ETM 620 - 09U1 ETM 620 - 09U1

2k factorialsRecall our example from last time …

Estimate the effectsDetermine significant effectsDevelop regression modelExamine residuals

ETM 620 - 09U2 ETM 620 - 09U2

Example: 22 factorial Look at the effect of oven temperature and

reaction time on the yield (in percent) of a process …Oven Temp. Reaction Time 110° 50 min. 130° 70 min.Take 2 observations at each combination with the following result:

 Observation

s

Temp R.T.Interacti

on #1 #2 SUM(1) -1 -1 1 55.5 54.5 110

a 1 -1 -1 60.2 61121.

2

b -1 1 -1 64.5 63.9128.

4

ab 1 1 1 67.7 68.7136.

4

ETM 620 - 09U3

From last week …Effects:

Temp = 4.8Time = 8.4Interaction = -0.8

Which are significant?Determine S.E.

If Effect + 2S.E. contains 0, ____________________Define the regression model,

433.0375.02*21

21

222

2 Sn k

2

...ˆ 22110

ii

nn

effectwhere

xxxy

ETM 620 - 09U4

Calculate residuals, , and investigatenormal probability plots residuals vs fitted valuesresiduals vs factorsresiduals vs order

ii yy ˆ

ETM 620 - 09U5

3 or more factors…Note: design of 2-factor (22) looks like …

while 3-factor (23) looks like …

(4-factor and higher hard to draw, but …)

(1) a

abb

B

A

ETM 620 - 09U6

Example – 23 designAs a consultant to the manufacturer of a reflective paint used in the design of safety signs, you are testing the effect of the reflectance of the background, the percent reflective chemical in the paint, and the ambient light level on the readability of the signs. You design a 23 factorial experiment with the following parameters.

You ask test subjects to read the signs and count the number of errors per hundred trials. You run two replications at each level.

  Background  Ambien

t

  Reflectance% Chemical Light

Low (-) 20% 5% 30 lx

High (+) 75% 15% 750 lx

ETM 620 - 09U7

An approach …1. Use Minitab to design the 23 experiment with 2

replications.2. The data are found in the file DOE examples 2.xls on

the website. Copy the design provided by Minitab into that file. Put the experimental design in standard order (that is, sort on the “standard order” column).

3. Create a column next to the design called “Errors/100 trials” and copy the results provided into that column. Re-sort the data (including the results) by run (i.e., sort the table of design and results by the column “run order”.) Copy the “Errors/100 Trials” column into the Minitab worksheet.

4. Analyze the factorial in Minitab. Response is “Errors/100 Trials”.

ETM 620 - 09U8

What if you can only afford 1 replication?Recall that we are most interested in main and

lower-order interaction effects.Higher-order interactions tend to be negligible.Therefore, higher-order interactions can be

pooled (combined) and used to estimate the error.

Example:In the previous example, assume that only the

first column of data is available – analyze the data assuming the 3rd order interactions are negligible.

ETM 620 - 09U9

Confounding in the 2k designIf the experiment must be run in blocks, then

certain interactions are confounded with (i.e., indistinguishable from) the blocks …

Any treatment combination that has at least 1 plus and 1 minus level in each block will not be affected by blocking.

In Minitab, use either default generators (highest order interactions) or specify the generators.

Example:In the reflective paint example, assume that we

can only run 4 trials per day. Set up a design using the default generators for 2 blocks.

ETM 620 - 09U10

Fractional factorialsIf a 25 design is used for the experiment, its 31

degrees of freedom would be allocated as follows:

Using effect hierarchy principle, one would argue that 4th and 5th order interactions (and maybe even 3rd order) are not likely to be important. There are 10+5+1 = 16 such effects, half of the total runs! Using a 25 design can be wasteful (unless 32 runs cost about the same as 16 runs.)

Main Interactions

Effects

2-Facto

r

3-Facto

r

4-Facto

r

5-Facto

r5 10 10 5 1

ETM 620 - 09U11

Example of a 25-1 factorialLeaf spring experiment:

y = free height of spring, target = 8.0 inches.Goal : get y as close to 8.0 as possible

Five factors at two levels, use a 16-run design with three replicates for each run (i.e., a 25−1 design, or ½ fraction of the 25 design.)Level

Factor − +B. high heat temp (deg F) 1840 1880C. heating time (sec.) 23 25D. transfer time (sec.) 10 12E. hold down time (sec.) 2 3Q. quench oil temp (deg F)

130-150

150-170

ETM 620 - 09U12

Leaf spring example …Create the 25-1 design (3 replications at each

level) in MinitabInsert the data from the examples file.Analyze the data and draw conclusions.