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# Advanced Research Methods II 03/25/2009 - PowerPoint PPT Presentation

Advanced Research Methods II 03/25/2009. General Linear Models (GLM). Topic Overview. The Basic Equation for GLM Analysis Methods Subsumed Under GLM ANOVA (One-way and Factorial) ANCOVA Regression MANOVA and Discriminant Analysis Repeated Measures ANOVA Multivariate Regression

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General Linear Models (GLM)

### Topic Overview

• The Basic Equation for GLM

• Analysis Methods Subsumed Under GLM

• ANOVA (One-way and Factorial)

• ANCOVA

• Regression

• MANOVA and Discriminant Analysis

• Repeated Measures ANOVA

• Multivariate Regression

• Canonical Correlation

• Conducting GLM by SPSS

• GLM versus Generalized Linear Models

### The General Linear Models (GLM)

Basic Equation:

• YM = XB + E

Notes: n: Number of subjects (observations/cases)

p: Number of dependent variables (DV)

p’: Number of linear composites formed by the DVs

k: Number of independent variables (IV)

• Y: Dependent variables n x p Matrix

• M: Coefficients determining the linear combinations of Y

p x p’ Matrix

• X: Independent variablesn x (k+1) Matrix

• B: Regression Coefficients (k +1) x p’ Matrix

• E: Error n x p’ Matrix

### The General Linear Models (GLM)

• Oneway ANOVA:

YM = XB + E

n: Number of subjects (observations/cases)

p: Number of dependent variables = 1

k: Number of independent variables = a - 1

(a = Number of categories for factor A)

• Y: n x 1 Matrix

• M: 1 x 1 Matrix = Scalar (1)

• B: (k+1)x 1 Matrix (k = a -1; )

• X: n x(k+1) Matrix (k = a -1)

• E: n x 1 Matrix

### The General Linear Models (GLM)

• One-Way ANOVA

Example:

ACTnx1 = (Race)n x 3*B3 x 1 + En x 1

ACTnx1= y1 Racenx3 = 1 1 0 B3x1 = β0 Enx1=ε1

y2 1 1 0 β1 ε2

..… ……β2 ..

..1 0 1..

.. 1 0 1 ..

..……….…

1 -1 -1

yn1 -1 -1 εn

### The General Linear Models (GLM)

• Factorial ANOVA:

YM = XB + E

n: Number of subjects (observations/cases)

p: Number of dependent variables = 1

k: Number of independent variables = ab - 1

(a = number of categories for factor A; b = number of categories for factor B)

• Y: n x 1 Matrix

• M: 1 x 1 Matrix = Scalar (1)

• B: (k+1) x 1 Matrix (k = ab - 1)

• X: n x (k+1) Matrix (k = ab - 1)

• E: n x 1 Matrix

### The General Linear Models (GLM)

• Factorial ANOVA

Example:

ACTnx1 = (Race, College, Race*College)n x 6*B6 x 1 + En x 1

ACTnx1= y1 Xnx6 = 1 1 0 1 1 0 B6x1 = β0

y2 1 1 0 -1 -1 0β1 (race1)

..… …………β2 (race2) ..1 0 1 1 0 1 β3 (college)

.. 1 0 1 -1 0 -1β4 (race1 x college)

..………….β5 (race2 x college)1 -1 -1 1 -1 -1

yn1 -1 -1 -1 1 1

### The General Linear Models (GLM)

• ANCOVA:

YM = XB + E

n: Number of subjects (observations/cases)

p: Number of dependent variables = 1

k: Number of independent variables = ab + c – 1

(a = number of categories for factor A; b = number of categories for factor B; c = number of covariates)

• Y: n x 1 Matrix

• M: 1 x 1 Matrix = Scalar (1)

• B: k+1 x 1 Matrix (k = ab + c - 1)

• X: n x k+1 Matrix (k = ab + c - 1)

• E: (n x 1) Matrix

### The General Linear Models (GLM)

• ANCOVA

Example:

ACTnx1 = (Race, College, Race*College, Father_edu)n x 7*B7 x 1 + En x 1

ACTnx1= y1 Xnx6 = 1 1 0 1 1 0 4 B7x1 = β0

y2 1 1 0 -1 -1 0 7β1 (race1)

..… …………β2 (race2) ..1 0 1 1 0 1 3β3 (college)

.. 1 0 1 -1 0 -1 2β4 (race1 x college)

..………….β5 (race2 x college)1 -1 -1 1 -1 -1 5β6 (Father_edu)

yn1 -1 -1 -1 1 1 3

### The General Linear Models (GLM)

• Regression:

YM = XB + E

n: Number of subjects (observations/cases)

p: Number of dependent variables = 1

k: Number of independent variables

(k may include variables reflecting interaction effects, and/or curvilinear effects)

