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General Linear Models -- #1. things to remember b weight interpretations 1 quantitative predictor 1 2-group predictor 1 k-group predictor 1 quantitative & a 2-group predictors 1 quantitative & a k-group predictors 2 quantitative predictors 2x2 – main effects 2x2 with interactions

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General linear models 1
General Linear Models -- #1

  • things to remember

  • b weight interpretations

  • 1 quantitative predictor

  • 1 2-group predictor

  • 1 k-group predictor

  • 1 quantitative & a 2-group predictors

  • 1 quantitative & a k-group predictors

  • 2 quantitative predictors

  • 2x2 – main effects

  • 2x2 with interactions

  • 2x3 – main effects

  • 2x3 with interactions


  • A few important things to remember…

  • we plot and interpret the model of the data, not the data

  • if the model fits the data poorly, then we’re carefully describing and interpreting nonsense

  • the interpretation of regression weights in a main effects model (without interactions) is different than in a model including interactions

    • regression weights reflect “main effects” in a main effects model

    • regression weights reflect “simple effects” in a model including interactions


b weight interpretations

Constant

the expected value of y when the value of all predictors = 0

Centered quantitative variable

the direction and extent of the expected change in the value of y for a 1-unit increase in that predictor, holding the value of all other predictors constant at 0

Dummy Coded binary variable

the direction and extent of expected mean difference of the Target group from the Comparison group, holding the value of all other predictors constant

Dummy Coded k-group variable

the direction and extent of the expected mean difference of the Target group for that dummy code from the Comparison group, holding the value of all other predictors constant.


b weight interpretations

Interaction between quantitative variables

the direction and extent of the expected change in the slope of the linear relationship between y and one predictor for each 1-unit change in the other predictor, holding the value or all other predictors constant at 0

Interaction between quantitative & Dummy Coded binary variables

the direction and extent of expected change in the slope of the linear relationship between y and the quantitative variable of the Target group from the slope of the Comparison group, holding the value of all other predictors constant at 0

Interaction between quantitative & Dummy Coded k-group variables

the direction and extent of expected change in the slope of the linear relationship between y and the quantitative variable of the Target groupfor that dummy code from the slope of the Comparison group, holding the value of all other predictors constant at 0

Interaction between Dummy Coded Variables

the direction and extend of expected change in the mean difference between the IVx Target & Comparison groups of the IVz Target group from mean difference between the IVx Target & Comparison groups of the IVz Comparison group,

holding the value of all other predictors constant at 0.


Single quantitative predictor (X)  Bivariate Regression

X = X – Xmean

y’ = b0 + b1 X

b0 = ht of line

b1 = slp of line

0 10 20 30 40 50 60

b1

b0

-20 -10 0 10 20  X


2-group predictor(Tx Cx)  2-grp ANOVA

X Tx = 1 Cx = 0

X = Tx vs. Cx

y’ = b0 + b1X

b0 = ht Cx

b1 = htdif Cx & Tx

Tx

0 10 20 30 40 50 60

b1

Cx

b0


3-group predictor (Tx1 Tx2 Cx)  k-grp ANOVA

X1 Tx1=1 Tx2=0 Cx=0 X2 Tx1=0 Tx2=1 Cx=0

X1 = Tx1 vs. Cx

X2 = Tx2 vs. Cx

y’ = b0 + b1X1 + b2X2

b0 = ht Cx

Tx1

b1 = htdif Cx & Tx1

b1

b2 = htdif Cx & Tx2

Cx

0 10 20 30 40 50 60

b0

b2

Tx2


quantitative (X) & 2-group (Tz Cz) predictors  2-grp ANCOVA

Z Tz = 1 Cz = 0

X = X – Xmean

Z = Tz vs. Cz

y’ = b0 + b1X + b2Z

b0 = ht of Cz line

b1 = slp of Cz line

b2 = htdif Cz & Tz

0 10 20 30 40 50 60

Tz

b1

b2

Z-lines all have same slp

(no interaction)

b0

Cz

-20 -10 0 10 20  X


quantitative (X) & 2-group (Tz Cz) predictors w/ interaction

Z Tz = 1 Cz = 0

X = X – Xmean

XZ = Xcen * Z

Z = Tz vs. Cz

y’ = b0 + b1X + b2Z + b3XZ

b0 = ht of Cz line

b3

b1 = slp of Cz line

b2 = htdif Cz & Tz

0 10 20 30 40 50 60

b3 = slpdif Cz & Tz

b1

b2

Tz

b0

Cz

-20 -10 0 10 20  Xcen


quantitative (X) & 3-group (Tz1 Tz2 Cz) predictors  3-grp ANCOVA

X = X – Xmean

Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0

Z1 = Tz1 vs. Cz

Z2 = Tz2 vs. Cz

y’ = b0 +b1X+ b2Z1 + b3Z2

b0 = ht of Cz line

b1 = slp of Cz line

b2

Tz1

b2 = htdif Cz & Tz1

0 10 20 30 40 50 60

b1

Cz

b3 = htdif Cz & Tz2

b3

Tz2

Z-lines all have same slp

(no interaction)

