Cheat Sheet 11-1(1).docx - SLR describe mean of the response σ{Y|X SD of Y on X μ{Y|X = β0 β1X least squares(LS -“best fitting” straight line

Cheat Sheet 11-1(1).docx - SLR describe mean of the...

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SLR : describe mean of the response. σ{Y|X}: SD of Y on X μ{Y|X} = β 0 + β 1 X least squares (LS) best fitting” straight line. Estimates are unbiased: E(b 0 )= β 0 and E(b 1 )= β 1 Residual resi= Y i ˆ-Y i β 0 ˆ β 1 ˆX 1 -Y i Estimates are unbiased: E(b 0 )= β 0 and E(b 1 )= β 1 Fitted/Predicted Value Sampling dist. of b 1 Mean = β 1 , Sampling dist. of b 0 Mean = β 0 df=n-2 t-test: b1=10.74, SE(b1)=0.63, σˆ=6.68. TS= (10.74 – 0)/0.63 = 17.15~T dis (if |TS|>1.96 or p-value<0.05/reject H0) p-value corresponding to this TS is 0. Reject the null. Conclude β1 ≠ 0, mean of the response is linearly related to explanatory variable. y CI: CI for mean of response b 0 +b 1 X 1 = 33.8 + 10.7*3 = 65.9 we are 95% confident that the mean of yy at xx lies in (64.46, 67.34) 1.predict(fit,xnew,interval='confidence', level=0.95) 2. summary$coefficients[1, 1]+-qt(0.975,summary$df[2])*summary$coef[1,2] y PI : CI(Plausible values for the mean of the response variable, for a particular value of the explanatory variable (X0). PI(Plausible values for a future value of the response variable, for..)pred b 0 +b 1 X 1 The existence of a statistical relation between the response and the explanatory variable (i.e. reject null and conclude β1 ≠ 0) does not imply that the response depends causally on the explanatory variable. 4 Assumptions: Linearity (straight-line), Normality : (There is a normally distributed population of responses for each value of the explanatory variable (Q-Q plot: Not fit on a straight line/ have outliers)) Constant variance (The population standard deviations are all equal: σ{Y|X} = σ) Independence (The selection of an observation from any of the populations is independent of the selection of any other observations) Assumption Violations: Linearity : Can cause the estimated means and predictions to be biased. Normality CV : SE inaccurately measure uncertainty. Independence : Can seriously affect standard errors. R-squared : SST (Total sum of squares )=SSR (Regression SS )+SSE (Residual or error SS unexplained) R 2 is the % of the total response variation explained by the explanatory variable, it tells us the percentage of the variation in the response that is explained by the linear regression. A “good” R 2 depends on the context. For SLR, R 2 is identical to the square of the sample correlation coefficient. Compare 2 models: Adjusted R 2 , 大好 , 但不能完 依赖它判断,还需其他条件。 MLR : µ(Y|X 1 , X 2 )=β 0 1 X 1 2 X 2 , β 1 gives the increase in µ(Y|X 1 , X 2 ) for a unit increase in X 1 with X 2 held constant. Indicator variables & Interaction: (Categorical 0/1: 已有变量数 -1) µ(Y|X)=β 0 1 X 1 2 X 2 + β 3( X 1* X 2 ) µ(Y|X)= β 0 1 X 1 (gender = male) µ(Y|X)= β 0 1 X 1 2 + β 3 X 1 (gender = female) 通过 β 2 是否 Sig 检测 Female Y Male Y 的差是否 Sig, 通过 β 3 是否 Sig 检测两个变量是否关联, p 0.05 reject, 无关联 ( 加上 interaction 变量后若 slope 改变的多,则 interaction important. (p 个变量 , p+1:#of parameters, n-p-1:dof) F-test: Null: β 2 3 =0, Alt: at least one of ≠0 extra sum of squares =SSR reduced - SSR full
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