Stat 501 Lab 09 1 Sequential or extra sums of squares This exercise reviews the concept of sequential or extra sums of squares Sequential sums of squares are useful because they can be used to test whether one slope parameter is 0 for example H0 1 0 whether a subset more than two but less than all of the slope parameters are 0 for example H0 2 3 0 Again what is a sequential sum of squares It can be viewed in either of two ways It is the reduction in the error sum of squares SSE when one or more predictor variables are added to the model Or it is the increase in the regression sum of squares SSR when one or more predictor variables are added to the model 1 1 Brain size and body size study Recall that the iqsize txt data set contains data on the intelligence based on the performance IQ y P IQ scores from the revised Wechsler Adult Intelligence Scale brain size x1 brain based on the count from MRI scans given as count 10000 and body size measured by height in inches x2 height and weight in pounds x3 weight on 38 college students 1 Fit the linear regression model with x1 brain as the only predictor Take note of the error sum of squares and since x1 is the only predictor in the model denote this value as SSE X1 What is the value of SSE X1 Take note of the regression sum of squares and since x1 is the only predictor in the model denote this value as SSR X1 What is the value of SSR X1 Take note of the total sum of squares and since x1 is the only predictor in the model denote this value as SST O X1 What is the value of SST O X1 2 Now fit the linear regression model with the predictors in order x1 brain and x2 height in the model Take note of the error sum of squares and since x1 and x2 are predictors in the model denote this value as SSE X1 X2 What is the value of SSE X1 X2 Take note of the regression sum of squares and since x1 and x2 are predictors in the model denote this value as SSR X1 X2 What is the value of SSR X1 X2 Take note of the total sum of squares and since x1 and x2 are predictors in the model denote this value as SST O X1 X2 What is the value of SST O X1 X2 3 Now let s use the above definitions to calculate the sequential sum of squares of adding X2 to the model in which X1 is the only predictor We denote this quantity as SSR X2 X1 The bar is read as given According to the alternative definitions SSR X2 X1 is the reduction in the error sum of squares when X2 is added to the model in which X1 is the only predictor That is SSR X2 X1 SSE X1 SSE X1 X2 What is the value of SSR X2 X1 calculated this way 1 Alternatively we can think of the SSR X2 X1 as the increase in the regression sum of squares when X2 is added to the model in which X1 is the only predictor That is SSR X2 X1 SSR X1 X2 SSR X1 What is the value of SSR X2 X1 calculated this way Did you get the same answer as above You should ignoring small round o error 4 Note that because you fit a multiple regression model Minitab automatically displays a column of sequential sum of squares named Seq SS The sequential sums of squares you get depends on the order in which you enter the predictors in the model Since you entered x1 brain first the number Minitab displays for the Seq SS for brain is SSR X1 What is the value Minitab displays for SSR X1 and is it consistent with the value of SSR X1 you obtained in question 1 In words how would you describe the sequential sum of squares SSR X1 Since you entered x2 height second the number Minitab displays for Seq SS for height is SSR X2 X1 What is the value Minitab displays for SSR X2 X1 and is it consistent with the value of SSR X2 X1 you obtained in question 3 5 Let s make sure we see how the sequential sums of squares that we get depends on the order in which we enter the predictors in the model Fit the linear regression model with the two predictors in the reverse order That is when fitting the model indicate x2 height first and x1 brain second Since you entered x2 height first the number Minitab displays for the Seq SS for height is SSR X2 What is the value Minitab displays for SSR X2 Since you entered x1 brain second the number Minitab displays for the Seq SS for brain is SSR X1 X2 What is the value Minitab displays for SSR X1 X2 You can and should verify the values Minitab displays for SSR X2 and SSR X1 X2 by fitting the linear regression model with x2 height as the only predictor Is the value of SSR X2 determined this way consistent with the value you obtained under the Seq SS column Calculate SSR X1 X2 using either of the two definitions Is your calculation consistent with the value Minitab displays under the Seq SS column 6 Sequential sum of squares can be obtained for any number of predictors that are added sequentially to the model To see this now fit the linear regression model with the predictors in order x1 brain and x2 height and x3 weight First Take note of the error sum of squares and since x1 and x2 and x3 are predictors in the model denote this value as SSE X1 X2 X3 What is the value of SSE X1 X2 X3 Take note of the regression sum of squares and since x1 and x2 and x3 are predictors in the model denote this value as SSR X1 X2 X3 What is the value of SSR X1 X2 X3 Take note of the total sum of squares and since x1 and x2 and x3 are predictors in the model denote this value as SST O X1 X2 X3 What is the value of SST O X1 X2 X3 Now consider the sequential sums of squares Minitab displays The first two values SSR X1 and SSR X2 X1 should be consistent with their previous values because you entered x1 brain first and x2 height second Are they Since you entered x3 weight third the number Minitab displays for the Seq SS for weight is SSR X3 X1 X2 What is the value Minitab displays for SSR X3 X1 X2 Calculate SSR X3 X1 X2 using either of the two definitions Is your calculation consistent with the value Minitab displays under the Seq SS column 2 7 All of the sequential sums of squares we considered so far are one degree of freedom sequential sums of squares because we …
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