ETM U k factorials Recall our example from last time … Estimate the effects Determine significant effects Develop regression model Examine residuals

ETM U 2 2 Example: 2 2 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: Observations TempR.T.Interaction#1#2SUM (1) a b ab

ETM U 3 From last week … Effects: Temp = 4.8 Time = 8.4 Interaction = -0.8 Which are significant? Determine S.E. If Effect + 2S.E. contains 0, ____________________ Define the regression model,

ETM U 4 Calculate residuals,, and investigate normal probability plots residuals vs fitted values residuals vs factors residuals vs order

ETM U 5 3 or more factors… Note: design of 2-factor (2 2 ) looks like … while 3-factor (2 3 ) looks like … (4-factor and higher hard to draw, but …) (1) a ab b B A

ETM U 6 Example – 2 3 design As 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 2 3 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 Ambient Reflectance% ChemicalLight Low (-) 20%5%30 lx High (+) 75%15%750 lx

ETM U 7 An approach … 1. Use Minitab to design the 2 3 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 U 8 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 3 rd order interactions are negligible.

ETM U 9 Confounding in the 2 k design If 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 U 10 Fractional factorials If a 2 5 design is used for the experiment, its 31 degrees of freedom would be allocated as follows: Using effect hierarchy principle, one would argue that 4 th and 5 th order interactions (and maybe even 3 rd order) are not likely to be important. There are = 16 such effects, half of the total runs! Using a 2 5 design can be wasteful (unless 32 runs cost about the same as 16 runs.) MainInteractions Effects2-Factor3-Factor4-Factor5-Factor

ETM U 11 Example of a factorial Leaf 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 2 5−1 design, or ½ fraction of the 2 5 design.) Level Factor−+ B. high heat temp (deg F) C. heating time (sec.)2325 D. transfer time (sec.)1012 E. hold down time (sec.)23 Q. quench oil temp (deg F)

ETM U 12 Leaf spring example … Create the design (3 replications at each level) in Minitab Insert the data from the examples file. Analyze the data and draw conclusions.