Simple linear regressionWhat is simple linear regression?A deterministic (or functional) relationshipOther deterministic relationshipsA statistical relationshipOther statistical relationshipsWhich is the “best fitting line”?NotationSlide 9Prediction error (or residual error)The “least squares criterion”Slide 12w = -331.2 + 7.1 hw = -266.5 + 6.1 hThe least squares regression lineFitted line plot in MinitabRegression analysis in MinitabPrediction of future responsesWhat do the “estimated regression coefficients” b0 and b1 tell us?So, the estimated regression coefficients b0 and b1 tell us…What do b0 and b1 estimate?Slide 22The simple linear regression modelSlide 24Slide 25What about (unknown) σ2?Will this thermometer yield more precise future predictions …?… or this one?Recall the “sample variance”Estimating σ2 in regression settingEstimating σ2 from Minitab’s fitted line plotEstimating σ2 from Minitab’s regression analysisInference for (or drawing conclusions about) β0 and β1Relationship between state latitude and skin cancer mortality?(1-α)100% t-interval for slope parameter β1Hypothesis test for slope parameter 1Inference for slope parameter β1 in Minitab(1-α)100% t-interval for intercept parameter β0Hypothesis test for intercept parameter 0Inference for intercept parameter β0 in MinitabWhat assumptions?Basic regression analysis output in MinitabSimple linear regression Linear regression with one predictor variableWhat is simple linear regression?•A way of evaluating the relationship between two continuous variables.•One variable is regarded as the predictor, explanatory, or independent variable (x).•Other variable is regarded as the response, outcome, or dependent variable (y).A deterministic (or functional) relationship5040302010013012011010090807060504030CelsiusFahrenheitOther deterministic relationships•Circumference = π×diameter•Hooke’s Law: Y = α + βX, where Y = amount of stretch in spring, and X = applied weight.•Ohm’s Law: I = V/r, where V = voltage applied, r = resistance, and I = current.•Boyle’s Law: For a constant temperature, P = α/V, where P = pressure, α = constant for each gas, and V = volume of gas.A statistical relationshipA relationship with some “trend”, but also with some “scatter.”27 30 33 36 39 42 45 48100150200Mortality (Deaths per 10 million)Latitude (at center of state)Skin cancer mortality versus State latitudeOther statistical relationships•Height and weight•Alcohol consumed and blood alcohol content•Vital lung capacity and pack-years of smoking•Driving speed and gas mileageWhich is the “best fitting line”?74706662210200190180170160150140130120110heightweightw = -266.5 + 6.1 hw = -331.2 + 7.1 hNotationiyis the observed response for the ith experimental unit.ixis the predictor value for the ith experimental unit.iyˆis the predicted response (or fitted value) for the ith experimental unit.Equation of best fitting line:iixbby10ˆ74706662210200190180170160150140130120heightweightw = -266.5 + 6.1 h 1 64 121 126.3 2 73 181 181.5 3 71 156 169.2 4 69 162 157.0 5 66 142 138.5 6 69 157 157.0 7 75 208 193.8 8 71 169 169.2 9 63 127 120.110 72 165 175.4ixiyiyˆiPrediction error (or residual error)In using iyˆto predict the actual response iywe make a prediction error (or a residual error)iiiyyeˆof sizeA line that fits the data well will be one for which the n prediction errors are as small as possible in some overall sense.The “least squares criterion”Choose the values b0 and b1 that minimize the sum of the squared prediction errors.Equation of best fitting line:iixbby10ˆThat is, find b0 and b1 that minimize: 21ˆniiiyyQWhich is the “best fitting line”?74706662210200190180170160150140130120110heightweightw = -266.5 + 6.1 hw = -331.2 + 7.1 hw = -331.2 + 7.1 h 1 64 121 123.2 -2.2 4.84 2 73 181 187.1 -6.1 37.21 3 71 156 172.9 -16.9 285.61 4 69 162 158.7 3.3 10.89 5 66 142 137.4 4.6 21.16 6 69 157 158.7 -1.7 2.89 7 75 208 201.3 6.7 44.89 8 71 169 172.9 -3.9 15.21 9 63 127 116.1 10.9 118.81 10 72 165 180.0 -15.0 225.00 ------ 766.51iyixiyˆ iiyyˆ 2ˆiiyy iw = -266.5 + 6.1 h 1 64 121 126.271 -5.3 28.09 2 73 181 181.509 -0.5 0.25 3 71 156 169.234 -13.2 174.24 4 69 162 156.959 5.0 25.00 5 66 142 138.546 3.5 12.25 6 69 157 156.959 0.0 0.00 7 75 208 193.784 14.2 201.64 8 71 169 169.234 -0.2 0.04 9 63 127 120.133 6.9 47.61 10 72 165 175.371 -10.4 108.16 ------ 597.28iyixiyˆ iiyyˆ 2ˆiiyy iThe least squares regression lineUsing calculus, minimize (take derivative with respect to b0 and b1, set to 0, and solve for b0 and b1): 2110niiixbbyQand get the least squares estimates b0 and b1: niiniiixxyyxxb1211xbyb10Fitted line plot in Minitab65 70 75120130140150160170180190200210heightweightweight = -266.534 + 6.13758 heightS = 8.64137 R-Sq = 89.7 % R-Sq(adj) = 88.4 %Regression PlotRegression analysis in MinitabThe regression equation isweight = - 267 + 6.14 heightPredictor Coef SE Coef T PConstant -266.53 51.03 -5.22 0.001height 6.1376 0.7353 8.35 0.000S = 8.641 R-Sq = 89.7% R-Sq(adj) = 88.4%Analysis of VarianceSource DF SS MS F PRegression 1 5202.2 5202.2 69.67 0.000Residual Error 8 597.4 74.7Total 9 5799.6Prediction of future responsesA common use of the estimated regression line.htiwtixy,,14.6267ˆ 24.1386614.6267ˆ,wtiyPredict mean weight of 66"-inch tall people.Predict mean weight of 67"-inch tall people. 38.1446714.6267ˆ,wtiyWhat do the “estimated regression coefficients” b0 and b1 tell us?•We can expect the mean response to increase or decrease by b1 units for
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