Engineering Statistics Handbook Episode 6 Part 13 docx

Hyper-Graeco-Latin Square Designs for 4- and 5-Level Factors Designs for 4-level factors there are no 3-level factor Hyper-Graeco Latin square designs 4-Level Factors row blocking factor

5 Process Improvement

5.3 Choosing an experimental design

5.3.3 How do you select an experimental design?

5.3.3.2 Randomized block designs

5.3.3.2.3 Hyper-Graeco-Latin square

designs

These designs

handle 4

nuisance

factors

Hyper-Graeco-Latin squares, as described earlier, are efficient designs

to study the effect of one treatment factor in the presence of 4 nuisance factors They are restricted, however, to the case in which all the

factors have the same number of levels

Randomize as

much as

design allows

Designs for 4- and 5-level factors are given on this page These designs show what the treatment combinations should be for each run

When using any of these designs, be sure to randomize the treatment units and trial order, as much as the design allows.

For example, one recommendation is that a hyper-Graeco-Latin square design be randomly selected from those available, then randomize the run order

Hyper-Graeco-Latin Square Designs for 4- and 5-Level Factors

Designs for

4-level factors

(there are no

3-level factor

Hyper-Graeco

Latin square

designs)

4-Level Factors

row blocking factor

column blocking factor

blocking factor

blocking factor

treatment factor

5.3.3.2.3 Hyper-Graeco-Latin square designs

3 2 1 4 3

with

k = 5 factors (4 blocking factors and 1 primary factor)

L1 = 4 levels of factor X1 (block)

L2 = 4 levels of factor X2 (block)

L3 = 4 levels of factor X3 (primary)

L4 = 4 levels of factor X4 (primary)

L5 = 4 levels of factor X5 (primary)

N = L1 * L2 = 16 runs

This can alternatively be represented as (A, B, C, and D represent the treatment factor and 1, 2, 3, and 4 represent the blocking factors):

Designs for

5-level factors

5-Level Factors

row blocking factor

column blocking factor

blocking factor

blocking factor

treatment factor

5.3.3.2.3 Hyper-Graeco-Latin square designs

3 4 1 3 5

with

k = 5 factors (4 blocking factors and 1 primary factor)

L1 = 5 levels of factor X1 (block)

L2 = 5 levels of factor X2 (block)

L3 = 5 levels of factor X3 (primary)

L4 = 5 levels of factor X4 (primary)

L5 = 5 levels of factor X5 (primary)

N = L1 * L2 = 25 runs

This can alternatively be represented as (A, B, C, D, and E represent the treatment factor and 1, 2, 3, 4, and 5 represent the blocking factors):

Further

information

More designs are given in Box, Hunter, and Hunter (1978)

5.3.3.2.3 Hyper-Graeco-Latin square designs

5 Process Improvement

5.3 Choosing an experimental design

5.3.3 How do you select an experimental design?

5.3.3.3 Full factorial designs

5.3.3.3.1 Two-level full factorial designs

Description

Graphical

representation

of a two-level

design with 3

factors

Consider the two-level, full factorial design for three factors, namely the 23 design This implies eight runs (not counting replications or center point runs) Graphically, we can represent the 23 design by the cube shown in Figure 3.1 The arrows show the direction of increase of the factors The numbers `1' through `8' at the corners of the design box reference the `Standard Order' of runs (see Figure 3.1)

FIGURE 3.1 A 2 3 two-level, full factorial design; factors X1, X2,

X3

5.3.3.3.1 Two-level full factorial designs

The design

matrix

In tabular form, this design is given by:

TABLE 3.3 A 2 3 two-level, full factorial design table showing runs in `Standard Order'

The left-most column of Table 3.3, numbers 1 through 8, specifies a (non-randomized) run order called the `Standard Order.' These numbers are also shown in Figure 3.1 For example, run 1 is made at the `low' setting of all three factors

