Provide Information Regarding Statistics & Econometrics : Statistical Model of 2^2 Factorial Experiment & Analysis Lecture

Statistical Model of 2^2 Factorial Experiment

Analysis

Lecture - 24

The 2^2 factorial experiment is represented by the following statistical model (CRD Model).

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

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

μ is the overall effect.

αj is the effect of j^th level of factor A, ∑αj = 0

βk is the effect of k^th level of factor B, ∑βk = 0

(αβ)jk is the interaction effect of j^th level of factor A and k^th level of factor B.∑(αβ)j = 0
∑(αβ)k = 0
ϵijk is the random error follow normal distribution with mean 0 and variance σ2.
that's

ϵijk ~ N (0, σ2)

A factorial experiment is not considered as an experimental design because of the fact that the

basic designs namely, CRD, RCBD, and LSD are used to carry out the factorial experiments. For this purpose of analysis of variance, the basic sums of squares are computed in a usual manner, with addition that the treatment sum of squares is further partitioned into component parts of main effects and interaction effects.

Analysis Factorial Experiment

Let Yijk is used to denote the effect of j^th level of factor A and k^th level of factor B in the i^th block and assume RCB design.

Add the observations in each cell denoted by and express in the following as:

Where:

The data for blocks can be arranged as:

The following term is called total variation:

ANOVA Table:

S.V	d.f	S S	M S	F
Block	r - 1	SS Block	MS Block
Treatment	t - 1	SST	MST	F

A	a - 1	SSA	MSA	FA
B	b - 1	SSB	MSB	FB
AB	(a – 1) (b – 1)	SSAB	MSAB	FAB
Error	(r – 1)(ab- 1)	SSE	MSE
Total	rab - 1	SS Total

Where a = 2 & b = 2

Conventional procedure of ANOVA to compute various sums of squares:

Error will be equivalent to Interaction sum of squares in table – II:

SSAB = SST - SSA - SSB

Where SST in table - II

Alternative procedure to compute sums of squares:

The simplest form of 2^2 factorial experiment is given as follows:

SV		d.f	SS	MS	F
A	1		SSA	MSA	FA
B	1		SSB	MSB	FB
AB	1		SSAB	MSAB	FAB
Error	4 (r – 1)		SSE	MSE
Total	4 r - 1		SS Total

Interpretation of Factorial Experiment:

The interpretation of factorial experiment depends on interaction effect(s). if the interaction effect is significant then there is no real mean whether the main effect is significant or not.

Example - 1: A 2^2 factorial experiment i.e. with 2 varieties (factors) and 2 manures (level) was carried out in a randomized complete block design with 3 replications. The yield given in the following table:

Write the statistical model and perform ANOVA and test the significance of varieties and manures?

Method – I:

Solution: The given 2^2 factorial experiment in RCBD can be represented by the following statistical model:

Yijk = μ + ρi + αj + βk + (αβ)jk + ϵijk i=1, 2, 3. j=k = 0, 1

The effect of four treatment combinations (T1 = a0b0, T2 = a0b1, T3 = a1b0, and T4 = a1b1) is considered as treatment and setup hypothesis as:

i. H0 :The effect of four treatments is non significant. Vs. H0: The effect of four treatments is significant.

a. H01: The effect of factor A is non significant Vs. H11: The effect of factor A is significant.

b. H02: The effect of factor B is non significant Vs. H12: The effect of factor A is non significant.

c. H03: There interaction b/w factor A & B is zero. Vs. H13: There interaction effect is not zero

ii. The significance level; α = 0.05

iii. The test statistic:

v. Computation:

ANOVA Table:

SV	d.f	S.S	MS	F
Block	2	18.50	9.25
Treatment	3	38.25	12.75	4.36
A	1	36.75		12.58
B	1	0.75		0.25
AB	1	0.75		0.25
Error	1	17.50
Total	11	74.25

vi. Remarks:

The calculated F value falling in the acceptance region, it is concluded that treatments (factors combination) are insignificant at . There is no need to check the significance difference between factor A and B.