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- Three-Level and Mixed-Level Factorial and Fractional Factorial Designs

## Douglas C. Montgomery

## Chapter 9

## Three-Level and Mixed-Level Factorial and Fractional Factorial Designs - all with Video Answers

## Educators

Chapter Questions

The effects of developer strength $(A)$ and development time (B) on the density of photographic plate film are being studied. Three strengths and three times are used, and four replicates of a $3^2$ factorial experiment are run. The data from this experiment follow. Analyze the data using the standard methods for factorial experiments.

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Raymond Matshanda

Numerade Educator

Compute the $I$ and $J$ components of the two-factor interaction in Problem 9.1.

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An experiment was performed to study the effect of three different types of 32 -ounce botles $(A)$ and three different shelf types ( $B$ )-smooth permanent shelves, end-aisle displays with grilled shelves, and beverage coolers--on the time it takes to stock ten 12 -bottle cases on the shelves. Three workers (factor $C$ ) were employed in the experiment, and two replicates of a $3^3$ factorial design were run. The observed time data are shown in the following table. Analyze the data and draw conclusions.

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Dominador Tan

Numerade Educator

A medical researcher is studying the effect of lidocaine on the enzyme level in the heart muscle of beagle dogs. Three different commercial brands of lidocaine ( $A$ ), three dosage levels ( $B$ ), and three dogs $(C)$ are used in the experiment, and two replicates of a $3^3$ factorial design are run. The observed enzyme levels follow. Analyze the data from this experiment.

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Compute the $I$ and $J$ components of the two-factor interactions for Example 10.1.

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An experiment is run in a chemical process using a $3^2$ factorial design. The design factors are temperature and pressure, and the response variable is yield. The data that result from this experiment are as follows.

Pressure, psig

$$

\begin{aligned}

&\text { Temper- }\\

&\begin{array}{cccc}

\text { ature, }{ }^{\circ} \mathrm{C} & 100 & 120 & 140 \\

\hline 80 & 47.58,48.77 & 64.97,69.22 & 80.92 .72 .60 \\

90 & 51.86 .82 .43 & 88.47,84.23 & 93.95 .88 .54 \\

100 & 71.18 .92 .77 & 96.57,88.72 & 76.58,83.04

\end{array}

\end{aligned}

$$

(a) Analyze the data from this experiment by conducting an analysis of varnance. What conclusions can you draw?

(b) Graphically analyze the residuals. Are there any concerns about underlying assumptions or model adequacy?

(c) Verify that if we let the low, medium, and high levels of both factors in this design take on the levels $-1,0$, and +1 , then a least squares fit to a second-order model for yield is

$$

\begin{aligned}

y= & 86.81+10.4 x_1+8.42 x_2 \\

& -7.17 x_1^2-7.84 x_2^2-7.69 x_1 x_2

\end{aligned}

$$

(d) Confirm that the model in part (c) can be written in terms of the natural variables temperature $(T)$ and pressure $(P)$ as

$$

\begin{aligned}

\hat{y}= & -1335.63+18.56 T+8.59 P \\

& -0.072 T^2-0.0196 P^2-0.0384 T P

\end{aligned}

$$

(e) Construct a contour plot for yield as a fuaction of pressure and temperature. Based on examination of this plot, where would you recommend running this process?

Victor Salazar

Numerade Educator

(a) Confound a $3^3$ design in three blocks using the $A B C^2$ component of the three-factor interaction. Compare your results with the design in Figure 9.7.

(b) Confound a $3^3$ design in three blocks using the $A B^2 C$ component of the three-factor interaction. Compare. your results with the design in Figure 9.7.

(c) Confound a $3^3$ design in three blocks using the ABC component of the three-factor interaction. Compare your results with the design in Figure 9.7.

(d) After looking at the designs in parts (a), (b), and (c) and Figure 9.7, what conclusions can you draw?

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Confound a $3^4$ design in three blocks using the $A B^2 C D$ component of the four-factor interaction.

Raymond Matshanda

Numerade Educator

Consider the data from the first replicate of Problem 9.3. Assuming that not all 27 observations could be run on the same day, set up a design for conducting the experiment over three days with $A B^2 C$ confounded with blocks. Analyze the data.

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Outline the analysis of variance table for the $3^4$ design in nine blocks. Is this a practical design?

Raymond Matshanda

Numerade Educator

Consider the data in Problem 9.3. If $A B C$ is confounded in replicate 1 and $A B C^2$ is confounded in replicate II, perform the analysis of variance.

Raymond Matshanda

Numerade Educator

Consider the data from replicate & of Problem 9.3. Suppose that only a one-third fraction of this design with $I=$ $A B C$ is run. Construct the design, determine the alias structure, and analyze the data.

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From examining Figure 9.9, what type of design would remain if after completing the first nine suns, one of the three factors could be dropped?

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Construct a $3_{\mathrm{NV}}^{+-1}$ design with $I=A B C D$. Write out the alias structure for this design.

Khoobchandra Agrawal

Numerade Educator

Verify that the design in Problem 9.14 is a resolution IV design.

Khoobchandra Agrawal

Numerade Educator

Construct a $3^{5-2}$ design with $l=A B C$ and $l=C D E$. Write out the alias structure for this design. What is the resolution of this design?

Khoobchandra Agrawal

Numerade Educator

Construct a $3^{9-6}$ design and verify that it is a resolution III design.

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Construct a $4 \times 2^3$ design confounded in two blocks of 16 observations each. Outline the analysis of variance for this design.

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Outline the analysis of variance table for a $2^2 3^2$ factorial design. Discuss how this design may be confounded in blocks.

