General Linear Models -- #1
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General Linear Models -- #1 • things to remember • b weight interpretations • 1 quantitative predictor • 1 quantitative predictor & non-linear component • 1 2-group predictor • 1 k-group predictor • 1 quantitative & a 2-group predictors • 1 quantitative & a k-group predictors • 2 quantitative predictors
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General Linear Models -- #1. things to remember b weight interpretations 1 quantitative predictor 1 quantitative predictor & non-linear component 1 2-group predictor 1 k-group predictor 1 quantitative & a 2-group predictors 1 quantitative & a k-group predictors 2 quantitative predictors. - PowerPoint PPT Presentation
Transcript of General Linear Models -- #1
General Linear Models -- #1
• things to remember• b weight interpretations• 1 quantitative predictor• 1 quantitative predictor & non-linear component• 1 2-group predictor• 1 k-group predictor• 1 quantitative & a 2-group predictors • 1 quantitative & a k-group predictors• 2 quantitative predictors
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 maineffects model
• regression weights reflect “simple effects” in a modelincluding 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 Non-linear term
the direction and extent of the expected change in the slope of the linear relationship between y and that predictor for 1-unit increase in that predictor, holding the value of all other predictors constant at 0
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 variablesthe 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 group for that dummy code from the slope of the Comparison group, holding the value of all other predictors constant at 0
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y’ = b0 + b1 X
b0
b1
-20 -10 0 10 20 X
b0 = ht of line
b1 = slp of line
X = X – Xmean
Single quantitative predictor (X) Bivariate Regression
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y’ = b0 + b1 X
-20 -10 0 10 20 X
b0 = ht of line
b1 = slp of line
X = X – Xmean
Single quantitative predictor (X) Bivariate Regression
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y’ = b0 + b1Xcen + b2Xq
b0
b1
Linear & nonlinear quantitative predictor Nonlinear regression
-20 -10 0 10 20 Xcen
Xquadratic = Xcen* Xcen
b2
Xcen = X – Xmean
b0 = ht of line
b1 = slp of line
b2 = dif from linearity
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y’ = b0 + b1Xcen + b2Xq
Linear & nonlinear quantitative predictor Nonlinear regression
-20 -10 0 10 20 Xcen
Xquadratic = Xcen* Xcen Xcen = X – Xmean
b0 = ht of line
b1 = slp of line
b2 = dif from linearity
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Cx
Tx
2-group predictor (Tx Cx) 2-grp ANOVA
b0 = ht Cx
b1 = htdif Cx & Tx
X Tx = 1 Cx = 0
X = Tx vs. Cx
b0
b1
y’ = b0 + b1X
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602-group predictor (Tx Cx) 2-grp ANOVA
b0 = ht Cx
b1 = htdif Cx & Tx
X Tx = 1 Cx = 0
X = Tx vs. Cx
y’ = b0 + b1X
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b2
Cx
Tx2
Tx1
b0 = ht Cx
b1 = htdif Cx & Tx1
b2 = htdif Cx & Tx2
3-group predictor (Tx1 Tx2 Cx) k-grp ANOVA
y’ = b0 + b1X1 + b2X2
X1 Tx1=1 Tx2=0 Cx=0 X2 Tx1=0 Tx2=1 Cx=0
X1 = Tx1 vs. Cx X2 = Tx2 vs. Cx
b0
b1
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b1 = htdif Cx & Tx1
b2 = htdif Cx & Tx2
3-group predictor (Tx1 Tx2 Cx) k-grp ANOVA
y’ = b0 + b1X1 + b2X2
X1 Tx1=1 Tx2=0 Cx=0 X2 Tx1=0 Tx2=1 Cx=0
X1 = Tx1 vs. Cx X2 = Tx2 vs. Cx
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y’ = b0 + b1X + b2Z
b0
b1b2
Cz
Tz
quantitative (X) & 2-group (Tz Cz) predictors 2-grp ANCOVA
-20 -10 0 10 20 X
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
X = X – Xmean Z Tz = 1 Cz = 0
Z = Tz vs. Cz
Z-lines all have same slp(no interaction)
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y’ = b0 + b1X + b2Z
quantitative (X) & 2-group (Tz Cz) predictors 2-grp ANCOVA
-20 -10 0 10 20 X
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
