1. i) Two way ANOVA without replication
CHAPTER 4
Analysis of Variance
One-way ANOVA
Two-way ANOVA
i) Two way ANOVA without replication
ii) Two way ANOVA with replication

2. Key Concepts
ANOVA can be used to analyze the data obtained from experimental or observational studies.
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.
Variation between treatment groups captures the effect of the treatment.
Variation within treatment groups represents random error not explained by the experimental treatments.

3. One-way ANOVA
The one-way analysis of variance specifically allows us to compare several groups of observations whether or not their population mean are equal.
One way ANOVA is also known as Completely Randomized Design (CRD).
The application of one way ANOVA requires that the following assumptions hold true:
(i) The populations from which the samples are drawn are
(approximately) normally distributed.
(ii) The populations from which the samples are drawn have the same
variance.
(iii) The samples drawn from different populations are random and
independent.

4. Each observation may be written as: Or alternatively written as: Where

5. The is the total of all observations from the treatment, while
is the grand total of all N observations.
Then the hypothesis can be written as:
Note: This hypothesis is use for model
Treatment 1 2 … i k .
Total

6. For model the hypothesis is as follows: The computations for an analysis of variance problem are usually summarized in tabular form as shown in table below. This table is of the referred to as the ANOVA table.
Source of Variation
Sum of Squares
Degree of freedom
Mean Square
F Calculated
Treatment
(Between levels)
SSTR
k - 1
Error
(within levels)
SSE
N - k
Total
SST
N - 1

7. where We reject if and conclude that some of the data is due to differences in the treatment levels.

8. Example 4.1 Three different types of acid can be used in a particular chemical process. The resulting yield (in %) from several batches using the different types of acid are given below: Test whether or not the three populations appear to have equal means using = 0.05.
Acid A B C 93 95 76 97 77 74 87 84

9. Solution: 1. Construct the table of calculation: 2
Solution: 1. Construct the table of calculation: 2. Set up the hypothesis:
Acid A B C 93 95 76 97 77 74 87 84

10. 3. Construct ANOVA table:

11. Treatment (Between levels)
Source of Variation
Sum of Squares
Degrees of Freedom
Mean Square
F Calculated
Treatment (Between levels)
3 – 1 = 2
Error (within levels)
9 – 3 = 6
Total
9 – 1 = 8
4. At = 0.05, from the statistical table for f distribution, we have
5. Since , thus we failed to reject
and conclude that there is no difference in the three types of acid at
significance at = 0.05

12. Exercise: Four catalyst that may affect the concentration of one component in a three-component liquid mixture are being investigated. The following concentrations are obtained. Compute a one-way analysis of variance for this experiment and test the hypothesis at 0.05 level of significance and state your conclusion concerning the effect of catalyst on the concentration of one component in three-component liquid mixture. 1 2 3 4
58.2
56.3
50.1
52.9
57.2
54.5
54.2
49.9
58.4
57.0
55.4
50.0
55.8
55.3
51.7
54.9

13. Two-way ANOVA Without Replication
Also known as Randomized Block Design (RBD)
In RBD there is one factor or variable that is
of primary interest. However, there are also
several other nuisance factors.
Nuisance factors are those that may affect the
measured result, but are not of primary interest.
The way to control nuisance factor is by blocking
them to reduce or eliminate the contribution to
experimental error contributed by nuisance factors

14. The model for a randomized block design with one nuisance variable can be written as:
: j th observation from i th treatment,
:Overall mean,
: i th effect of treatment,
: j th effect of block
: random error.

15. We can use the following layout for this kind of two-way classification:
Blocks
Total
Treatment 1
Treatment 2
Treatment i
Treatment k

16. In the two way analysis of variance where each treatment is represented once in each block, the major objective is to test:
The effect of treatment:
The effect of block:

18. We reject if:
The effect of treatment:
2. The effect of block:

20. Solution; 1. Construct the table of calculation, we have k = 4 and n = 3: 2. Set up the hypothesis:
Engine 1
Engine 2
Engine 3
Detergent A
Detergent B
Detergent C
Detergent D
Totals
182
176
207
139
145
153
128
565 45 43 51 47 46 52 48 50 55 42 37 49

21. 3. Construct ANOVA table:

22. Source of Variation
Sum of Squares
Degree of Freedom
Mean Square
F Calculated
Treatments
111 3
Blocks
135 2
Error 19 6
Total
265 11
4. At = 0.01, from the statistical table for f distribution, we have (treatments) and (blocks). 5. Since , thus we reject and conclude that there are differences in the effectiveness of the 4 detergents at = 0.01 and also since , thus we reject and conclude that there are differences among the results obtained for the 3 engines are significant

23. Exercise: Consider the hardness testing experiment described below
Exercise: Consider the hardness testing experiment described below. There are 4 tips and 4 available metal coupons. Each tip is tested once on each coupon, resulting in a randomized block design. The data obtained are represented in the below. Analyze the data by test the hypothesis at the 0.05 level of significance and draw the appropriate conclusion.
Coupon (Block)
Type of tip 1 2 3 4
9.3
9.4
9.6
10.0
9.8
9.9
9.2
9.5
9.7
10.2