Standard Order for a 2k Level Factorial Design

Rule for

writing a 2 k

full factorial

in "standard

order"

We can readily generalize the 23 standard order matrix to a 2-level full

factorial with k factors The first (X1) column starts with -1 and

alternates in sign for all 2k runs The second (X2) column starts with -1

repeated twice, then alternates with 2 in a row of the opposite sign until all 2k places are filled The third (X3) column starts with -1 repeated 4 times, then 4 repeats of +1's and so on In general, the i-th column (X i) starts with 2i-1 repeats of -1 folowed by 2i-1 repeats of +1

Example of a 2 3 Experiment

5.3.3.3.1 Two-level full factorial designs

matrix for the

3-factor

complete

factorial

An engineering experiment called for running three factors; namely,

Pressure (factor X1), Table speed (factor X2) and Down force (factor

X3), each at a `high' and `low' setting, on a production tool to

determine which had the greatest effect on product uniformity Two replications were run at each setting A (full factorial) 23 design with 2 replications calls for 8*2=16 runs

TABLE 3.4 Model or Analysis Matrix for a 2 3 Experiment

Variables

I X1 X2 X1*X2 X3 X1*X3 X2*X3 X1*X2*X3

Rep 1

Rep 2

The block with the 1's and -1's is called the Model Matrix or the

Analysis Matrix The table formed by the columns X1, X2 and X3 is

called the Design Table or Design Matrix.

Orthogonality Properties of Analysis Matrices for 2-Factor Experiments

Eliminate

correlation

between

estimates of

main effects

and

interactions

When all factors have been coded so that the high value is "1" and the low value is "-1", the design matrix for any full (or suitably chosen fractional) factorial experiment has columns that are all pairwise

orthogonal and all the columns (except the "I" column) sum to 0 The orthogonality property is important because it eliminates correlation between the estimates of the main effects and interactions 5.3.3.3.1 Two-level full factorial designs

representation

of the factor

level settings

We want to try various combinations of these settings so as to establish the best way to run the polisher There are eight different ways of combining high and low settings of Speed, Feed, and Depth These eight are shown at the corners of the following diagram

FIGURE 3.2 A 2 3 Two-level, Full Factorial Design; Factors X1, X2, X3 (The arrows show the direction of increase of the factors.)

2 3 implies 8

runs

Note that if we have k factors, each run at two levels, there will be 2 k different combinations of the levels In the present case, k = 3 and 23 = 8

Full Model Running the full complement of all possible factor combinations means

that we can estimate all the main and interaction effects There are three main effects, three two-factor interactions, and a three-factor interaction, all of which appear in the full model as follows:

A full factorial design allows us to estimate all eight `beta' coefficients

Standard order

5.3.3.3.2 Full factorial example

variables in

standard order

The numbering of the corners of the box in the last figure refers to a standard way of writing down the settings of an experiment called

`standard order' We see standard order displayed in the following tabular representation of the eight-cornered box Note that the factor settings have been coded, replacing the low setting by -1 and the high setting by 1

Factor settings

in tabular

form

TABLE 3.6 A 2 3 Two-level, Full Factorial Design Table Showing Runs in `Standard Order'

Replication

Replication

provides

information on

variability

Running the entire design more than once makes for easier data analysis because, for each run (i.e., `corner of the design box') we obtain an average value of the response as well as some idea about the dispersion (variability, consistency) of the response at that setting

Homogeneity

of variance

One of the usual analysis assumptions is that the response dispersion is uniform across the experimental space The technical term is

`homogeneity of variance' Replication allows us to check this assumption and possibly find the setting combinations that give inconsistent yields, allowing us to avoid that area of the factor space

5.3.3.3.2 Full factorial example

Factor settings

in standard

order with

replication

We now have constructed a design table for a two-level full factorial in three factors, replicated twice

TABLE 3.7 The 2 3 Full Factorial Replicated Twice and Presented in Standard Order

