STAT 511
SPRING 2001
Assignment 6
Name _______________
Reading assignment: Neter, Kutner, Nachtsheim & Wasserman, Chapter 13
Bates & Watts, Chapter 3 (one of the supplemental readings for the course)
Written assignment:
Due Friday, April 20, in class.
Final Exam: Monday, April 30, 9:45 am-11:45 am.
1.
2.
For each of the following models, state whether it is a linear model, an intrinsically linear
model, or a nonlinear model. In the case of an intrinsically linear model, indicate how it can be
transformed to a linear model by describing a suitable transformation. In each case, εi denotes a
random error
(a)
Yi = exp(β0 + β1 X iβ 2 ) εi
(b)
Yi = β0 + β1X1i + β2 exp(X1i + 1) + εi
(c)
Yi = exp(εi exp(β0 + β1X1i )) [1–exp(εi exp(β0 + β1X1i ))]-1
(d)
Yi = exp(β0 + β1 log(X1i)) εi
(e)
Yi = β0 X 1i β 1 X 2i β 2 + εi
In an enzyme kinetics study, the velocity of a reaction (Y) is expected to be related to the
concentration (X) of a substance as follows:
Yi =
β0 X i
β1 + X i
+ εi
Observations were made at n = 18 concentrations and the observed velocity of the reaction was
recorded. That data are stored in the file kinetics.dat with one line for each run and Xi preceded
by Yi.
(a)
To obtain starting values, ignore the additive error and note that
Yi
−1
≈
1
β
+ 1 Xi
β0
β0
−1
.
Use least squares regression analysis to obtain estimates of (β0)-1 and (β1/β0) , and list
your starting values for β0 and β1.
2
3.
(b)
Using the starting values from Part (a), find the least squares estimates of β0 and β1 in
the original non-linear model.
(c)
Make a plot showing the estimated mean velocity and the observed data. Does the
model appear to be adequate?
(d)
Plot the residuals against the concentration (X). What does this plot suggest?
(e)
Construct a normal probability plot for the residuals. What does this plot suggest?
There is no need to print and submit the plots for parts (c), (d) and (e). Just comment on
what you see.
(f)
Assuming that the proposed model is correct, use the large sample distribution of the
parameter estimates to construct approximate 95% confidence intervals for β0 and β1.
(g)
Estimate the mean velocity of the reaction when the concentration of the substance is X
= 12. Construct an approximate 95% confidence interval for the mean velocity at X =
12.
(h)
Estimate the concentration (X) at which the expected velocity of the reaction is 30.
Report an approximate 95% confidence interval for the concentration.
(i)
Construct an approximate 95% prediction interval for the velocity that would be
observed if the reaction is run at a concentration of X = 15.
A certain chemical reaction can be described by the nonlinear model


 1
1  
Y = exp  − β1 X1 exp - β2 
  + ε
−
620  

 X2

where β1 and β2 are parameters to be estimated, Y is the fraction of original material remaining
after the reaction occurs, X1 is the reaction time in minutes, and X2 is the temperature in
degrees Kelvin. Data from n = 38 runs of this chemical reaction under various lengths of time
and temperature are stored in the file bhexane.dat on the course web page. There is one line of
data in the file for each run with four numbers on each line corresponding to the run number,
X1, X2 and Y, respectively. S-PLUS code for fitting this model is posted in the file bhexane.ssc.
(a)
What does this model imply about E(Y), the expected fraction of original material that is
not consumed in the reaction, as the reaction time (X1) becomes very large? What about
very short reaction times? What happens when the temperature approaches zero?
(b)
Give an interpretation of β1.
(c)
Find least squares estimates for β1 and β2 and approximate standard errors. Explain how
you obtained initial estimates of β1 and β2 to start the iterative process for finding the
least squares estimates.
3
(d)
Assuming that the random errors are independent and identically distributed, each with a
normal distribution with mean zero and variance σ 2 , find maximum likelihood estimates
for β1 and β2 and the corresponding standard errors.
(e)
Use the results from maximizing the likelihood in part (d) to construct a 95 percent
confidence interval for the expected fraction of original material remaining after the
process is run for 100 minutes at 630 degrees Kalvin.