Overview of our study of the multiple linear regression modelIs brain and body size predictive of intelligence?Scatter matrix plotSlide 4Slide 5A multiple linear regression model with three quantitative predictorsSome research questionsSlide 8Baby bird breathing habits in burrows?Slide 10Three-dimensional scatter plotA first order model with two quantitative predictorsSlide 13Slide 14Is baby’s birth weight related to smoking during pregnancy?Slide 16A first order model with one binary predictorEstimated first order model with one binary predictorSlide 19Slide 20Compare three treatments (A, B, C) for severe depressionSlide 22A second order model with one quantitative predictor, a three-group qualitative variable, and interactionsThe estimated regression functionPotential research questionsSlide 26How is the length of a bluegill fish related to its age?Scatter plotA second order polynomial model with one quantitative predictorEstimated regression functionSlide 31Slide 32The good news!New things we need to learn!Slide 35Overview of our study of the multiple linear regression model Regression models with more than one slope parameterIs 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 1Scatter matrix plotExample 1100.72886.28373.2565.75170.5127.5130.591.5100.72886.28373.2565.75PIQBrainHeightWeightScatter matrix plotExample 1100.72886.28373.2565.75170.5127.5130.591.5100.72886.28373.2565.75Brain Height WeightPIQBrainHeightScatter matrix plot•Illustrates the marginal relationships between each pair of variables without regard to the other variables.•The challenge is how the response y relates to all three predictors simultaneously.A multiple linear regression model with three quantitative 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 Example 1and … the independent error terms i follow a normal distribution with mean 0 and equal variance 2.Some research questions•Which predictors – brain size, height, or weight – explain some variation in PIQ?•What is the effect of brain size on PIQ, after taking into account height and weight?•What is the PIQ of an individual with a given brain size, height, and weight?Example 1Example 1The regression equation isPIQ = 111 + 2.06 Brain - 2.73 Height + 0.001 WeightPredictor Coef SE Coef T PConstant 111.35 62.97 1.77 0.086Brain 2.0604 0.5634 3.66 0.001Height -2.732 1.229 -2.22 0.033Weight 0.0006 0.1971 0.00 0.998S = 19.79 R-Sq = 29.5% R-Sq(adj) = 23.3%Analysis of VarianceSource DF SS MS F PRegression 3 5572.7 1857.6 4.74 0.007Residual Error 34 13321.8 391.8Total 37 18894.6Source DF Seq SSBrain 1 2697.1Height 1 2875.6Weight 1 0.0Baby bird breathing habits in burrows?•Experiment with n = 120 nestling bank swallows•Response (y): % increase in “minute ventilation”, Vent, i.e., total volume of air breathed per minute•Potential predictor (x1): percentage of oxygen, O2, in the air the baby birds breathe•Potential predictor (x2): percentage of carbon dioxide, CO2, in the air the baby birds breatheExample 2Scatter matrix plotExample 217.514.56.752.25484.7552.2517.514.5VentO2CO2Three-dimensional scatter plot13-20001420040015161718VentO2400600864CO2218019Example 2A first order model with two quantitative predictors iiiixxy22110where …• yi is percentage of minute ventilation• xi1 is percentage of oxygen • xi2 is percentage of carbon dioxideand … the independent error terms i follow a normal distribution with mean 0 and equal variance 2.Example 2Some research questions•Is oxygen related to minute ventilation, after taking into account carbon dioxide?•Is carbon dioxide related to minute ventilation, after taking into account oxygen?•What is the mean minute ventilation of all nestling bank swallows whose breathing air is comprised of 15% oxygen and 5% carbon dioxide?Example 2Example 2The regression equation isVent = 86 - 5.33 O2 + 31.1 CO2Predictor Coef SE Coef T PConstant 85.9 106.0 0.81 0.419O2 -5.330 6.425 -0.83 0.408CO2 31.103 4.789 6.50 0.000S = 157.4 R-Sq = 26.8% R-Sq(adj) = 25.6%Analysis of VarianceSource DF SS MS F PRegression 2 1061819 530909 21.44 0.000Residual Error 117 2897566 24766Total 119 3959385Source DF Seq SSO2 1 17045CO2 1 1044773Is baby’s birth weight related to smoking during pregnancy?•Sample of n = 32 births•Response (y): birth weight in grams of baby•Potential predictor (x1): smoking status of mother (yes or no)•Potential predictor (x2): length of gestation in weeksExample 3Scatter matrix plot40360.750.253252.52697.54036WeightGestSmokingExample 3A 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.Example 3Estimated first order modelwith one binary predictor0 1 4241403938373635343700320027002200Gestation (weeks)Weight (grams)The regression equation isWeight = - 2390 + 143 Gest - 245 SmokingExample 3Some research questions•Is baby’s birth weight related to smoking during pregnancy?•How is birth weight related to gestation, after taking into account smoking status?Example 3Example 3The regression equation isWeight = - 2390 + 143 Gest - 245 SmokingPredictor Coef SE Coef T PConstant -2389.6 349.2 -6.84 0.000Gest 143.100
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