Raymond Matshanda

Numerade Educator

Starting with a $16-\operatorname{run} 2^4$ design. show how two threelevel factors can be incorporated in this experiment. How many two-level factors can be included if we want some information on two-factor interactions?

Raymond Matshanda

Numerade Educator

Starting with a $16-\mathrm{run} 2^4$ design, show how one threelevel factor and three two-level factors can be accommodated and still allow the estimation of two-factor interactions.

Raymond Matshanda

Numerade Educator

In Problem 8.26, you met Harry and Judy PetersonNedry, two friends of the author who have a winery and vineyard in Newberg, Oregon. That problem described the application of two-level fractional factorial designs to their 1985 Pinot Noir product. In [987, they wanted to conduct another Pinot Noir experiment. The variables for this experiment were

Variable

Clone of Pinot Noir

Beny size

Fermentation temperature

Whole berty

Maceration time

Yeast type

Oak type

Levels

Wadenswil, Pommard

Small, large

$80^{\circ}, 85^{\circ}, 90 / 80^{\circ}$, and $90^{\circ} \mathrm{F}$

None, $10 \%$

10 and 21 days

Assmanhau, Champagne

Tronçais, Allier

Harry and Judy decided to use a 16-run two-level fractional factonial design, treating the four levels of fermentation temperature as two two-level variables. As in Problem 8.27 , they used the rankings from a taste-test panel as the response variable. The design and the resulting average ranks are as follows.

$$

\begin{array}{|c|c|c|c|c|c|c|c|c|}

\hline \text { Run } & \text { Clone } & \begin{array}{l}

\text { Berry } \\

\text { Size }

\end{array} & \begin{array}{l}

\text { Ferm. } \\

\text { Teinp. }

\end{array} & \begin{array}{l}

\text { Whole } \\

\text { Berry }

\end{array} & \begin{array}{l}

\text { Macer, } \\

\text { Time }

\end{array} & \begin{array}{l}

\text { Yeast } \\

\text { Type }

\end{array} & \begin{array}{l}

\text { Oak } \\

\text { Type }

\end{array} & \begin{array}{l}

\text { Average } \\

\text { Rank }

\end{array} \\

\hline \text { । } & \text { - } & \text { - } & \cdots & - & - & \text { - } & \text { - } & 4 \\

\hline 2 & + & - & \text { - } & \text { - } & + & + & + & 10 \\

\hline 3 & \text { - } & \text { t } & - & + & \text { — } & + & + & 6 \\

\hline 4 & \text { + } & \text { + } & \cdots & + & + & \text { - } & \text { - } & 9 \\

\hline 5 & \text { - } & \text { - } & +- & + & + & + & = & \text { II } \\

\hline 6 & 4 & - & +- & + & \text { - } & \text { - } & t & 1 \\

\hline 7 & - & + & +- & - & + & \text { - } & + & 15 \\

\hline 8 & + & t & +- & - & \text { - } & + & \text { - } & 5 \\

\hline 9 & - & - & -+ & + & + & \text { - } & + & 12 \\

\hline 10 & + & - & -\div & + & \text { - } & + & - & 2 \\

\hline \text { (1 } & - & + & -+ & \text { - } & + & + & \text { - } & 16 \\

\hline 12 & + & + & -+ & \text { - } & \text { - } & - & + & 1 \\

\hline 13 & - & - & ++ & \text { - } & \text { - } & + & + & 8 \\

\hline 14 & + & - & ++ & \text { - } & + & - & \text { - } & 14 \\

\hline 15 & - & \text { † } & ++ & + & \text { - } & = & \text { - } & 7 \\

\hline 16 & + & + & ++ & + & \text { † } & + & + & 13 \\

\hline

\end{array}

$$

(a) Describe the aliasing in this design.

(b) Analyze the data and draw conclusions.

(c) What comparisons can you make between this experiment and the 1985 Pinot Noir experiment from Problem 8.27 ?

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An article by W. D. Baten in the 1956 volume of Industrial Qualing Control described an experiment to study the effect of three factors on the lengths of steel bars. Each bar was subjected to one of two heat treatment processes and was cut on one of four machines at one of three times during the day ( 8 A.M. II A.M. or 3 p.M.). The coded length data are as follows:

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(a) Analyze the data from this experiment, assuming that the four observations in each cell are replicates.

(b) Analyze the residuals from this experiment. Is there any indication that there is an outlier in one cell? If you find an outlier. remove it and repeat the analysis from part (a). What are your conclusions?

(c) Suppose that the observations in the cells are the lengths (coded) of bars processed together in heat treatment and then cut sequentially (that is, in order) on the four machines. Analyze the data to determine the effects of the three factors on mean length.

(d) Calculate the log variance of the observations in each cell. Analyze this response. What conclusions can you draw?

(e) Suppose the time at which a bar is cut really cannot be controlled during routine production. Analyze the average length and the log variance of the length for each of the 12 bars cut at each machine/heat treatment process combination. What conclusions can you draw?

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Reconsider the experiment in Problem 9.23 . Suppose that it was necessary to estimate all main effects and twofactor interactions, but the full factorial with 24 runs (nol counting replication) was too expensive. Recommend an altemative design.

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Suppose there are four three-level categorical factor and a style two-level continuous factor. What is the minimum number of nuns required to estimate all main effects and twofactor interactions? Construct this design.

Raymond Matshanda

Numerade Educator

Reconsider the experiment in Problem 9.25. Construct a design with $N=48$ runs and compare it to the design you constructed in Problem 9.25.

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Reconsider the experiment in Problem 9.25. Suppose that you are only interested in main effects. Construct a design with $N=12$ runs for this experiment.

Raymond Matshanda

Numerade Educator