Part VThe Generalized Linear Model
Chapter 16 Introduction
t-testANOVA
Simple Linear Regression
Multiple Linear Regression
ANCOVA
GENERAL LINEAR MODELSε ~ Normal R: lm()
t-testANOVA
Simple Linear Regression
Multiple Linear Regression
ANCOVA
PoissonBinomial
Negative Binomial Gamma
Multinomial
GENERALIZED LINEAR MODELS
Inverse Gaussian
Exponential
GENERAL LINEAR MODELSε ~ Normal
Linear combination of parameters
R: lm()
R: glm()
Generalized Linear Model (GzLM)Introduction
• Assumptions of GLM not always met using biological data
Generalized Linear Model (GzLM)Introduction
Generalized Linear Model (GzLM)Introduction
Generalized Linear Model (GzLM)Introduction
• Assumptions of GLM not always met using biological data– Transformations typically recommended– We can randomize…• Assumes parameter estimates (means, slopes, etc.) are
correct– But a few large counts or many zeros will influence skew our
estimates
Generalized Linear Model (GzLM)Introduction
Generalized Linear Model (GzLM)Introduction
Generalized Linear Model (GzLM)Introduction
• Assumptions of GLM not always met using biological data– Transformations typically recommended– We can randomize…• Assumes parameter estimates (means, slopes, etc.) are
correct– But a few large counts or many zeros will influence skew our
estimates
– Best to use an appropriate error structure under the Generalized Linear Model framework
Generalized Linear Model (GzLM)Introduction
Poisson error structure
Generalized Linear Model (GzLM)Introduction
Binomial error structure
Generalized Linear Model (GzLM)Advantages
• Assumptions more evident• Decouples assumptions• Improves quality• Greater flexibility
Generalized Linear Model (GzLM)Advantages
• Assumptions more evident• Decouples assumptions• Improves quality• Greater flexibility
Part VThe Generalized Linear Model
Chapter 16.1 Goodness of Fit
Goodness of Fit - The Chi-square statistic
• Have to learn a new concept to apply GzLM:– Goodness of Fit
• Chi-square statistic• G-statistic
Classic Chi-square Statistic Example
Gregor Mendel’s Peas
Purple: White:
χ 2=∑ (𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑−𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 )2𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑
χ2 = 0.3907df = 1p = 0.532
Classic Chi-square Statistic Example
Gregor Mendel’s Peas
χ2 = 0.3907df = 1p = 0.532
Classic Chi-square Statistic Example
Gregor Mendel’s Peas
• Deviation from genetic model (3:1) not significant
Goodness of Fit - The G-statistic
• Can deal with complex models• Based in likelihood
Goodness of Fit - The G-statistic
Smaller deviation smaller G-statistic
G-statistic p-value = 0.53
Improvement in Fit - ΔG
• Next time we will…– Compare G values (ΔG) to assess improvement in
fit of one model over another