Plotting Non-linear & Complex Main Effects Models

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Presentation transcript:

Plotting Non-linear & Complex Main Effects Models Plotting single-predictor non-linear models Plotting 2-predictor non-linear models – 2xQ, kxQ & QxQ Plotting complex non-linear main effects models

Models with a single centered quantitative predictor Xcen = X – Xmean y’ = bXcen + a a  regression constant expected value of y when x = 0 (re-centered to mean=0) height of y-x regression line b  regression weight expected direction and extent of change in y for a 1-unit increase in x slope of y-x regression line X will be Xcen in all of the following models

Graphing & Interpreting Non-linear Models with a single centered quantitative predictor Xcen = X – Xmean Xcen2 = (X – Xmean)2 y’ = b1Xcen + b2Xcen2 + a a = ht of line b1 = slp of line at X=0 b2 = curve of line 0 10 20 30 40 50 60 +b2 +b1 a -20 -10 0 10 20  Xcen

Graphing & Interpreting Non-linear Models with a single centered quantitative predictor Xcen = X – Xmean Xcen2 = (X – Xmean)2 y’ = b1Xcen + b2Xcen2 + a a = ht of line b1 = slp of line at X=0 b2 = curve of line 0 10 20 30 40 50 60 +b2 b1 = 0 a -20 -10 0 10 20  Xcen

Graphing & Interpreting Non-linear Models with a single centered quantitative predictor Xcen = X – Xmean Xcen2 = (X – Xmean)2 y’ = b1Xcen + b2Xcen2 + a a = ht of line b1 = slp of line at X=0 -b1 b2 = curve of line a 0 10 20 30 40 50 60 -b2 -20 -10 0 10 20  Xcen

Plotting 2-Predictor Non-linear Models Now we get to the funer part – plotting multiple regression equations involving multiple variables. Three kinds (for now, more later) … centered quant variable & dummy-coded binary variable centered quant variable & dummy-coded k-category variable 2 centered quant variables What we’re trying to do is to plot these models so that we can see how both of the predictors are related to the criterion. Like when we’re plotting data from a factorial design, we have to represent 3 variables -- the criterion & the 2 predictors X & Z -- in a 2-dimensional plot. We’ll use the same solution.. We’ll plot the relationship between one predictor and the criterion for different values of the other predictor

Non-linear Models with a centered quantitative predictor Non-linear Models with a centered quantitative predictor & a dummy coded binary predictor y’ = b1X + b2X2 + b3Z + a This is called a main effects model  there are no interaction terms. a  regression constant expected value of y if X=0 (mean) and Z=0 (comparison group) mean of the control group height of control group quant-criterion regression line b1  regression weight for centered quant predictor – linear main effect of X expected direction and extent of change in y for a 1-unit increase in x , after controlling for the other variable(s) in the model expected direction and extent of change in y for a 1-unit increase in x , for the comparison group (coded 0) slope of quant-criterion regression line for the group coded 0 (comp)

Non-linear Models with a centered quantitative predictor Non-linear Models with a centered quantitative predictor & a dummy coded binary predictor y’ = b1X + b2X2 + b3Z + a This is called a main effects model  there are no interaction terms. b2  weight for centered & squared quant predictor – non-linear main effect of X expected direction and extent of change in y-x slope for a 1-unit increase in x , after controlling for the other variable(s) in the model expected direction and extent of change in y-x slope for a 1-unit increase in x , for the comparison group (coded 0) curve of quant-criterion regression line for the group coded 0 (comp) b3  regression weight for dummy coded binary predictor – main effect of Z expected direction and extent of change in y for a 1-unit increase in x, after controlling for the other variable(s) in the model direction and extent of y mean difference between groups coded 0 & 1, after controlling for the other variable(s) in the model group mean/reg line height difference (when X = 0, the centered mean)

