simple - Simple linear regression Linear regression with...

Info iconThis preview shows pages 1–15. Sign up to view the full content.

View Full Document Right Arrow Icon
Simple linear regression Linear regression with one predictor variable 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
A deterministic relationship (model) 50 40 30 20 10 0 130 120 110 100 90 80 70 60 50 40 30 Celsius Fahrenheit 2
Background image of page 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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 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
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 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
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 10
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
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 12
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
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 14
Image of page 15
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 52

simple - Simple linear regression Linear regression with...

This preview shows document pages 1 - 15. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online