Section 3.pdf - section3 Simulating the regression model#What does it mean that the estimates are unbiased#try different types of noisel

Section 3.pdf - section3 Simulating the regression...

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section3 Simulating the regression model #What does it mean that the estimates are unbiased? #try different types of noisel rt(df=10),rchisq(2) and so on. n= 50 x= runif ( 50 ) ##the x s stay constant howmany= 1000 b0vals= 1 : 1000 b1vals= 1 : 1000 for (i in 1 : howmany) { y= 20 + 2 * x + rnorm (n, 0 ,. 1 ) fit= lm (y ~ x) b0vals[i] = coef (fit)[ 1 ] b1vals[i] = coef (fit)[ 2 ] } #what should the mean equal? mean (b1vals) ## [1] 2.000419 hist (b1vals) Histogram of b1vals b1vals Frequency 1.8 1.9 2.0 2.1 2.2 0 100 200 300 1
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Model Assumptions Recall the simple linear regression model statement, Y i = β 0 + β 1 x i 1 + + i where i N (0 , σ 2 ) . Often, the assumptions of linear regression , are stated as, L inearity: the response can be written as a linear combination of the predictors. (With noise about this true linear relationship.) I ndependence: the errors are independent. N ormality: the distribution of the errors should follow a normal distribution. E qual Variance: the error variance is the same at any set of predictor values. The linearity assumption is encoded as β 0 + β 1 x i 1 , while the remaining three, are all encoded in i N (0 , σ 2 ) , since the i are iid normal random variables with constant variance. If these assumptions are met, great! We can perform inference, and it is valid . If these assumptions are not met, we can still “perform” a t -test using R , but the results are not valid . The distributions of the parameter estimates will not be what we expect. Hypothesis tests will then accept or reject incorrectly. Essentially, garbage in, garbage out. Checking Assumptions We’ll now look at a number of tools for checking the assumptions of a linear model. To test these tools, we’ll use data simulated from three models: Model 1: Y = 3 + 5 x + , N (0 , 1) Model 2: Y = 3 + 5 x + , N (0 , x 2 ) Model 3: Y = 3 + 5 x 2 + , N (0 , 25) sim_ 1 = function ( sample_size = 500 ) { x = runif ( n = sample_size) * 5 y = 3 + 5 * x + rnorm ( n = sample_size, mean = 0 , sd = 1 ) data.frame (x, y) } sim_ 2 = function ( sample_size = 500 ) { x = runif ( n = sample_size) * 5 y = 3 + 5 * x + rnorm ( n = sample_size, mean = 0 , sd = x) data.frame (x, y) } sim_ 3 = function ( sample_size = 500 ) { x = runif ( n = sample_size) * 5 y = 3 + 5 * x ^ 2 + rnorm ( n = sample_size, mean = 0 , sd = 5 ) data.frame (x, y) } 2
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Fitted versus Residuals Plot Probably our most useful tool will be a Fitted versus Residuals Plot . It will be useful for checking both the linearity and constant variance assumptions. Data generated from Model 1 above should not show any signs of violating assumptions, so we’ll use this to see what a good fitted versus residuals plot should look like. First, we’ll simulate observations from this model. set.seed ( 42 ) sim_data_ 1 = sim_1 () #head(sim_data_1) We then fit the model and add the fitted line to a scatterplot. plot (y ~ x, data = sim_data_ 1 , col = "grey" , pch = 20 , main = "Data from Model 1" ) fit_ 1 = lm (y ~ x, data = sim_data_ 1 ) abline (fit_ 1 , col = "darkorange" , lwd = 3 ) 0 1 2 3 4 5 5 10 15 20 25 30 Data from Model 1 x y We now plot a fitted versus residuals plot. Note, this is residuals on the y -axis despite the ordering in the name. Sometimes you will see this called a residuals versus fitted, or residuals versus predicted plot.
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