# Steffen grønneberg bi lecture 5 gra6036 4th february

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Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 13 / 61

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Contents 1 Linear regression Review of statistical inference Analysis of variance Simple linear regression Some conceptual comments Multiple linear regression Interaction, part I Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 14 / 61
Simple linear regression Boys Girls Age Birth weight Age Birth weight 40 2968 40 3317 38 2795 36 2729 40 3163 40 2935 35 2925 38 2754 36 2625 42 3210 37 2847 39 2817 41 3292 40 3126 40 3473 37 2539 37 2628 36 2412 38 3176 38 2991 40 3421 39 2875 38 2975 40 3231 Means 38.33 3024.00 38.75 2911.33 How does gestational age influence birth weight? Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 15 / 61

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Simple linear regression Let us for the moment ignore gender, and plot gestational age versus birth weight. There is a clear linear trend. How should we best describe this observation? Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 16 / 61
Simple linear regression A basic observation is that gestational age and birth weight are positively correlated . High X -values correspond to high Y -values: Birth weight increases with gestational age. Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 17 / 61

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Simple linear regression Gestational age and birth weight are correlated by 74 . 4 % , which is a fairly strong linear relationship. While r measures strength of linear relationship, let us try to estimate the actual linear relationship. Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 18 / 61
Simple linear regression Recall that the equation for a straight line is y = β 0 + β 1 x where β 0 is the intersect with the y -axis, β 1 is the slope of the line. Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 19 / 61

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Simple linear regression Suppose we observe ( Y 1 , X 1 ) , ( Y 2 , X 2 ) , . . . , ( Y n , X n ) and plot the observations Often, scatter plots looks like a line distorted by some noise. That is, Y i = β 0 + β 1 X i | {z } straight line + ε i |{z} noise where ε 1 , . . . , ε n are “small disturbance terms”. Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 20 / 61
Simple linear regression Suppose Y i = β 0 + β 1 X i + ε i with ε i “small disturbance terms”. What is the best fitting line to the data? Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 21 / 61

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Simple linear regression Suppose Y i = β 0 + β 1 X i + ε i with ε i “small disturbance terms”. What is the best fitting line to the data? Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 22 / 61
Simple linear regression Suppose Y i = β 0 + β 1 X i + ε i with ε i “small disturbance terms”. What is the best fitting line to the data? We need to define what we mean by “best”! Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 23 / 61

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Simple linear regression theory Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 24 / 61
Simple linear regression The ordinary least squares (OLS) estimate ˆ Y = ˆ β 0 + ˆ β 1 X is intuitive, but we can show that if Y i = β 0 + β 1 X i + ε i for ε i N ( 0 , σ 2 ) , the OLS estimates are the optimal approximations of β 0 , β 1 (under several definitions of “best”).

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