Chapter 13, Introduction to Nonlinear Regression Video Solutions, Introduction to Linear Regression Analysis

Problem 1

Consider the Michaelis-Menten model introduced in Eq. $(13.23) .$ Graph the expectation function for this model for $\theta_{1}=200$ and $\theta_{2}=0.04,0.06$ $0.08,0.10 .$ Overlay these curves on the same set of $x-y$ axes. What effect does the parameter $\theta_{2}$ have on the behavior of the expectation function?

Robin Corrigan

Numerade Educator

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Problem 2

Consider the Michaelis-Menten model introduced in Eq. $(13.23) .$ Graph the expectation function for $\theta_{1}=100,150,200,250$ for $\theta_{2}=0.06 .$ Overlay these curves on the same set of $x-y$ axes. What effect does the parameter $\theta_{1}$ have on the behavior of the expectation function?

Victor Salazar

Numerade Educator

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Problem 3

Graph the expectation function for the logistic growth model (13.34) for $\theta_{1}=10, \theta_{2}=2,$ and values of $\theta_{3}=0.25,1,2,3,$ respectively. Overlay these plots on the same set of $x-y$ axes. What effect does the parameter $\theta$, have on the expectation function?

Victor Salazar

Numerade Educator

00:22

Problem 4

Sketch the expectation function for the logistic growth model (13.34) for $\theta_{1}=1, \theta_{3}=1,$ and values of $\theta_{2}=1,4,8,$ respectively. Overlay these plots on the same $x-y$ axes. Discuss the effect of $\theta_{2}$ on the shape of the function.

Amrita Bhasin

Numerade Educator

02:28

Problem 5

Consider the Gompertz model in Eq. (13.35). Graph the expectation function for $\theta_{1}=1, \theta_{3}=1,$ and $\theta_{2}=\frac{1}{8}, 1,8,64$ over the range $0 \leq x \leq 10$
a. Discuss the behavior of the model as a function of $\theta_{2}$
b. Discuss the behavior of the model as $x \rightarrow \infty$
c. What is $E(y)$ when $x=0 ?$

Victor Salazar

Numerade Educator

01:08

Problem 6

For the models shown below, determine whether it is a linear model, an intrinsically linear model, or a nonlinear model. If the model is intrinsically linear, show how it can be linearized by a suitable transformation.
a. $y=\theta_{1} e^{\theta_{2}+f_{3} x}+\varepsilon$
b. $y=\theta_{1}+\theta_{2} x_{1}+\theta_{2} x_{2}^{\theta_{3}}+\varepsilon$
c. $y=\theta_{1}+\theta_{2} / \theta_{1} x+\varepsilon$
d. $y=\theta_{1}\left(x_{1}\right)^{9_{2}}\left(x_{2}\right)^{\theta_{3}}+\varepsilon$
e. $y=\theta_{1}+\theta_{2} e^{b_{3} x}+\varepsilon$

Carson Merrill

Numerade Educator

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Problem 7

Reconsider the regression models in Problem $13.6,$ parts a-e. Suppose the error terms in these models were multiplicative, not additive. Rework the problem under this new assumption regarding the error structure.

Rashmi Sinha

Numerade Educator

04:46

Problem 8

Consider the following observations:
$$\begin{array}{rrr}
\hline x & & y & \\
\hline 0.5 & 0.68 & 1.58 \\
1 & 0.45 & 2.66 \\
2 & 2.50 & 2.04 \\
4 & 6.19 & 7.85 \\
8 & 56.1 & 54.2 \\
9 & 89.8 & 90.2 \\
10 & 147.7 & 146.3 \\
\hline
\end{array}$$
a. Fit the nonlinear regression model
$$y=\theta_{1} e^{\theta_{2} x}+\epsilon$$
to these data. Discuss how you obtained the starting values.
b. Test for significance of regression.
c. Estimate the error variance $\sigma^{2}$
d. Test the hypotheses $H_{0}: \theta_{1}=0$ and $H_{0}: \theta_{2}=0 .$ Are both model parameters different from zero? If not, refit an appropriate model
e. Analyze the residuals from this model. Discuss model adequacy.

