15.1 The Regression Model: Analysis of Residuals
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1 15.1 The Regression Model: Analysis of Residuals Tom Lewis Fall Term 2009 Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
2 Outline 1 The regression model 2 Estimating the common standard deviation, σ 3 Testing our assumptions Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
3 Example It will be helpful to have a simple example in mind as we work this material. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
4 Example It will be helpful to have a simple example in mind as we work this material. Suppose that we collect information on the number of years of education (Y ) and the yearly income (I ) for a sample of individuals. To what extent can we explain I through a linear relationship with Y. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
5 Example It will be helpful to have a simple example in mind as we work this material. Suppose that we collect information on the number of years of education (Y ) and the yearly income (I ) for a sample of individuals. To what extent can we explain I through a linear relationship with Y. We strongly suspect that I β 1 Y + β 0, for some coefficients β 1 and β 0. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
6 Example It will be helpful to have a simple example in mind as we work this material. Suppose that we collect information on the number of years of education (Y ) and the yearly income (I ) for a sample of individuals. To what extent can we explain I through a linear relationship with Y. We strongly suspect that I β 1 Y + β 0, for some coefficients β 1 and β 0. We suspect, however, that there are other explanatory factors that also contribute to an individual s income. These other contributing factors might be, among other things, location, age, gender, race, intelligence, etc. We might express this as I = β 1 Y + b 0 + ε where ε represents the collective contributions from the other factors. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
7 The regression model Consider the relationship between an explanatory variable x and a response variable y. A powerful way to model the relationship between x and y is to begin with the assumption that y = β 1 x + β }{{} 0 + }{{} ε Part I Part II The the value of the response variable, y, is composed of two parts: Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
8 The regression model Consider the relationship between an explanatory variable x and a response variable y. A powerful way to model the relationship between x and y is to begin with the assumption that y = β 1 x + β }{{} 0 + }{{} ε Part I Part II The the value of the response variable, y, is composed of two parts: Part I: β 1 x + β 0 is the part of y that can be explained from x alone. This part is the best predicted value of y from the value of x. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
9 The regression model Consider the relationship between an explanatory variable x and a response variable y. A powerful way to model the relationship between x and y is to begin with the assumption that y = β 1 x + β }{{} 0 + }{{} ε Part I Part II The the value of the response variable, y, is composed of two parts: Part I: β 1 x + β 0 is the part of y that can be explained from x alone. This part is the best predicted value of y from the value of x. Part II: ε, the error, is the part of y that is due to factors other than x, independent of the value of x. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
10 Assumptions about the error, ε In a regression model, we make certain assumptions concerning the error term, ε. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
11 Assumptions about the error, ε In a regression model, we make certain assumptions concerning the error term, ε. 1 For each value of x, ε is a normal random variable; the value of ε is independent of x. In other words, the particular value of x exerts no influence on the value of ε. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
12 Assumptions about the error, ε In a regression model, we make certain assumptions concerning the error term, ε. 1 For each value of x, ε is a normal random variable; the value of ε is independent of x. In other words, the particular value of x exerts no influence on the value of ε. 2 ε has a mean value of 0. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
13 Assumptions about the error, ε In a regression model, we make certain assumptions concerning the error term, ε. 1 For each value of x, ε is a normal random variable; the value of ε is independent of x. In other words, the particular value of x exerts no influence on the value of ε. 2 ε has a mean value of 0. 3 ε has standard deviation σ. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
14 Assumptions about the error, ε In a regression model, we make certain assumptions concerning the error term, ε. 1 For each value of x, ε is a normal random variable; the value of ε is independent of x. In other words, the particular value of x exerts no influence on the value of ε. 2 ε has a mean value of 0. 3 ε has standard deviation σ. 4 The error terms corresponding to independent trials of the explanatory variable are independent of one another. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
15 Assumptions about the error, ε In a regression model, we make certain assumptions concerning the error term, ε. 1 For each value of x, ε is a normal random variable; the value of ε is independent of x. In other words, the particular value of x exerts no influence on the value of ε. 2 ε has a mean value of 0. 3 ε has standard deviation σ. 4 The error terms corresponding to independent trials of the explanatory variable are independent of one another. Consequences for the response variable Each of our four assumptions about ε has an effect on the value of the response variable, y. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
16 Consequences for the response, y Consider the model y = β 1 x + β 0 + ε. Given our assumptions about ε, we may conclude that: Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
17 Consequences for the response, y Consider the model y = β 1 x + β 0 + ε. Given our assumptions about ε, we may conclude that: 1 For each value of x, y is a normal random variable. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
18 Consequences for the response, y Consider the model y = β 1 x + β 0 + ε. Given our assumptions about ε, we may conclude that: 1 For each value of x, y is a normal random variable. 2 For each value of x, y has a mean value of β 1 x + β 0. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
19 Consequences for the response, y Consider the model y = β 1 x + β 0 + ε. Given our assumptions about ε, we may conclude that: 1 For each value of x, y is a normal random variable. 2 For each value of x, y has a mean value of β 1 x + β 0. 3 For every value of x, y has the same standard deviation, σ. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
