ANALYSIS OF VARIANCE ANOVA

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ANALYSIS OF VARIANCE(ANOVA)

• BCT2053
• CHAPTER 6

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CONTENT

• 6.1 Analysis of Variance Purpose
• and Procedure
• 6.2 One-Way ANOVA
• 6.3 Two-Way ANOVA

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OBJECTIVE

• After completing this chapter you should be able
  to
• Explain the purpose of ANOVA
• Identify the assumptions that underlie the ANOVA
  technique
• Describe the ANOVA hypothesis testing procedure
• Use the one-way ANOVA technique to determine if
  there is a significance difference among three or
  more means
• Use the two-way ANOVA technique to determine if
  there is an effect of interaction between two
  factors experiment

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6.1 ANALYSIS OF VARIANCE PURPOSE AND
PROCEDURE

• ANOVA
• the approach that allows us to use sample data to
  see if the values of three or more unknown
  population means are likely to be different
• Also known as factorial experiments
• this name is derived from the fact that in order
  to test for statistical significance between
  means, we are actually comparing (i.e.,
  analyzing) variances. (so F-distribution will be
  used)
• Example of problems
• A manager want to evaluate the performance of
  three (or more) employees to see if any
  performance different from others
• A marketing executive want to see if theres a
  difference in sales productivity in the 5 company
  region
• A teacher wants to see if theres a difference in
  students performance if he use 3 or more
  approach to teach.

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The Procedural Steps for an ANOVA Test

• State the Null and Alternative hypothesis
• Select the level of significance
• Determine the test distribution to use
• Define rejection or critical region
• State the decision rule
• Compute the test statistic
• Make the statistical decision

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6.2 One-Way ANOVA

• Only one classification factor (variable) is
  considered

Response/ outcome/ dependent variable (samples)
(The level of the factor)
Replicates (1, j) The object to a given
treatment
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The resulting input grid of factorial experiment
where,   i 1, 2, a is the number of levels
being tested. j 1, 2,    is the number of
replicates at each level.  
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Assumptions

• To use the one-way ANOVA test, the following
  assumptions
• must be true
• The population under study have normal
  distribution
• The samples are drawn randomly, and each sample
  is independent of the other samples.
• All the populations from which the samples values
  are obtained, have the same unknown population
  variances, that is for k number of populations,

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The Null and Alternative hypothesis
(All population means are equal)
If Ho is true we have k number of normal
populations with
(Not all population means are equal)
Or H1 At least one mean is different from others
If H1 is true we may have k number of normal
populations with
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The format of a general one-way ANOVA table
T k n

• Reject Ho if

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Example 1

• The data shows the Maths test score for 4 group
  of student with 3 different methods of study.
  Test the hypothesis that theres no difference
  between the Maths score at significance level
  0.05.

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Example 2

• An experiment was performed to determine whether
  the annealing temperature of ductile iron affects
  its tensile strength. Five specimens were
  annealed at each of four temperatures. The
  tensile strength (in ksi) was measured for each
  temperature. The results are presented in the
  following table. Can you conclude that there are
  differences among the mean strengths at a 0.05?

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Example 3

• Three random samples of times (in minutes) that
  commuters are stuck in traffic are shown below.
  At a 0.05, is there a difference in the mean
  times among the three cities?

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Solve one-way ANOVA by EXCEL

• Excel key in data

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Solve one-way ANOVA by EXCEL

• Tools Add Ins Analysis Toolpak Data
  Analysis ANOVA single factor enter the data
  range set a value for a - ok
• Reject H0 if P-value a or F gt F crit

P-value lt 0.05 so Reject H0
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6.3 Two-Way ANOVA

• Two classification factor is considered

• Example
• A researcher whishes to test the effects of two
  different types of plant food and two different
  types of soil on the growth of certain plant.

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Some types of two way ANOVA design
B1 B2
B1 B2
A1 A2 A3
A1 A2
B1 B2 B3
B1 B2 B3
A1 A2 A3 A4
A1 A2 A3
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Assumptions

• The standard two-way ANOVA tests are valid under
  the following conditions
• The design must be complete
• Observations are taken on every possible
  treatment
• The design must be balanced
• The number of replicates is the same for each
  treatment
• The number of replicates per treatment, k must be
  at least 2
• Within any treatment, the observations
  
• are a simple random sample from a normal
  population
• The sample observations are independent of each
  other (the samples are not matched or paired in
  any way)
• The population variance is the same for all
  treatments.

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Null Alternative Hypothesis
interaction effect
H0 there is no interaction effect between factor
A and factor B. H1 there is an interaction
effect between factor A and factor B.
Row effect
H0 there is no difference in means of factor
A. H1 there is a difference in means of factor A.
Column effect
H0 there is no difference in means of factor
B. H1 there is a difference in means of factor B.
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The format of a general two-way ANOVA table
Reject if
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Procedure for Two-Way ANOVA
Ho No interaction between two factors
Yes (Reject Ho)
No (Accept Ho)
Ho No effects from the row factor A (the row
means are equal)
Ho No effects from the column factor B (the
column means are equal)
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Example 1

• A chemical engineer is studying the effects of
  various reagents and catalyst on the yield of a
  certain process. Yield is expressed as a
  percentage of a theoretical maximum. 4 runs of
  the process were made for each combination of 3
  reagents and 4 catalysts. Construct an ANOVA
  table and test is there an interaction effect
  between reagents and catalyst.

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Example 2

• A study was done to determine the effects of two
  factors on the lather ability of soap. The two
  factors were type of water and glycerol. The
  outcome measured was the amount of foam produced
  in mL. The experiment was repeated 3 times for
  each combination of factors. The result are
  presented in the following table. Construct an
  ANOVA table and test is there an interaction
  effect between factors.

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Solve two-way ANOVA by EXCEL

• Excel key in data

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Solve ANOVA by EXCEL

• tools Data Analysis ANOVA two factor with
  replication enter the data range set a value
  for a - ok
• Reject H0 if P-value a or F gt F crit

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Summary

• The other name for ANOVA is experimental design.
• ANOVA help researchers to design an experiment
  properly and analyzed the data it produces in
  correctly way.

Thank You