Speed, X1 Feed, X2 Depth, X3

Randomization

No

randomization

and no center

points

If we now ran the design as is, in the order shown, we would have two deficiencies, namely:

no randomization, and

1

no center points

2

5.3.3.3.2 Full factorial example

provides

protection

against

extraneous

factors

affecting the

results

The more freely one can randomize experimental runs, the more insurance one has against extraneous factors possibly affecting the results, and

hence perhaps wasting our experimental time and effort For example, consider the `Depth' column: the settings of Depth, in standard order, follow a `four low, four high, four low, four high' pattern

Suppose now that four settings are run in the day and four at night, and that (unknown to the experimenter) ambient temperature in the polishing shop affects Yield We would run the experiment over two days and two nights and conclude that Depth influenced Yield, when in fact ambient temperature was the significant influence So the moral is: Randomize experimental runs as much as possible

Table of factor

settings in

randomized

order

Here's the design matrix again with the rows randomized (using the RAND function of EXCEL) The old standard order column is also shown for comparison and for re-sorting, if desired, after the runs are in

TABLE 3.8 The 2 3 Full Factorial Replicated Twice with Random Run Order Indicated Random

Order

Standard

5.3.3.3.2 Full factorial example

Table showing

design matrix

with

randomization

and center

points

This design would be improved by adding at least 3 centerpoint runs placed at the beginning, middle and end of the experiment The final design matrix is shown below:

TABLE 3.9 The 2 3 Full Factorial Replicated Twice with Random Run Order Indicated and

Center Point Runs Added Random

Order

Standard

5.3.3.3.2 Full factorial example

representation

of blocking

scheme

FIGURE 3.3 Blocking Scheme for a 2 3 Using Alternate Corners

Three-factor

interaction

confounded

with the block

effect

This works because we are in fact assigning the `estimation' of the (unwanted) blocking effect to the three-factor interaction, and because

of the special property of two-level designs called orthogonality That

is, the three-factor interaction is "confounded" with the block effect as will be seen shortly

Orthogonality Orthogonality guarantees that we can always estimate the effect of one

factor or interaction clear of any influence due to any other factor or interaction Orthogonality is a very desirable property in DOE and this

is a major reason why two-level factorials are so popular and successful

5.3.3.3.3 Blocking of full factorial designs

showing

blocking

scheme

Formally, consider the 23 design table with the three-factor interaction column added

TABLE 3.10 Two Blocks for a 2 3 Design SPEED

X1

FEED

X2

DEPTH

X3 X1*X2*X3

BLOCK

Block by

assigning the

"Block effect"

to a

high-order

interaction

Rows that have a `-1' in the three-factor interaction column are assigned to `Block I' (rows 1, 4, 6, 7), while the other rows are assigned to `Block II' (rows 2, 3, 5, 8) Note that the Block I rows are the open circle corners of the design `box' above; Block II are

dark-shaded corners

Most DOE

software will

do blocking

for you

The general rule for blocking is: use one or a combination of high-order interaction columns to construct blocks This gives us a formal way of blocking complex designs Apart from simple cases in which you can design your own blocks, your statistical/DOE software will do the blocking if asked, but you do need to understand the

principle behind it

Block effects

are

confounded

with

higher-order

interactions

The price you pay for blocking by using high-order interaction columns is that you can no longer distinguish the high-order interaction(s) from the blocking effect - they have been `confounded,'

or `aliased.' In fact, the blocking effect is now the sum of the blocking effect and the high-order interaction effect This is fine as long as our assumption about negligible high-order interactions holds true, which

it usually does

Center points

within a block

Within a block, center point runs are assigned as if the block were a separate experiment - which in a sense it is Randomization takes place within a block as it would for any non-blocked DOE

5.3.3.3.3 Blocking of full factorial designs

5.3.3.3.3 Blocking of full factorial designs

5.3.3.4 Fractional factorial designs