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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.

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General Linear Models -- #1

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  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

  2. 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

  3. 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.

  4. 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 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

  5. 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

  6. 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 -20 -10 0 10 20  X

  7. Linear & nonlinear quantitative predictor  Nonlinear regression Xcen = X – Xmean Xquadratic = Xcen* Xcen y’ = b0 + b1Xcen + b2Xq b0 = ht of line b1 = slp of line b2 b2 = dif from linearity 0 10 20 30 40 50 60 b1 b0 -20 -10 0 10 20  Xcen

  8. Linear & nonlinear quantitative predictor  Nonlinear regression Xcen = X – Xmean Xquadratic = Xcen* Xcen y’ = b0 + b1Xcen + b2Xq b0 = ht of line b1 = slp of line b2 = dif from linearity 0 10 20 30 40 50 60 -20 -10 0 10 20  Xcen

  9. 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

  10. 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 0 10 20 30 40 50 60

  11. 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

  12. 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 b1 = htdif Cx & Tx1 b2 = htdif Cx & Tx2 0 10 20 30 40 50 60

  13. 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

  14. 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 Z-lines all have same slp (no interaction) -20 -10 0 10 20  X

  15. 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

  16. 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 b1 = slp of Cz line b2 = htdif Cz & Tz 0 10 20 30 40 50 60 b3 = slpdif Cz & Tz -20 -10 0 10 20  Xcen

  17. 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

  18. 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 = htdif Cz & Tz1 0 10 20 30 40 50 60 b3 = htdif Cz & Tz2 Z-lines all have same slp (no interaction) -20 -10 0 10 20  X

  19. 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

  20. 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 b0 = ht of Cz line b1 = slp of Cz line b2 = htdif Cz & Tz1 0 10 20 30 40 50 60 b4 = slpdif Cz & Tz1 b3 = htdif Cz & Tz2 b5 = slpdif Cz & Tz2 -20 -10 0 10 20  Xcen

  21. 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

  22. 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 b2 = htdifs among Z-lines 0 10 20 30 40 50 60 Z-lines all have same slp (no interaction) -20 -10 0 10 20  X

  23. 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

  24. 2 quantitative predictors w/ interaction Xcen = X – Xmean Zcen = Z – Xmean ZX = Xcen * Zcen y’ = b0 + b1Xcen + b2Zcen + b3XZ a = ht of Zmean line b1 = slope of Zmean line b2 = htdifs among Z-lines 0 10 20 30 40 50 60 b3 = slpdifs among Z-lines -20 -10 0 10 20  Xcen

  25. 3 quantitative predictors  multiple regression X = X – Xmean Z = Z – Zmean y’ = b0 + b1X + b2Z + b3V b0 = ht of Zmean line b1 = slope of Zmean line +1std Z b2 = htdifs among Z-lines b2 0 10 20 30 40 50 60 b3 = whole graph for one value of V Z=0 b1 b2 -1std Z b0 Z-lines all have same slp (no interaction) -20 -10 0 10 20  X

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