For the Comparison Group coded Z = 0 To plot the model we need to get separate regression formulas for each Z group. We start with the multiple regression model… Model  y’ = b1X + b2X2 + b3Z + a For the Comparison Group coded Z = 0 Substitute the 0 in for Z Simplify the formula y’ = b1X + b2X2 + b30 + a y’ = b1X + b2X2 + a slope curve height For the Target Group coded Z = 1 Substitute the 1 in for Z Simplify the formula y’ = b1X + b2X2 + b31 + a y’ = b1X + b2X2 + ( b3 + a) slope curve height

Plotting & Interpreting Non-linear Models with a centered quantitative predictor & a dummy coded binary predictor y’ = b1X + b2X2 + b3Z + a This is called a main effects model  no interaction  the regression lines are parallel. Xcen = X – Xmean Xcen2 = (X – Xmean)2 Z = Tx(1) vs. Cx(0) a = ht of Cx line  mean of Cx b1 = slp of Cx line b2 Cx slp = Tx slp No interaction Tx 0 10 20 30 40 50 60 b2 = curve of Cx line b1 b3 Cx curv = Tx curv No interaction Cx a b3 = htdif Cx & Tx  Cx & Tx mean dif -20 -10 0 10 20  Xcen

Plotting & Interpreting Non-linear Models with a centered quantitative predictor & a dummy coded binary predictor y’ = b1X + b2X2 + b3Z + a This is called a main effects model  no interaction  the regression lines are parallel. Xcen = X – Xmean Xcen2 = (X – Xmean)2 Z = Tx(1) vs. Cx(0) a = ht of Cx line  mean of Cx a Cx -b3 b1 = slp of Cx line b1 Tx Cx slp = Tx slp No interaction b2 0 10 20 30 40 50 60 b2 = curve of Cx line Cx curv = Tx curv No interaction b3 = htdif Cx & Tx  Cx & Tx mean dif -20 -10 0 10 20  Xcen

Plotting & Interpreting Non-linear Models with a centered quantitative predictor & a dummy coded binary predictor y’ = b1X + b2X2 + b3Z + a This is called a main effects model  no interaction  the regression lines are parallel. Xcen = X – Xmean Xcen2 = (X – Xmean)2 Z = Tx(1) vs. Cx(0) a = ht of Cx line  mean of Cx b1 = slp of Cx line Cx slp = Tx slp No interaction Tx b1 = 0 b2 = 0 0 10 20 30 40 50 60 b2 = curve of Cx line b3 Cx curv = Tx curv No interaction Cx a a b3 = htdif Cx & Tx  Cx & Tx mean dif -20 -10 0 10 20  Xcen

Non-linear models with a centered quantitative predictor & Non-linear models with a centered quantitative predictor & a dummy coded k-category predictor y’ = b1X + b2X2 + b3 Z1 + b4 Z2 + a This is called a main effects model  there are no interaction terms. a  regression constant expected value of y if Z1 & Z2 = 0 (the control group) mean of the control group height of control group quant-criterion regression line Group Z1 Z2 1 1 0 2 0 1 3* 0 0 b1  regression weight for centered quant predictor – main effect of X expected direction and extent of change in y for a 1-unit increase in x slope of quant-criterion regression line (for both groups) b2  weight for centered & squared quant predictor – non-linear main effect of X expected direction and extent of change in y-x slope for a 1-unit increase in x , after controlling for the other variable(s) in the model expected direction and extent of change in y-x slope for a 1-unit increase in x , for the comparison group (coded 0) curve of quant-criterion regression line for the group coded 0 (comp)

Non-linear models with a centered quantitative predictor & Non-linear models with a centered quantitative predictor & a dummy coded k-category predictor This is called a main effects model  there are no interaction terms. y’ = b1X + b2X2 + b3 Z1 + b4 Z2 + a Group Z1 Z2 1 1 0 2 0 1 3* 0 0 b3  regression weight for dummy coded comparison of G1 vs G3 – main effect expected direction and extent of change in y for a 1-unit increase in x direction and extent of y mean difference between groups 1 & 3 group height difference between comparison & target groups (3 & 1) (X = 0) b4  regression weight for dummy coded comparison of G2 vs. G3 – main effect 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 0 & 1 group height difference between comparison & target groups (3 & 2) (X = 0)

