Three-Factor Experiments

14.4 Three-Factor Experiments

In this section, we consider an experiment with three factors, A, B, and C, at a, b, and c levels, respectively, in a completely randomized experimental design. Assume again that we have n observations for each of the abc treatment combinations. We shall proceed to outline significance tests for the three main effects and interactions involved. It is hoped that the reader can then use the description given here to generalize the analysis to k > 3 factors.

Model for the The model for the three-factor experiment is Three-Factor Experiment

y ijkl =μ+α i +β j +γ k + (αβ) ij + (αγ) ik + (βγ) jk + (αβγ) ijk +ǫ ijkl ,

i = 1, 2, . . . , a; j = 1, 2, . . . , b; k = 1, 2, . . . , c; and l = 1, 2, . . . , n, where α i , β j , and γ k are the main effects and (αβ) ij , (αγ) ik , and (βγ) jk are the two- factor interaction effects that have the same interpretation as in the two-factor experiment.

The term (αβγ) ijk is called the three-factor interaction effect, a term that represents a nonadditivity of the (αβ) ij over the different levels of the factor C. As before, the sum of all main effects is zero and the sum over any subscript of the two- and three-factor interaction effects is zero. In many experimental situations, these higher-order interactions are insignificant and their mean squares reflect only random variation, but we shall outline the analysis in its most general form.

Again, in order that valid significance tests can be made, we must assume that the errors are values of independent and normally distributed random variables,

each with mean 0 and common variance σ 2 .

The general philosophy concerning the analysis is the same as that discussed for the one- and two-factor experiments. The sum of squares is partitioned into eight terms, each representing a source of variation from which we obtain independent

estimates of σ 2 when all the main effects and interaction effects are zero. If the effects of any given factor or interaction are not all zero, then the mean square will estimate the error variance plus a component due to the systematic effect in question.

Sum of Squares

(¯ y ij.. − ¯y i... − ¯y .j.. +¯ y .... ) 2 Three-Factor

for a SSA = bcn

(¯ y i...

− ¯y 2 .... ) SS(AB) = cn

.j.. − ¯y .... )

SS(AC) = bn

(¯ y i.k. − ¯y i... − ¯y ..k. +¯ y .... )

j=1

SSC = abn

(¯ y ..k. − ¯y .... ) 2 SS(BC) = an

(¯ y .jk. − ¯y .j.. − ¯y ..k. +¯ y .... ) 2

k=1

SS(ABC) = n

(¯ y

ijk. − ¯y ij.. − ¯y i.k. − ¯y .jk. +¯ y i... +¯ y .j.. +¯ y ..k. − ¯y .... )

SST =

(y ijkl − ¯y .... ) 2 SSE =

(y ijkl − ¯y ijk. ) 2

kl

580 Chapter 14 Factorial Experiments (Two or More Factors) Although we emphasize interpretation of annotated computer printout in this

section rather than being concerned with laborious computation of sums of squares, we do offer the following as the sums of squares for the three main effects and interactions. Notice the obvious extension from the two- to three-factor problem.

The averages in the formulas are defined as follows: ¯ y .... = average of all abcn observations,

¯ y i... = average of the observations for the ith level of factor A, ¯ y .j.. = average of the observations for the jth level of factor B, ¯ y ..k. = average of the observations for the kth level of factor C, ¯ y ij.. = average of the observations for the ith level of A and the jth level of B, ¯ y i.k. = average of the observations for the ith level of A and the kth level of C, ¯ y .jk. = average of the observations for the jth level of B and the kth level of C, ¯ y ijk. = average of the observations for the (ijk)th treatment combination.

The computations in an analysis-of-variance table for a three-factor problem with n replicated runs at each factor combination are summarized in Table 14.7.

Table 14.7: ANOVA for the Three-Factor Experiment with n Replications Source of

Mean Computed Variation

Sum of

Degrees of

f Main effect:

s 2 f s 3 = 2 3 3 2 s Two-factor interaction:

C SSC

c−1

2 s AB 2 SS(AB) (a − 1)(b − 1) s

AC SS(AC)

(a − 1)(c − 1)

BC SS(BC)

s 2 s (b − 1)(c − 1) 2 6 f 6 = 6 s 2

Three-factor interaction: ABC

SS(ABC) (a − 1)(b − 1)(c − 1)

Error

SSE

abc(n − 1)

For the three-factor experiment with a single experimental run per combination, we may use the analysis of Table 14.7 by setting n = 1 and using the ABC interaction sum of squares for SSE. In this case, we are assuming that the (αβγ) ijk interaction effects are all equal to zero so that

a b SS(ABC) c n

E =σ 2 +

(αβγ) 2 =σ ijk 2 .

