Simple and Multiple Linear Regression
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1 Sta. 113 Chapter 12 and 13 of Devore March 12, 2010
2 Table of contents 1 Simple Linear Regression 2
3 Model Simple Linear Regression A simple linear regression model is given by Y = β 0 + β 1 x + ɛ where Y is the reponse x is the predictor β 0 is the unknown intercept of the line β 1 is the unknown slope of the line ɛ N(0, σ 2 ) is the noise with unknown variance σ 2
4 Model Simple Linear Regression Notice that Y is a random quantity due to ɛ only E(Y ) = β 0 + β 1 x V(Y ) = σ 2 Y N(β 0 + β 1 x, σ 2 )
5 Assumptions Simple Linear Regression Notice that A linear underlying relationship between the response and the predictor Normality of random noise Constant variance of random noise all throughout the data Independence of random noise
6 Least Squares Find the line passing through the data points such that the sum of squared vertical distances from this line to the data points is minimized. min b 0,b 1 n (y i b 0 b 1 x i ) 2 i=1 Since this is a minimization problem, taking the derivatives with respect to b 0 and b 1 and setting them equal to zero will result in two equations which are called the normal equations. nb 0 + ( x i )b 1 = 0 ( x i )b 0 + ( x 2 i )b 1 = x i y i
7 Least Squares If we solve this system we obtain b 1 = ˆβ 1 = (xi x)(y i ȳ) (xi x) 2 b 0 = ˆβ 0 = ȳ b 1 x.
8 How does LSE relate to MLE Notice that there is nothing probabilistic about least squares estimation. It s merely an optimization problem where the sum of squared vertical distances from actual points to a line is minimized. There is no underlying distribution assumption. In fact, nothing is treated as random. We just have a cloud of points and we pass a line through them. In the beginning we made certain assumptions about the response. We said Y i N(β 0 + β 1 x i, σ 2 ). Assuming that the responses are distributed normally with mean β 0 + β 1 x i and variance σ 2 will yield a likelihood over the unknown model parameters β 0, β 1 and σ 2. Maximizing this likelihood will yield the MLE. It turns out that under the assumptions we made earlier, the maximum likelihood estimators for β 0 and β 1 are identical to the least squares estimators.
9 Estimating the error variance The maximum likelihood estimator for the error variance σ 2 is easily obtained as ˆσ 2 = n i=1 (y i b 0 b 1 x i ) 2. n Recall that this is a biased estimator for σ 2. To correct for the bias we have to subtract the number of parameters estimated prior to the estimation of σ 2 from n. Thus, the unbiased estimator is obtained as n s 2 i=1 = (y i b 0 b 1 x i ) 2. n 2
10 Example - Murder rate vs unemployment percentage
11 Example - Murder rate vs unemployment percentage
12 Example - Murder rate vs unemployment percentage
13 The coefficient of determination, R 2 The coefficient of determination, denoted by R 2, is given by R 2 = 1 SSE n SST = 1 i=1 (y i b 0 b 1 x i ) 2 n i=1 (y i ȳ) 2. It is interpreted as the proportion of observed variation in y that is explained by the simple linear regression model.
14 s about β 1 It can be shown that b 1 = ˆβ 1 is normally distributed with mean E(b 1 ) = β 1 and variance V(b 1 ) = σ2 S xx where S xx = (x i x) 2. Thus the quantity z = b 1 β 1 σ/ S xx would be standard normally distributed. Since we don t know σ 2, if we replace is by its estimator s 2 t = b 1 β 1 s/ S xx has a t distribution with n 2 df.
15 Confidence interval and hypothesis test for β 1 A 100(1 α)% CI for the slope β 1 of the true regression line is given by s b 1 ± t α/2,n 2. S xx We usually test the null hypothesis H 0 : β 1 = 0 vs H a : β 1 0 where the test statistic is t = b 1 s/ S xx. Since under the null hypothesis t = b 1 s/ S xx has a t distribution with n 2 degrees of freedom, the null hypothesis is rejected if t t alpha/2,n 2 or t t alpha/2,n 2. This can easily be turned into a one-sided test.
16 s on µ Y,x Simple Linear Regression and the prediction of future Y values Notice that once b 0 and b 1 are calculated, b 0 + b 1 x is a point estimate of µ Y,x (the expected ot true average value of Y when x = x ). The point estimate or prediction by itself gives no information concerning how precisely µ Y,x has been estimated or Y predicted. This can be remedied by developing a CI for µ Y,x and a prediction interval (PI) for a single Y value.
17 s on µ Y,x Simple Linear Regression and the prediction of future Y values A 100(1 α)% CI for µ Y,x, the expected value of Y when x = x, is given by b 0 + b 1 x 1 ± t α/2,n 2 s n + (x x) 2. S xx A 100(1 α)% PI for a future Y observation to be made when x = x is given by b 0 + b 1 x ± t α/2,n 2 s n + (x x) 2. S xx
18 Model Simple Linear Regression A multiple linear regression model is given by Y = β 0 + β 1 x 1 + β 3 x ɛ where Y is the reponse x 1, x 2, x 3,... are the predictors β 0, β 1, β 2,... are unknown regression coefficients ɛ N(0, σ 2 ) is the noise with unknown variance σ 2
19 Model Simple Linear Regression When we have n observations from such a model, i.e. y = (y 1, y 2,..., y n ), with we define X as the design matrix 1 x 11 x 12 x 1p 1 x 21 x 22 x 2p X = x n1 x n2 x np
20 Least Squares The least squares solution ˆβ = (β 0, β 1, β 2,..., β p ) is given by ˆβ = (X X) 1 X y. Just like in the simple linear regression case, this is equivalent to the MLE under aforementioned assumptions.
21 Example - Cirrhosis data
22 Example - Cirrhosis data
23 Example - Cirrhosis data
24 Example - Cirrhosis data
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