Model  y’ = b1X + b2X2 + b3 Z1 + b4 Z2 + a To plot the model we need to get separate regression formulas for each Z group. We start with the multiple regression model… Group Z1 Z2 1 1 0 2 0 1 3* 0 0 Model  y’ = b1X + b2X2 + b3 Z1 + b4 Z2 + a For the Comparison Group - coded Z1= 0 & Z2 = 0 Substitute the Z code values y’ = b1X + b2X2 + b2*0 + b3*0 + a Simplify the formula y’ = b1X + b2X2 + a slope curve height For the Target Group 1 - coded Z1= 1 & Z2 = 0 Substitute the Z code values y’ = b1X + b2X2 + b2*1 + b3*0 + a Simplify the formula y’ = b1X + b2X2 + (b2 + a) slope curve height For the Target Group 2- coded Z1= 0 & Z2 = 1 Substitute the Z code values y’ = b1X + b2X2 + b2*0 + b3*1 + a Simplify the formula y’ = b1X + b2X2 + (b3 + a) slope curve height

Plotting & Interpreting Non-linear Models with a centered quantitative predictor & a dummy coded k-category predictor This is called a main effects model  no interaction  the regression lines are parallel. y’ = b1X + b2X2 + b3 Z1 + b4 Z2 + a Z1 = Tx1 vs. Cx(0) a = ht of Cx line  mean of Cx Xcen = X – Xmean Z2 = Tx2 vs. Cx (0) a Cx b1 = slp of Cx line -b3 Cx slp = Tx1 slp = Tx1 slp No interaction b1 -b4 Tx1 b2 = curve of Cx line Cx crv = Tx1 crv = Tx1 crv No interaction 0 10 20 30 40 50 60 Tx2 b2 b2 = htdif Cx & Tx1  Cx & Tx1 mean dif b3 = htdif Cx & Tx2  Cx & Tx2 mean dif -20 -10 0 10 20  Xcen

Plotting & Interpreting Non-linear Models with a centered quantitative predictor & a dummy coded k-category predictor This is called a main effects model  no interaction  the regression lines are parallel. y’ = b1X + b2X2 + b3 Z1 + b4 Z2 + a Z1 = Tx1 vs. Cx(0) a = ht of Cx line  mean of Cx Xcen = X – Xmean Z2 = Tx2 vs. Cx (0) Tx1 b1 = slp of Cx line Cx Cx slp = Tx1 slp = Tx1 slp No interaction a b2 = curve of Cx line Tx2 Cx crv = Tx1 crv = Tx1 crv No interaction 0 10 20 30 40 50 60 b3 b2 = htdif Cx & Tx1  Cx & Tx1 mean dif -b4 b1 b2 b3 = htdif Cx & Tx2  Cx & Tx2 mean dif -20 -10 0 10 20  Xcen

Plotting & Interpreting Non-linear Models with a centered quantitative predictor & a dummy coded k-category predictor This is called a main effects model  no interaction  the regression lines are parallel. y’ = b1X + b2X2 + b3 Z1 + b4 Z2 + a Z1 = Tx1 vs. Cx(0) a = ht of Cx line  mean of Cx Xcen = X – Xmean Z2 = Tx2 vs. Cx (0) b1 = slp of Cx line b2 = 0 Cx slp = Tx1 slp = Tx1 slp No interaction -b1 b2 = curve of Cx line b4 b3 Cx crv = Tx1 crv = Tx1 crv No interaction 0 10 20 30 40 50 60 Tx1 Tx2 b2 = htdif Cx & Tx1  Cx & Tx1 mean dif a Cx b3 = htdif Cx & Tx2  Cx & Tx2 mean dif -20 -10 0 10 20  Xcen

a dummy coded binary predictor or a dummy coded k-category predictor weights for non-linear main effects models w/ centered quantitative predictor & a dummy coded binary predictor or a dummy coded k-category predictor Constant “a” the expected value of y when the value of all predictors = 0 height of the Y-X regression line for the comparison group b for a centered quantitative variable – main effect 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 slope of the Y-X regression line for the comparison group b for a centered & squared quantitative variable – main effect the direction and extent of the expected change in the y-x for a 1-unit increase in x, holding the value of all other predictors constant at 0 curve of the Y-X regression line for the comparison group b for a dummy coded binary variable -- main effect the direction and extent of expected mean difference of the Target group from the Comparison group, holding the value of all other predictors constant at 0 difference in height of the Y-X regression lines for the comparison & target groups comparison & target groups Y-X regression lines have same slope – no interaction b for a dummy coded k-group variable – main effect direction and extent of expected mean difference of Target group for that dummy code from Comparison group, holding the value of all other predictors constant at 0

