random variable X takes on
values within a range.
b
ò
lt
lt


b
X
a

P
dx
)
x
(
f
a
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0
)
x
(
f

ò

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dx
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x
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-
a b
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Some rules of probability
assuming independence
A
B
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Real data Histograms
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Histograms and PDFs
Probability density functions approximate the
distribution of finite data sets.
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Uses of Frequency Distributions

• Empirical (frequentist)
• Make predictions about the frequency of a
  particular event.
• Judge whether an observation belongs to a
  population.
• Theoretical
• Predictions about the distribution of the data
  based on some basic assumptions about the nature
  of the forces acting on a particular biological
  system.
• Describe the randomness in the data.

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Some useful distributions

• Discrete
• Binomial Two possible outcomes.
• Poisson Counts.
• Negative binomial Counts.
• Multinomial Multiple categorical outcomes.
• Continuous
• Normal.
• Lognormal.
• Exponential
• Gamma
• Beta

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An example Seed predation
x no seeds taken
t2 ( )
t1
0 to N
Assume each seed has equal probability (p) of
being taken. Then
Normalization constant
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Zero-inflated binomial
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Binomial distribution Discrete events that can
take one of two values
n 20 p 0.5
Ex np Variance np(1-p) n number of sites p
prob. of survival
Example Probability of survival derived from pop
data
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Binomial distribution
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Poisson Distribution Counts (or getting hit in
the head by a horse)
400
0.4
300
0.3
Count
Proportion per Bar
k number of seedlings ? arrival rate
200
0.2
100
0.1
0
0.0
0
1
2
3
4
5
6
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Alt param ?rt
POISSON
Number of Seedlings/quadrat
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Poisson distribution
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Example Number of seedlings in census quad.
60
0.4
Alchornea latifolia
50
0.3
40
Count
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Proportion per Bar
0.2
20
0.1
10
0
0.0
0
10
20
30
40
50
60
70
80
90
100
Number of seedlings/trap
(Data from LFDP, Puerto Rico)
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Clustering in space or time
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Negative binomialTable 4.2 4.3 in HM Bycatch
Data
EX0.279 VarianceX1.56
Suggests temporal or spatial aggregation in the
data
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Negative Binomial Counts
0.2
100
90
80
70
60
Count
50
0.1
Proportion per Bar
40
30
20
10
0.0
0
0
10
20
30
40
50
NEGBIN
Number of Seeds
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Negative Binomial Counts
0.2
100
90
80
70
60
Count
50
0.1
Proportion per Bar
40
30
20
10
0.0
0
0
10
20
30
40
50
NEGBIN
Number of Seeds
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Negative binomial
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Negative Binomial Count data
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0.2
Prestoea acuminata
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Count
0.1
Proportion per Bar
10
0
0.0
0
10
20
30
40
50
60
70
80
90
100
No seedlings/quad.
(Data from LFDP, Puerto Rico)
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Normal Distribution
Ex m Variance d2
Normal PDF with mean 0
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Normal Distribution with increasing variance
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Lognormal One tail and no negative values
0.8
0.7
0.6
f(x)
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
60
70
x
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Lognormal Radial growth data
(Data from Date Creek, British Columbia)
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Exponential
Count
Variable
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Exponential Growth data (negatives assumed 0)
(Data from BCI, Panama)
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Gamma One tail and flexibility
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Gamma raw growth data
Alseis blackiana
Cordia bicolor
Growth (mm/yr)
(Data from BCI, Panama)
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Beta distribution
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Beta Light interception by crown trees
(Data from Luquillo, PR)
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Mixture models

• What do you do when your data dont fit any known
  distribution?
• Add covariates
• Mixture models
• Discrete
• Continuous

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Discrete mixtures
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Discrete mixtureZero-inflated binomial
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Continuous (compounded) mixtures
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The Method of Moments

• You can match up the sample values of the moments
  of the distributions and match them up with the
  theoretical moments.
• Recall that
• The MOM is a good way to get a first (but biased)
  estimate of the parameters of a distribution. ML
  estimators are more reliable.

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MOM Negative binomial