Ch12: Analysis of Variance (ANOVA)

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Ch12 Analysis of Variance(ANOVA)

• 12.1 INTRO
• This is an extension of Chap 11 (2-sample design)
  to more than two. The name ANOVA is misleading
  because it compares the means of data and not
  their variances.
• One-way and Two-way Layouts will be applied to
  parametric and nonparametric methods.

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12.2 The One-Way Layout

• The One-Way Layout is an experimental design in
  which independent measurements are made under
  each of several (more than 2) treatments.
• Thus, the One-Way Layout ANOVA focuses on
  comparing more than 2 popns or treatment means.
• Terminology
• The characteristic that differentiates the popns
  or treatments from one another is called the
  factor under study.
• The different popns or treatments are termed as
  the levels of the factor.

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12.2.1 Normal Theory the F-test

• This section deals with ANOVA and F-test for the
  case of I samples (treatments or levels) of same
  size , J.
• Notation

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12.2.1 (contd)
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12.2.1 (contd)
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12.2.2 Multiple Comparisons

• The F-test in Ex. A of Sec. 12.2.1 states that
  the means of measurements from the 7 different
  labs are NOT all equal, but how much do they
  differ which pairs are significantly different?
  
• These questions will be addressed in this
  section.
• Our main focus is to compare pairs or groups of
  treatments to estimate the treatment means and
  their differences.
• Naïve approach compare all pairs thru t-tests
  (?)
• New approaches Tukey Bonferroni methods

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12.2.2.1 Tukeys method

• It is used to construct CIs for the differences
  of all pairs of means in such a way (unlike the
  naïve approach) that the intervals simultaneously
  have a set coverage probability. Using the
  duality between CI HT, one can determine which
  particular pairs are significantly different.
• Tukeys procedure depends on the so-called
  studentized range distn (A14-A19, textbook)
  characterized by 2 parameters Ithe number of
  samples being compared I(J-1)the degrees of
  freedom in the pooled sample std deviation.

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12.2.2.1 Tukeys method (contd)
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Tukeys method (steps)read example A on page 452
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12.2.2.2 The Bonferroni method

• This method was briefly introduced in Section
  11.4.8
• If null hypotheses are to be tested, then a
  desired overall type I error rate of at most
  can be guaranteed by testing each null hypothesis
  at level .
• By duality between CI HT, one can say that
• If confidence intervals are each formed to
  have confidence level ,
  then they all hold
• simultaneously with confidence level at least
• Nice results are obtained for not too large .

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12.2.3 The Kruskal-Wallis Test(a nonparametric
method)

• The Kruskal-Wallis test is a generalization of
  the Mann-Witney test seen in Section 11.2.3
• Thus, such the Kruskal-Wallis test makes no
  normality assumption and has a wider range of
  applicability than does the F test.
• The Kruskal-Wallis is especially useful for
  small-sample size problems.
• Data are replaced by their ranks but outliers
  will have less influence in the Kruskal-Wallis
  test (nonparametric) than they do on its
  counterpart F test (parametric).

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12.3 The Two-Way Layout

• Here the experimental design involves 2 factors
  (each factor has 2 or more levels).
• Goal We would to assess the effect of 2 factors
  (Temperature Humidity) on a variable of
  interest (yield of a chemical reaction).
• I4 levels for factor 1
• J2 levels for factor 2
• ? IJ8 combinations (cells)
• Take K independent obs. in cells
• Another situation an agricultural scientist may
  be interested in the corn yield using 3 different
  fertilizers with 4 different types of soils.
  What are the effects?

T1 T2 T3 T4
H1
H2
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12.3.1 Additive Parametrization
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12.3.1Normal Theory for the 2-Way Layout
Assume the number of observations per cell
Kgt1. If K is the same for each cell, then the
design is to be said balanced.
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12.2.1 (contd)
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12.2.1 (contd)