8.0
8.1
8.2
8.3
Analysis of Variance
One way ANOVA
Multiple Comparison of Means
Two Way ANOVA
(with Replication and Without Replication)
Introduction to Design of Experiments
• Experimental Design
• A plan and a structure to test hypotheses in which
the researcher controls or manipulates one or more variables.
• Independent Variable
• Treatment variable - one that the experimenter controls
or modifies in the experiment.
• Classification variable - a characteristic of the experimental subjects that was
present prior to the experiment, and is not a result of the experimenter’s
manipulations or control.
• Levels or Classifications - the subcategories of the independent variable used
by the researcher in the experimental design.
• Independent variables are also referred to as factors.
• Manipulation of the independent variable depends on the
concept being studied
• Researcher studies the phenomenon being studied under
conditions of the aspects of the variable
• Dependent Variable
• the response to the different levels of the independent variables.
• Analysis of Variance (ANOVA) – a group of statistical techniques
used to analyse experimental designs.
• ANOVA begins with notion that individual items being studied
are all the same
Three Types of Experimental Designs
1. Completely Randomized Design
2. Randomized Block Design
3. Factorial Experiments
One Way Anova
• Completely Randomized Design – subjects are assigned randomly to
treatments; single independent variable.
• Randomized Block Design – includes a blocking variable; single
independent variable.
Completely Randomized Design
• The completely randomized design contains only
one independent variable with two or more
treatment levels.
• If two treatment levels of the independent variable
are present, the design is the same used to test the difference in
means of two independent populations presented in chapter 10
which used the t test to analyze the data.
Test procedure
1. State the hypothesis
H 0 : 1  2  ...  k  all the population mean are equal 
H1: i   j for at least one i, j  at least one of the mean is not equal 
@
H 0 :  1   2  ...   k  0  there is no treatment effect 
H1 :  i  0 for at least one i  there is exist treatment effect 
Where k = number of treatment groups or levels
2. Compute test statistic (Using SPSS output)
The computations CRD problem in SPSS will be summarized in tabular
form as shown in table below. This table is known as ANOVA table.
Sum of
Squares
df
Between
Groups
SSTR
k 1
Within
Groups
SSE
N k
Total
SST
N 1
Mean Square
SSTR
MST 
k 1
MSE 
SSE
N k
F
MST
MSE
Sig.
P-value
(Computer
generated)
SPSS output
k-1/N-k
SSTR
k-1
SSE
N-k
SST
N-1
MST/MSE
N-k/N-1
Look the p value in output
3.Decision: Reject H0 if p-value < α
4.Conclusion
Example
Solution
Step 1
State the Hypothesis
H 0 : 1  2  3
H1 : At least one of the means is different from the others
Step 2
Read the SPSS ouput
How to use spss
• Step 1 copy all data an input into spss
• 1=plant 1
• 2=plant 2
• 3=plant 3
• Dependent list>Age
• Factor>Plant
• Click option
• Tick descriptive
• Tick homogeneity of variance
Before we look for the p-value WE MUST CHECK
HOMOEGENEITY OF VARIACE 1ST.
H 0 : Equal variance is assumed
H1 : Equal variance is not assumed
P  value  0.393    0.01
Fail to reject H 0
Hence, equal variance is assumed
p-value = 0.000
Step 3
Compare with 𝛼 . If p-value < 𝛼 we reject 𝐻0
0.000< 0.01
Decision = reject 𝐻0
Step 4
Conclusion. There is significant difference in the mean age of
workers at three plants.
Multiple Comparison of Means
• Multiple Comparison techniques are used to identify which pairs of
means are significantly different given that the ANOVA test reveals
overall significance.
• Several post-hoc tests are available, but for this chapter, we are going
to illustrates Tukey’s HSD post-hoc test using the same data set.
• As for our previous analysis of CRD, having obtained a significant
result, using Tukey’s HSD test you can go further and determine
where the significance lies:
• Which plant is there actually a significant difference in age?
• Analysis in SPSS
Click “Post
Hoc”
Click on the check box for
“Tukey”. Continue and ok
Analyze  Compare Means  One 
Way ANOVA
• For the significance difference look a sig value. ANY values < 𝛼 is significant
difference.
• In this problem we can say there is significant difference between:
• plant 1 and 2
and
• plant 2 and 3
Exercise