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
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.
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.
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.
Figure 8a. Sampling experiment:
Process and Terminology
8.2 Experimental Design
8.2.1
8.2.2
8.2.3
Complete Randomization
The Randomized Block
Steps for Conducting a ANOVA
for a Randomized Block Design
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.
Table 8a.
Complete Randomization
D
A
C
C
B
D
B
A
D
C
B
D
A
B
C
A
•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.
8.2.2
The Randomized Block
Design
Consist of two-step procedures:
1. 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.
2. One experimental unit from each block
is randomly assigned to each
treatment, resulting in a total of n = bp
responses.
Table 8b. Randomized Block
I
II
III
C
B
A
D
A
B
D
C
B
C
D
A
IV
A
D
C
B
• 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.
Figure 8b. Partitioning of the Total Sum of
Squares for the Randomized
Block Design
8.2.3 Steps for Conducting an
ANOVA for a Randomized
Block Design
1. 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.
2. If possible, check the assumptions of normality and
equal variances for all block-treatment
combinations.[Note:This maybe difficult to do since
the design will likely have only one observation for
each block-treatment combination.]
3.
4.
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.
5.
If the F-test leads to the non-rejection of the null
hypothesis that the treatment means are equal, several
possibilities exist:
a. The treatment means are equal-that is, the null
hypothesis is true.
b. 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.
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.
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.
Table 8c.
Schematic Layout of TwoFactor Factorial Experiment
Factor B at b levels
Levels
1
2
Factor A
3
at a levels

a
1
Trt. 1
Trt. b  1
Trt. 2b  1

Trt. (a - 1)b  1
2
Trt. 2
Trt. b  2
Trt. 2b  2

Trt. (a - 1)b  2
3
Trt. 3
Trt. b  3
Trt. 2b  3

Trt. ( a - 1)b  3





b
Trt. b
Trt. 2b
Trt. 3b1

Trt. ab
Figure 8c. Illustration of possible treatment
effect: Factorial experiment
Figure 8d.Partitioning the Total Sum of
Squares for a two-factor factorial
8.5.1 Procedures for Analysis of Two-Factor
Factorial Experiment
1.
2.
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.
a. 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.
b. If the result in rejection of the null hypothesis, then
proceed to step 3.
3.
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.
4.
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.
a. If the test results in nonrejection of the null hypothesis,
proceed to step 5.
b. 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.
5.
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.
a. If one or both tests result in rejection of the null hypothesis,
conclude that the factor affect the mean response. Proceed
to step 6b.
b. 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.
6. Compare the means:
a. If the test for interaction (step 4) is significant, use a
multiple comparisons procedure to compare any or all
pairs of the treatment means.
b. 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).
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
MST
Test statistic : F 
MSE
Rejection region: F  F based on ( ab  1) numerator and (n  ab)
denominator degrees of freedom [Note: n = abr].
2.
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
MS ( AB)
Test statistic : F 
MSE
Rejection region: F  F based on (a  1)(b  1) numerator
( n  ab) and denominator degrees of freedom.
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
MS ( A)
Test statistic : F 
MSE
Rejection region: F  F based on ( a  1) numerator and ( n  ab)
denominator degrees of freedom.
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
MS ( B)
Test statistic : F 
MSE
Rejection region: F  F based on (b  1) numerator
and ( n  ab) denominator degrees of freedom.
5.
Assumptions For All Test
1.
2.
3.
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.