Introduction to the multiple linear regression modelIs brain and body size predictive of intelligence?Scatter plot matrixSlide 4Marginal response plotsSlide 6A potential multiple linear regression modelPotential research questionsPredictorsTermsTypes of termsSimple linear regression model with a transformed predictorVisualizing simple linear regression model with a transformed predictorA first order model with two predictorsVisualizing a first order model with two predictorsA first order model with more than 2 predictorsVisualizing a first order model with more than 2 predictorsA second order polynomial model with one predictorVisualizing a second order polynomial model with one predictorA second order polynomial model with 2 predictorsVisualizing second order polynomial model with 2 predictorsA first order model with one binary predictorVisualizing a first order model with one binary predictorIntroduction to the multiple linear regression model Regression models with more than one predictor (or term)Is brain and body size predictive of intelligence?•Sample of n = 38 college students•Response (Y): intelligence based on PIQ (performance) scores from the (revised) Wechsler Adult Intelligence Scale.•Potential predictor (x1): Brain size based on MRI scans (given as count/10,000).•Potential predictor (x2): Height in inches.•Potential predictor (x3): Weight in pounds.Example 1130.591.5100.72886.28373.2565.75130.591.5170.5127.5100.72886.28373.2565.75170.5127.5PIQMRIHeightWeightScatter plot matrixExample 1Scatter plot matrix•Tells us about 2D marginal relationships between each pair of variables without regard to other variables.•The challenge is how the 2D relationships relate to how the response y depends on all 3 predictors simultaneously.130.591.5100.72886.28373.2565.75130.591.5170.5127.5100.72886.28373.2565.75170.5127.5PIQMRIHeightWeightMarginal response plotsExample 1Marginal response plots•Scatter plot of response y vs. each predictor.•Suggest how response y depends on each predictor without regard to other predictors.•Provide a visual lower bound for the goodness-of-fit that can be achieved by the full regression model.A potential multiple linear regression model iiiiixxxY3322110where …• Yi is intelligence (PIQ) of student i • xi1 is brain size (MRI) of student i • xi2 is height (Height) of student i • xi3 is weight (Weight) of student i Example 1and … the independent error terms i follow a normal distribution with mean 0 and equal variance 2.Potential research questions•Which predictors explain some of the variation in PIQ?•What is the effect of brain size on PIQ?•What is the PIQ of an individual with a given brain size, height, and weight?Example 1Predictors•As before, the x variable. Also, called explanatory variables or independent variables.•Most often numerical measurements, such as age, weight, length, and temperature.•But, can be categorical, such as gender, race, and species.TermsTerms are functions of the predictor variables, such as:211xxu 212xu 23log xue14xu iiuuuuY443322110Linear regression model as function of terms:ieixxxxxY14232122110logTypes of terms•The predictors themselves.•Powers of predictors.•Transformations of predictors.•Interactions.•Binary (or categorical) predictors.Simple linear regression model with a transformed predictor iiixY1010logwhere …• Yi is proportion of items correctly recalled for person i• xi is time since person i initially memorized the list and … the independent error terms i follow a normal distribution with mean 0 and equal variance 2.Visualizing simple linear regression model with a transformed predictor0 1 2 3 40.00.10.20.30.40.50.60.70.80.9log10timepropprop = 0.846415 - 0.182427 log10timeS = 0.0233881 R-Sq = 99.0 % R-Sq(adj) = 98.9 %Regression PlotA first order model with two predictors iiiixxY22110where …• Yi is life of power cell i (number of cycles)• xi1 is charge rate of power cell i (amperes) • xi2 is ambient temperature of power cell i (celsius)and … the independent error terms i follow a normal distribution with mean 0 and equal variance 2.Visualizing a first order model with two predictorsA first order model with more than 2 predictors iiiiixxxY3322110where …• Yi is intelligence (PIQ) of student i • xi1 is brain size (MRI) of student i • xi2 is height (Height) of student i • xi3 is weight (Weight) of student i and … the independent error terms i follow a normal distribution with mean 0 and equal variance 2.Visualizing a first order model with more than 2 predictorsA second order polynomial model with one predictor iiiixxY21110where …• Yi is length of bluegill (fish) i (in mm)• xi is age of bluegill (fish) i (in years)and … the independent error terms i follow a normal distribution with mean 0 and equal variance 2.Visualizing a second order polynomial model with one predictor654321200150100agelengthS = 10.9061 R-Sq = 80.1 % R-Sq(adj) = 79.6 %length = 13.6224 + 54.0493 age - 4.71866 age**2Regression PlotA second order polynomial model with 2 predictors iiiiiiiixxxxxxY21122222211122110where …• Yi is grade point average of student i• xi1 is verbal test score of student i • xi2 is math test score of student iand … the independent error terms i follow a normal distribution with mean 0 and equal variance 2.Visualizing second order polynomial model with 2 predictorsA first order modelwith one binary predictor iiiixxY22110where …• Yi is birth weight of baby i• xi1 is length of gestation of baby i • xi2 = 1, if mother smokes and xi2 = 0, if notand … the independent error terms i follow a normal distribution with mean 0 and equal variance 2.Visualizing a first order modelwith one binary predictor0 1 4241403938373635343700320027002200Gestation (weeks)Weight (grams)The regression equation isWeight = - 2390 + 143 Gest - 245
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