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									    Chapter 10




Analysis of Variance
Chapter 10 - Chapter Outcomes
After studying the material in this chapter, you
should be able to:
Recognize the applications that call for the use
of analysis of variance.
Understand the logic of analysis of variance.
Be aware of several different analysis of
variance designs and understand when to use each
one.
Perform a single factor hypothesis test using
analysis of variance manually and with the aid of
Excel or Minitab software.
Chapter 10 - Chapter Outcomes
                  (continued)

After studying the material in this chapter, you
should be able to:
Conduct and interpret post-analysis of
variance pairwise comparisons procedures.
Recognize when randomized block analysis of
variance is useful and be able to perform the
randomized block analysis.
Perform two factor analysis of variance tests
with replications using Excel or Minitab and
interpret the output.
      One-Way Analysis of
           Variance

One-way analysis of variance is a
design in which independent samples
are obtained from k levels of a single
factor for the purpose of testing
whether the k levels have equal means.
     One-Way Analysis of
          Variance

A factor refers to a quantity under
examination in an experiment as a
possible cause of variation in the
response variable.
     One-Way Analysis of
          Variance

Levels refer to the categories,
measurements, or strata of a factor of
interest in the current experiment.
         One-Way Analysis of
              Variance

 Null hypothesis in an ANOVA experiment:

    H 0 : 1   2  3     k

Alternative hypothesis in an ANOVA experiment:

H A : At least two population means are different
       One-Way Analysis of
            Variance
  COMPLETELY RANDOMIZED DESIGN

An experiment is completely randomized
if it consists of the independent random
selection of observations representing
each level of one factor.
   One-Way Analysis of
        Variance
      BALANCED DESIGN

An experiment is said to have a
balanced design if the factor
levels have equal sample sizes.
      One-Way Analysis of
           Variance
The ANOVA test is based on three assumptions:
All populations are normally distributed.
The population variances are equal.
The observations are independent,
 meaning that any one individual value is
 not dependent on the value of any other
 observation.
     One-Way Analysis of
          Variance
          Total Variation
The total variation in the data refers
to the aggregate dispersion of the
individual data values across the
various factor levels.
     One-Way Analysis of
          Variance
     Within-Sample Variation
The dispersion that exists among
the data values within a particular
factor level is called the within-
sample variation.
 One-Way Analysis of
      Variance
  Between-Sample Variation
The dispersion among the
factor sample means is called
the between-sample variation.
     One-Way Analysis of
          Variance

   PARTITIONED SUM OF SQUARES

         TSS  SSB SSW
where:
          TSS = Total Sum of Squares
          SSE = Sum of Squares Between
          SSW = Sum of Squares Within
         One-Way Analysis of
              Variance

 Null hypothesis in an ANOVA experiment:

    H 0 : 1   2  3     k

Alternative hypothesis in an ANOVA experiment:

H A : At least two population means are different
   One-Way Analysis of
        Variance
         F-TEST STATISTIC

                       2
                  s
           FMax       max
                       2
                  s    min
where:
     s2max = Largest sample variance
     s2min = Smallest sample variance
           One-Way Analysis of
                Variance
           TOTAL SUM OF SQUARES
                      k   nj

             TSS   ( xij  x ) 2
                      i 1 j 1
where:
      TSS = Total sum of squares
        k = Number of populations (levels)
        ni = Sample size from population i
       xij = jth measurement from population i
       x   = Grand Mean (mean of all the data values)
        One-Way Analysis of
             Variance
       SUM OF SQUARES BETWEEN
                     k
             SSB   ni ( xi  x )   2

                    i 1
where:
      SSB = Sum of Squares Between Samples
       k = Number of populations (levels)
       ni = Sample size from population i
       x i = Sample mean from population i
       x   = Grand Mean
       One-Way Analysis of
            Variance
        SUM OF SQUARES WITHIN
             SSW = TSS - SSB
or                 k    nj

            SSW           ( xij  xi )   2

where:             i 1  j 1
      SSW = Sum of Squares Within Samples
       k = Number of populations
       ni = Sample size from population i
       x = Sample mean from population i
        i
       xij = jth measurement from population i
        One-Way ANOVA Table
                       (Table 10-3)