Non-linear models with 2 centered quantitative predictors This is called a main effects model  there are no interaction terms. y’ = b1X1 + b2X12 + b3 X2 + b4X22 + a Same idea … we want to plot these models so that we can see how both of the predictors are related to the criterion Different approach … when the second predictor was binary or had k-categories, we plotted the Y-X regression line for each Z group now, however, we don’t have any groups – both the X1 & X2 variables are centered quantitative variables what we’ll do is to plot the Y-X1 regression line for different values of X2 the most common approach is to plot the Y-X1 regression line for… the mean of X2 +1 std above the mean of X2 -1 std below the mean of X2 We’ll plot 3 lines

Non-linear Models with 2 centered quantitative predictors This is called a main effects model  there are no interaction terms. y’ = b1X1 + b2X12 + b3 X2 + b4X22 + a a  regression constant expected value of y if X1=0 (mean) X2=0 (mean) mean criterion score of those having X1=0 (mean) X2=0 (mean) height of quant-criterion regression line for those with X2=0 (mean) b1  regression weight for X1 centered quant predictor expected direction and extent of change in y for a 1-unit increase in X1, after controlling for the other variable(s) in the model expected direction and extent of change in y for a 1-unit increase in X2 , for those with X2=0 (mean) slope of quant-criterion regression line when X2=0 (mean) b2  weight for centered & squared quant predictor – non-linear main effect of X1 expected direction and extent of change in y-x slope for a 1-unit increase in X1, after controlling for the other variable(s) in the model expected direction and extent of change in y-x slope for a 1-unit increase in X1 , for those with X2=0 (mean) curve of quant-criterion regression line when X2 =0 (mean)

Non-linear Models with 2 centered quantitative predictors This is called a main effects model  there are no interaction terms. y’ = b1X1 + b2X12 + b3 X2 + b4X22 + a b3  regression weight for X2 centered quant predictor expected direction and extent of change in y for a 1-unit increase in X2, after controlling for the other variable(s) in the model expected direction and extent of change in y for a 1-unit increase in X2, for those with X1=0 (mean) expected direction and extent of change of the height of Y-X regression line for a 1-unit change in X2 when X1=0 (mean) b4  weight for centered & squared quant predictor – non-linear main effect of X2 expected direction and extent of change in y-x slope for a 1-unit increase in X2, after controlling for the other variable(s) in the model expected direction and extent of change in y-x slope for a 1-unit increase in X2, for those with X1=0 (mean) curve of quant-criterion regression line when X1 =0 (mean)

For X2 = 0 (the mean of centered X2) Substitute the 0 in for X2 To plot the model we need to get separate regression formulas for each chosen value of Z. Start with the multiple regression model.. y’ = b1X1 + b2X12 + b3 X2 + b4X22 + a For X2 = 0 (the mean of centered X2) Substitute the 0 in for X2 Simplify the formula y’ = b1X1 + b2X12 + b3 0 + b40 + a y’ = b1X + b2X12 + a height curve slope For X2 = +1 std Substitute the std for X2 Simplify the formula y’ = b1X + b2X12 + b2*std + b4std2 + a y’ = b1X + b2X12 + ( b2*std + b4std2 + a) slope curve height For X2 = -1 std y’ = b1X + b2X12 - b2*std - b4std2 + a y’ = b1X + b2X12 + ( -b2*std - b4std2 + a) Substitute the std for X2 Simplify the formula slope curve height

