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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.

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Presentation on theme: "General Linear Models -- #1 things to remember b weight interpretations 1 quantitative predictor 1 quantitative predictor & non-linear component 1 2-group."— Presentation transcript:

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 interpretingnonsense 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 group for that dummy code from the slope of the Comparison group, holding the value of all other predictors constant at 0

5 0 10 20 30 40 50 60 y’ = b 0 + b 1 X b0b0 b1b1 -20 -10 0 10 20  X b 0 = ht of line b 1 = slp of line X = X – X mean Single quantitative predictor (X)  Bivariate Regression

6 0 10 20 30 40 50 60 y’ = b 0 + b 1 X -20 -10 0 10 20  X b 0 = ht of line b 1 = slp of line X = X – X mean Single quantitative predictor (X)  Bivariate Regression

7 0 10 20 30 40 50 60 y’ = b 0 + b 1 X cen + b 2 X q b0b0 b1b1 Linear & nonlinear quantitative predictor  Nonlinear regression -20 -10 0 10 20  X cen X quadratic = X cen * X cen b2b2 X cen = X – X mean b 0 = ht of line b 1 = slp of line b 2 = dif from linearity

8 0 10 20 30 40 50 60 y’ = b 0 + b 1 X cen + b 2 X q Linear & nonlinear quantitative predictor  Nonlinear regression -20 -10 0 10 20  X cen X quadratic = X cen * X cen X cen = X – X mean b 0 = ht of line b 1 = slp of line b 2 = dif from linearity

9 0 10 20 30 40 50 60 Cx Tx 2-group predictor (Tx Cx)  2-grp ANOVA b 0 = ht Cx b 1 = htdif Cx & Tx X Tx = 1 Cx = 0 X = Tx vs. Cx b0b0 b1b1 y’ = b 0 + b 1 X

10 0 10 20 30 40 50 60 2-group predictor (Tx Cx)  2-grp ANOVA b 0 = ht Cx b 1 = htdif Cx & Tx X Tx = 1 Cx = 0 X = Tx vs. Cx y’ = b 0 + b 1 X

11 0 10 20 30 40 50 60 b2b2 Cx Tx 2 Tx 1 b 0 = ht Cx b 1 = htdif Cx & Tx 1 b 2 = htdif Cx & Tx 2 3-group predictor (Tx 1 Tx 2 Cx)  k-grp ANOVA y’ = b 0 + b 1 X 1 + b 2 X 2 X1 Tx1=1 Tx2=0 Cx=0 X2 Tx1=0 Tx2=1 Cx=0 X 1 = Tx 1 vs. CxX 2 = Tx 2 vs. Cx b0b0 b1b1

12 0 10 20 30 40 50 60 b 0 = ht Cx b 1 = htdif Cx & Tx 1 b 2 = htdif Cx & Tx 2 3-group predictor (Tx 1 Tx 2 Cx)  k-grp ANOVA y’ = b 0 + b 1 X 1 + b 2 X 2 X1 Tx1=1 Tx2=0 Cx=0 X2 Tx1=0 Tx2=1 Cx=0 X 1 = Tx 1 vs. CxX 2 = Tx 2 vs. Cx

13 0 10 20 30 40 50 60 y’ = b 0 + b 1 X + b 2 Z b0b0 b1b1 b2b2 Cz Tz quantitative (X) & 2-group (Tz Cz) predictors  2-grp ANCOVA -20 -10 0 10 20  X b 0 = ht of Cz line b 1 = slp of Cz line b 2 = htdif Cz & Tz X = X – X mean Z Tz = 1 Cz = 0 Z = Tz vs. Cz Z-lines all have same slp (no interaction)

14 0 10 20 30 40 50 60 y’ = b 0 + b 1 X + b 2 Z quantitative (X) & 2-group (Tz Cz) predictors  2-grp ANCOVA -20 -10 0 10 20  X b 0 = ht of Cz line b 1 = slp of Cz line b 2 = htdif Cz & Tz X = X – X mean Z Tz = 1 Cz = 0 Z = Tz vs. Cz Z-lines all have same slp (no interaction)

15 0 10 20 30 40 50 60 b0b0 b1b1 b2b2 Cz Tz -20 -10 0 10 20  X cen XZ = X cen * Z b3b3 b 0 = ht of Cz line b 1 = slp of Cz line b 2 = htdif Cz & Tz b 3 = slpdif Cz & Tz quantitative (X) & 2-group (Tz Cz) predictors w/ interaction y’ = b 0 + b 1 X + b 2 Z + b 3 XZ X = X – X mean Z Tz = 1 Cz = 0 Z = Tz vs. Cz

16 0 10 20 30 40 50 60 -20 -10 0 10 20  X cen XZ = X cen * Z b 0 = ht of Cz line b 1 = slp of Cz line b 2 = htdif Cz & Tz b 3 = slpdif Cz & Tz quantitative (X) & 2-group (Tz Cz) predictors w/ interaction y’ = b 0 + b 1 X + b 2 Z + b 3 XZ X = X – X mean Z Tz = 1 Cz = 0 Z = Tz vs. Cz

