Case Study

13.12 Case Study

Case Study 13.1: Chemical Analysis: Personnel in the Chemistry Department of Virginia Tech were called upon to analyze a data set that was produced to compare 4 different methods of analysis of aluminum in a certain solid igniter mixture. To get a broad range of analytical laboratories involved, 5 laboratories were used in the experiment. These laboratories were selected because they are generally adept in doing these types of analyses. Twenty samples of igniter material containing 2.70% aluminum were assigned randomly, 4 to each laboratory, and directions were given on how to carry out the chemical analysis using all 4 methods. The data retrieved are as follows:

D 2.65 2.69 2.60 2.64 2.73 2.662 The laboratories are not considered as random effects since they were not se-

lected randomly from a larger population of laboratories. The data were analyzed as a randomized complete block design. Plots of the data were sought to determine if an additive model of the type

y ij =μ+m i +l j +ǫ ij

552 Chapter 13 One-Factor Experiments: General is appropriate: in other words, a model with additive effects. The randomized

block is not appropriate when interaction between laboratories and methods exists. Consider the plot shown in Figure 13.12. Although this plot is a bit difficult to interpret because each point is a single observation, there appears to be no appreciable interaction between methods and laboratories.

2.60 3 A B C D

Method

Figure 13.12: Interaction plot for data of Case Study 13.1.

Residual Plots

Residual plots were used as diagnostic indicators regarding the homogeneous vari- ance assumption. Figure 13.13 shows a plot of residuals against analytical methods. The variability depicted in the residuals seems to be remarkably homogeneous. For completeness, a normal probability plot of the residuals is shown in Figure 13.14.

Standard Normal Quantile

Figure 13.13: Plot of residuals against method for Figure 13.14: Normal probability plot of residuals the data of Case Study 13.1.

for the data of Case Study 13.1. The residual plots show no difficulty with either the assumption of normal

errors or the assumption of homogeneous variance. SAS PROC GLM was used

Exercises 553 to conduct the analysis of variance. Figure 13.15 shows the annotated computer

printout. The computed f- and P-values do indicate a significant difference between an- alytical methods. This analysis can be followed by a multiple comparison analysis to determine where the differences are among the methods.

Exercises

13.37 Testing patient blood samples for HIV antibod- Operator ies, a spectrophotometer determines the optical density

1 2 3 4 of each sample. Optical density is measured as the

170.1 175.2 absorbance of light at a particular wavelength. The

173.4 175.7 blood sample is positive if it exceeds a certain cutoff

175.7 180.1 value that is determined by the control samples for that

170.7 183.7 run. Researchers are interested in comparing the lab- (a) Perform a random effects analysis of variance at

oratory variability for the positive control values. The the 0.05 level of significance. data represent positive control values for 10 different (b) Compute an estimate of the operator variance com- runs at 4 randomly selected laboratories.

ponent and the experimental error variance compo- Laboratory

nent.

Run

13.40 Five “pours” of metals have had 5 core samples 2 0.983

each analyzed for the amount of a trace element. The 3 1.047

data for the 5 randomly selected pours are as follows: 4 1.087

5 1.16 0.99 1.05 0.94 1.41 (a) Write an appropriate model for this experiment.

(a) The intent is that the pours be identical. Thus, (b) Estimate the laboratory variance component and

test that the “pour” variance component is zero. the variance within laboratories.

Draw conclusions. (b) Show a complete ANOVA along with an estimate

13.38 An experiment is conducted in which 4 treat- of the within-pour variance. ments are to be compared in 5 blocks. The data are given below.

13.41 A textile company weaves a certain fabric on

a large number of looms. The managers would like Treatment

Block

1 2 3 4 5 the looms to be homogeneous so that their fabric is of 1 12.8 10.6 11.7 10.7 11.0 uniform strength. It is suspected that there may be 2 11.7 14.2 11.8 9.9 13.8 significant variation in strength among looms. Con- 3 11.5 14.7 13.6 10.7 15.9 sider the following data for 4 randomly selected looms. 4 12.6 16.5 15.4 9.6 17.1 Each observation is a determination of strength of the fabric in pounds per square inch.

(a) Assuming a random effects model, test the hypoth- esis, at the 0.05 level of significance, that there is

Loom no difference between treatment means.

(b) Compute estimates of the treatment and block vari- ance components.

13.39 The following data show the effect of 4 oper- ators, chosen randomly, on the output of a particular (a) Write a model for the experiment. machine.

(b) Does the loom variance component differ signifi-

554 Chapter 13 One-Factor Experiments: General

The GLM Procedure Class Level Information Class

Number of Observations Read

Number of Observations Used

Dependent Variable: Response

Sum of

Source

DF Squares

Mean Square

F Value

Corrected Total 19

R-Square Coeff Var

Root MSE

Response Mean

DF Type III SS

Mean Square

F Value

Observation Observed

Figure 13.15: SAS printout for data of Case Study 13.1.

Review Exercises 555

Review Exercises

13.42 An analysis was conducted by the Statistics (b) Perform the analysis of variance and give conclu- Consulting Center at Virginia Tech in conjunction with

sions concerning the laboratories. the Department of Forestry. A certain treatment was (c) Do a normal probability plot of residuals. applied to a set of tree stumps in which the chemical Garlon was used with the purpose of regenerating the

roots of the stumps. A spray was used with four lev- 13.46 An experiment was designed for personnel in els of Garlon concentration. After a period of time, the Department of Animal Science at Virginia Tech to study urea and aqueous ammonia treatment of wheat the height of the shoots was observed. Perform a one- straw. The purpose was to improve nutritional value

factor analysis of variance on the following data. Test for male sheep. The diet treatments were control, urea to see if the concentration of Garlon has a significant at feeding, ammonia-treated straw, and urea-treated impact on the height of the shoots. Use α = 0.05.

straw. Twenty-four sheep were used in the experiment, and they were separated according to relative weight.

Garlon Level There were four sheep in each homogeneous group (by 1 2 3 4 weight) and each of them was given one of the four 2.87 2.31 3.27 2.66 2.39 1.91 3.05 0.91 diets in random order. For each of the 24 sheep, the 3.91 2.04 3.15 2.00 2.89 1.89 2.43 0.01 percent dry matter digested was measured. The data follow.

13.43 Consider the aggregate data of Example 13.1. Group by Weight (block) Perform Bartlett’s test, at level α = 0.1, to determine

1 2 3 4 5 6 gates.

if there is heterogeneity of variance among the aggre-

Diet

32.68 36.22 36.36 40.95 34.99 33.89 13.44 Three catalysts are used in a chemical process;

Control

Urea at

35.90 38.73 37.55 34.64 37.36 34.35 are yield data from the process:

a control (no catalyst) is also included. The following

1 2 3 Urea 74.5 77.5 81.5 78.1 treated

75.9 80.6 81.4 81.5 (a) Use a randomized complete block type of analy- 78.1 84.9 79.5 83.0 sis to test for differences between the diets. Use 76.2 81.0 83.0 82.1 α = 0.05.

Use Dunnett’s test at the α = 0.01 level of significance (b) Use Dunnett’s test to compare the three diets with to determine if a significantly higher yield is obtained

the control. Use α = 0.05. with the catalysts than with no catalyst.

(c) Do a normal probability plot of residuals. 13.45 Four laboratories are being used to perform

chemical analysis. Samples of the same material are 13.47 In a study that was analyzed for personnel sent to the laboratories for analysis as part of a study in the Department of Biochemistry at Virginia Tech,

to determine whether or not they give, on the average, three diets were given to groups of rats in order to study the same results. The analytical results for the four the effect of each on dietary residual zinc in the blood- laboratories are as follows:

stream. Five pregnant rats were randomly assigned to each diet group, and each was given the diet on day 22

Laboratory of pregnancy. The amount of zinc in parts per million A B C D was measured. The data are as follows:

58.2 60.3 58.1 62.3 Determine if there is a significant difference in resid- ual dietary zinc among the three diets. Use α = 0.05.

(a) Use Bartlett’s test to show that the within- Perform a one-way ANOVA. laboratory variances are not significantly different at the α = 0.05 level of significance.

556 Chapter 13 One-Factor Experiments: General 13.48 An experiment was conducted to compare

Gasoline Brand three types of paint for evidence of differences in their

A B C wearing qualities. They were exposed to abrasive ac-

Model

A 32.4 35.6 38.7 tion and the time in hours until abrasion was noticed

B 28.8 28.6 29.9 was observed. Six specimens were used for each type

C 36.5 37.6 39.1 of paint. The data are as follows.

D 34.4 36.2 37.9 Paint Type

(a) Discuss the need for the use of more than a single

1 2 3 model of car.

(b) Consider the ANOVA from the SAS printout in 315 220 115

Figure 13.17 on page 558. Does brand of gasoline matter?

(a) Do an analysis of variance to determine if the evi- (c) Which brand of gasoline would you select? Consult dence suggests that wearing quality differs for the

the result of Duncan’s test. three paints. Use a P-value in your conclusion.

(b) If significant differences are found, characterize 13.51 Four different locations in the northeast were what they are. Is there one paint that stands out? used for collecting ozone measurements in parts per

Discuss your findings. million. Amounts of ozone were collected in 5 samples (c) Do whatever graphical analysis you need to deter- at each location. mine if assumptions used in (a) are valid. Discuss

Location your findings.

1 2 3 4 (d) Suppose it is determined that the data for each

0.09 0.15 0.10 0.10 treatment follow an exponential distribution. Does

0.10 0.12 0.13 0.07 this suggest an alternative analysis? If so, do the

0.08 0.17 0.08 0.05 alternative analysis and give findings.

13.49 A company that stamps gaskets out of sheets of rubber, plastic, and cork wants to compare the mean (a) Is there sufficient information here to suggest that number of gaskets produced per hour for the three

there are differences in the mean ozone levels across types of material. Two randomly selected stamping

locations? Be guided by a P-value. machines are chosen as blocks. The data represent the (b) If significant differences are found in (a), charac- number of gaskets (in thousands) produced per hour.

terize the nature of the differences. Use whatever The data is given below. In addition, the printout anal-

methods you have learned. ysis is given in Figure 13.16 on page 557.

13.52 Show that the mean square error Machine

A 4.31 4.27 4.40 3.36 3.42 3.48 4.01 3.94 3.89 SSE B 3.94 3.81 3.99 3.91 3.80 3.85 3.48 3.53 3.42

k(n − 1) (a) Why would the stamping machines be chosen as

blocks? for the analysis of variance in a one-way classification 2 (b) Plot the six means for machine and material com- is an unbiased estimate of σ . binations.

13.53 Prove Theorem 13.2.

(c) Is there a single material that is best? (d) Is there an interaction between treatments and

13.54 Show that the computing formula for SSB, in blocks? If so, is the interaction causing any seri- the analysis of variance of the randomized complete

ous difficulty in arriving at a proper conclusion? block design, is equivalent to the corresponding term Explain.

in the identity of Theorem 13.3. 13.55 For the randomized block design with k treat-

13.50 A study is conducted to compare gas mileage for 3 competing brands of gasoline. Four different au- ments and b blocks, show that tomobile models of varying size are randomly selected.

The data, in miles per gallon, follow. The order of b 2 E(SSB) = (b − 1)σ 2 +k β j testing is random for each model. .

j=1

Review Exercises 557

The GLM Procedure

Dependent Variable: gasket

Sum of

Source

DF Squares

Mean Square

F Value

Corrected Total

R-Square Coeff Var

Root MSE

gasket Mean

Pr > F material

DF Type III SS

Mean Square

F Value

23.04 0.0004 material*machine

87.39 <.0001 Level of

Level of

------------gasket-----------

material machine

Level of

------------gasket-----------

material N

Level of

------------gasket-----------

machine N

Figure 13.16: SAS printout for Review Exercise 13.49.

558 Chapter 13 One-Factor Experiments: General

The GLM Procedure

Dependent Variable: MPG

Sum of

Source

DF Squares

Mean Square

F Value

Corrected Total

R-Square Coeff Var

Root MSE

MPG Mean

DF Type III SS

Mean Square

F Value

Duncan’s Multiple Range Test for MPG NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.

Alpha

Error Degrees of Freedom

Error Mean Square

Number of Means

Critical Range

Means with the same letter are not significantly different. Duncan Grouping

B A 34.5000

B 33.0250

Figure 13.17: SAS printout for Review Exercise 13.50.

13.56 Group Project: It is of interest to determine each team would be similar. which type of sports ball can be thrown the longest dis- (b) Each team should be gender mixed. tance. The competition involves a tennis ball, a base- ball, and a softball. Divide the class into teams of five (c) The experimental design for each team should be a individuals. Each team should design and conduct a

randomized complete block design. The five indi- separate experiment. Each team should also analyze

viduals throwing are the blocks. the data from its own experiment. For a given team, (d) Be sure to incorporate the appropriate randomiza-

each of the five individuals will throw each ball (after tion in conducting the experiment. sufficient arm warmup). The experimental response (e) The results should contain a description of the ex-

will be the distance (in feet) that the ball is thrown. periment with an ANOVA table complete with a P - The data for each team will involve 15 observations.

value and appropriate conclusions. Use graphical Important points:

techniques where appropriate. Use multiple com- (a) This is not a competition among teams. The com-

parisons where appropriate. Draw practical conclu- petition is among the three types of sports balls.

sions concerning differences between the ball types. One would expect that the conclusion drawn by

Be thorough.

13.13 Potential Misconceptions and Hazards; Relationship to Material in Other Chapters 559