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SLIDES
. BY
John Loucks
St. Edward’s
University
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 1
An Introduction to Experimental Design
and Analysis of Variance
In this chapter three types of experimental designs
are introduced.
•
•
•
a completely randomized design
a randomized block design
a factorial experiment
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 2
An Introduction to Experimental Design
and Analysis of Variance
A factor is a variable that the experimenter has
selected for investigation.
A treatment is a level of a factor.
Experimental units are the objects of interest in the
experiment.
A completely randomized design is an experimental
design in which the treatments are randomly
assigned to the experimental units.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 3
Analysis of Variance: A Conceptual Overview
Analysis of Variance (ANOVA) can be used to test
for the equality of three or more population means.
Data obtained from observational or experimental
studies can be used for the analysis.
We want to use the sample results to test the
following hypotheses:
H0: 1 = 2 = 3 = . . . = k
Ha: Not all population means are equal
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 4
Analysis of Variance: A Conceptual Overview
H0: 1 = 2 = 3 = . . . = k
Ha: Not all population means are equal
If H0 is rejected, we cannot conclude that all
population means are different.
Rejecting H0 means that at least two population
means have different values.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 5
Analysis of Variance: A Conceptual Overview
Assumptions for Analysis of Variance
For each population, the response (dependent)
variable is normally distributed.
The variance of the response variable, denoted 2,
is the same for all of the populations.
The observations must be independent.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 6
Analysis of Variance: A Conceptual Overview
Sampling Distribution of x Given H0 is True
Sample means are close together
because there is only
one sampling distribution
when H0 is true.
x2
x2
x1
2
n
x3
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 7
Analysis of Variance: A Conceptual Overview
Sampling Distribution of x Given H0 is False
Sample means come from
different sampling distributions
and are not as close together
when H0 is false.
x3
3
x1 1
2
x2
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 8
Analysis of Variance and
the Completely Randomized Design
Between-Treatments Estimate of Population Variance
Within-Treatments Estimate of Population Variance
Comparing the Variance Estimates: The F Test
ANOVA Table
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 9
Between-Treatments Estimate
of Population Variance 2
The estimate of 2 based on the variation of the
sample means is called the mean square due to
treatments and is denoted by MSTR.
k
MSTR
2
n
(
x
x
)
j j
Denominator is the
degrees of freedom
associated with SSTR
j 1
k1
Numerator is called
the sum of squares due
to treatments (SSTR)
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 10
Within-Treatments Estimate
of Population Variance 2
The estimate of 2 based on the variation of the
sample observations within each sample is called the
mean square error and is denoted by MSE.
k
MSE
Denominator is the
degrees of freedom
associated with SSE
2
(n j 1) s 2j
j1
nT k
Numerator is called
the sum of squares
due to error (SSE)
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 11
Comparing the Variance Estimates: The F Test
If the null hypothesis is true and the ANOVA
assumptions are valid, the sampling distribution of
MSTR/MSE is an F distribution with MSTR d.f.
equal to k - 1 and MSE d.f. equal to nT - k.
If the means of the k populations are not equal, the
value of MSTR/MSE will be inflated because MSTR
overestimates 2.
Hence, we will reject H0 if the resulting value of
MSTR/MSE appears to be too large to have been
selected at random from the appropriate F
distribution.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 12
Comparing the Variance Estimates: The F Test
Sampling Distribution of MSTR/MSE
Sampling Distribution
of MSTR/MSE
Reject H0
Do Not Reject H0
F
Critical Value
MSTR/MSE
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 13
ANOVA Table
for a Completely Randomized Design
Source of
Variation
Sum of Degrees of
Squares Freedom
SSTR
k-1
Error
SSE
nT - k
Total
SST
nT - 1
Treatments
SST is partitioned
into SSTR and SSE.
Mean
Square
pValue
F
SSTR MSTR
k-1
MSE
SSE
MSE
nT - k
MSTR
SST’s degrees of freedom
(d.f.) are partitioned into
SSTR’s d.f. and SSE’s d.f.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 14
ANOVA Table
for a Completely Randomized Design
SST divided by its degrees of freedom nT – 1 is the
overall sample variance that would be obtained if we
treated the entire set of observations as one data set.
With the entire data set as one sample, the formula
for computing the total sum of squares, SST, is:
k
nj
SST ( xij x )2 SSTR SSE
j 1 i 1
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 15
Test for the Equality of k Population Means
Hypotheses
H0: 1 = 2 = 3 = . . . = k
Ha: Not all population means are equal
Test Statistic
F = MSTR/MSE
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 16
Test for the Equality of k Population Means
Rejection Rule
p-value Approach:
Reject H0 if p-value <
Critical Value Approach:
Reject H0 if F > F
where the value of F is based on an
F distribution with k - 1 numerator d.f.
and nT - k denominator d.f.
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 17
End of Chapter 13
© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 18
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