STA 242/ENV 255: February 10, 1998

Lack of Fit test and Multiple Regression

Assignment: Due Thurday -- February 19 -- by 5pm

Part 1. The susceptibility of catfish to a certain chemical was determined by immersing individual fish in 2 liters of an emulsion containing the pollutant and measuring the survival time in minutes. Fit a straight line regression using $\log$ base 10 of survival time $(Y)$ as the dependent variable, and $\log$ base 10 of the concentration as $X$ (measured in parts per million). The datafile already contains the transformed data. Does the straight line model seem appropriate? Carry out a formal lack of fit test to see if we should reject the straight line model.

Part 2: Carry out the regressions and analyses to answer questions 1-9 on pages 102-103 in RWG. Briefly summarize your conclusions about what you think is the "best model" and what it implies for estimating the number of damaged trees.

Suggestions for Part 2.

• Create a SAS dataset for data on page 102. The data in the file are location ( LOC), elevation, (ELEV) and % damaged (DAMAGE). Before your datalines, add a line to create the "interaction" variable for location*elevation, i.e.

     LOCELEV = ELEV*LOC;	
  

  This is for the regression with the slope dummy variable in problem 6. Note: You could also do this within SAS INSIGHT later.
• You may want to use a different symbol for the two different locations. You can change the symbol using the Edit:Windows:Tools menu. You can also color observations.

  Pull down the menu: Edit-> Windows-> Tools Click on the "Circle" button in the Tools Window. A "Mark Observation" window will appear. To make all of the South datapoints appear as Circles, highlight LOC, =, 0, and press OK.

  (You can change the symbol for the north points also (by default they will be squares). To change colors, click on a color, and repeat the same steps you just did.

• To carry out Multiple regression in SAS INSIGHT, proceed as before using the Analyze:Fit menu. Select DAMAGE as Y and select ELEV and LOC as X.
• The regular SAS procedure can be used to obtain confidence intervals for the mean or prediction intervals at the observed X values. Submit the following commands from the program editor:

  	proc reg data=sasuser.rwg102;	
  	model DAMAGE = LOC ELEV / cli clm;
  	run;
  

  The output will show up in the SAS Output Window.
• For a new X value you will need to carry out some matrix alegebra (don't panic!) to get teh Standard error. To get the matrix S = Se^2*(X'X)^-1 in SAS Insight, GO to the Tables menu and select "Estimated Cov Matrix". To get the CI, the vector C = (1 1 1500) assuming your regression has Loc as X1 and Elev as X2 (switch the 2nd and 3rd values if you have them in the other order). The SE for Yhat is calculated using equation 3.26,

  	Var(Yhat Loc = North ELEV = 1500) = [C(Se^2*(X'X)^-1) C'] = CSC'
  

  to get the SE, just take the square root with your calculator.

  Read Appendix 3 for more on Matrix multiplication. We will discuss this in class, so be prepared.

• Look at a 3-dimensional plot of Y=DAMAGE, X=ELEV and Z=LOC. (under the menu ANALYZE:ROTATE .) Click (slowly several times or hold the mouse button down to spin continuously) on the "horizontal right arrow" key to spin the plot. What do you learn from this? Does this visually confirm the statistical tests for adding the slope and intercept dummy variables?