Heather Duong

Numerade Educator

03:09

Problem 9

Reconsider the data in the previous problem. The response measurements in the two columns were collected on two different days. Fit a new model
$$y=\theta_{3} x_{2}+\theta_{1} e^{\theta_{2} x_{1}}+\varepsilon$$
to these data, where $x_{1}$ is the original regressor from Problem 13.8 and $x_{2}$ is an indicator variable with $x_{2}=0$ if the observation was made on day 1 and $x_{2}=1$ if the observation was made on day 2 , Is there any indication that there is a difference between the two days (use $\theta_{30}=0$ as the starting value).

James Kiss

Numerade Educator

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Problem 10

Consider the model
$$y=\theta_{1}-\theta_{2} e^{-\theta_{3} x}+\varepsilon$$
This is called the Mitcherlich equation, and it is often used in chemical engineering. For example, $y$ may be yield and $x$ may be reaction time.
a. Is this a nonlinear regression model?
b. Discuss how you would obtain reasonable starting values of the parameters $\theta_{1}, \theta_{2},$ and $\theta_{3}$
c. Graph the expectation function for the parameter values $\theta_{1}=0.5$ $\theta_{2}=-0.10,$ and $\theta_{3}=0.10 .$ Discuss the shape of the function.
d. Graph the expectation function for the parameter values $\theta_{1}=0.5$ $\theta_{2}=0.10,$ and $\theta_{3}=0.10 .$ Compare the shape with the one obtained in part c.

Victor Salazar

Numerade Educator

05:29

Problem 11

The data below represent the fraction of active chlorine in a chemical product as a function of time after manufacturing.
a. Construct a scatterplot of the data.
b. Fit the Mitcherlich law (see Problem 13.10 ) to these data. Discuss how you obtained the starting values.
c. Test for significance of regression.
d. Find approximate $95 \%$ confidence intervals on the parameters $\theta_{1}, \theta_{2}$ and $\theta_{3}$. Is there cvidence to support the claim that all three parameters are different from zero?
e. Analyze the residuals and comment on model adequacy.

Raymond Matshanda

Numerade Educator

02:14

Problem 12

Consider the data below.
These data were collected in an experiment where $x_{1}=$ reaction time inminutes and $x_{2}=$ temperature in degrees Celsius. The response variable $y$ is concentration (grams per liter). The engincer is considering the model
$$y=\theta_{1}\left(x_{1}\right)^{\theta_{2}}\left(x_{2}\right)^{\theta_{2}}+\varepsilon$$
a. Note that we can linearize the expectation function by taking logarithms. Fit the restulting linear regression model to the data.
b. Test for significance of regression. Does it appear that both variables $x_{1}$ and $x_{2}$ have important effects?
c. Analyze the residuals and comment on model adequacy.

Raymond Matshanda

Numerade Educator

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Problem 13

Continuation of Problem 13.12
a. Fit the nonlinear model given in Problem 13.12 using the solution you obtained by linearizing the expectation function as the starting values.
b. Test for significance of regression. Does it appear that both variables $x_{1}$ and $x_{2}$ have important effects?
c. Analyze the residuals and comment on model adequacy.
d. Which model do you prefer, the nonlinear model or the linear model from Problem $13.12 ?$

Rashmi Sinha

Numerade Educator

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Problem 14

Continuation of Problem 13.12 . The two observations in each cell of the data table in Problem 13.12 are two replicates of the experiment. Each replicate was run from a unique batch of raw material. Fit the model
$$y=\theta_{4} x_{3}+\theta_{1}\left(x_{1}\right)^{\theta_{2}}\left(x_{2}\right)^{9_{3}}+\varepsilon$$
where $x_{3}=0$ if the observation comes from replicate 1 and $x_{3}=1$ if the observation comes from replicate 2 , Is there an indication of a difference between the two batches of raw material?

Victor Salazar

Numerade Educator