8'ANALYSIS OF VARIANCE

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8. ANALYSIS OF VARIANCE

• 8.1 Elements of a Designed Experiment
• 8.2 Experimental Design
• 8.3 Multiple Comparisons of Means
• 8.4 Factorial Experiments

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8.1 Elements of a Designed Experiment

• Definition 1
• The response variable is the variable of interest
  to be measured in the experiment. We also refer
  to the response as the dependent variable.

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• Definition 2
• Factors are those variables whose effect on the
  response is of interest to the experimenter.
  Quantitative factors are measured on a numerical
  scale, whereas qualitative factors are those that
  are not (naturally) measured on a numerical
  scale.

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• Definition 3
• Factor levels are the values of the factor
  utilized in the experiment.
• Definition 4
• The Treatments of an experiment are the
  factor-level combination utilized
• Definition 5
• An experimental unit is the object on which the
  response of factors are observed or measured.

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Figure 8a. Sampling experiment Process and
Terminology
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8.2 Experimental Design

• 8.2.1 Complete Randomization
• 8.2.2 The Randomized Block
• 8.2.3 Steps for Conducting a ANOVA for a
  Randomized Block Design

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8.2.1 The Completely Randomized Design

• Definition
• A completely randomized design is a design for
  which independent random samples of experimental
  units are selected for each treatment.
• We use completely randomized design to refer to
  both designed and observational experiments.

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Table 8a. Complete Randomization

• We could divide the land into 4 X 4 16 plot and
  assign each treatment to four blocks chosen
  completely at random.
• Purposes to eliminate various source of error.

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8.2.2 The Randomized Block Design

• Consist of two-step procedures
• Matched sets of experimental units called block,
  are formed, each block consisting of p
  experimental units (where p is the number of
  treatments). The b blocks should consist of
  experimental units that are the similar as
  possible.

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• 2. One experimental unit from each block is
  randomly assigned to each treatment, resulting in
  a total of n bp responses.

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Table 8b. Randomized Block
I
II
III
IV

• The treatment A, B, C and D are introduced in
  random order within each block I, II, III, and
  IV, but it is necessary to have complete set of
  treatment for each block.
• Purposes Used when it is desired to control one
  source of error or variability, namely, the
  difference in block.

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Figure 8b. Partitioning of the Total Sum of
Squares for the Randomized Block Design
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8.2.3 Steps for Conducting an ANOVA for a
Randomized Block Design

• Be sure the design consists of blocks
  (preferably, blocks of homogeneous experimental
  units) and that each treatments randomly assigned
  to one experimental unit in each block.
• If possible, check the assumptions of normality
  and equal variances for all block-treatment
  combinations.NoteThis maybe difficult to do
  since the design will likely have only one
  observation for each block-treatment combination.

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• Create an ANOVA summary table that specifies the
  variability attributable to Treatments, Blocks
  and Error, and that leads to the calculation of
  the F statistic to test the null hypothesis that
  the treatment means are equal in the population.
  Use a statistical software package or the
  calculation formulas to obtain the necessary
  numerical ingredients.
• If the F-test leads to the conclusion that the
  means differ, use the Bonferroni, Tukey, or
  similar procedure to conduct multiple comparisons
  of as many of the pairs of means as you wish. Use
  the results to summarize the statistically
  significantly differences among the treatment
  means. Remember that, in general, the randomized
  block design cannot be used to form confidence
  intervals for individual treatment means.

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• 5. If the F-test leads to the non-rejection of
  the null hypothesis that the treatment means are
  equal, several possibilities exist
• The treatment means are equal-that is, the null
  hypothesis is true.
• The treatment means really differ, but other
  important factors affecting the ii response are
  not accounted for by the randomized block design,
  These factors inflate the sampling variability,
  as measured by MSE, resulting in smaller values
  of the F statistic. Either increase the sample
  size for each treatment, or conduct an experiment
  that accounts for the other factors affecting the
  response. Do not automatically reach the former
  conclusions, since the possibility of a type II
  error must be considered if you accept H0.
• 6. If desired, conduct the F-test of the null
  hypothesis that the block means are equal.
  Rejection of this hypothesis lends statistical
  support to the utilization of the randomized
  block design.

