PDB_Stat_100_Lecture_27_Printable

# PDB_Stat_100_Lecture_27_Printable - STA 100 Lecture 27 Paul...

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STA 100 Lecture 27 Paul Baines Department of Statistics University of California, Davis March 11th, 2011

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Admin for the Day I Final project due today, 3pm I Oﬃce hours Friday – 9.30-11.30am I Practice Final Exam Questions Posted I Review session on Monday References for Today: Rosner, Ch 11, Ch 12 (7th Ed.) References for Friday: Rosner, Ch 11, Ch 12 (7th Ed.)
Understanding R 2 We can decompose the total variability in our data: n X i =1 ( y i - ¯ y ) 2 = n X i =1 y i - ¯ y ) 2 + n X i =1 ( y i - ˆ y i ) 2 Total SS = Regression SS + Residual SS Using this, it turns out that R 2 also has a nice interpretation: R 2 = r 2 = n i =1 y i - ¯ y ) 2 n i =1 ( y i - ¯ y ) 2 = Regression SS Total SS

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Understanding R 2 Using this deﬁnition, we get a nice interpretation for R 2 as: the proportion of variability in y that can be explained by a linear relationship with x . Sometimes we use a slightly modiﬁed version of R 2 , known as adjusted R 2 . Details will be omitted, but you should use adjusted R 2 (with the same interpretation as above).
10 11 12 13 2 4 6 8 Histogram of y y

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10.00 10.05 10.10 10.15 10.20 10.25 10.30 Total Variability in the Response Variable (Zoom In) Response Variable (y) Mean of y y 1 y 2 y 3 y 4 y 5 y 1 - y y 2 - y y 3 - y y 4 - y y 5 - y
2 4 6 8 10 9.0 9.5 10.0 10.5 11.0 Total Variability in the Response Variable (Zoom In) Explanatory Variable (x) Response Variable (y) ( x 1 , y 1 29 ( x 2 , y 2 29 ( x 3 , y 3 29 ( x 4 , y 4 29 ( x 5 , y 5 29

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2 4 6 8 10 9.0 9.5 10.0 10.5 11.0 Residual Variability in the Response After Regression (Zoom In) Explanatory Variable (x) Response Variable (y) ( x 1 , y 1 29 ( x 2 , y 2 29 ( x 3 , y 3 29 ( x 4 , y 4 29 ( x 5 , y 5 29
2 4 6 8 10 9.0 9.5 10.0 10.5 11.0 Regression Variability in the Response (Zoom In) Explanatory Variable (x) Response Variable (y) ( x 1 , y 1 29 ( x 2 , y 2 29 ( x 3 , y 3 29 ( x 4 , y 4 29 ( x 5 , y 5 29

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2 4 6 8 10 9.0 9.5 10.0 10.5 11.0 Decomposing Variability in the Response (Zoom In) Explanatory Variable (x) Response Variable (y) ( x 1 , y 1 29 ( x 2 , y 2 29 ( x 3 , y 3 29 ( x 4 , y 4 29 ( x 5 , y 5 29
Interpreting R 2 Heuristically, we interpret this decomposition as: (Total Variability of Response Variable) = (Variability of Points on the Regression Line) + (Variability of Points around Regression Line) Or, put another way: (Variation ignoring the explanatory variable) = (Variation explained by the linear relationship) + (Variation not explained by the linear relationship) Thus: R 2 = ± Variation in Response Variable Explained by the Linear Relationship with the Explanatory Variable ² Total Variation in Response Variable

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Doing Linear Regression For the brain-body weight example we get: > RegModel.2 <- lm(log(brain)~log(body), data=animals) > summary(RegModel.2) ... Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.99064 0.47051 6.356 1.18e-06 *** log(body) 0.43580 0.08751 4.980 3.93e-05 *** --- Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1 Residual standard error: 1.655 on 25 degrees of freedom Multiple R-squared: 0.498, Adjusted R-squared: 0.4779 F-statistic: 24.8 on 1 and 25 DF, p-value: 3.926e-05 Therefore, 49.8% of the variability in log-brain weight can be explained by the linear relationship with log-body weight.
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## This note was uploaded on 01/17/2012 for the course STAT 100 taught by Professor drake during the Fall '10 term at UC Davis.

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PDB_Stat_100_Lecture_27_Printable - STA 100 Lecture 27 Paul...

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