# Steffen grønneberg bi lecture 5 gra6036 4th february

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Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 39 / 61

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Some conceptual comments Note that we cannot observe ε i ! We only observe e i = Y i - ˆ Y i . To diagnose assumptions on ε i we must use e i . If the relation between X and Y is linear, then ˆ β i β i , so e i ε i . If Y i = f ( X i ) + ε i for some non-linear function f , the residual e i = ˆ Y i - Y i may be quite far away from ε i Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 40 / 61
Some conceptual comments If we see some systematic components in the residuals e i = ˆ Y i - Y i , this indicates that we are missing something in our model. For our birth-weight model: There is a weak parabolic tendency in the residuals, as should be expected from the actual non-linear relationship between birth-weight and gestational age. Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 41 / 61

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Some conceptual comments Statistical inference for linear regression is based on either large sample theory, or concrete distributional assumptions (here: i N ( 0 , σ 2 ) independently of each other) We have no reason to be concerned with independence among the i ’s, as the babies in our study are not related in any way. We have a small sample, so in order to use the inference tools, we must check if the distributional assumptions are reasonable: The residuals are approximations for the unobservable error terms i , so we can use them to assess if we have reason to doubt that i N ( 0 , σ 2 ) ? The higher the sample-size, the larger degree of non-normality is tolerated. Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 42 / 61
Some conceptual comments To get Stata to compute the residuals, use predict res, r after a regression command is completed. To assess Normality, we can request a PP-plot through pnorm res and a histogram via hist res . Residuals display slight deviations from Normality, but not enough to cause concern (relative to the number of observations and the number of parameters we estimate) The sample size required relative to the degree of non-normality depends on the type of deviation from Normality and the number of parameters in the model. Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 43 / 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 44 / 61
Multiple 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 If we take gestational age into account, is gender important in explaining birth weight? Steffen Grønneberg (BI) Lecture 5, GRA6036 4th February 2016 45 / 61

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Multiple linear regression In our original t -test, we concluded that gender difference was due to chance. However, when taking age into account, it appears that boys are heavier than girls.
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