Chapter 3 Experiments with a Single Factor: The Analysis of Variance

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Chapter 3 Experiments with a Single Factor The
Analysis of Variance
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3.1 An Example

• Chapter 2 A signal-factor experiment with two
  levels of the factor
• Consider signal-factor experiments with a levels
  of the factor, a ? 2
• Example
• The tensile strength of a new synthetic fiber.
• The weight percent of cotton
• Five levels 15, 20, 25, 30, 35
• a 5 and n 5

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• Does changing the cotton weight percent change
  the mean tensile strength?
• Is there an optimum level for cotton content?

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3.2 The Analysis of Variance

• a levels (treatments) of a factor and n
  replicates for each level.
• yij the jth observation taken under factor level
  or treatment i.

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• Models for the Data
• Means model
• yij is the ijth observation,
• ?i is the mean of the ith factor level
• ?ij is a random error with mean zero
• Let ?i ? ?i , ? is the overall mean and ?i is
  the ith treatment effect
• Effects model

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• Linear statistical model
• One-way or Signal-factor analysis of variance
  model
• Completely randomized design the experiments are
  performed in random order so that the environment
  in which the treatment are applied is as uniform
  as possible.
• For hypothesis testing, the model errors are
  assumed to be normally and independently
  distributed random variables with mean zero and
  variance, ?2, i.e. yij N(??i, ?2)
• Fixed effect model a levels have been
  specifically chosen by the experimenter.

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3.3 Analysis of the Fixed Effects Model

• Interested in testing the equality of the a
  treatment means, and E(yij) ?i ? ?i, i
  1,2, , a
• H0 ?1 ?a v.s.
• H1 ?i ? ?j, for at least one
  pair (i, j)
• Constraint
• H0 ?1 ?a 0 v.s. H1 ?i ? 0, for at least
  one i

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• Notations
• 3.3.1 Decomposition of the Total Sum of Squares
• Total variability into its component parts.
• The total sum of squares (a measure of overall
  variability in the data)
• Degree of freedom an 1 N 1

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• SSTreatment sum of squares of the differences
  between the treatment averages (sum of squares
  due to treatments) and the grand average, and a
  1 degree of freedom
• SSE sum of squares of the differences of
  observations within treatments from the treatment
  average (sum of squares due to error), and N a
  degrees of freedom.

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• A large value of SSTreatments reflects large
  differences in treatment means
• A small value of SSTreatments likely indicates
  no differences in treatment means
• dfTotal dfTreatment dfError
• If there are no differences between a treatment
  means,

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• Mean squares
• 3.3.2 Statistical Analysis
• Assumption ?ij are normally and independently
  distributed with mean zero and variance ?2

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• SST/?2 Chi-square (N 1), SSE/?2 Chi-square
  (N a), SSTreatments/?2 Chi-square (a 1),
  and SSE/?2 and SSTreatments/?2 are independent
  (Theorem 3.1)
• H0 ?1 ?a 0 v.s. H1 ?i ? 0, for at least
  one i

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• Reject H0 if F0 gt F?, a-1, N-a
• Rewrite the sum of squares
• See page 71

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Response Strength ANOVA for Selected
Factorial Model Analysis of variance table
Partial sum of squares Sum of Mean F Source
Squares DF Square Value Prob gt
F Model 475.76 4 118.94 14.76 lt
0.0001 A 475.76 4 118.94 14.76 lt 0.0001 Pure
Error 161.20 20 8.06 Cor Total 636.96 24 Std.
Dev. 2.84 R-Squared 0.7469 Mean 15.04 Adj
R-Squared 0.6963 C.V. 18.88 Pred
R-Squared 0.6046 PRESS 251.88 Adeq
Precision 9.294
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• 3.3.3 Estimation of the Model Parameters
• Model yij ? ?i ?ij
• Estimators
• Confidence intervals

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• Example 3.3 (page 75)
• Simultaneous Confidence Intervals (Bonferroni
  method) Construct a set of r simultaneous
  confidence intervals on treatment means which is
  at least 100(1-?) 100(1-?/r) C.I.s
• 3.3.4 Unbalanced Data
• Let ni observations be taken under treatment i,
  i1,2,,a, N ?i ni,

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• 1. The test statistic is relatively insensitive
  to small departures from the assumption of equal
  variance for the a treatments if the sample sizes
  are equal.
• 2. The power of the test is maximized if the
  samples are of equal size.

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3.4 Model Adequacy Checking

• Assumptions yij N(??i, ?2)
• The examination of residuals
• Definition of residual
• The residuals should be structureless.

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• 3.4.1 The Normality Assumption
• Plot a histogram of the residuals
• Plot a normal probability plot of the residuals
• See Table 3-6

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• May be
• Slightly skewed (right tail is longer than left
  tail)
• Light tail (the left tail of error is thinner
  than the tail part of standard normal)
• Outliers
• The possible causes of outliers calculations,
  data coding, copy error,.
• Sometimes outliers are more informative than the
  rest of the data.

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• Detect outliers Examine the standardized
  residuals,
• 3.4.2 Plot of Residuals in Time Sequence
• Plotting the residuals in time order of data
  collection is helpful in detecting correlation
  between the residuals.
• Independence assumption

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• 3.4.3 Plot of Residuals Versus Fitted Values
• Plot the residuals versus the fitted values
• Structureless

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• Nonconstant variance the variance of the
  observations increases as the magnitude of the
  observation increase, i.e. yij ? ?2
• If the factor levels having the larger variance
  also have small sample sizes, the actual type I
  error rate is larger than anticipated.
• Variance-stabilizing transformation

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• Statistical Tests for Equality Variance
• Bartletts test
• Reject null hypothesis if

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• Example 3.4 the test statistic is
• Bartletts test is sensitive to the normality
  assumption
• The modified Levene test
• Use the absolute deviation of the observation in
  each treatment from the treatment median.
• Mean deviations are equal gt the variance of the
  observations in all treatments will be the same.
• The test statistic for Levenes test is the ANOVA
  F statistic for testing equality of means.