(a − 1)(b − 1)(c − 1)

(a − 1)(b − 1)(c − 1) i=1 j=1 k=1

14.4 Three-Factor Experiments 581 That is, SS(ABC) represents variation due only to experimental error. Its mean

square thereby provides an unbiased estimate of the error variance. With n = 1 and SSE = SS(ABC), the error sum of squares is found by subtracting the sums of squares of the main effects and two-factor interactions from the total sum of squares.

Example 14.4: In the production of a particular material, three variables are of interest: A, the operator effect (three operators): B, the catalyst used in the experiment (three catalysts); and C, the washing time of the product following the cooling process (15 minutes and 20 minutes). Three runs were made at each combination of factors. It was felt that all interactions among the factors should be studied. The coded yields are in Table 14.8. Perform an analysis of variance to test for significant effects.

Table 14.8: Data for Example 14.4

Washing Time, C

15 Minutes

20 Minutes

Catalyst, B

Catalyst, B

Operator, A

14.5 11.5 11.5 12.7 10.9 12.2 Solution : Table 14.9 shows an analysis of variance of the data given above. None of the

interactions show a significant effect at the α = 0.05 level. However, the P-value for BC is 0.0610; thus, it should not be ignored. The operator and catalyst effects are significant, while the effect of washing time is not significant.

Impact of Interaction BC

More should be discussed regarding Example 14.4, particularly about dealing with the effect that the interaction between catalyst and washing time is having on the test on the washing time main effect (factor C). Recall our discussion in Section

14.2. Illustrations were given of how the presence of interaction could change the interpretation that we make regarding main effects. In Example 14.4, the BC interaction is significant at approximately the 0.06 level. Suppose, however, that we observe a two-way table of means as in Table 14.10.

It is clear why washing time was found not to be significant. A non-thorough analyst may get the impression that washing time can be eliminated from any future study in which yield is being measured. However, it is obvious how the

582 Chapter 14 Factorial Experiments (Two or More Factors)

Table 14.9: ANOVA for a Three-Factor Experiment in a Completely Randomized Design Source

df Sum of Squares Mean Square F-Value P-Value

A 2 13.98 6.99 11.64 0.0001

B 2 10.18 5.09 8.48 0.0010 AB 4 4.77 1.19 1.99 0.1172

C 1 1.19 1.19 1.97 0.1686 AC 2 2.91 1.46 2.43 0.1027 BC 2 3.63 1.82 3.03 0.0610

Table 14.10: Two-Way Table of Means for Example 14.4 Washing Time, C

Catalyst, B

effect of washing time changes from a negative effect for the first catalyst to what appears to be a positive effect for the third catalyst. If we merely focus on the data for catalyst 1, a simple comparison between the means at the two washing times will produce a simple t-statistic:

which is significant at a level less than 0.02. Thus, an important negative effect of washing time for catalyst 1 might very well be ignored if the analyst makes the incorrect broad interpretation of the insignificant F-ratio for washing time.

Pooling in Multifactor Models

We have described the three-factor model and its analysis in the most general form by including all possible interactions in the model. Of course, there are many situations where it is known a priori that the model should not contain certain interactions. We can then take advantage of this knowledge by combining or pooling the sums of squares corresponding to negligible interactions with the

error sum of squares to form a new estimator for σ 2 with a larger number of degrees of freedom. For example, in a metallurgy experiment designed to study the effect on film thickness of three important processing variables, suppose it is known that factor A, acid concentration, does not interact with factors B and C. The

14.4 Three-Factor Experiments 583

Table 14.11: ANOVA with Factor A Noninteracting

Source of

Computed Variation

Sum of

Degrees of

Mean

f Main effect:

Squares Freedom

3 f 3 = c−1 s 3 s 2 Two-factor interaction:

C 2 SSC s 2

s 2 s 4 2 f 4 = 4 s 2 Error

BC SS(BC)

