Yates Algorithm for 3^2 Factorial Experiment lecture - 32

 

Yates Algorithm

 for 

3^2 Factorial Experiment 

lecture - 32

The Yates algorithm is used to calculate single degree of freedom contrast.

The order of treatment combination of 2^3 factorial experiment is given below:

a0b0	a0b1	a0b2	a1b0	a1b1	a1b2	a2b0	a2b1	a2b2
00	01	02	10	11	12	20	21	22

The standard order of treatment combination of 2^3 factorial experiment is given below:

a0b0	a1b0	a2b0	a0b1	a1b1	a2b1	a0b2	a1b2	a2b2
00	10	20	01	11	21	02	12	22

OR

we can writes as:

1	a	a^2	b	ab	a^2 b	b^2	a b^2	a^2 b^2
00	10	20	01	11	21	20	12	22

Procedure:

1.     1.    Develop a table and write down the treatment combination in a standard order and yield totals.

2.      2.   Column – I obtained as:

3. Repeat the same procedure and obtain column – II.

4. Further mention effect in the next column after column – II.

5. The entries in the divisor can be obtained by using the relation

Where:

k:  Number of factors in the effect considered.

t: Number of factors in the experiment minus number of linear terms in the effect

r: Number of replicates

6. The number of columns is depended on the number of factors.

      K columns will be developed for 3^k factorial experiment.

Example:

An experiment was conducted to assess the effects of 3 raw material sources (Suppliers) and 3 mixtures (Compositions) on the crushing strength of concrete blocks, 18 blocks are selected, 2 at random from these manufactured by each of the 9 treatments and the experiment was conducted as random complete block with 2 replicates. The results are as follow:

Suppliers	Mixtures
	A	B	C
1	57 46	65 73	93 92
2	26 38	44 67	81 90
3	39 40	57 60	96 99

Solution: The statistical model of factorial experiment using RCBD:

$$

Yijk = μ + ρi + αj + βk + (αβ)jk + ϵijk

$$\begin{gathered} \text{ \\ ρi : Block effect} \end{gathered}$$

$$\begin{gathered} \text{ \\ αj : Suppliers effect} \end{gathered}$$

$$\begin{gathered} \text{ \\ βk : Mixture effect. \\ } \end{gathered}$$

$Computation:$

ANOVA Table

SV	df	SS	MS	F	F0.05(v, 8)
Block	1	122.72	122.72	2.75	5.32
Treatment	8	8602.78	1075.35	24.11	3.44
A	2	536.11	268.225	6.01	4.46
B	2		3934.725	88.22	4.46
AB	4	197.23	49.31	1.106	3.84
Error	8	356.78	44.60
Total	17

Now transform into single degree of freedom contrast by Yates Algorithm:

Example:

$$\begin{gathered} \text{ \\ } \end{gathered}$$

The effects of developer strength (A) and development time (B) on the density of photographic plate film are being studied. Three strengths and three times are used, and four replicates of a 3^2 factorial experiment are run. The data from this experiment follow. Analyze the data using the standard methods for factorial experiments.

Developer Strength	Development Time (minutes)
	10	14	18
1	0 2 5 4	1 3 4 2	2 5 4 6
2	4 6 7 5	6 8 7 7	9 10 8 5
3	7 10 8 7	10 10 7 7	12 10 9 8

Solution: 

The experiment consist 2 factors each at 3 levels i.e. 2^3 factorial experiment.

The statistical CRD model is used to represent the above experiment:

$$

Yijk =μ + αj + βk + (αβ)jk + ϵijk   

 i = 1, 2, 3, 4.

j, k = 0, 1, 2

Where:

αj : Developer Strength

βk : Development Time.

i. Statement of Hypotheses

H01:  αj = 0  Vs.  H11 : αj ≠ 0

H02 : βk = 0 Vs. H12 : βk ≠ 0

H03 : (αβ)jk = 0 Vs. H13 : (αβ)jk ≠0

ii. The significance level; α = 0.05

iii. The Test statistic:

F1 = MSA / MSE ~ F(2, 18)

F2 = MSB / MSE ~ F(2, 18)

F3 = MSAB / MSE ~F(4, 18)

iv. Critical Region:

      Reject H01, when F > F0.05 (2, 18) = 3.55

       Reject H02, when F > F0.05 (2, 18) = 3.55

        Reject H03, when F > F0.05 (4, 18) = 2.93

v. Computation:

Developer Strength	Development Time (minutes)
	10	14	18
1	0 2 5 4	1 3 4 2	2 5 4 6
2	4 6 7 5	6 8 7 7	9 10 8 5
3	7 10 8 7	10 10 7 7	12 10 9 8

ANOVA Table

vi. Remarks

     The interaction effect of AB is insignificant.

Now using Yates method

Example: 

The data of 2^3 factorial experiment is given below:

Factor A	Factor B
	0	1	2
0	6.3	8.0	7.3
	5.4	8.0	7.8
2	6.9	7.5	8.6
	6.5	7.5	8.3
3	7.2	8.6	9.0
	7.0	8.0	9.0

Test the significance at 5 %.

Solution:

ANOVA Table:

SV	df	SS	MS	F	F tab
SST	8	14.975	1.872	16.57	2.47
SSA	2	3.03	1.515	13.43	3.01
SSB	2	10.51	5.255	46.58	3.01
SSAB	4	1.395	0.3487	3.09	2.69
SSE	9	1.015	0.1128
SSToal	17	15.99

Now using Yates method

Presentation in the ANOVA Table:

Example:

$$

The ANOVA table is constructed for Nitrogen and Phosphorus each at 3 levels. the computer operator compiles the results but not properly saved.  

SV	d.f	SS	MS	F
N	2	350.00
P	?	300.00	150
NP	?	200.00	50
Error	18	150.00
Total	30	1000.00

The computer operator is unable to complete it. How you complete the ANOVA using your statistical knowledge and answer the following questions:

(i). What type of experiment it was?

(ii). How many replicates were used?

(iii). What are your conclusions about interaction and the two main effects?

(iv). If this experiment had been run in blocks what will be the degrees of freedom for blocks.

(v). What will be the estimate of the standard deviation of the response variable?

 Solution:

SV	d.f	SS	MS	F	F tab	Conclusion
N	2	350.00	175	21.00	3.55	Significant
P	2	300.00	150	18.00	3.55	Significant
NP	4	200.00	50	6.00	2.93	Significant
Error	18	150.00	8.33
Total	30	1000.00

(i). It was 3^2 factorial experiment.

(iv)  d.f  fort total = 30

d.f for error = 18

d.f for N = 2

d.f for P = 2

d.f for NP = 4

d.f for replication = d.f  fort total - d.f for N - d.f for P – d.f for error

d.f for replication = 30 – 2 – 2 – 4 – 18

d.f for replication = 4

 Number of replications = 5

The number of blocks = 5

(v).  The estimate of the standard deviation of the response variable

Example: 

The two factors A and B each three levels in an experiment. The data obtained as given below:

Factor A	Factor B
	0	1	2
0	449	413	326
1	409	358	291
2	341	278	312

Test the significance at 5 % level of significance.

Solution:

In single replicate the SSE can obtained by combining the interaction of smaller contribution.

MSE = 20 + 21 + 02 + 12 + 22

MSE = 0.0017 + 2.10 + 0.17 + 6.52 + 0.60

MSE = 9.3917

ANOVA Table:

The linear effects of factor A and B are significant.

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