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8.3 Multiple Comparisons of Means

• The choice of a multiple comparisons method in
  ANOVA will depend on the type of experimental
  design used and the comparisons of interest to
  the analyst. For example, Turkey ( 1949)
  developed his procedure specifically for pairwise
  comparisons when the sample sizes of the
  treatments are equal. The Bonferroni method (see
  Miller, 1981), like the Tukey procedure, can be
  applied when pair wise comparisons are of
  interest however, Bonferroni's method does not
  require equal sample sizes. Scheffe (1953)
  developed a more general procedure for comparing
  all possible linear combinations of treatment
  means ( called contrasts). Consequently, when
  making pairwise comparisons, the confidence
  intervals produced by Scheffe's method will
  generally be wider than the Tukey or Bonferroni
  confidence intervals.

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8.4 Factorial Experiments

• Definition
• A complete factorial experiment is one in which
  every factor-level combination is utilized. That
  is, the number of treatments in the experiments
  equals the total number of factor-level
  combinations.

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Table 8c. Schematic Layout of Two-
Factor Factorial Experiment
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Figure 8c. Illustration of possible treatment
effect Factorial experiment
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Figure 8d.Partitioning the Total Sum of Squares
for a two-factor factorial
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8.5.1 Procedures for Analysis of Two-Factor
Factorial Experiment

• Partition the Total Sum of Squares into the
  Treatment and Error components (stage 1 of Figure
  8d). Use either a statistic at software package
  or the calculation formulas to accomplish the
  partitioning.
• Use the F-ratio of Mean Square for Treatments to
  Mean Square for Error to test the null hypothesis
  that the treatment means are equal.
• If the test results in nonrejection of the null
  hypothesis, consider refining the experiment by
  increasing the number of replications or
  introducing other factors. Also consider the
  possibility that the response is unrelated to
  the two factors.
• If the result in rejection of the null
  hypothesis, then proceed to step 3.

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• Partition the Treatment Sum of Squares into the
  Main Effect and Interaction Sum of Squares (stage
  2 of Figure 8b). Use either a statistical
  software package or the calculation formulas to
  accomplish the partitioning.
• Test the null hypothesis that factors A and B do
  not interact to affect the response by computing
  the F-ratio of the Mean Square for Interaction to
  the Mean Square for Error.
• If the test results in nonrejection of the null
  hypothesis, proceed to step 5.
• If the test results in rejection of the null
  hypothesis, conclude that the two factors
  interact to affect the mean response. Then
  proceed to step 6a.

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• Conduct tests of two null hypotheses that the
  mean response is the same at each level of factor
  A and factor B. Compute two F-ratios by comparing
  the Mean Square for each Factor Main Effect to
  the Mean Square for Error.
• If one or both tests result in rejection of the
  null hypothesis, conclude that the factor affect
  the mean response. Proceed to step 6b.
• If both tests result in nonrejection, an apparent
  contradiction has occurred. Although the
  treatment means apparently differ (step 2 test),
  the interaction (step 4) and main effect (step 5)
  tests have not supported that result, Further
  experimentation is advised.

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• Compare the means
• If the test for interaction (step 4) is
  significant, use a multiple comparisons procedure
  to compare any or all pairs of the treatment
  means.
• If the test for one or both main effects (step 5)
  is significant, use the multiple comparisons
  procedure to compare the pairs of means
  corresponding to the levels of the significant
  factor (s).

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8.5.2 Test Conducted in Analysis of
Factorial Experiments Completely Randomized
Design, r Replicates per Treatment

• 1. Test for Treatment Means
• H0 No difference among the ab treatment means
• Ha At least two treatment means differ
• Rejection region based on numerator
  and
• denominator degrees of freedom Note n abr.

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• Test For Factor Interaction
• H0 Factor A and B do not interact to affect the
  response mean
• Ha Factor A and B do interact to affect the
  response mean
• Rejection region based on numerator
  and denominator degrees of freedom.

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• 3. Test For Main Effect Of Factor A
• H0 No difference among the a mean levels of
  factor A
• Ha At least two factor A mean levels differ
• Rejection region based on
  numerator and
• denominator degrees of freedom.

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• 4. Test For Main Effect Of Factor B
• H0 No difference among the b mean levels of
  factor B
• Ha At least two factor B mean levels differ
• Rejection region based on numerator
  and denominator degrees
  of freedom.

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• Assumptions For All Test
• The response distribution for each factor-level
  combination (treatment) is normal
• The response variance is constant for all
  treatments
• Random and independent samples of experimental
  units are associated with each treatment.