Introduction to Factorial Experiment lecture - 23

 

Introduction to Factorial Experiment

lecture - 23

Treatment

A treatment is something that researcher administered to experimental units and their effect is to be investigated on dependent variable (variable of interest). The selection of treatments is depending on the nature of the study or the researcher choice which based on in-depth theoretical study and logical argumentation.

Example: 
Suppose a researcher desired to investigate which of three fertilizers increase the yield capabilities of corn.
To achieve the above objective, divide the corn field into three parts and treated each with each fertilizer.

Factor

A factor is a control categorical independent variable whose levels are set by the experimenter. The combination of levels of different factors is consider as a treatment. The sub division of a factor is called levels. Let a factor A have two levels then it is denoted by a0 , a1 or 0, 1. 0 or a0 is called lower level of and 1 or a1 is called upper level of factor A.

Example

Let us it is desired to study the effect of anti-allergic medicine (A) with types of dosages 75 mg and 100 mg and analgesic medicine (B) with two types dosages simple and forte on runny nose and head ache. The number of hours relief provides to patients is our response variable. Further assume that the a0 = 75 mg and a1 = 100 mg are the levels of factor A and b0 simple and b1 = 100 mg are levels of factor B.

Factorial Experiment

An experiment involves two or more factors and each factor is at two or more levels. The possible combinations of factors levels are considered as treatments and the effect of these treatments is investigated on response variable. The factorial experiments not only investigate the main effects of factors but also their interaction effect on response variable. Generally, factors are representing by capital letters A, B, …and levels are representing by small letters a, b, …. with subscripts 0, 1, 2, …., depending on the levels of the factors.

Let we have two factors A and B, each with two levels that’s a0, a1, b0, and b1.

The possible treatment combinations are:

a0 b0, a1 b0,   a0 b1,     a1 b1

Standard order for Treatment Combination in Factorial Experiment

General notation for representing the factors is to use capital letters, e.g., A, B, C etc. and levels of a factor are represented in small letters. For example, if there are two levels of A, they are denoted by a0 and  a1. Similarly, the two levels of B are represented by b0 and b1. Another alternative representation to indicate the two levels of A is 0 (for a0 ) and 1 (for a1 ). The factors of B are then 0 (for b0 ) and 1 (for b1). This experiment is represented by  factorial experiment which gives 4 groups.

Types of Factorial Experiments

A factorial experiment can be divided into two distinct types.

1.      Symmetrical Factorial Experiments

2.      Asymmetrical Factorial Experiments

Symmetrical Factorial Experiment

A factorial experiment where each factor appears at the same levels. e.g. if we have k factors all must have the same levels neither factor will be different levels. if we have k factors and each factor is at p levels, then it denoted p ^k. The term 2 X 2 X …...X 2 = 2 ^k refer to experiment where each of the k factors has two levels and 3 ^k refer to k factors each at three level factorial experiment.

Asymmetrical Factorial Experiment

A factorial experiment in which all factors occur with the varying number of levels is called asymmetrical or mixed factorial experiment. The asymmetrical factorial experiment the arrangement describes by product of levels.

For example, a factorial experiment consists 3 factors i.e. A, B, and C. factor A at 2 levels, factor B at 3 levels and factor C at 4 levels respectively then this factorial experiment is represented by 2 X 3 X 4 factorial experiment.

Effects in Factorial Experiment

Consider the two factors A, B and each at two levels.

Factor A	Factor B
	B1	B2
A1	1	b	b - 1
A2	a	ab	ab - 1
	a - 1	ab - b

Simple Effect

The simple effect of a factor is the difference between its responses for a fixed level of other factors.

Similarly,

Main Effect & Total Effect

The average of simple effect is called main effect or the main effect of factor A is the average effect of A at two levels of B and denoted by A. the total effect of A is denoted by

Consider two factors A and B each have 2 levels given below:

Similarly,

Interaction Effect

The effects of one factor depend on the level of another factor. The interaction effect of factors {AB} is denoted by .

Plus – Minus Table for 2^2 Factorial Experiment

In the following table I represent the total of all observations. 1 denotes that both factors are at lower levels  and this is called as the control treatment.  Factor A has plus sign for a treatment combination which include “a” and negative sign which not include “a”. Similarly, for factor B. interaction AB is obtained by multiplying the signs of A and B.

• Read:2^2 Factorial Experiment