DOE (Design of Experiment)

Made By:ISHA JAIN

NIDHI GAHLOTDivision of MPAE

Some Typical Applications of Experimental Design

• Characterising: also known as “screening”. To determine which factors affect the output.

• Optimising: to determine the region in the important factors that leads to the best possible response.

Strategy of Experimentation

• One factor at a time approach: keep all other factors constant and change any one (say A). This gives “main effect A ONLY”.

• Factorial: gives main effects as well as interaction.

Basic Principles of ExperimentationBasic principles What do they mean? Why do we do them?

replication Repetition of basic experiment. NOT same as repeated measurements

•Improves validity of DOE•Reflects variability b/w runs

randomisation Allocation of experimental material and order of runs in random

Assists in averaging out the effects of extraneous factors

blocking A design technique to improve precision with which comparisons among factors on interest are made

Reduces effect of nuisance factors

What is Factorial Design?• factors• levels• x y = (no. of levels) (no. of factors) • Main effect• Interaction

let us consider simplest factorial design possible.22 full factorial

Regression Model• Refers to the equation establishing

quantitative relationship b/w factors of interest (A & B) and response (y)

Types of Plots Obtained from DOE

• Interaction Plots• Normal Probability Plots/ Half Normal Plots

Interaction Plots

• One factor interaction plot• Two factor interaction plot

Let us study “two factor interaction plot”

Two factor interaction plot• Plots that help us realise interaction AB.• A significant interaction will often “mask” the

significance of main effects.

Normal Probability Plots

• The effects that are negligible are normally distributed, with mean zero & variance ^2 & will tend to fall along a straight line on this plot, whereas significant effects will have non zero means and hence will not lie along a straight line.

Normal Probability Plots Vs Half Normal Plots

Take only +ve half of bell shaped curves!

Analysis of Variance Table (ANOVA)

Source of variation

Sum of squares

Degrees of freedom

Mean square

F - Value P - Value

A SSA (a-1) MSA FA PA

B SSB (b-1) MSB FB PB

AB SSAB (a-1)(b-1) MSAB FC PC

Error SSE ab(n-1) MSE 1

Total SST abn-1

•A “P value” less than 0.005 implies variation is significant•A “P value” more than 0.005 implies variation is NOT significant

When no. of factors increase…• 23 = 8

Fractional Factorial Designs

• When do we use fractional factorial?Too many no. of runsCharacterising/ screening• Properties of fractional factorial?Sparsity of effectsProjective property*Sequential experimentation*

*later

The One Half Fraction on 2k Design

• Consider 23= 8half of 8= 4for fractional factorial, we will perform 4 runsONLY.

2(3-1) Design way of representing one half fractional factorial on 23

Which 4 runs to choose and which 4 runs to reject?

Which 4 runs to choose and which 4 runs to reject?

combination I A B C AB AC BC ABC

a + + - - - - + +b + - + - - + - +

c + - - + + - - +

abc + + + + + + + +

ab + + + - + - - -

ac + + - + - + - -

bc + - + + - - + -

1 + - - - + + + -

Hence. 2 relations are possible.I= ABC or I= -ABCHence. 2 one half fractional factorials can be obtained from one one full factorial.

• I = ABC• Known as “principle fraction”

lA A + BC

lB B + AC

lc C + AB

• I= -ABC• Known as “alternate fraction”

lA’ A - BC

lB’ B - AC

lc’ C - AB

Add to obtain A, B and C.Subtract to obtain AB, BC and AC.

Sequential experimentation

Projection Property

A one half fractional factorial design for 23 factorial can be perceived as 22 full factorial design.