20 Consequences for the response, y Consider the model y = β 1 x + β 0 + ε. Given our assumptions about ε, we may conclude that: 1 For each value of x, y is a normal random variable. 2 For each value of x, y has a mean value of β 1 x + β 0. 3 For every value of x, y has the same standard deviation, σ. 4 The values of y corresponding to independent trials of the explanatory variable are independent of one another. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
21 Consequences for the response, y Consider the model y = β 1 x + β 0 + ε. Given our assumptions about ε, we may conclude that: 1 For each value of x, y is a normal random variable. 2 For each value of x, y has a mean value of β 1 x + β 0. 3 For every value of x, y has the same standard deviation, σ. 4 The values of y corresponding to independent trials of the explanatory variable are independent of one another. Problem Explain the meaning of each of these items for the income/years of school model. What does this mean for individuals with 12, 16, and 20 years of schooling? Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
22 Estimating the common standard deviation, σ Estimating σ We have two tasks before us: Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
23 Estimating the common standard deviation, σ Estimating σ We have two tasks before us: 1 Assuming that our model is correct, how can we estimate σ, the standard deviation of the error term? Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
24 Estimating the common standard deviation, σ Estimating σ We have two tasks before us: 1 Assuming that our model is correct, how can we estimate σ, the standard deviation of the error term? 2 How can we be sure that our model is accurate. In other words, how can we be sure that the conditions on ε, the error term, are satisfied? Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
25 Estimating the common standard deviation, σ Some data Consider the following data set: x y For this data set, we have S xx = 82.5, S yy = , and S xy = The regression equation is ŷ = x We have SST = , SSR = , SSE = , and r 2 = Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
26 Estimating the common standard deviation, σ The error terms The last column of this data set gives the error between the actual value of y and its predicted value, ŷ. x y ŷ e = y ŷ Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
27 Estimating the common standard deviation, σ Estimating σ Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
28 Estimating the common standard deviation, σ Estimating σ Our estimate for σ (the standard deviation of ε) comes from looking at the variation of the error terms, e. s e = variation in e n 2 = SSE n 2. This term is called the residual standard error. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
29 Estimating the common standard deviation, σ Estimating σ Our estimate for σ (the standard deviation of ε) comes from looking at the variation of the error terms, e. s e = variation in e n 2 = SSE n 2. This term is called the residual standard error. You will notice that there is an n 2 where we would normally expect to see an n 1. This is due to the fact that we lose a degree of freedom in using the data set to build the regression line. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
30 Estimating the common standard deviation, σ The estimate of σ in our example For our example, SSE = ; thus, s e = = This gives us an estimate for the common standard deviation of the error term, ε. Note that R Commander will compute, among other things, the residual standard error through the command. Statistics Fit Models Linear regression... Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
31 Testing our assumptions Testing our assumptions The validity of our model depends on whether or not the conditions on ε are in place. How can test these conditions with our data set? R Commander provides diagnostic tools for this task. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
32 Testing our assumptions Testing our assumptions The validity of our model depends on whether or not the conditions on ε are in place. How can test these conditions with our data set? R Commander provides diagnostic tools for this task. First run Statistics Fit Models Linear Regression...; this will, among other things, load the model into R Commander. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
33 Testing our assumptions Testing our assumptions The validity of our model depends on whether or not the conditions on ε are in place. How can test these conditions with our data set? R Commander provides diagnostic tools for this task. First run Statistics Fit Models Linear Regression...; this will, among other things, load the model into R Commander. To obtain the diagnostic plots, following the menus Models Graphs basic diagnostic plots will give you four plots. We will consider the top two. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
34 Testing our assumptions Testing our assumptions The validity of our model depends on whether or not the conditions on ε are in place. How can test these conditions with our data set? R Commander provides diagnostic tools for this task. First run Statistics Fit Models Linear Regression...; this will, among other things, load the model into R Commander. To obtain the diagnostic plots, following the menus Models Graphs basic diagnostic plots will give you four plots. We will consider the top two. The Residual vs Fitted shows how the size of the residual (the error) varies with the value of the fitted variable, ŷ. These values should be independent, hence we should see a fairly random pattern of residual values across the spectrum of fitted values. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
35 Testing our assumptions Testing our assumptions The validity of our model depends on whether or not the conditions on ε are in place. How can test these conditions with our data set? R Commander provides diagnostic tools for this task. First run Statistics Fit Models Linear Regression...; this will, among other things, load the model into R Commander. To obtain the diagnostic plots, following the menus Models Graphs basic diagnostic plots will give you four plots. We will consider the top two. The Residual vs Fitted shows how the size of the residual (the error) varies with the value of the fitted variable, ŷ. These values should be independent, hence we should see a fairly random pattern of residual values across the spectrum of fitted values. The Normal Q-Q gives a normal probability plot of the residual values. As we discussed earlier, if this plot is linear, then the assumption of normality is justified. Tom Lewis () 15.1 The Regression Model: Analysis of Residuals Fall Term / 12
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