# simple - Simple linear regression Linear regression with...

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Simple linear regression Linear regression with one predictor variable 1

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A deterministic relationship (model) 50 40 30 20 10 0 130 120 110 100 90 80 70 60 50 40 30 Celsius Fahrenheit 2
Other deterministic relationships Circumference = π × diameter Hooke’s Law: Y = α + β X , 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. What if the relationships are not exact??? 3

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A statistical relationship A relationship with some “ trend ”, but also with some “ scatter .” 27 30 33 36 39 42 45 48 100 150 200 Mortality (Deaths per 10 million) Latitude (at center of state) Skin cancer mortality versus State latitude 4
Other statistical relationships Height and weight Alcohol consumed and blood alcohol content Vital lung capacity and pack-years of smoking Driving speed and gas mileage 5

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Simple linear regression A way of evaluating the linear relationship between two continuous (quantitative) variables . One variable is regarded as the predictor , or independent variable (x). Other variable is regarded as the response , or dependent variable (y). 6
Simple linear regression A first order probabilistic model: y = β 0 + β 1 x + ε . Deterministic component: E(y) = β 0 + β 1 x. Random error component: ε. β 0 is the y-intercept of the line. The β 1 is the slope of the line 7

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Fitting the model: which is the “best regression line”? 74 70 66 62 210 200 190 180 170 160 150 140 130 120 110 height weight w = -266.5 + 6.1 h w = -331.2 + 7.1 h 8
Notation i y is the observed response for the i th experimental unit. i x is the predictor value for the i th experimental unit. i y ˆ is the predicted response (or fitted value ) for the i th experimental unit. Fitted model: ˆ y i   0   1 x i 9

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74 70 66 62 210 200 190 180 170 160 150 140 130 120 height weight w = -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 63 127 120.1 10 72 165 175.4 i x i y i y ˆ i 10 Weight-height example
Prediction error (or residual error) In using i y ˆ to predict the actual response i y we make a prediction error (or a residual error ) i i i y y e ˆ of size A line that fits the data well will be one for which the n prediction errors are as small as possible in some overall sense. 11

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Least square principal The sum of errors Choose the values β 0 and β 1 that minimize the sum of the squared errors (SSE). That is, find β 0 and β 1 that minimize:   n i i i y y Q 1 2 ˆ 12 SE i i 1 n y i ˆ y i i 1 n 0
Weight-height example 74 70 66 62 210 200 190 180 170 160 150 140 130 120 110 height weight w = -266.5 + 6.1 h w = -331.2 + 7.1 h 13

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w = -331.2 + 7.1 h (dashed line) 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
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## This note was uploaded on 09/28/2011 for the course STAT METHO 33:623:385 taught by Professor Faridalizadeh during the Spring '11 term at Rutgers.

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simple - Simple linear regression Linear regression with...

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