+linear +linear & inv-U It can be a challenge to “see” the shape of the variable represented with the different lines for different values… 5040 30 20 10 0 5040 30 20 10 0 1std mean -1std X1 +linear -1std mean 1std X1 1std mean -1std X1 5040 30 20 10 0 5040 30 20 10 0 +linear & inv-U -1std mean 1std X1

+linear & U -linear & inv-U 5040 30 20 10 0 5040 30 20 10 0 X1 1std mean -1std 5040 30 20 10 0 5040 30 20 10 0 -1std mean 1std X1 X1 Mean -1std 1std -linear & inv-U -1std mean 1std X1

Plotting & Interpreting Non-linear Models with 2 centered quantitative predictors This is called a main effects model  no interaction  the regression lines are parallel. a = ht of X2mean line y’ = b1X1cen + b2X12cen + b3X2cen + b4X22cen + a b1 = slp of X2mean line X1cen = X1 – X1mean X2cen = X2 – X2mean 0 slp = +1std slp = -1std slp No interaction a b2 = curve of X2=mean line +1std X2 0 crv = +1crv slp = -1crv slp No interaction b1 X2=0 b3 & -b4 b3 = linear htdifs among X2-lines 0 10 20 30 40 50 60 -1std X2 Lin ht dif same all X1 values No interaction b2 b4 = quad htdifs among X2-lines quad ht dif same all X1 values No interaction -20 -10 0 10 20  X2cen

Plotting & Interpreting Non-linear Models with 2 centered quantitative predictors This is called a main effects model  no interaction  the regression lines are parallel. a = ht of X2mean line y’ = b1X1cen + b2X12cen + b3X2cen + b4X22cen + a b1 = slp of X2mean line X1cen = X1 – X1mean X2cen = X2 – X2mean 0 slp = +1std slp = -1std slp No interaction b2 = curve of X2=mean line +1std X2 0 crv = +1crv slp = -1crv slp No interaction a X2=0 b3 b3 = linear htdifs among X2-lines b1 b4 = 0 0 10 20 30 40 50 60 b3 -1std X2 Lin ht dif same all X1 values No interaction b2 = 0 b4 = quad htdifs among X2-lines quad ht dif same all X1 values No interaction -20 -10 0 10 20  X2cen

Plotting Complex Linear Main Effects Models Sometimes we have a model with more than 2 predictors, and sometimes we want to plot these more complex models. There is only so much that can be put into a single plot and reasonably hope the reader will be able to understand it. Remember, with main effects models (with no interactions), all lines are parallel!!! So, our plots are a series of lines, each for a different combination of variable groups/values. The most common version of these plots is to show how 2 variables relate to the criterion, for a given set of values for the other variables (usually their mean) Here’s an example!

So, we have this model… perf’ = b1age + b2 mar1 + b3mar2 + b4sex + b5exp + a age & exp(erience) are centered quantitative variables sex is dummy coded 0 = male 1 = female mar(ital status) is dummy coded mar1 single = 1 mar2 divorced = 1 married = 0 To plot this we have to decide what we want to show, say… How are experience & marital status related to performance for 25-year-old females? All we do is fill in the values of the “selection variables” and simplify the formula to end up with a plotting function for each regression line… For this plot: performance (the criterion) is on the Y axis experience (a quantitative predictor) is on the X axis we’ll have 3 regression lines, one each for single, married & divorced

How are experience & marital status related to perf for 25-year-old females? perf’ = b1age + b2 mar1 + b3mar2 + b4sex + b5exp + a sex 0 = male 1 = female mar mar1 single = 1 mar2 divorced = 1 married = 0 For married  perf’ = b1*25 + b2*0 + b3*0 + b4*1 + b5exp + a we plot perf’ = b5exp + ( b4*1 + b1*25 + a) height slope For singles  perf’ = b1*25 + b2*1 + b3*0 + b4*1 + b5exp + a we plot perf’ = b5exp + (b2 + b4*1 + b1*25 + a) height slope For divorced  perf’ = b1*25 + b2*0 + b3*1 + b4*1 + b5exp + a we plot perf’ = b5exp + (b3 + b4*1 + b1*25 + a) height slope