Source of
Variation        SS             df            MS           F Ratio

Between                       k-1            MSB
Samples       SSB                            MSW
Within        SSW             N-k             MSW
Samples
Total         TSS             N-1

        k = Number of populations
        N = Sum of the sample sizes from all populations
        df = Degrees of freedom
    One-Way ANOVA Table
            (Table 10-3)



                            SSB
MSB  Mean square between 
                            k-1

                          SSW
MSW  Mean square within
                          N-k
        Example of a One-Way
            ANOVA Table
                      (Table 10-4)

Source of
Variation       SS            df           MS        F Ratio

Between                                    7.333      7.333
                22             3
Samples                                              7.10286
Within        198.88          28          7.10286 = 1.03245
Samples
Total         220.88          31
                                     22
         MSB  Mean square between      7.3333
                                     3
                                  198.88
        MSW  Mean square within         7.10286
                                    28
                One-Way Analysis of
                     Variance
                               (Figure 10-5)
H0: 1= 2 = 3 = 4
HA: At least two population means are different
 = 0.05                                       MSB   7.333
                                            F             1.03244
Degrees of freedom:                            MSW 7.10286
D1 = k - 1 = 4 - 1 = 3,
D2 = N - k = 32 - 4 = 28
                                                      Rejection Region




                           0                      F  2.95       F
           Since F=1.03244  F= 2. 95, do not reject H0
The Tukey-Kramer Procedure


The Tukey-Kramer procedure is a
method for testing which populations
have different means, after the one-
way ANOVA null hypothesis has
been rejected.
  The Tukey-Kramer Procedure
      and One-Way ANOVA

The experiment-wide error rate is the
proportion of experiments in which at
least one of the set of confidence intervals
constructed does not contain the true
value of the population parameter being
estimated.
   The Tukey-Kramer Procedure
       and One-Way ANOVA
     TUKEY-KRAMER CRITICAL RANGE

                                MSW  1 1 
                                      
     Critical Range  q
                                 2  ni n j 
                                           
where:
      q = Value from standardized range table
              with k and N - k degrees of freedom for
              the desired level of .
    MSW = Mean Square Within
 ni and nj = Sample sizes from populations (levels) i
              and j, respectively.
Randomized Complete Block
         ANOVA

A treatment is a combination of one
level of each factor in an experiment
associated with each observed value
of the response variable.
   Randomized Complete Block
            ANOVA
     SUM OF SQUARES PARTTIONING -
  RANDOMIZED COMPLETE BLOCK DESIGN

       TSS  SSB  SSBL  SSW
where:
      TSS = Total sum of squares
      SSB = Sum of squares between factor levels
      SSBL= Sum of squares between blocks
      SSW = Sum of squares within levels
   Randomized Complete Block
            ANOVA
         SUM OF SQUARES FOR BLOCKING
                          n
            SSBL   k ( x j  x )         2

where:                     j
          k = Number of levels for the factor
          n = Number of blocks
         x j= The mean of the jth block
         x = Grand Mean
Randomized Complete Block
         ANOVA

    SUM OF SQUARES WITHIN

  SSW  TSS  (SSB  SSBL)
 Randomized Complete Block
          ANOVA
The randomized block design requires the
 following assumptions:
The populations are normally distributed.
The populations have equal variances.
The observations are independent.
   Randomized Block ANOVA
           Table
                      (Table 10-7)
Source of
Variation       SS            df           MS            F Ratio

Between                      b- 1                         MSBL
              SSBL                         MSBL           MSW
Blocks
Between       SSB            k-1                          MSB
                                           MSB
Samples                                                   MSW
Within        SSW        (k - 1)(b - 1)    MSW
Samples
Total         TSS            N-1
        k = Number of levels
        b = Number of blocks
        df = Degrees of freedom and N = Combined sample size
Randomized Block ANOVA Table
             (From Table 10-7)

where:
                                  SSB
   MSB  Mean square between 
                                   k 1
                                 SSBL
   MSBL  Mean square blocking 
                                 (b  1)
                                SSW
 MSW  Mean square within 
                            (k  1)(b  1)
            Randomized Block ANOVA
                               (Figure 10-12)
H0: 1= 2 = 3
HA: At least two population means are different
 = 0.05

Degrees of freedom:                                 F  8.544
D1 = k - 1 = 3 - 1 = 2,
D2 = (n - 1)(k - 1)
   = (4)(2) = 8                                         Rejection Region