(b − 1)(c − 1)

sums of squares SSA, SSB, SSC, and SS(BC) are computed using the methods described earlier in this section. The mean squares for the remaining effects will

now all independently estimate the error variance σ 2 . Therefore, we form our new mean square error by pooling SS(AB), SS(AC), SS(ABC), and SSE, along with the corresponding degrees of freedom. The resulting denominator for the significance tests is then the mean square error given by

s 2 = SS(AB) + SS(AC) + SS(ABC) + SSE . (a − 1)(b − 1) + (a − 1)(c − 1) + (a − 1)(b − 1)(c − 1) + abc(n − 1)

Computationally, of course, one obtains the pooled sum of squares and the pooled degrees of freedom by subtraction once SST and the sums of squares for the existing effects are computed. The analysis-of-variance table would then take the form of Table 14.11.

Factorial Experiments in Blocks

In this chapter, we have assumed that the experimental design used is a completely randomized design. By interpreting the levels of factor A in Table 14.11 as dif- ferent blocks, we then have the analysis-of-variance procedure for a two-factor experiment in a randomized block design. For example, if we interpret the op- erators in Example 14.4 as blocks and assume no interaction between blocks and the other two factors, the analysis of variance takes the form of Table 14.12 rather than that of Table 14.9. The reader can verify that the mean square error is also

which demonstrates the pooling of the sums of squares for the nonexisting inter- action effects. Note that factor B, catalyst, has a significant effect on yield.

584 Chapter 14 Factorial Experiments (Two or More Factors)

Table 14.12: ANOVA for a Two-Factor Experiment in a Randomized Block Design Source of

Computed Variation

Sum of

Degrees of

Mean

f P-Value Blocks

Main effect:

B 10.18 2 5.09 6.88 0.0024

C 1.18 1 1.18 1.59 0.2130 Two-factor interaction: BC 3.64 2 1.82 2.46 0.0966 Error

Example 14.5: An experiment was conducted to determine the effects of temperature, pressure, and stirring rate on product filtration rate. This was done in a pilot plant. The experiment was run at two levels of each factor. In addition, it was decided that two batches of raw materials should be used, where batches were treated as blocks. Eight experimental runs were made in random order for each batch of raw materials. It is thought that all two-factor interactions may be of interest. No interactions with batches are assumed to exist. The data appear in Table 14.13. “L” and “H” imply low and high levels, respectively. The filtration rate is in gallons per hour.

(a) Show the complete ANOVA table. Pool all “interactions” with blocks into

error. (b) What interactions appear to be significant? (c) Create plots to reveal and interpret the significant interactions. Explain what

the plot means to the engineer.

Table 14.13: Data for Example 14.5 Batch 1

High Stirring Rate Temp. Pressure L Pressure H

Low Stirring Rate

Temp. Pressure L Pressure H L

High Stirring Rate Temp. Pressure L Pressure H

Low Stirring Rate

Temp. Pressure L Pressure H L

49 57 L

H 70 76 H 103

14.4 Three-Factor Experiments 585 Solution : (a) The SAS printout is given in Figure 14.7.

(b) As seen in Figure 14.7, the temperature by stirring rate (strate) interaction appears to be highly significant. The pressure by stirring rate interaction also appears to be significant. Incidentally, if one were to do further pooling by combining the insignificant interactions with error, the conclusions would remain the same and the P-value for the pressure by stirring rate interaction would become stronger, namely 0.0517.

(c) The main effects for both stirring rate and temperature are highly significant, as shown in Figure 14.7. A look at the interaction plot of Figure 14.8(a) shows that the effect of stirring rate is dependent upon the level of temperature. At the low level of temperature the stirring rate effect is negligible, whereas at the high level of temperature stirring rate has a strong positive effect on mean filtration rate. In Figure 14.8(b), the interaction between pressure and stirring rate, though not as pronounced as that of Figure 14.8(a), still shows a slight inconsistency of the stirring rate effect across pressure.

Source

F Value Pr > F batch

DF Type III SS Mean Square

1 5292.562500 5292.562500 5340.24 <.0001 pressure*temp

1 1040.062500 1040.062500 1049.43 <.0001 pressure*strate

5.11 0.0583 temp*strate

pressure*temp*strate 1 1.562500

Corrected Total

Figure 14.7: ANOVA for Example 14.5, batch interaction pooled with error.

80 80 High Pressure

Low 70

Filtration Rate 60

Filtration Rate 60

High Stirring Rate

High

Low

Stirring Rate (a) Temperature versus stirring rate.