b0

-20 -10 0 10 20  X


Models with quant (X) & 3-group (Tz1 Tz2 Cz) predictors w/ interaction

Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0

XZ1 = Xcen * Z1

X = X – Xmean

XZ2 = Xcen * Z2

Z1 = Tz1 vs. Cz

Z2 = Tz2 vs. Cz

y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2

b4

b0 = ht of Cz line

b1 = slp of Cz line

b1

b2

b2 = htdif Cz & Tz1

Tx2

b3

0 10 20 30 40 50 60

b5

b4 = slpdif Cz & Tz1

Tx1

Cx

b3 = htdif Cz & Tz2

b0

b5 = slpdif Cz & Tz2

-20 -10 0 10 20  Xcen


2 quantitative predictors  multiple regression

X = X – Xmean

Z = Z – Zmean

y’ = b0 + b1X + b2Z

b0 = ht of Zmean line

b1 = slope of Zmean line

+1std Z

b2

b2 = htdifs among Z-lines

0 10 20 30 40 50 60

Z=0

b1

b2

-1std Z

Z-lines all have same slp

(no interaction)

b0

-20 -10 0 10 20  X


2 quantitative predictors w/ interaction

Xcen = X – Xmean

Zcen = Z – Xmean

ZX = Xcen * Zcen

y’ = b0 + b1Xcen + b2Zcen + b3XZ

a = ht of Zmean line

b3

b1 = slope of Zmean line

b1

b2 = htdifs among Z-lines

b2

+1std Z

b3

0 10 20 30 40 50 60

b3 = slpdifs among Z-lines

-b2

Z=0

-1std Z

b0

-20 -10 0 10 20  Xcen


2-group (Tx Cx) & 2-group (Tz Cz) predictors Main Effects Model

Z Tz = 1 Cz = 0

X Tx = 1 Cx = 0

X

C T

X = Tx vs. Cx

Z = Tz vs. Cz

TxTz

CxTz

T

Z

C

TxCz

CxCz

y’ = b0 + b1X + b2Z

TxTz

b0 = mean of CxCz

CxTz

b1 = htdif of CxCz & TxCz

TxCz

b2

0 10 20 30 40 50 60

b1

b2 = htdifs of CxCz

& CxTz

CxCz

b0

= simple effects

(no interaction)

0 1  X


Models with 2-group (Tx Cx) & 2-group (Tz Cz) predictors Model 2x2 ANOVA

Z Tz = 1 Cz = 0

XZ = X * Z

X Tx = 1 Cx = 0

X = Tx vs. Cx

Z = Tz vs. Cz

y’ = b0 + b1X + b2Z + b3XZ

b0 = mean of CxCz

b1 = htdif of CxCz & TxCz

CxTz

b3

TxTz

vs.

b2 = htdifs of CxCz

& CxTz

b2

0 10 20 30 40 50 60

TxCz

b1

b3 = dif htdifs of

CxCz - TxCz &

CxTz - TxTz

CxCz

b0

0 1  X


Z C T Model1 T2

Models with 2-group (Tx Cx) &

3-group (Tz1 Tz2 Cz) predictors  ME model

CxTz2

CxTz1

CxCz

C

X

T

TxCz

TxTz1

TxTz2

X Tx = 1 Cx = 0

Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0

Z1 = Tz1 vs. Cz

Z2 = Tz2 vs. Cz

X = Tx vs. Cx

y’ = b0 + b1X + b2Z1 + b3Z2

b0 = mean of CxCz

TxTz2

b1 = htdif of CxCz & TxCz

TxTz1

CxTz2

b3

b2 = htdifs of CxCz

& CxTz1

CxTz1

TxCz

0 10 20 30 40 50 60

b1

b2

b3 = htdifs of CxCz

& CxTz2

CxCz

= simple effects

(no interaction)

b0

0 1  X


Models with 2-group (Tx Cx) & 3-group (Tz Model1 Tz2 Cz) predictors  2x3 ANOVA

X Tx = 1 Cx = 0

Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0

XZ1 = X * Z1

Z1 = Tz1 vs. Cz

Z2 = Tz2 vs. Cz

XZ2 = X * Z2

X = Tx vs. Cx

b0 = mean of CxCz

y’ = b0 + b1X + b2Z1 + b3Z2 +b4XZ1 + b5XZ2

b1 = htdif of CxCz & TxCz

TxTz2

b2 = htdifs of CxCz

& CxTz1

b5

CxTz2

vs.

b3 = htdifs of CxCz

& CxTz2

b3

TxTz1

b4

CxTz1

TxCz

vs.

0 10 20 30 40 50 60

b4 = dif htdifs of

CxCz - TxCz &

CxTz1 – TxTz1

b2

b1

CxCz

b5 = dif htdifs of

CxCz - TxCz &

CxTz2 – TxTz2

b0

0 1  X


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