                          0                       F  4.4589       F
            Since F=8.544 > F= 4.4589, reject H0
    Fisher’s Least Significant
      Difference (LSD) Test
 FISHER’S LEAST SIGNIFICANT DIFFERENCE
      FOR COMPLETE BLOCK DESIGN

                                      2
              LSD  t / 2 MSW
                                      n
where:
    t/2 = Upper-tailed value from Student’s t-
             distribution for /2 and (k -1)(n - 1)
             degrees of freedom
   MSW = Mean square within from ANOVA table
        n = Number of blocks
        k = Number of levels
  Two-Factor Analysis of
        Variance


Two-factor ANOVA is a technique
used to analyze two factors in an
analysis of variance framework.
Two-Factor Analysis of
      Variance
       (Figure 10-13)

                        SSA   Factor A



                        SSB   Factor B

 SST                          Interaction
                    SSAB      Between A
                              and B

                              Inherent
                        SSE   Variation
                              (Error)
     Two-Factor Analysis of
           Variance
The necessary assumptions for the two
 factor ANOVA are:
The population values for each
 combination of pairwise factor levels are
 normally distributed.
The variances for each population are
 equal.
The samples are independent.
The observations are independent.
 Randomized Block ANOVA Table
                          (Table 10-9)

Source of
Variation           SS              df             MS      F Ratio
                                                           MSA
Factor A           SSA            a- 1             MSA     MSE
 Factor B                                                  MSB
                   SSB            b-1              MSB     MSE
    AB                                                      MSAB
                   SSAB        (a - 1)(b - 1)      MSAB
Interaction                                                 MSE
 Error            SSE              N - ab           MSE
   Total           TSS            N-1
           a = Number of levels of factor A
           b = Number of levels of factor B
           N = Total number of observations in all cells
Randomized Block ANOVA Table
              (From Table 10-9)

where:
                                       SS A
   MSA  Mean square factor A 
                                      a 1
                                       SS B
   MSB  Mean square factor B 
                                      b 1
                                        SS AB
 MS AB  Mean square interactio n 
                                    (a  1)(b  1)
                              SSE
   MSE  Mean square within
                             N  ab
    Two Factor ANOVA Equations
Total Sum of Squares:               a       b          n
                        TSS   ( xijk  x )                   2

                                    i 1 j 1 k 1
Sum of Squares Factor A:                         a
                           SS A  bn ( xi  x ) 2
                                             i 1
Sum of Squares Factor B:                       b
                           SS B  an ( x j  x ) 2
Sum of Squares Interaction                      j 1

Between A and B:                a       b
                   SS AB   n  ( xij  xi  x j  x ) 2
                               i 1 j 1
Sum of Squares Error:                       a          b    n
                             SSE   ( xijk  xij )                2

                                            i 1 j 1 k 1
   Two Factor ANOVA Equations
                                a         b   n
where:
                                x
                                i 1 j 1 k 1
                                                   ijk

                      x                                  Grand Mean
       b      n                          abn
        x
       j 1 k 1
                    ijk

xi                        Mean of each level of factor A
             bn      a   n

                     x            ijk
             xj    i 1 k 1
                                           Mean of each level of factor B
                          an
        n xijk
xij  
                                                             a = Number of levels of factor A
                 Mean of each cell                          b = Number of levels of factor B
      k 1 n                                                n’ = Number of replications in
                                                                     each cell
                Differences Between Factor
                     Level Mean Values:
                       No Interaction
Mean Response




                                        Factor B Level 1
                                        Factor B Level 4
                                        Factor B Level 3
                                        Factor B Level 2

                1          2           3
                     Factor A Levels
                Differences Between Factor
                     Level Mean Values:
                    Interaction Present
Mean Response




                                        Factor B Level 1

                                        Factor B Level 2

                                        Factor B Level 3


                                        Factor B Level 4
                1          2           3
                     Factor A Levels
  A Caution About Interaction
When conducting hypothesis tests for a two
 factor ANOVA:
Test for interaction.
If interaction is present conduct a one-
 way ANOVA to test the levels of one of
 the factors using only one level of the
 other factor.
If no interaction is found, test factor A
 and factor B.
            Key Terms
Balanced Design     Levels
Between-Sample      One-Way Analysis
 Variation            of Variance
Completely          Total Variation
 Randomized Design   Treatment
Experiment-Wide     Within-Sample
 Error Rate           Variation
Factor

								
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