Statistics 242 -- Applied Regression Analysis

Statistic 242 -- Lab 1 -- 1/19/2000

Topics

  1. Starting S-Plus
  2. Reading data into S-Plus
  3. Scatterplots
  4. Summary Statistics
  5. Simple Linear Regression

Assignment

For problem 7.24 (Old Faithful) find the prediction intervals as asked. Turn in a brief report (one page max write up including any graphical displays) suitable for park rangers to use for giving 95% prediction interval estimates for the next eruption time to tourists. Be sure to give interpretation of the slope and the prediction intervals using language that a visiting tourist can understand.

Solutions for problems should be neatly and clearly written up (typing helps :-). All computer output that is turned in should be clearly labeled, and any extraneous material should be removed using an editor. These are due by 5 pm Tuesday 1/26 in the box outside 219A. You may also turn in the assignment in class or give it to the TA.

Lab Preliminaries

We will need to download the data for the exercises. The data sets are available in the Datasets Folder under Assignments, and have names such as EX0714.ASC for exercise 14, chapter 7. (Case studies from the chapter are have names like CASE0301.ASC for Case study 3.1 in chapter 3.) Save the files EX0116.ASC, EX0724.ASC and EX0725.ASC to the PC. Note: you may want to save them to your own floppy so that you may continue working on another computer at a later stage (if need be). Remember where they are!

Introduction to S-Plus

Start up S-Plus using the Start menu. The program may be listed under Programs > Statistics & Mathematics > S-PLUS 2000. If you have questions, please check with the TA.

You can create your own workspace directory for saving your class work, this will help if you need to move files to different computers. See Chapter 7 of the Online Guide to Statistics (under the Help Menu) or follow the instruction on here for creating a workspace.

The next thing we need to do is read in the data. S-Plus stores data in objects called "dataframes". A dataframe is like a matrix or table of numbers, with columns corresponding to variables and rows corresponding to observations. Data in dataframes can be continuous or discrete (numeric) or categorical. For example, the planet names in EX016.ASC would be a categorical variable, while the distance and order are numeric.

Creating a Dataframe

When you start S-Plus, a "Select Data" window may be open. Select "Import data", and click "OK". If the window was not opened at startup (or you need to create a new dataframe at a later point in the session), go to the File menu and select Import data > From file. In the "Import Data" window, enter or select the file Ex0116.asc. The default name for the dataframe will be Ex0116; you may change it if you prefer. (just be sure you use your name in place of Ex0116 in any commands below). Click on Open (or hit Enter). You should see a "spreadsheet" that represents the dataframe Ex0116, with 3 columns for the variables planet, order, and distance, and with 10 rows.

You can repeat at some point in time the above steps to create dataframes for the data for Ex 24 and Ex 25.

Note for future reference: S-Plus assumes that the first row of the data file contains the variable names.

Transformations of Variables

In some of the exercises, you are asked to use log transformations. To create a variable for the log(distance) for the dataframe Ex0116,
  1. go to the Data menu, and select Transform
  2. in the pop-up window, make sure that Ex0116 is the dataframe name
  3. add a name for the new variable in the target column field, say log.distance. Avoid using "-", "(" or ")" or other math symbols in names!
  4. in the box for Expression, enter the expression or "code" for the transformation

    log(distance)

    ( you could also select distance from the Variable field, and log from the Function field box, and then click on Add to create the expression, but that requires more effort in my view :-) so just ignore that part of the dialog box.
  5. Click on OK, and if all is correct you should see a new column labeled "log.distance" in your dataframe!
For other transformations, use the same procedure, ie to take the square root of a varialbe enter sqrt(distance); for the inverse, enter 1/distance for the expression.
Now we are ready to make the scatter plots, calculate summary statistics, and fit regressions as asked in problems 1.16, 7.15, and 7.16.

Creating a Scatterplot

Before fitting a regression it is always a good idea to first look at a scatterplot of the data.

In exercise 1.16 you are asked to draw a scatterplot of distance versus order.

  1. Go to the Graph menu, and select 2DGraph
  2. Select Scatter Plot, then click OK
  3. In the pop-up window, make sure that Ex0116 is the Dataset
  4. Specify order for the x Column (horizontal or x-axis)
  5. Specify distance for the y Column (vertical or y-axis)
  6. Then click on OK
You should see a Graph menu. Clicking on the clipboard icon on the far right of the menu bar will send the current graph to the clipboard where you may paste it into another application, such as Word, Wordpad, etc.

To add a title to the graph, go to the Insert menu (on the main Menu bar, not the graph menu bar) and select Titles, and Main. Enter in the text. Make sure that the title is clear and informative.

By default S-Plus use the variable names for the axis titles, which may not be as meaningful, for example, log.distance. To modify, click twice on the axis title. Change the @Auto label to the desired text. Click outside the edit box to indicate you are finished. If you want to change the size or type of font, with the text selected change the font or other characteristics. The text may cover multiple lines (just hit return).

