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