Let’s look at the plotting formulas again How are experience & marital status related to perf for 25-year-old females? perf’ = b1age + b2 mar1 + b3mar2 + b4sex + b5exp + a sex 0 = male 1 = female mar mar1 single = 1 mar2 divorced = 1 married = 0 Let’s look at the plotting formulas again For married  perf’ = b5exp + (b4*1 + b1*25 + a) For singles  perf’ = b5exp + (b2 + b4*1 + b1*25 + a) For divorced  perf’ = b5exp + (b3 + b4*1 + b1*25 + a) Notice that each group’s regression line has the same slope (b5), because there is no interaction. The difference between the group’s regression lines are their height – which differ based on the relationship between marital status and perf !!!

perf’ = b125 + b2 mar1 + b3mar2 + b4female + b5exp + a How are experience & marital status related to perf for 25-year-old females? perf’ = b125 + b2 mar1 + b3mar2 + b4female + b5exp + a The selection of the value of Age (b1) & Gender (b4) determine the height of the set of lines! Single b2 0 10 20 30 40 50 60 -b3 Married b5 Divorced a -20 -10 0 10 20  Expcen

Plotting Complex Non-Linear Main Effects Models Sometimes we have a non-linear model with more than 2 predictors, and sometimes we want to plot these more complex models. There is only so much that can be put into a single plot and reasonably hope the reader will be able to understand it. Remember, with main effects models (with no interactions), all lines are parallel!!! So, our plots are a series of lines, each for a different combination of variable groups/values. The most common version of these plots is to show how 2 variables relate to the criterion, for a given set of values for the other variables (usually their mean) Here’s an example!

So, we have this model… perf’ = b1age + b2 mar1 + b3mar2 + b4sex + b5exp + a age & exp(erience) are centered quantitative variables sex is dummy coded 0 = male 1 = female mar(ital status) is dummy coded mar1 single = 1 mar2 divorced = 1 married = 0 To plot this we have to decide what we want to show, say… How are experience & marital status related to performance for 25-year-old females? All we do is fill in the values of the “selection variables” and simplify the formula to end up with a plotting function for each regression line… For this plot: performance (the criterion) is on the Y axis experience (a quantitative predictor) is on the X axis we’ll have 3 regression lines, one each for single, married & divorced

How are experience & marital status related to perf for 25-year-old females? perf’ = b1age + b2age2 + b3 mar1 + b4mar2 + b5sex + b6exp + b7exp2 + a sex 0 = male 1 = female mar mar1 single = 1 mar2 divorced = 1 married = 0 For married  perf’ = b125 + b2252 + b3 0 + b40 + b51+ b6exp + b7exp2 + a we plot perf’ = b6exp + b7exp2 + ( b5*1 + b1*25 + b2252 + a) slope curve height For singles  perf’ = b125 + b2252 + b3 1 + b40 + b51+ b6exp + b7exp2 + a we plot perf’ = b6exp + b7exp2 + (b3 + b5*1 + b1*25 + b2252 + a) slope curve height For divorced  perf’ = b125 + b7252 + b3 0 + b41 + b51+ b6exp + b7exp2 + a we plot perf’ = b6exp + b7exp2 + (b4 + b5*1 + b1*25 + b2252 + a) slope curve height

Let’s look at the plotting formulas again How are experience & marital status related to perf for 25-year-old females? perf’ = b1age + b2age2 + b3 mar1 + b4mar2 + b5sex + b6exp + b7exp2 + a sex 0 = male 1 = female mar mar1 single = 1 mar2 divorced = 1 married = 0 Let’s look at the plotting formulas again For married  perf’ = b6exp + b7exp2 + ( b5*1 + b1*25 + b2252 + a) For singles  perf’ = b6exp + b7exp2 + (b3 + b5*1 + b1*25 + b2252 + a) For divorced  perf’ = b6exp + b7exp2 + (b4 + b5*1 + b1*25 + b2252 + a) Notice that each group’s regression line has the same slope (b6), because there is no interaction. Notice that each group’s regression line has the same curve (b7), because there is no interaction. The difference between the group’s regression lines are their height – which differ based on the relationship between marital status and perf !!!