CHAPTER 11 EXPERIMENTAL DESIGN AND ANALYSIS OF VARIANCE

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CHAPTER 11EXPERIMENTAL DESIGN AND ANALYSIS OF
VARIANCE
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I. Introduction

• This chapter introduces an alternative approach
  to testing the difference among more than 2
  means, which avoids the problem of inflating the
  long-run probability of making a Type I Error and
  incorrectly rejecting a hypothesis of equal means

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EXAMPLE
If we are interested in testing the difference of
5 sample means to determine whether the
differences are truly significant or whether they
are attributable to random error, the 2 means at
a time testing procedures discussed in chapters 8
and 9 are terribly inefficient. Those procedures
would require us to run a total of 10 such tests
by comparing 2 means at a time.
5C2 5!/(2!)(3!) 10
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SO WHAT?
The procedure would involve an unacceptable error
probability. If we chose a significance level of
5, then the long-run probability of reaching the
correct conclusion in a test that compares 2
means is 95, and the probability of making a
Type I Error is 5. Yet the corresponding
probability of reaching the correct conclusion in
all 10 comparisons of means then equals (0.95)10
or 0.5987 (59.87)! This implies a long-run
probability of 40.13 of getting at least one
wrong set of test results when using the pairwise
comparison procedure.
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Analysis of Variance (ANOVA) does not require
pairwise comparisons and involves a single test
only. We will use the variances to test
differences amongmeans because the variance is
thesquare of the standard deviation, andthe
standard deviation is the standard measure of
dispersion about the mean.
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II. The Design of Experiments

• A. Controlled Experiment experiment in which
  persons or objects are subdivided into at least
  one experimental group that is exposed to some
  stimulus and one control group that is not so
  exposed

• B. Experimental Units elementary units the
  recipients of treatments in controlled experiments

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• C. Experimental Group a set of experimental
  units that is exposed to something new

• D. Control Group a set of experimental units
  that is not exposed to anything new

• E. Treatment a stimulus applied to experimental
  units in a controlled experiment

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• F. Randomization a procedure that lets
  extraneous factors operate during an experiment
  but that assures, by virtue of the random
  assignment of experimental units to experimental
  and control groups, that each treatment has an
  equal chance to be enhanced or handicapped by
  those factors

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• G. Blocking a procedure that eliminates the
  effects of extraneous factors during an
  experiment by forming blocks of experimental
  units within each of which all units are as alike
  as possible with respect to those factors

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• H. Experimental Design a plan for assigning
  treatments to experimental units under controlled
  conditions and, thus, for generating valid data

• I. Randomized Group Design (Completely Randomized
  Design) an experimental plan that creates on
  treatment group for each treatment and then
  assigns each experimental unit to one of these
  groups by a random process

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• J. Randomized Block Design an experimental plan
  that divides the available experimental units
  into blocks of fairly homogeneous units, each
  block containing as many units as there are
  treatments or some multiple of that number, and
  then matches each treatment with one or more
  units within each block by a random process

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• K. Matched-Pairs Design a randomized block
  design with blocks of two experimental units

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III. Errors in Experimental Data

• A. Random Error (Experimental Error) the
  difference between the value of a variable
  obtained by taking a single random sample (or by
  performing a single experiment) and the value
  obtained by taking a census or by averaging the
  results of all possible random samples of like
  size (or by averaging the results of a large
  number of identical experiments)

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• B. Systematic Error (Bias) equals the
  difference between the value of a variable
  obtained by averaging the results of a large
  number of identical experiments and the true
  value in controlled experiments

• C. Double-Blind Experiments experiments in
  which response bias is controlled by letting
  neither the subjects nor the judges know who is
  receiving which type of treatment

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IV. The Nature of ANOVA
A. ASSUMPTIONS OF THE ANOVA TEST

• 1. Sampled populations are normally distributed

• 2. Sampled populations have identical variances
  (Homoscedasticity)

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V. Alternative Versions of ANOVA

• A. One-Way ANOVA (One-Factor ANOVA) when
  extraneous factors are controlled by use of the
  randomized group design

• B. Two-Way ANOVA (Two-Factor ANOVA) when
  extraneous factors are controlled by use of the
  randomized block design (can be used with
  interaction or without interaction)

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VI. One-Way ANOVA
Consider a 10-year study in which a sample of 15
people has been observed while using toothpaste
1, 2, or 3 respectively. Let us assume that
five of the participants have been randomly
assigned to each of the treatments and that the
study has provided the following data.
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Number of Cavities Observed During 10-Year Period
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Ho The mean number of cavities for all users of
toothpaste 1 is the same as that for all users of
toothpaste 2 or 3 that is, Mu1 Mu2 Mu3.
HA At least one of the population means is
different from the others.
Significance Level 0.05
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• A. Explained Variation (Treatments) the
  variation among the sample means, which summarize
  the data associated with each of the treatments
  because it is attributable not to chance, but to
  inherent differences among the treatment
  populations

• B. Grand Mean the mean of all sample means

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• C. Treatments Sum of Squares (TSS) the sum of
  the squared deviations between each sample mean
  and the grand mean, multiplied by the number of
  observations made for each treatment

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• D. Treatments Mean Square (TMS) or Explained
  Variance the treatments sums of squares divided
  by columns 1
• TMS TSS / c-1
• 96.4 / 2 48.2

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• E. Unexplained Variation (Residual Variation or
  Error) the variation of the sample data within
  each of the columns (samples) about the
  respective sample means it is attributable to
  chance

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• F. Error Sum of Squares (ESS) the sum of each
  samples sum of squared deviations of individual
  observations from the samples means
• ESS EE(X Sample Mean)2

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• G. Error Mean Square (EMS) or Unexplained
  Variance - the error sum of squares divided by
  (rows 1) columns
• EMS ESS/(r 1) c
• EMS 71.2 / (5 1) 3 5.93

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The ANOVA Table
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F Statistic

• s2a/s2w
• TMS/EMS
• Explained variance/unexplained variance

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Figure 11.4
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VII. Discriminating Among Different Population
Means

• A. Establishing Confidence Intervals for
  Individual Population Means

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Mu1 18.4 /- 2.179 (square root of
5.93/5) 16.03 lt Mu1 lt 20.77
Mu2 21.8 /- 2.179 (square root of
5.93/5) 19.43 lt Mu2 lt 24.17
Mu3 15.6 /- 2.179 (square root of
5.93/5) 13.23 lt Mu3 lt 17.97
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• B. Establishing Confidence Intervals for
  Difference Between 2 Population Means

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Mu1Mu2(18.4-21.8) /- 2.179 (square root of
(2)5.93/5) -6.76 lt Mu1-Mu2 lt -0.04
Mu1Mu3(18.4-15.6) /- 2.179 (square root of
(2)5.93/5) -0.56 lt Mu1-Mu3 lt 6.16
Mu2Mu3(21.8-15.6) /- 2.179 (square root of
(2)5.93/5) 2.84 lt Mu2-Mu3 lt 9.56
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CAUTION
The stated confidence level of 95 applies to
each test individually, but not to the series of
all three. We cannot state with a 95 degree of
confidence that toothpaste 1 is better than
toothpaste 2, which is worse than toothpaste 3,
which is as good as toothpaste 1. The chance of
making at least one erroneous rejection (Type I
Error) in the series of null hypotheses exceeds
5.
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And you thought you had it rough studying for
this class!