Suggestion: It is a good idea to include the units of measurements if possible in the axis title. If possible, indicate the source of the data at the bottom of the figure.

Summary Statistics

To obtain summary statistics, go to the Statistics menu and select Data Summaries and then Summary Statistics

Specify the dataframe, Ex0116. Click OK if you want summaries of everything. If not, just click to select the summaries that you want. To select variables, click on the first one, and then use Ctrl-Click to select additional variables. Now click OK.

The summaries will appear in the Report Window. You may copy/paste them to a Word document.

Fitting the Simple Linear Regression

To fit a simple linear regression model, go to the Statistics menu and select Regression and then Linear.

Specify the dataframe, Ex0116.

In order to specify the model, we will need to specify a formula. You may either type it in directly, or click on the button "Create Formula". Let's do the latter. You should see a pop-up window titled Formula.

First we need to specify the response variable. Select log.distance and then below, click on the Response button. In the formula window you should see log.distance~

The "~" or tilde character is used to separate the response variable from the predictor or explanatory variables.

Next we need to specify the explanatory variable for our model. Select order in the variables window, and then click the button "Main Effect (+)" to add order to the formula. The formula should now read

log.distance ~ order

Click on OK to return to the Linear Regression Window. The formula should now be in place. (now that you see the form, you can just directly type it in or click on the variable names for Dependent (Y) and Independent (X's) variables.

To save the fitted values and residuals, click on the Results "tab" at the top of the Linear Regression Window. Under "Saved Results" enter the name of the dataframe, Ex0116, and then click the boxes for Fitted values and Residuals. These variables will be calculated and added as new columns to the dataframe, Ex0116.

Right now we will ignore all plots, so click on the Plot tab and de-select all plots.

Click on OK.

In the Report Window, you should have the output with the estimates of the coefficients and their standard errors. In the dataframe, you should have the fitted values and residuals. From this output you should be able to answer all the questions for problems 7.15 and 7.16.

Tips for Ex 7.24

Fit the regression model. Calculate the prediction intervals using duration times of 2, 3, 4, and 5 minutes by hand. To use the expression on page 184, you will need the mean and variance of durations (see summary statistics above) and the estimate of sigma (this is the residual standard error which should be in the report output). That will give you the SE of the mean which can be plugged into the expression on page 185 to get the SE of the prediction. You can also use S-Plus to calculate the SE of the estimated mean. When you have the regression dialog box open, click on the "Predict" tab. Specify Ex0724 (or the name of the dataframe) in the New Data field and in the Save In field. Click the boxes for Predictions and Standard Errors. These are the standard errors for the estimated mean; you will still need to use the expression on page 185 to get the standard error for the predictions.

To graph the data, regression line, and prediction intervals, go to the Graph menu and select 2D Plots. This time scroll up to select "Fit - Linear Least Squares". Click on OK.

Specify duration as the x-Column, interval as the y-column. Then click on the "By Conf Bound" tab. Choose "Confidence 0.95" at the bottom. Then choose any easy to view line style, color and width. Click on OK

. To add an equation for the regression line to the graph, click anywhere on the regression line. A small green box should appear at the bottom left corner of the graph. Now go to the Insert menu and select "Curve Fit Equation". The equation "object will appear in the screen. You can drag it to another location if need be. As with the axis titles, double click on the text in order to edit it. For example, change x to duration.

Tips for Ex 7.25

Create the variables log(force) and log(height).

To create the three regressions, follow the instructions for regression from before, but now in the main dialog box we will need to specify the subset of the data to use. The expression "code == 1", will return an indicator that is true for all rows where code equals 1, and is false elsewhere. You can use that expression to specify the subset in order to fit the regression to only species H. nudus. In the Subset Rows with field, enter code == 1.00. Repeat for the regression using code == 2.00 and code == 3.00. The output you need to complete the questions will be in the Report window.

(optional) Here is code for how to create a plot with all three species for log force (y-axis) versus log height (x-axis). Follow the instructions for the scatterplot given above, but now specify that the z-variable is the column for code. We'll use this to specify different plotting symbols based on code. Click on the Vary Symbols tab in the Lines/Scatter Plot window. Select the z-column for the Vary Style by field. (You can vary color as well). Click on OK to create the plot. To get really fancy, add a Legend to the plot. Go to the Insert menu and select Legend. In the pop-up window specify 3 for the number of items (one for each symbol/species). You can specify the location, or just drag it later to your preferred location. Click on OK. To change any of the legend items, double click on the each item to bring up a Legend Item dialog. Change the text to reflect the species name, i.e. H. nudus. (You may need to click the box to override some defaults). Add a title. To be really complete, add text at the bottom indicating the source of the data. Go to the Insert menu, and select Text. (it will add a box with "Your text:") Move it to the bottom of the page (or wherever you prefer). Click twice to highlight and then replace the text with the source info. You may wish to use a smaller font. Here is an example: