Design and Analysis of Single-Factor Experiments:

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CHAPTER 13

• Design and Analysis of Single-Factor Experiments
  
• The Analysis of Variance

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Learning Objectives

• Design and conduct engineering experiments
• Understand how the analysis of variance is used
  to analyze the data
• Use multiple comparison procedures
• Make decisions about sample size
• Understand the difference between fixed and
  random factors
• Estimate variance components
• Understand the blocking principle
• Design and conduct experiments involving the
  randomized complete block design

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Engineering Experiments

• Experiments are a natural part of the engineering
  decision-making process
• Designed to improve the performance of a subset
  of processes
• Processes can be described in terms of
  controllable variables
• Determine which subset has the greatest influence
• Such analysis can lead to
• Improved process yield
• Reduced variability in the process and closer
  conformance to nominal or target requirements
• Reduced cost of operation

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Steps In Experimental Design

• Usually designed sequentially
• Determine which variables are most important
• Used to refine the information to determine the
  critical variables for improving the process
• Determine which variables result in the best
  process performance

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Single Factor Experiment

• Assume a parameter of interest
• Consist of making up several specimens in two
  samples
• Analyzed them using the statistical hypotheses
  methods
• Can say an experiment with single factor
• Has two levels of investigations
• Levels are called treatments
• Treatment has n observations or replicates

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Designing Engineering Experiments

• More than two levels of the factor
• This chapter shows
• ANalysis Of VAriance (ANOVA)
• Discuss randomization of the experimental runs
• Design and analyze experiments with several
  factors

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Linear Statistical Model

• Following linear model
• Yij µ?i?ij
• i1, 2,,a, and j1, 2,,n
• Yij is (ij)th observation
• µ called the overall mean
• ?i called the ith treatment effect
• ?ij is a random error component with mean zero
  and variance ?2

• Each treatment defines a population
• Mean µi consisting of the overall mean µ
• Plus an effect ?i

Pg. 471 Fig 13-1b
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Completely Randomized Design

• Table shows the underlying model
• Following observations are taken in random order
• Treatments are used as uniform as possible
• Called completely randomized design

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Fixed-effects and Random Models

• Chosen in two different ways
• Experimenter chooses the a treatments
• Called the fixed-effect model
• Experimenter chooses the treatments from a larger
  population
• Called random-effect model

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Development of ANOVA

• Total of the observations and the average of the
  observations under the ith treatment
• Grand total of all observations and the grand
  mean
• Nan is the total number of observations
• dot subscript notation implies summation

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Hypothesis Testing

• Interested in testing the equality of the
  following a treatment means
• ?1,?2 .. ?a
• Equivalent
• H0 ?1?2?a0
• H1 ?a0 for at least one i
• If the null hypothesis is true, changing the
  levels of the factor has no effect on the mean
  response

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Components of Total Variability

• Total variability in data is described by the
  total sum of squares
• Partitions this total variability into two parts
• Measure the differences between treatments
• Measure the random error effect

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Computational Formulas

• Mean square for treatments
• MSTreatmentsSSTreatments /(a-1)
• Error mean square
• MSESSE /a(n-1)

• Efficient formulas
• Total sum of squares
• Treatment sum of squares
• Error sum of squares
• SSESST - SSTreatment

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ANOVA TABLE
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Using Computer Software

• Packages have the capability to analyze data from
  designed experiments
• Presents the output from the Minitab one-way
  analysis of variance routine

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Example

• The tensile strength of a synthetic fiber is of
  interest to the manufacturer. It is suspected
  that strength is related to the percentage of
  cotton in the fiber. Five levels of cotton
  percentage are used, and five replicates are run
  in random order, resulting in the data below. Use
  a0.05.
• a) Does cotton percentage affect breaking
  strength?

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Solution

• Use the general steps in hypothesis testing
• Parameter of interest is the cotton percentage
• H0 ?1?2 ?3?4?50
• H1 ?i 0 for at least one I
• a 0.05
• Test statistic
• Fo MSTR /MSE
• 6. Reject Ho if fo gt fa,(a-1)n(a-1)
• Computations

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Initial calculations

• Compute the last two columns

Conc 1 2 3 4 5
15 7 7 15 11 9 49 9.8
20 12 17 12 18 18 77 15.4
25 14 18 18 19 19 88 17.6
30 19 25 22 19 23 108 21.6
35 7 10 11 15 11 54 10.8
376
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Solution - Cont.

• Compute SST, SSTR, SSE , MSTR, and MSE
• (7)2 (7)2 .(376)2/25
  636.96
• ((49)2 (77)2
  ..(54)2)/5 -376/5 475.7
• SSE 636.96-475.75 161.20
• MSTR SSTR/a-1 475.76/4 118.9
• MSE SSE/a(n-1)161.20/5(5-1) 8.0
• Hence, the test statistic
• Fo MSTR /MSE 118.96/8.06 14.75
• 8. Since fo14.75gt f0.05,4,20 2.87, reject Ho

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Solution

• ANOVA results
• Source DF SS MS F
  P
• COTTON 4 475.76 118.94 14.76 0.000
• Error 20 161.20 8.06
• Total 24 636.96
• Reject H0 and conclude that cotton percentage
  affects breaking strength

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Multiple Comparisons Following the ANOVA

• When H0?1?2?a0 is rejected
• Know that some of the treatment are different
• Doesnt identify which means are different
• Called multiple comparisons methods
• Called Fishers least significant difference
  (LSD) method

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Fishers Least Significant Difference (LSD) Method

• Compares all pairs of means with the H0
  for all i j
• Test statistic
• Pair of means i and j would be different
• Least significant difference, LSD, is

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Example

• Use Fishers LSD method with a 0.05 test to
  analyze the means of five different levels of
  cotton percentage content in the previous example
• Recall H0 was rejected and concluded that cotton
  percentage affects the breaking strength
• Apply the Fishers LSD method to determine which
  treatment means are different

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Solution

• Summarize
• a 5 means, n5, MSE 8.06, and t0.025,202.086
• Treatment means are

9.8
15.4
17.6
21.6
10.8
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Solution Cont.

• Value of LSD
  
• Comparisons
• 5 Vs. 1I10.89.8I1
• 5 Vs. 2I10.8-15.4I4.6gt3.74
• 5 Vs. 3I10.8-17.6I6.8gt3.74
• 5 Vs. 4I10.8-21.6I10.8gt3.74
• 4 Vs. 1I21.6-9.8I11.8gt3.74
• 4 Vs. 2I21.6-15.4I6.2 gt 3.74
• 4 Vs. 3I21.6 17.6I4gt3.74
• 3 Vs. 1I17.6-9.8I7.8gt3.74
• 3 Vs. 2I17.6 -15.4I2.2
• 2 Vs. 1I15.4-9.8I5.6gt3.74
• From this analysis, we see that there are
  significant differences between all pairs of
  means except 5 vs. 1 and 3 vs. 2

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C.I. on Treatment Means

• Confidence interval on the mean of the ith
  treatment µi
• Confidence interval on the difference in two
  treatment means

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Determining Sample Size

• Choice of the sample size to use is important
• OC curves provide guidance in making this
  selection
• Power of the ANOVA test is
• 1-ßP( Reject H0 H0 is false)
• P(F0gt fa, a-1, a(n-1) H0 is false)
• Plot ß against a parameter ?

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Sample OC Curves
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Example

• Suppose that four normal populations have common
  variance ?225 and means µ1 50, µ 260, µ350,
  and µ460. How many observations should be taken
  on each population so that the probability of
  rejecting the hypothesis of equality of means is
  at least 0.90? Use a0.05

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Solution

• Average mean
• ?1 -5, ?2 5, ?3 -5, ?4 5
• Various choices
• Therefore, n 5 is needed

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The Random-effects Model

• A large number of possible levels
• Experimenter randomly selects a of these levels
  from the population of factor levels
• Called random-effect model
• Valid for the entire population of factor levels

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Linear Statistical Model

• Following linear model
• Yij µ?i?ij
• 1,2,.a, j1,2,n
• Yij is the (ij)th observation
• ?i and ?ij are independent random variables
• Identical in structure to the fixed-effects case
• Parameters have a different interpretation
• ?ij are with mean 0 and variance ?2
• ?i are with mean zero and variance ??2

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Testing the Hypothesis

• Testing the hypothesis that the individual
  treatment effects are zero is meaningless
• Appropriate to test a hypothesis about the
  variance of the treatment effect
• H0 ??2 0 vs. H1 ??2 gt0
• ??2 0, all treatments are identical
• There is variability between them
• Total variability
• SSTSSTreatments SSE
• Expected values of the MS
• E(MSTreatments) ?2 n??2 and E(MSE) ?2
• Computational procedure and construction of the
  ANOVA table are identical to the fixed-effects
  case

Eq 13 -21,22
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Randomized Complete Block Design

• Desired to design an experiment so that the
  variability arising from a nuisance factor can be
  controlled
• Recall about the paired t-test
• When all experimental runs cannot be made under
  homogeneous conditions
• See the paired t-test as a method for reducing
  the noise in the experiment by blocking out a
  nuisance factor effect
• Randomized block design can be viewed as an
  extension of the paired t-test
• Factor of interest has more than two levels
• More than two treatments must be compared

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Randomized Complete Block Design

• General procedure for a randomized complete block
  design consists of selecting b blocks
• Data that result from running a randomized
  complete block design for investigating a single
  factor with a levels and b blocks

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Linear Statistical Model

• Following linear model
• Yij is the (ij)th observation
• ?j is the effect of the jth block
• µ called the overall mean
• ?i called the ith treatment effect
• ?ij is a random error component with mean zero
  and variance ?2

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Hypothesis Testing

• Interested in testing the equality of the a
  treatment means
• ?1,?2 .. ?a
• Equivalent
• H0 ?1?2?a0
• H1 ?a0 for at least one i
• If the null hypothesis is true, changing the
  levels of the factor has no effect on the mean
  response

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Displaying Data
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Components of Total Variability

• Total variability in data
• Partitions this total variability into three
  parts
• Or symbolically,
• SSTSStreatmentsSSblocksSSerrors

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Computational Formulas

• Computing formulas for the sums of squares
• Error sum of squares
• SSerrorsSST-SStreatments-SSblocks
• Computer software package will be used to perform
  the analysis of variance

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Analysis of Variance
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Next Agenda

• Ends our discussion with the analysis of variance
  when there are more then two levels of a single
  factor
• In the next chapter, we will show how to design
  and analyze experiments with several factors with
  more than two levels