(b) Pressure versus stirring rate.

Figure 14.8: Interaction plots for Example 14.5.

586 Chapter 14 Factorial Experiments (Two or More Factors)

Exercises

tions (4 levels); B, the analysis time (2 levels); and 14.16 Consider an experimental situation involving

C, the method of loading propellant into sample hold- factors A, B, and C, where we assume a three-way ers (hot and room temperature). The following data,

fixed effects model of the form y ijkl =μ+α i +β j + which represent the weight percent of ammonium per- γ k +(βγ) jk +ǫ ijkl . All other interactions are considered chlorate in a particular propellant, were recorded. to be nonexistent or negligible. The data are presented here.

Method of Loading, C

B 1 B 2 Hot

Room Temp. C 1 C 2 C 3 C 1 C 2 C 3 A B 1 B 2 B 1 B 2 A 1 4.0 3.4 3.9 4.4 3.1 3.1 1 38.62 38.45 39.82 39.82 4.9 4.1 4.3 3.4 3.5 3.7 37.20 38.64 39.15 40.26 A 2 3.6 2.8 3.1 2.7 2.9 3.7 38.02 38.75 39.78 39.72 3.9 3.2 3.5 3.0 3.2 4.2 2 37.67 37.81 39.53 39.56 A 3 4.8 3.3 3.6 3.6 2.9 2.9 37.57 37.75 39.76 39.25 3.7 3.8 4.2 3.8 3.3 3.5 37.85 37.91 39.90 39.04 A 4 3.6 3.2 3.2 2.2 2.9 3.6 3 37.51 37.21 39.34 39.74 3.9 2.8 3.4 3.5 3.2 4.3 37.74 37.42 39.60 39.49

(a) Perform a test of significance on the BC interaction 37.58 37.79 39.62 39.45 at the α = 0.05 level.

(b) Perform tests of significance on the main effects A, 37.51 37.91 39.67 39.00 B, and C using a pooled mean square error at the

α = 0.05 level. (a) Perform an analysis of variance with α = 0.01 to test for significant main and interaction effects.

14.17 The following data are measurements from an (b) Discuss the influence of the three factors on the experiment conducted using three factors A, B, and C,

weight percent of ammonium perchlorate. Let your all fixed effects:

discussion involve the role of any significant inter- action.

B 1 B 2 B 3 B 1 B 2 B 3 B 1 B 2 B 3 14.19 Corrosion fatigue in metals has been defined A 1 15.0 14.8 15.9

15.8 15.5 19.2 as the simultaneous action of cyclic stress and chem- 18.5 13.6 14.8

14.3 13.7 13.5 ical attack on a metal structure. In the study Effect 22.1 12.2 13.6

13.0 12.6 11.1 of Humidity and Several Surface Coatings on the Fa- A 2 11.3 17.2 16.1

12.7 17.3 7.8 tigue Life of 2024-T351 Aluminum Alloy, conducted by 14.6 15.5 14.7

14.2 15.8 11.5 the Department of Mechanical Engineering at Virginia 18.2 14.2 13.4

15.9 14.6 12.2 Tech, a technique involving the application of a protec- tive chromate coating was used to minimize corrosion (a) Perform tests of significance on all interactions at fatigue damage in aluminum. Three factors were used

the α = 0.05 level. in the investigation, with 5 replicates for each treat- (b) Perform tests of significance on the main effects at ment combination: coating, at 2 levels, and humidity the α = 0.05 level.

and shear stress, both with 3 levels. The fatigue data, (c) Give an explanation of how a significant interaction recorded in thousands of cycles to failure, are presented has masked the effect of factor C.

here. (a) Perform an analysis of variance with α = 0.01 to

14.18 The method of X-ray fluorescence is an impor- test for significant main and interaction effects. tant analytical tool for determining the concentration (b) Make a recommendation for combinations of the

of material in solid missile propellants. In the paper three factors that would result in low fatigue dam- An X-ray Fluorescence Method for Analyzing Polybu-

age.

tadiene Acrylic Acid (PBAA) Propellants (Quarterly Report, RK-TR-62-1, Army Ordinance Missile Com- mand, 1962), it is postulated that the propellant mix- ing process and analysis time have an influence on the homogeneity of the material and hence on the accu- racy of X-ray intensity measurements. An experiment was conducted using 3 factors: A, the mixing condi-

