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Regression Models with 2 quantitative variables (& Interaction)

This document delves into regression modeling involving two quantitative variables, focusing on both main effects and interaction terms. It outlines methods for centering variables to better understand their relationships and slopes without interaction, and how including an interaction term can affect the analysis. We explore the implications of having centered quantitative variables and the significance of regression coefficients, providing a detailed view of how these models operate and the interpretations of their slopes in different scenarios.

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Regression Models with 2 quantitative variables (& Interaction)

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  1. Regression Models with 2 quantitative variables (& Interaction) 2 quantitative variables main effects model with no interaction interaction model

  2. 2 centered quant vars y’ = b0+ b1x1+ b2x2 “X1” is a centered quantitative variable X1 X1 – X1mean “X2” is a centered quantitative variable X1 X1 – X1mean

  3. 2 centered quant vars y’ = b0+ b1x1+ b2x2 b0 mean of those in Cz with X=0 (mean) b1 slope of Y-X1 regression - slope same for all values of X2 no interaction b2 slope of Y-X2 regression - slope same for all values of X1 no interaction

  4. 3a Xs are centered quant variables (both are standardized std = 1) y’ = b0 + b1X1 + b2X2 b0 = ht of X2mean line b1 = slope of X2mean line +1std X2 b2 = htdifs among X2-lines 0 10 20 30 40 50 60 X2=0 X2-lines all have same slp (no interaction) -1std X2 -2 -1 0 1 2  X

  5. 2 centered quant var & their product term/interaction y’ = b0+ b1x1+ b2x2+ b3xz “X1” is a centered quantitative variable X1 X1 – X1mean “X2” is a centered quantitative variable X1 X1 – X1mean “XZ” represents the interaction of “X1” and”X2”

  6. 2 centered quant vars & their interaction y’ = b0+ b1x1+ b2x2+ b3xz b0 mean of those in Cz with X=0 (mean) b1 slope of Y-X1 regression - Simple Y-X1 slope for X2 mean=0 b2 slope of Y-X2 regression - Simple Y-X2 slope for X1 mean=0 b3  interaction of X1 & X2 - how slope of Y-X1 reg lines change with X2 value - how slope of Y-X2 reg lines change with X1 value

  7. 3b 2 quantitative predictors w/ interaction (both are standardized std = 1) 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 -2 -1 0 1 2  Xcen

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