17 0 10 20 30 40 50 60 b0b0 b1b1 b2b2 Cz Tz2 Tz1 b3b3 -20 -10 0 10 20  X b 0 = ht of Cz line b 2 = htdif Cz & Tz 1 b 3 = htdif Cz & Tz 2 b 1 = slp of Cz line y’ = b 0 +b 1 X + b 2 Z 1 + b 3 Z 2 Z 1 = Tz 1 vs. CzZ 2 = Tz 2 vs. Cz X = X – X mean Z 1 Tz 1 =1 Tz 2 =0 Cx=0 Z 2 Tz 1 =0 Tx 2 =1 Cx=0 quantitative (X) & 3-group (Tz 1 Tz 2 Cz) predictors  3-grp ANCOVA Z-lines all have same slp (no interaction)

18 0 10 20 30 40 50 60 -20 -10 0 10 20  X b 0 = ht of Cz line b 2 = htdif Cz & Tz 1 b 3 = htdif Cz & Tz 2 b 1 = slp of Cz line y’ = b 0 +b 1 X + b 2 Z 1 + b 3 Z 2 Z 1 = Tz 1 vs. CzZ 2 = Tz 2 vs. Cz X = X – X mean Z 1 Tz 1 =1 Tz 2 =0 Cx=0 Z 2 Tz 1 =0 Tx 2 =1 Cx=0 quantitative (X) & 3-group (Tz 1 Tz 2 Cz) predictors  3-grp ANCOVA Z-lines all have same slp (no interaction)

19 0 10 20 30 40 50 60 y’ = b 0 + b 1 X cen + b 2 Z 1 + b 3 Z 2 + b 4 XZ 1 + b 5 XZ 2 b0b0 b1b1 b2b2 Cx Tx1 Tx2 b3b3 -20 -10 0 10 20  X cen b 0 = ht of Cz line b 2 = htdif Cz & Tz 1 b 3 = htdif Cz & Tz 2 b 1 = slp of Cz line b 4 = slpdif Cz & Tz 1 b 5 = slpdif Cz & Tz 2 XZ 1 = X cen * Z 1 XZ 2 = X cen * Z 2 b4b4 b5b5 Models with quant (X) & 3-group (Tz 1 Tz 2 Cz) predictors w/ interaction Z 1 = Tz 1 vs. CzZ 2 = Tz 2 vs. Cz X = X – X mean Z 1 Tz 1 =1 Tz 2 =0 Cx=0 Z 2 Tz 1 =0 Tx 2 =1 Cx=0

20 0 10 20 30 40 50 60 y’ = b 0 + b 1 X cen + b 2 Z 1 + b 3 Z 2 + b 4 XZ 1 + b 5 XZ 2 -20 -10 0 10 20  X cen b 0 = ht of Cz line b 2 = htdif Cz & Tz 1 b 3 = htdif Cz & Tz 2 b 1 = slp of Cz line b 4 = slpdif Cz & Tz 1 b 5 = slpdif Cz & Tz 2 XZ 1 = X cen * Z 1 XZ 2 = X cen * Z 2 Models with quant (X) & 3-group (Tz 1 Tz 2 Cz) predictors w/ interaction Z 1 = Tz 1 vs. CzZ 2 = Tz 2 vs. Cz X = X – X mean Z 1 Tz 1 =1 Tz 2 =0 Cx=0 Z 2 Tz 1 =0 Tx 2 =1 Cx=0

21 0 10 20 30 40 50 60 y’ = b 0 + b 1 X + b 2 Z b0b0 b1b1 b 2 Z=0 +1 std Z -1 std Z b2b2 Z = Z – Z mean 2 quantitative predictors  multiple regression -20 -10 0 10 20  X b 0 = ht of Z mean line b 1 = slope of Z mean line b 2 = htdifs among Z-lines X = X – X mean Z-lines all have same slp (no interaction)

22 0 10 20 30 40 50 60 y’ = b 0 + b 1 X + b 2 Z Z = Z – Z mean 2 quantitative predictors  multiple regression -20 -10 0 10 20  X b 0 = ht of Z mean line b 1 = slope of Z mean line b 2 = htdifs among Z-lines X = X – X mean Z-lines all have same slp (no interaction)

23 0 10 20 30 40 50 60 y’ = b 0 + b 1 X cen + b 2 Z cen + b 3 XZ b0b0 b1b1 -b 2 Z=0 +1 std Z -1 std Z b2b2 Z cen = Z – X mean 2 quantitative predictors w/ interaction -20 -10 0 10 20  X cen a = ht of Z mean line b 1 = slope of Z mean line b 2 = htdifs among Z-lines X cen = X – X mean ZX = X cen * Z cen b3b3 b3b3 b 3 = slpdifs among Z-lines

24 0 10 20 30 40 50 60 y’ = b 0 + b 1 X cen + b 2 Z cen + b 3 XZ Z cen = Z – X mean 2 quantitative predictors w/ interaction -20 -10 0 10 20  X cen a = ht of Z mean line b 1 = slope of Z mean line b 2 = htdifs among Z-lines X cen = X – X mean ZX = X cen * Z cen b 3 = slpdifs among Z-lines

25 0 10 20 30 40 50 60 y’ = b 0 + b 1 X + b 2 Z + b 3 V b0b0 b1b1 b 2 Z=0 +1 std Z -1 std Z b2b2 Z = Z – Z mean 3 quantitative predictors  multiple regression -20 -10 0 10 20  X b 0 = ht of Z mean line b 1 = slope of Z mean line b 2 = htdifs among Z-lines X = X – X mean Z-lines all have same slp (no interaction) b 3 = whole graph for one value of V

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