Advanced research methods ii 03 25 2009
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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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Advanced Research Methods II 03/25/2009

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Advanced research methods ii 03 25 2009

Advanced Research Methods II03/25/2009

General Linear Models (GLM)


Topic overview

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

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 glm1

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 glm2

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 glm3

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 glm4

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 glm5

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 glm6

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 glm7

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 glm8

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 glm9

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 glm10

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 glm11

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 glm12

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 glm13

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

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

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 models1

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)


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