Design of Engineering Experiments – Nested and Split-Plot Designs

Design and Analysis of Experiments 8E 2012 Montgomery

1Chapter 14

2Chapter 14

• Text reference, Chapter 14• These are multifactor experiments that have some

important industrial applications• Nested and split-plot designs frequently involve one

or more random factors, so the methodology of Chapter 13 (expected mean squares, variance components) is important

• There are many variations of these designs – we consider only some basic situations

3Chapter 14

Two-Stage Nested Design• Section 14.1• In a nested design, the levels of one factor (B) is

similar to but not identical to each other at different levels of another factor (A)

• Consider a company that purchases material from three suppliers– The material comes in batches– Is the purity of the material uniform?

• Experimental design – Select four batches at random from each supplier– Make three purity determinations from each batch

4Chapter 14

Two-Stage Nested Design

5Chapter 14

Two-Stage Nested DesignStatistical Model and ANOVA

( ) ( )

1,2,...,1, 2,...,1, 2,...,

: 1 1 ( 1) ( 1)

ijk i j i ij k

T A B A E

i ay j b

SS SS SS SS

df abn a a b ab n

6Chapter 14

Two-Stage Nested DesignExample 14.1

Three suppliers, four batches (selected randomly) from each supplier, three samples of material taken (at random) from each batch

Experiment and data, Table 14.3

Data is coded

JMP and Minitab balanced ANOVA will analyze nested designs

Mixed model, assume restricted form

7Chapter 14

8Chapter 14

9Chapter 14

10Chapter 14

Minitab Analysis

JMP Analysis (REML estimates of variance components)

Chapter 14

12Chapter 14

Practical Interpretation – Example 14.1

• There is no difference in purity among suppliers, but significant difference in purity among batches (within suppliers)

• What are the practical implications of this conclusion?

• Examine residual plots – plot of residuals versus supplier is very important (why?)

• What if we had incorrectly analyzed this experiment as a factorial? (see Table 14.5)

• Estimation of variance components (ANOVA method)

13Chapter 14

14Chapter 14

15Chapter 14

Variations of the Nested Design• Staggered nested designs

– Prevents too many degrees of freedom from building up at lower levels

– Can be analyzed in JMP or Minitab (General Linear Model) – see the supplemental text material for an example

• Several levels of nesting – The alloy formulation example– This experiment has three stages of nesting

• Experiments with both nested and “crossed” or factorial factors

16Chapter 14

17Chapter 14

18Chapter 14

Example 14.2 Nested and Factorial Factors

( ) ( ) ( )

1,2,31,2

( ) ( )1,2,3,4

ijkl i j k j ij ik j ijk l

19Chapter 14

Example 14.2 – Expected Mean Squares

Assume that fixtures and layouts are fixed, operators are random – gives a mixed model (use restricted form)

20Chapter 14

Example 14.2 – Minitab Analysis

21Chapter 14

22Chapter 14

23Chapter 14

The Split-Plot Design

• Text reference, Section 14.4 page 621• The split-plot is a multifactor experiment where it

is not possible to completely randomize the order of the runs

• Example – paper manufacturing– Three pulp preparation methods– Four different temperatures – Each replicate requires 12 runs– The experimenters want to use three replicates– How many batches of pulp are required?

24Chapter 14

25Chapter 14

The Split-Plot Design

• Pulp preparation methods is a hard-to-change factor

• Consider an alternate experimental design:– In replicate 1, select a pulp preparation method,

prepare a batch– Divide the batch into four sections or samples, and

assign one of the temperature levels to each– Repeat for each pulp preparation method– Conduct replicates 2 and 3 similarly

26Chapter 14

The Split-Plot Design• Each replicate (sometimes called blocks) has been

divided into three parts, called the whole plots• Pulp preparation methods is the whole plot

treatment• Each whole plot has been divided into four

subplots or split-plots• Temperature is the subplot treatment• Generally, the hard-to-change factor is assigned to

the whole plots• This design requires only 9 batches of pulp

(assuming three replicates)

27Chapter 14

The Split-Plot DesignModel and Statistical Analysis( ) ( ) ( )

1,2,...,( ) 1,2,...,

1, 2,...,

ijk i j ij k ik jk

ijk ijk

i rj ak b

There are two error structures; the

whole-plot error and the subplot error

28Chapter 14

Split-Plot ANOVA

Calculations follow a three-factor ANOVA with one replicate

Note the two different error structures; whole plot and subplot

29Chapter 14

Alternate Model for the Split-Plot1,2,...,

( ) ( ) 1, 2,...,1, 2,...,

ijk i j ij k jk ijk

i ry j a

30Chapter 14

“Inadvertent” Split-Plot and CRD Analysis

Chapter 14

32Chapter 14

Variations of the basic split-plot design

More than two factors – see page 627

A & B (gas flow & temperature) are hard to change; C & D (time and wafer position) are easy to change.

33Chapter 14

Unreplicated designs and fractional factorial design in a split-plot framework

34Chapter 14

35Chapter 14

36Chapter 14

37Chapter 14

38Chapter 14

39Chapter 14

40Chapter 14

• A split-split-plot design

• Two randomization restrictions present within each replicate

41Chapter 14

The strip-split-plot design

The “strips” are just another set of whole plots

Design of Engineering Experiments – Nested and Split-Plot Designs

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