X = X – Xmean Z Tz = 1 Cz = 0
Z = Tz vs. Cz
Z-lines all have same slp(no interaction)
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b0
b1b2
Cz
Tz
-20 -10 0 10 20 Xcen
XZ = Xcen * Z
b3
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
b3 = slpdif Cz & Tz
quantitative (X) & 2-group (Tz Cz) predictors w/ interaction
y’ = b0 + b1X + b2Z + b3XZ
X = X – Xmean Z Tz = 1 Cz = 0
Z = Tz vs. Cz
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-20 -10 0 10 20 Xcen
XZ = Xcen * Z
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
b3 = slpdif Cz & Tz
quantitative (X) & 2-group (Tz Cz) predictors w/ interaction
y’ = b0 + b1X + b2Z + b3XZ
X = X – Xmean Z Tz = 1 Cz = 0
Z = Tz vs. Cz
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b0
b1
b2
Cz
Tz2
Tz1
b3
-20 -10 0 10 20 X
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
y’ = b0 +b1X + b2Z1 + b3Z2
Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz
X = X – Xmean Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0
quantitative (X) & 3-group (Tz1 Tz2 Cz) predictors 3-grp ANCOVA
Z-lines all have same slp(no interaction)
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-20 -10 0 10 20 X
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
y’ = b0 +b1X + b2Z1 + b3Z2
Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz
X = X – Xmean Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0
quantitative (X) & 3-group (Tz1 Tz2 Cz) predictors 3-grp ANCOVA
Z-lines all have same slp(no interaction)
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y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2
b0
b1b2
Cx
Tx1
Tx2 b3
-20 -10 0 10 20 Xcen
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
b4 = slpdif Cz & Tz1
b5 = slpdif Cz & Tz2
XZ1 = Xcen * Z1
XZ2 = Xcen * Z2
b4
b5
Models with quant (X) & 3-group (Tz1 Tz2 Cz) predictors w/ interaction
Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz
X = X – Xmean Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0
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y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2
-20 -10 0 10 20 Xcen
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
b4 = slpdif Cz & Tz1
b5 = slpdif Cz & Tz2
XZ1 = Xcen * Z1
XZ2 = Xcen * Z2
Models with quant (X) & 3-group (Tz1 Tz2 Cz) predictors w/ interaction
Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz
X = X – Xmean Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0
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y’ = b0 + b1X + b2Z
b0
b1 b2Z=0
+1std Z
-1std Z
b2
Z = Z – Zmean
2 quantitative predictors multiple regression
-20 -10 0 10 20 X
b0 = ht of Zmean line
b1 = slope of Zmean line
b2 = htdifs among Z-lines
X = X – Xmean
Z-lines all have same slp(no interaction)
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y’ = b0 + b1X + b2Z
Z = Z – Zmean
2 quantitative predictors multiple regression
-20 -10 0 10 20 X
b0 = ht of Zmean line
b1 = slope of Zmean line
b2 = htdifs among Z-lines
X = X – Xmean
Z-lines all have same slp(no interaction)
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y’ = b0 + b1Xcen + b2Zcen + b3XZ
b0
b1
-b2Z=0
+1std Z
-1std Z
b2
Zcen = Z – Xmean
2 quantitative predictors w/ interaction
-20 -10 0 10 20 Xcen
a = ht of Zmean line
b1 = slope of Zmean line
b2 = htdifs among Z-lines
Xcen = X – Xmean ZX = Xcen * Zcen
b3
b3b3 = slpdifs among Z-lines
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y’ = b0 + b1Xcen + b2Zcen + b3XZ
Zcen = Z – Xmean
2 quantitative predictors w/ interaction
-20 -10 0 10 20 Xcen
a = ht of Zmean line
b1 = slope of Zmean line
b2 = htdifs among Z-lines
Xcen = X – Xmean ZX = Xcen * Zcen
b3 = slpdifs among Z-lines
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y’ = b0 + b1X + b2Z + b3V
b0
b1 b2Z=0
+1std Z
-1std Z
b2
Z = Z – Zmean
3 quantitative predictors multiple regression
-20 -10 0 10 20 X
b0 = ht of Zmean line
b1 = slope of Zmean line
b2 = htdifs among Z-lines
X = X – Xmean
Z-lines all have same slp(no interaction)
b3 = whole graph for one value of V