• Y: n x 1 Matrix

• M: 1 x 1 Matrix = Scalar (1)

• B: (k+1) x 1 Matrix

• X: n x (k+1) Matrix

• E: n x 1 Matrix

### The General Linear Models (GLM)

• MANOVA:

YM = XB + E

n: Number of subjects (observations/cases)

p: Number of dependent variables

k: Number of independent variables = ab - 1

(a = number of categories for factor A; b = number of categories for factor B)

• Y: n x p Matrix

• M: p x p Identity Matrix

• B: (k+1) x p Matrix (k = ab - 1)

• X: n x (k+1) Matrix (k = ab - 1)

• E: (n x p) Matrix

E.g. p=2

### The General Linear Models (GLM)

• MANOVA

Example:

(GPA, ACT)nx2 = (Race, College, Race*College)n x 6*B6 x 2 + En x 2

Ynx2= y11 y12 Xnx6 = 1 1 0 1 1 0 B6x1 = β01β02

y21 y221 1 0 -1 -1 0β11 β12

.. ..… …………β21 β22 .. ..1 0 1 1 0 1 β31 β32

.. ..1 0 1 -1 0 -1β41 β41

.. ..………….β51 β51 1 -1 -1 1 -1 -1

yn1 yn2 1 -1 -1 -1 1 1

M = 1 0

0 1

### The General Linear Models (GLM)

• Repeated Measures ANOVA (Within-subject factor):

YM = XB + E

n: Number of subjects (observations/cases)

p: Number of dependent variables = t-1 (t = number of levels/times associated with the DV)

k = 0

• Y: n x p Matrix

• M: p x (p - 1) Matrix

• B: 1x (p - 1) Matrix

• X: n x 1 Matrix (vector 1: all cells = 1)

• E: n x (p -1) Matrix

E.g. p=2

### The General Linear Models (GLM)

• Repeated-Measure ANOVA

Example:

ACTnx2 M2x1= (ACT1 - ACT2)nx1 = Xnx1*B1x1 + En x 1

ACTnx2= y11 y12 M = 1 ACT*M = y11 - y12 Xnx1 = 1 B3x1 = β0

y21 y22 -1 y21 -y22 1… …

... … … …

yn1 yn2yn1 -yn2 1

### The General Linear Models (GLM)

• Mutivariate Regression:

YM = XB + E

n: Number of subjects (observations/cases)

p: Number of dependent variables

k: Number of independent variables

(k may include variables reflecting interaction effects,

and/or curvilinear effects)

• Y: n x p Matrix

• M: p x p Identity Matrix

• B: (k+1) x p Matrix

• X: n x (k+1) Matrix

• E: (n x p) Matrix

### The General Linear Models (GLM)

• Multivariate Regression

Example:

(GPA, ACT)nx2 = (Father_Edu, Mother_Educ)n x 2*B2 x 2 + En x 2

Ynx2 = y11 y12 Xnx6 = 5 4 B6x1 = β01β02

y21 y227 6 β11 β12

.. ..… … .. ..3 2

.. ..1 4

.. ..……… 1 3

yn1 yn2 2 2

M = 1 0

0 1

### Conducting GLM by SPSS

• Fixed Factors = Categorical Variables

• Covariates = Continuous Variables

• Full Factorial Model (default) =

All main and interaction effects for Fixed factors plus main effects for covariates.

• Matrix M (default) = Identity matrix.

M can only be specified by syntax:

E.g. Dependent variable of interest: ACT – 10*HS_GPA

GLM ACT HS_GPA

By Race College

With Father_education Mother_education

/MMatrix = ACT 1 HS_GPA -10

/Intercept=Include

/Design= Race College Race*College Father_education Mother_education Father_education*Mother_education.

### General Linear Models (GLM) vs. Generalized Linear Models

Assumptions for GLM:

+ Normality (Distribution of the DV Y is multivariate normal)

+ Linearity

Y = XB + E

or E(Y’) = XB

When the assumptions cannot be satisfied (e.g., Y is a dichotomous variable)  Generalized Linear Models:

+ Normality: Assumes that Y follows a distribution belonging to the exponential family of distributions, which includes distributions such as the binomial, Poisson, exponential, and gamma distributions, in addition to the normal distribution.

### General Linear Models (GLM) vs. Generalized Linear Models

+ Linearity: E(Y) = XB

Generalized Linear Models extend GLM by suggesting a link function g such that:

g[E(Y)] = XB

For Logistic regression (Y = 0, 1); g = ln{(E(Y)/[1-E(Y)]}

For General linear models Y ~ N (normal); g = E(Y)