Exercises 587

Type Coating

Shear Stress (psi)

Gold Foil Goldent Uncoated

14.21 Electronic copiers make copies by gluing black (86–91% RH)

ink on paper, using static electricity. Heating and glu-

663 ing the ink on the paper comprise the final stage of

the copying process. The gluing power during this fi-

nal process determines the quality of the copy. It is Chromated

130 postulated that temperature, surface state of the glu- (20–25% RH)

841 ing roller, and hardness of the press roller influence the

gluing power of the copier. An experiment is run with

treatments consisting of a combination of these three

529 factors at each of three levels. The following data show the gluing power for each treatment combination. Per-

Medium

252 form an analysis of variance with α = 0.05 to test for (50–60% RH)

105 significant main and interaction effects.

State of

Hardness of the

Press Roller High

0.58 0.64 0.79 0.78 0.74 0.50 14.20 For a study of the hardness of gold dental fill-

ings, five randomly chosen dentists were assigned com- binations of three methods of condensation and two Medium Soft

0.46 0.40 0.31 0.49 0.56 0.42 types of gold. The hardness was measured. (See Temp.

0.58 0.37 0.48 0.66 0.49 0.49 Hoaglin, Mosteller, and Tukey, 1991.) Let the den-

Medium 0.60 0.43 0.66 0.57 0.64 0.54 tists play the role of blocks. The data are presented

0.52 0.44 0.54 0.52 0.65 0.49 (b) Is there a significant interaction between method Temp.

(a) State the appropriate model with the assumptions. High

Soft

0.57 0.53 0.65 0.56 0.65 0.52 of condensation and type of gold filling material?

Medium 0.53 0.65 0.53 0.45 0.49 0.48 (c) Is there one method of condensation that seems to

0.66 0.56 0.59 0.47 0.74 0.50 be best? Explain.

Gold Foil

14.22 Consider the data set in Exercise 14.21.

(a) Construct an interaction plot for any two-factor in-

teraction that is significant.

(b) Do a normal probability plot of residuals and com-

14.23 Consider combinations of three factors in the

588 Chapter 14 Factorial Experiments (Two or More Factors) removal of dirt from standard loads of laundry. The is believed that extrusion rate does not interact with

first factor is the brand of the detergent, X, Y , or Z. die temperature and that the three-factor interaction The second factor is the type of detergent, liquid or should be negligible. Thus, these two interactions may powder. The third factor is the temperature of the wa-

be pooled to produce a 2 d.f. “error” term. ter, hot or warm. The experiment was replicated three (a) Do an analysis of variance that includes the three times. Response is percent dirt removal. The data are

main effects and two two-factor interactions. De- as follows:

termine what effects influence the radius of the pro- Brand

Type Temperature

pellant grain.

X Powder

Hot

(b) Construct interaction plots for the powder temper-

ature by die temperature and powder temperature Liquid

by extrusion rate interactions.

(c) Comment on the consistency in the appearance of Y

the interaction plots and the tests on the two in-

teractions in the ANOVA. Liquid

14.25 In the book Design of Experiments for Qual- Z

ity Improvement, published by the Japanese Standards

Association (1989), a study is reported on the extrac- Liquid

tion of polyethylene by using a solvent and how the

amount of gel (proportion) is influenced by three fac- (a) Are there significant interaction effects at the α = tors: the type of solvent, extraction temperature, and 0.05 level?

Warm

extraction time. A factorial experiment was designed, and the following data were collected on proportion of

(b) Are there significant differences between the three gel. brands of detergent?

(c) Which combination of factors would you prefer to Time use?

Solvent Temp.

95.3 92.5 92.4 14.24 A scientist collects experimental data on the

radius of a propellant grain, y, as a function of pow-

94.6 94.5 93.6 94.1 91.1 91.0 der temperature, extrusion rate, and die temperature.

Toluene 120

80 95.4 95.4 95.6 96.0 92.1 92.1 Results of the three-factor experiment are as follows:

(a) Do an analysis of variance and determine what fac- Powder Temp

tors and interactions influence the proportion of gel.

(b) Construct an interaction plot for any two-factor in- Rate

Die Temp

Die Temp

teraction that is significant. In addition, explain what conclusion can be drawn from the presence of

the interaction.

(c) Do a normal probability plot of residuals and com- Resources are not available to make repeated experi-

ment.

mental trials at the eight combinations of factors. It