2^k Factorial Experiment lecture - 29

 

2^k Factorial Experiment

 lecture - 29

The 2^k factorial experiment is extension of  factorial experiment in which k factors each at two levels are analyzed.

The 2^k factorial experiment with replication is represented by the following statistical model.

Yijk...p = μ + ρi + αj + βk +γl +...+ (αβ)jk+ ···+ (αβγ....)jkl...p + ϵij...p

i = 1, 2, ...., r

j, k, .... = 0, 1

Where:

Yijk...p is the yield of j^th level of factor A and k^th level of factor B in i^th block.

 μ is the overall effect.

ρi is the effect of the i^th block

αj is the effect of j^th level of factor A,

 βk is the effect of k^th level of factor B,

 (αβ)jk is the interaction effect of j^th level of factor A and k^th level of factor B.

The standard order of any factor is obtained step by step by multiplying it with an additional letter to preceding standard order.

e.g. The standard order of 2^2 factorial experiment is given by;

1, a, b, ab

The standard form of 2^3 can be obtained as:

c x{1, a, b, ab} = a, b, ab, c, ac, bc, abc.

2^k Factorial Experiment with Single Replicate

In 2^k single replicate factorial experiment is also called un replicated factorial experiment and no internal error can be estimated. In such case the high order interaction effects are assumed to be negligible and the analysis based on main effects and lower order interaction effects. The degree of freedoms for main effect is 

and first order interactions is 

This phenomenon is called sparsity of effects principle. According to sparsity principle consider the main effects and first order interaction (assumed to be significant). The second and third order interactions (assumed insignificant) is diverted to error. The error degree of freedom will be

The factors and their interaction will drop, if their percent contribution is smaller but those interactions will be included in the ANOVA whose main effect is maximum contribution. If one of the factors is minimum contribution, then the interaction effect include minimum contribution will be dropped.

The contribution formula is given by;

SSTotal is the sum of all main and their interaction effects.

To explain the smaller contribution but excluded from the ANOVA table:

Let factor A and Factor B is maximum contribution but the interaction of AB is smaller contribution can not be excluded from ANOVA table. Similarly, A and B is maximum contribution but C is minimum contribution, the drop the interaction ABC.

Example

 The following data obtained 2^4 factorial experiment of an agriculture unit. 

No.	1	2	3	4	5	6	7	8
Treatment	1	a	b	ab	c	ac	bc	abc
Yield	44	70	49	66	68	60	80	65
No.	9	10	11	12	13	14	15	16
Treatment	d	ad	bd	abd	cd	acd	bcd	abcd
Yield	42	100	45	102	77	85	72	94

Find the effects and test the significance of second order and high order interactions.

Solution: Applying Frank Yates method.

TC	Yield	Column I	Column II	Column III	Column IV	SS	% SSE	Remarks
(1)	44	114	229	502	1119
a	70	115	273	617	165	1701.56	30.59
b	49	128	289	20	27	45.56	0.819	Dropped
ab	66	145	328	145	-2	0.25	0.004	Dropped
c	68	142	43	18	83	430.56	7.74
ac	60	147	-23	9	151	1425.06	25.62
bc	80	162	115	-16	15	14.06	0.252	Dropped
abc	65	166	30	14	18	20.25	0.36	Dropped
d	42	26	1	44	115	826.56	2.07
ad	100	17	17	39	125	976.56	17.54
bd	45	-8	5	-66	-9	5.06	0.09	Dropped
abd	102	-15	4	-85	30	56.25	1.10	Drooped
cd	77	58	-9	16	-11	7.56	0.11
acd	85	57	-7	-1	-19	22.56	0.39
bcd	72	8	-1	2	-17	18.06	0.32	Dropped
abcd	94	52	15	16	14	12.25	0.21	Dropped
						5562.16

Note: The contribution of AC is 0.11 but can not be dropped because A and C have maximum contribution. Similarly, the contribution of ACD is 0.39 can’t be dropped. BCD will de dropped because B is minimum contribution.

SV	df	SS	MS	F	F tab
A	1	1701.56	1701.56	79.26	5.32
C	1	430.56	430.56	20.05
D	1	826.56	826.56	38.50
AC	1	1425.06	1425.06	66.38
AD	1	976.56	976.56	45.49
CD	1	7.56	7.56	0.35
ACD	1	22.56	22.56	1.05
Error	8	171.74	21.4675
Total	15	5562.16

The three factors A, B, C and their interaction AC, AD are significant.

Q 1: Answer the following

i.                     Define Factorial experiments

ii.                   Define main and interaction effects with the help of  experiment.

iii.                 Calculate the main and interaction effects of the following:

Factor A	Factor B
	Low	High
Low	5	9
High	19	7

Q 2: The effects of the amount of curing agent and the curing time on the adhesion strength of dental brackets were studied using a full factorial experiment. The experiment and its results are summarized in the tables below:

Factor	Levels
A(amount of curing agent (mg))	50	100
B(curing time (seconds))	15	60

Write down appropriate statistical model of the above experiment. Using ANOVA, determine the effect of the curing agent, curing time, and the interaction between them on the adhesion strength. Which of these effects is most important?

Q 3: An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The results are as follows:

(a) Estimate the factor effects. Which effects appear to be large?

 (b) Use the analysis of variance to and determine which factors are important in explaining yield.

• Read more: 2^2 Factorial Experiment