3^2 Factorial Experiment lecture - 31

3^2 Factorial Experiment 

lecture - 31 

The 3^2 factorial experiment is a factorial experiment in which two factors at each of three levels are analyzed is called  factorial experiment. The following linear statistical model (CRD) represents the  factorial experiment:

                Yijk = μ + αj + βk + (αβ)jk + ϵijk    i = 1, 2, ..., r        j, k = 0, 1, 2

$$

In CRD the subscript i represent replications.

The RCBD model of 2^3 factorial experiment is given by;

Yijk = μ + ρi + αj + βk + (αβ) jk + ϵijk        i = 1, 2, ..., b        j, k = 0, 1, 2

The 3^2  factorial experiment consist 2 factors (i.e. A, B) and each factor at three levels (i.e. low, intermediate, and high). In 3^2  factorial experiment the total treatment combinations are 9. The 9 treatment combinations are:

Or we can write as:

Factor A	Factor B
	Low	Intermediate	High
Low	00	01	02
Intermediate	10	11	12
High	20	21	22

The number of main effects = 2C1 = 2 each with 2 d.f.

The number of 2 factors interaction = 2C2 = 1 with 1 d.f.

Factor A have two mutual orthogonal contrasts i.e. AL, AQ. Factor B have two mutual orthogonal contrasts i.e. BL, BQ and factor (AB) have four mutually orthogonal contrasts i.e. ALBL, ALBQ , AQBL AQBQ .

The main effect A is estimated by the contrasts among 3 sets of treatment combinations. The first group is containing all those treatment combination in which factor A occurs at 0 level. The second and third group contains treatment combination having levels of A as 1 and 2 respectively.

Hence the three sets of treatment combinations are;

Levels
0	1	2
Group I	Group II	Group III
00	10	20
01	11	21
02	12	22

Note:

 for A

First group is obtained by X1 = 0 (mod 3)

Second group is obtained by X1 = 1 (mod 3)

Third group is obtained by X1 = 2 (mod 3)

Similarly for B

First group is obtained by X2 = 0 (mod 3)

Second group is obtained byX2 = 1 (mod 3)

Third group is obtained by X2 = 2 (mod 3)

Since there are three groups, so there will be two independent contrasts each having 1 d.f. A have 2 d.f. so, it will have 2 contrasts that’s  AL, AQ.

Where:

Similarly for B

The interaction effect AB is split into 4 components on interaction between linear and quadratic effects of two factors. The contrasts of 4 components are:

ANOVA Table:

SV	d.f	SS	MS	F
Replication	r - 1	SSR
Treatments	3^2 -1	SST
A	2	SSA
B	2	SSB
AB	4	SSAB
Error	(3^2 -1) (r -1)	SSE
Total	3^2 -1	SS Total

 

Example: 

An engineer tests 3 plate materials for a new battery at 3 temperature levels (15°F, 70°F, and 125°F). Four batteries (replicates) are tested at each combination of plate material and temperature, and all 36 tests are run in random order.

Material Type	Temperature
	15°F		70°F		125°F
1	130	155	34	40	20	70
	74	160	80	75	82	58
2	150	188	136	122	25	70
	159	126	106	115	58	45
3	138	110	174	120	96	104
	168	160	150	139	82	60

i.                    1. What effects do material type and temperature have on the life of a battery?

ii.                  2. Is there a material that would give long life regardless of temperature?

State the statistical model of the above experiment and analyze the above.

Solution: The experiment consists 2 factors (i.e. Temp and Material Type) and each factor has 3 levels. Each factor combination is replicated four times. The above experiment is  factorial experiment using RCBD and represented as:

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

i = 0, 1, 2

j = 0, 1, 2

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

αj : Material Type Effect

βk : Temperature Effect

 τj  represent material types regardless temperature.

$$\begin{gathered} \text{ \\ i. Set up our hypotheses as:} \end{gathered}$$

$$\begin{gathered} \text{ \\ <br>} \end{gathered}$$

$$\begin{gathered} \text{ \\ H0 : τ1 =τ2 =τ3 =τ4 = τ5 = τ6 Vs. } \end{gathered}$$H1 : τ1 =τ2 =τ3 =τ4 = τ5 = τ6  

H01 : αj = 0           Vs    .   H11 : αj ≠ 0     i = 0, 1, 2

H02 : βk  = 0          Vs.      H12 : βk  ≠ 0    j = 0, 1, 2

H03 :  (αβ)jk  = 0   Vs.      H03 :  (αβ)jk ≠ 0

ii. The significance level; α =0.05

iii. The test statistic:

iv. Critical Region:

v. Computation:

The data can be arranged as:

For blocks effect:

ANOVA Table

vi. Remarks

 it is concluded that all effects and interaction effect are significant

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       i = 1, 2  j,k = 0, 1, 2

Where:

ρi =Block effect

αj : Suppliers Effect

βk : Mixture Effect.

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

Conclusion: The interaction effect is insignificant. 

Practice Question:

Q 1: The yield of a chemical process is being studied. The two most important variables are thought to be the pressure and the temperature. Three levels of each factor are selected, and a factorial experiment with two replicates is performed. The yield data are as follows:

Temperature (°C)	Pressure (psig)
	200	215	230
150	90.4 90.2	90.7 90.6	90.2 90.4
160	90.1 90.3	90.5 90.6	89.9 90.1
17-	90.5 90.7	90.8 90.9	90.4 90.1

(a) Is there any indication that either factor influences brightness? Use 0.05.

(b) Do the two factors interact? Use 0.05.

(c) Analyze the residuals from this experiment.

Q 2: An experiment with 3 levels of nitrogen fertilizer and 3 levels of phosphorus fertilizer, the data is the number of seeds that germinated total over 12 plots.

Write down appropriate statistical model of the above experiment. Using ANOVA and test the hypothesis about main effects of factors and their interaction levels.

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