Assignment

• Exercises 9:1-10 (do not turn in, self-check answers on page 253)
  
• Exercise 9:13
• Exercise 9:14

Lab Goals

• Creating and understanding a Scatterplot matrix
• Fitting multiple regression in S-Plus
• Continue viewing residual plots to assess Model adequacy

reads "the response y is modeled as a linear function of X". For multiple regression, we allow the mean of Y to be a function of multiple explanatory variables, so the model formula might be

which means that the response Y is modeled (the tilde) as a linear function of X1 and X2. The variables X1 and X2 can be anything. In exercise 13 we will examine quadratic regression, where we fit a model with time and time squared as variables; this is a special case of multiple regression. In exercise 14, we have 3 different explanatory variables. In Exercise 16 (covered in class) we created variables based on dummy variables and interaction variables

We will focus first on fitting multiple regressions, and then come back to interpreting the output in more detail in later labs.

Specific Tips for the Homework

Exercise 13

• The data are in file Case0702.asc. Download, and then read into S-PLUS.

• From the Data menu, select Transform and create the variables for the exercise; we will need time^2, log(time) and (log(time))^2. (this is also how to specify the expression for the transformation in the dialog box). Recall, if you click apply, then it will create the first transformation, and then you may continue with another. In what follows I will use the names time2, log.time and log.time2 as the names for these transformed variables.

• For (a): To fit the multiple regression, select the Statistics Menu, choose Regression and Linear. To enter the formula you may type it in directly: ph ~ time + time2 or click "create" to add the parts; select ph as the response and both time and time2 as Main effects.
  

• Select the residual plots that you would like to view, and then click on OK. The output will be in the Report Window.

• Locate the output that lists the coefficient estimates, standard errors, t-stats, and p-values. You will need the p-value for time2. This p-value corresponds to the null hypothesis that the regression coefficient for time^2 is 0 (vs not 0) given that time is included in the model. In other words, we are looking for evidence that we need to include additional terms in time to model possible curvilinear relationships between time and pH.

• For part b just substitute log.time and log.time2 for time and time2 in the above.

• In addition to the p-values, do the residual or normal probability plots indicate whether it is better to use the logarithm of time or leave it untransformed?

Exercise 14

• The data are in file Ex0914.asc. Download, and then read into S-PLUS.

• To create a scatterplot matrix, select the Graph menu and then choose "Scatterplot matrix". In the dialog box, select the variables bank, walk, talk, and heart for the X-axis.

  In this matrix or table, each entry corresponds to a simple scatterplot. For a particular column of the matrix, the variable for the column is plotted on the X-axis in all scatterplots in the column. For a particular row, the corresponding variable is plotted on the Y-axis. The diagonal elements would corresponds to plotting each variable against itself, which is not very interesting, so instead that space contains the variable name, so you can identify the variable for that row or column. The output using brush and spin is similar but includes only the lower half below the diagonal.

• To fit the multiple regression of heart on bank, walk, and talk, go to the Statistics menu, select <>Regression, Linear. The model formula should be heart ~ bank + walk + talk

• Make sure that the selected Plots include the residuals vs fitted values, and also check out the normal probability plot (normally part of the default output)

• The output in the Report Window contains the estimates of the coefficients and standard errors as needed for part (d).