Plotting & Interpreting Multiple Regression ModelsSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Plotting & Interpreting Multiple Regression Models•Why plot MR models?•Interpretive review & extensions •Plotting single-predictor models–centered quant, binary or k-category predictors•Plotting two-predictor models–centered quant & binary, k-category or centered quant•Plotting multiple-predictor modelsSo far we have emphasized the clear interpretation of regression weights for each type of predictor.Many of the “more complicated” regression models, such as ANCOVA or those with non-linear or interaction terms are plotted, because, as you know, “a picture is worth a lot of words”.Each term in a multiple regression model has an explicit representation in a regression plot as well as an explicit interpretation – usually with multiple parallel phrasing versions. So, we’ll start simple with simple models and learn the correspondence between:• interpretation of each regression model term• the graphical representation of that termCoding & Transforming predictors for MR models• Categorical predictors will be converted to dummy codes• comparison/control group coded 0• @ other group a “target group” of one dummy code, coded 1• Quantitative predictors will be centered, usually to the mean• centered = score – mean (like for quadratic terms)• so, mean = 0Why?Mathematically – 0s (as control group & mean) simplify the math & minimize collinearity complicationsInterpretively – the “controlling for” included in multiple regression weight interpretations is really “controlling for all other variables in the model at the value 0” – “0” as the comparison group & mean will make b interpretations simpler and more meaningfulVery important things to remember…1) 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 2) The interpretation of regression weights in a main effects model is different than in a model including interactions• regression weights reflect “main effects” in a maineffects model (without interactions)• regression weights reflect “simple effects” in a modelincluding interactionsy’ = bX + aModels with a single quantitative predictora  regression constant• expected value of y if x = 0• height of predictor-criterion regression lineb  regression weight• expected direction and extent of change in y for a 1-unit increase in x • slope of line0 10 20 30 40 50 60  Yy’ = bX + aa+ b0 10 20 30 40  X a = height of lineb = slope of lineGraphing & InterpretingModels with a single quantitative predictory’ = bXcen + aModels with a single centered quantitative predictora  regression constant• expected value of y when x = 0 (x = mean )• height of predictor-criterion regression lineb  regression weight• expected direction and extent of change in y for a 1-unit increase in x • slope of predictor-criterion regression line Xcen = X – XmeanX will be Xcen in all of the following modelsy’ = bXcen + aModels with a single centered quantitative predictor Xcen = X – XmeanSo, how do we plot this formula?Simple  pick 2 values of xcen, substitute them into the formula to get y’ values and plot the line defined by those two x-y pointsWhat x values?• doesn’t matter -- so keep it simple…• 0 & 1, 0 & 10, 0 & 100, depending on the x-scale• +/- 2 std isn’t as simple, but tells you what x & y ranges are needed on the plotCouple of things to remember…• “0” is the center of the x-axis – X has been centered !!!• the x-axis should extend about +/- 2 Std (include 96% of pop)y’ = 1.5X + 30Models with a single centered quantitative predictor0 10 20 30 40 50 60 70-20 -10 0 10 20  Xcen For xcen = 0  1.6*0 + 30 y’ = 30For xcen = 10  1.6*10 + 30 y’ = 46If X has mean = 42 & std = 7.5For +2std  1.6*15 + 30 y’ = 54For -2std  1.6*-15 + 30 y’ = 6Any set of x values will lead to the same plotted line!20 30 40 50 60  X We can even substitute the original x scale back into the graph !!0 10 20 30 40 50 60a (x = mean)+b-20 -10 0 10 20  Xcen a = ht of lineb = slp of lineGraphing & Interpreting Models with a single centered quantitative predictor Xcen = X – Xmeany’ = bXcen + a0 10 20 30 40 50 60b = 0-20 -10 0 10 20  Xcen a = ht of lineb = slp of linea (x = mean)Graphing & Interpreting Models with a single centered quantitative predictor Xcen = X – Xmeany’ = bXcen + a-20 -10 0 10 20  Xcen 0 10 20 30 40 50 60 -ba = ht of lineb = slp of linea (x = mean)Graphing & Interpreting Models with a single centered quantitative predictor Xcen = X – Xmeany’ = -bXcen + ay’ = bX + aModels with a single binary predictor coded 1-2a  regression constant• expected value of y if x = 0 (no one has this value – all coded 1 or 2)• height of lineb  regression weight• expected direction and extent of change in y for a 1-unit increase in x • direction and extent of y mean difference between groups coded 1 & 2• slope of line0 10 20 30 40 50 60y’ = bZ + a Control Tx1 Cx = 1 Tx = 2Z = Tx1 vs. CxPlotting & Interpretingmodels with a single binary predictor coded 1-2+b0 10 20 30 40 50 60y’ = bZ + a Control Tx1 Cx = 1 Tx = 2Z = Tx1 vs. CxPlotting & Interpretingmodels with a single binary predictor coded 1-2b = 00 10 20 30 40 50 60y’ = -bZ + a Control Tx1 Cx = 1 Tx = 2Z = Tx1 vs. CxPlotting & Interpretingmodels with a single binary predictor coded 1-2-by’ = bZ + aa  regression constant• expected value of y if z