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• Example 3.5
• Four methods of estimating flood flow frequency
  procedure (see Table 3.7)
• ANOVA table (Table 3.8)
• The plot of residuals v.s. fitted values (Figure
  3.7)
• Modified Levenes test F0 4.55 with P-value
  0.0137. Reject the null hypothesis of equal
  variances.

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• Let E(y) ? and ?y ? ??
• Find y y? that yields a constant variance.
• ? ? ???-1
• Variance-Stabilizing Transformations

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• How to find ?
• Use
• See Figure 3.8, Table 3.10 and Figure 3.9

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3.5 Practical Interpretation of Results

• Conduct the experiment gt perform the statistical
  analysis gt investigate the underlying
  assumptions gt draw practical conclusion
• 3.5.1 A Regression Model
• Qualitative factor compare the difference
  between the levels of the factors.
• Quantitative factor develop an interpolation
  equation for the response variable.

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• Regression analysis
• See Figure 3.1

Final Equation in Terms of Actual Factors
Strength 62.61143-9.01143 Cotton Weight
0.48143 Cotton Weight 2 -7.60000E-003
Cotton Weight 3 This is an empirical model of
the experimental results
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• 3.5.2 Comparisons Among Treatment Means
• If that hypothesis is rejected, we dont know
  which specific means are different
• Determining which specific means differ following
  an ANOVA is called the multiple comparisons
  problem
• 3.5.3 Graphical Comparisons of Means

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• 3.5.4 Contrast
• A contrast a linear combination of the
  parameters of the form
• H0 ? 0 v.s. H1 ? ? 0
• Two methods for this testing.

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• The first method

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• The second method

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• The C.I. for a contrast, ?
• Unequal Sample Size

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• 3.5.5 Orthogonal Contrast
• Two contrasts with coefficients, ci and di,
  are orthogonal if ?ci di 0
• For a treatments, the set of a 1 orthogonal
  contrasts partition the sum of squares due to
  treatments into a 1 independent
  single-degree-of-freedom components. Thus, tests
  performed on orthogonal contrasts are
  independent.
• See Example 3.6 (Page 94)

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• 3.5.6 Scheffes Method for Comparing All
  Contrasts
• Scheffe (1953) proposed a method for comparing
  any and all possible contrasts between treatment
  means.
• See Page 95 and 96

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• 3.5.7 Comparing Pairs of Treatment Means
• Compare all pairs of a treatment means
• Tukeys Test
• The studentized range statistic
• See Example 3.7

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• Sometimes overall F test from ANOVA is
  significant, but the pairwise comparison of mean
  fails to reveal any significant differences.
• The F test is simultaneously considering all
  possible contrasts involving the treatment means,
  not just pairwise comparisons.
• The Fisher Least Significant Difference (LSD)
  Method
• For H0 ?i ?j

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• The least significant difference (LSD)
• See Example 3.8
• Duncans Multiple Range Test
• The a treatment averages are arranged in
  ascending order, and the standard error of each
  average is determined as

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• Assume equal sample size, the significant ranges
  are
• Total a(a-1)/2 pairs
• Example 3.9
• The Newman-Keuls Test
• Similar as Duncans multiple range test
• The critical values

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• 3.5.8 Comparing Treatment Means with a Control
• Assume one of the treatments is a control, and
  the analyst is interested in comparing each of
  the other a 1 treatment means with the control.
  
• Test H0 ?i ?a v.s. H1 ?i ? ?a, i 1,2,, a
  1
• Dunnett (1964)
• Compute
• Reject H0 if
• Example 3.10

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3.7 Determining Sample Size

• Determine the number of replicates to run
• 3.7.1 Operating Characteristic Curves (OC Curves)
• OC curves a plot of type II error probability of
  a statistical test,

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• If H0 is false, then
• F0 MSTreatment / MSE noncentral F
• with degree of freedom a 1 and N a and
  noncentrality parameter ?
• Chart V of the Appendix
• Determine
• Let ?i be the specified treatments. Then
  estimates of ?i
• For ?2, from prior experience, a previous
  experiment or a preliminary test or a judgment
  estimate.

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• Example 3.11
• Difficulty How to select a set of treatment
  means on which the sample size decision should be
  based.
• Another approach Select a sample size such that
  if the difference between any two treatment means
  exceeds a specified value the null hypothesis
  should be rejected.

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• 3.7.2 Specifying a Standard Deviation Increase
• Let P be a percentage for increase in standard
  deviation of an observation. Then
• For example (Page 110) If P 20, then

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• 3.7.3 Confidence Interval Estimation Method
• Use Confidence interval.
• For example we want 95 C.I. on the difference
  in mean tensile strength for any two cotton
  weight percentages to be ? 5 psi and ? 3. See
  Page 110.

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3.9 The Regression Approach to the Analysis of
Variance

• Model yij ? ?i ?ij

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• The normal equations
• Apply the constraint
• Then estimations are
• Regression sum of squares (the reduction due to
  fitting the full model)

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• The error sum of squares
• Find the sum of squares resulting from the
  treatment effects

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• The testing statistic for H0 ?1 ?a