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Lecture slides from EKOJ203 Winter 2014 course at University of Jyväskylä (Finland). The course attempted to teach the basic/foundational concepts of statistical modeling for ecologists and evolutionary biologists. Lecture 1. Introduction. Sets the scene for the course, explaining the purpose of statistical models in empirical science and their basic structure. It also brushes up some basic concepts.
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Founda'ons in Sta's'cs Ecology and Evolu'onary Biology
Sara Calhim (sara.calhim@jyu.fi ) Andrés López-‐Sepulcre (alopez@biologie.ens.fr)
Founda'ons in Sta's'cs Ecology and Evolu'onary Biology
Sara Calhim (sara.calhim@jyu.fi ) Andrés López-‐Sepulcre (alopez@biologie.ens.fr)
Lecture
Discussion
Course Structure
Prac'cal Homework
Mostly Mondays 10-‐12
Mostly Wednesdays 10-‐12
Hand in by next lecture
Office Hours: (Mostly) Mondays 13-‐16h Sara: C410.2 Andrés: C423.1
1. Sta's'cs and the Scien'fic Process
What is Sta's'cs?
• Study of uncertainty Lindley 2000 The philosophy of sta's'cs. Sta$s$cian 49: 293-‐337
• Applied Philosophy of Science Kempthorne 1971 Probability, Sta's'cs, and the Knowledge Business. In Godambe and Spro3, pp 470-‐499
Realism vs. Pragma'sm
Regardless, what ma`ers is PREDICTION
What is Scien'fic Knowledge?
Models
Data
Data
Models
The Scien'fic Process
Charles S. Peirce
Abduc'on Data Model
Predic'ons
Deduc'on Induc'on
Hypothesis genera'on
Theory Inferen'al Sta's'cs
Sta's'cal Inference
Popula'on
Sample
STATISTIC PARAMETER
(Random) Selec'on
We want to know about these We have these to work with
Popula'on mean (μ) Popula'on standard devia'on(σ) Probability of survival Popula'on size
Sample mean (x) Sample standard devia'on (s) Propor'on survived? hmmm…
Es'mate
MODEL y = μ + N(0, σ)
Sta's'cal Models
weight = ! + ! ∙ length+ ! ∙ length! + ! ∙ length! + !(0,!)
weight = !(length!)+ !(0,!) Determinis'c Stochas'c
dependent variable
parameters
independent variable
Spherical cow
Sta's'cal Models weight = !(length!)+ !(0,!) weight = !(length!)+ !(0,!) weight = !(length)+ !(0,!)
Guppy males are lines and females are discs
…maybe except very pregnant females
Sta's'cal Models
• Determinis'c component – Describes a central tendency or propensity
• Stochas'c component – Inherent random process – Ignorance – Measurement error
Inferring Pa`erns or Processes?
Phenomenological Models
Mechanis'c Models
Phenomenological vs. Mechanis'c
Fryxell et al. 1994
Phenomenological Models ! = ! + !" + !
a = 0.34 b = 0.07
How high How steep
Phenomenological Models ! = ! + !" + !"! + !
How humped
a = 0.16 b = 0.22 c = -‐0.02
Phenomenological Models ! = ! ∙ !"# ! + 1 + !
a = 0.41
How steep and fast it saturates
Phenomenological Models
Mechanis'c Models
Mechanis'c Models ! = !"
1+ !ℎ! + !
a = 0.762 kg/day
h = 1.2 days/kg
Beavers found
and handled them for
PHENOMENOLOGICAL
• Pa`erns • No causality • Parameters describe shape
• Cannot predict beyond range of data
• Underlying processes • Model implies causality • Parameters describe processes
• Be`er for predic'on and extrapola'on
MECHANISTIC
Things to do with Sta's'cal Models
Integra'ng Models and Data
1. Model fihng and parameter es'ma'on: • Least squares, Maximum Likelihood, Bayesian • Parameter uncertainty
! = !"1+ !ℎ! + !
a = 0.76 ± 0.29 h = 1.2 ± 0.18
Integra'ng Models and Data
1. Model fihng and parameter es'ma'on:
2. Model Selec'on • Best Fit • Parsimony • Predic've power
Integra'ng Models and Data
1. Model fihng and parameter es'ma'on:
2. Model Selec'on
3. Hypothesis tes'ng • Sta's'cal significance
Integra'ng Models and Data
1. Model fihng and parameter es'ma'on:
2. Model Selec'on
3. Hypothesis tes'ng
4. Predic'on
Sta's'cal Models vs. Techniques
Models • Linear models, GLMs • GLMMs, repeated measures • Logis'c model • Phylo(gene'c) models • Popula'on models • Null hypothesis • Normal distribu'on et al. • … your own hypothesis!
Techniques • Least squares regression • ANOVA • Maximum Likelihood • Hypothesis tes'ng (p-‐value) • Model selec'on, AIC, etc. • Monte Carlo • Bayesian inference • …
THE BIOLOGY THE SCIENTIFIC METHOD
Course Emphasis
• Construc'on and biological interpreta'on of models
• Mechanis'c understanding of sta's'cal techniques (programming)
• Cri'cal evalua'on of methods and their assump'ons
• Think ‘out of the canned package’ • Foster independent learning by teaching the fundamental building blocks
Literature
Review of Sta's'cs 101
Sta's'cal Inference
Popula'on
Sample
STATISTIC PARAMETER
(Random) Selec'on
We want to know about these We have these to work with
Popula'on mean (μ) Popula'on variance (σ2)
Sample mean (x) Sample variance (s2)
Es'mate
How wrong?
BIAS (Accuracy)
ERROR (Precision)
Describing Distribu'ons
Variance
Standard Devia'on
Variances are addi've!!
Standard Error
Popula'on
Sample
STATISTIC PARAMETER
(Random) Selec'on
We want to know about these
We have these to work with
Popula'on mean (μ) Popula'on variance (σ2)
Sample mean (x) Sample variance (s2)
Es'mate
Repeat infinite 'mes
What is the standard devia'on of my infinite es'mates?
e.g. for the mean
COVARIANCE
CORRELATION
Correla'ons only describe linear rela'onships
Correla'on does not imply causa'on (maybe …we’ll see)
Rela'onships among variables
What is a Probability?
• Physical Probability – Frequen'st: long-‐run outcome – Propensity: property of the system
• Eviden'al Probability (Bayesian) – Measure of statement (un)certainty
Probability Cheat Sheet
Recommended