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Stat 232 Experimental Design Spring 2008

Stat 232 Experimental Design Spring 2008. Ching-Shui Cheng Office:  419 Evans Hall Phone:  642-9968 Email: cheng@stat.berkeley.edu Office Hours: Tu Th 2:00-3:00 and by appointment. Course webpage: http://www.stat.berkeley.edu/~cheng/232.htm. No textbook

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Stat 232 Experimental Design Spring 2008

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  1. Stat 232 Experimental Design Spring 2008

  2. Ching-Shui Cheng Office:  419 Evans HallPhone:  642-9968Email: cheng@stat.berkeley.edu Office Hours: Tu Th 2:00-3:00 and by appointment

  3. Course webpage: http://www.stat.berkeley.edu/~cheng/232.htm

  4. No textbook Recommended (for first half of the course): Design of Comparative Exeperimentsby R. A. Bailey, to appear in 2008 http://www.maths.qmul.ac.uk/~rab/DOEbook/ Experiments: Planning, Analysis, and Parameter Design Optimization by C. F. J. Wu and M. Hamada Statistics for Experimenters: Design, Innovation and Discovery by Box, Hunter and Hunter A useful software: GenStat

  5. Experimental Design Planning of experiments to produce valid information as efficiently as possible

  6. Comparative Experiments • Treatments (varieties) Varieties of grain, fertilizers, drugs, …. • Experimental units (plots): smallest division of the experimental material so that different units can receive different treatments Plots, patients, ….

  7. Design: How to assign the treatments to the experimental units Fundamental difficulty: variability among the units; no two units are exactly the same. Each unit can be assigned only one treatment. Different responses may be observed even if the same treatment is assigned to the units. Systematic assignments may lead to bias.

  8. R. A. Fisher worked at the Rothamsted Experimental Station in the United Kingdom to evaluate the success of various fertilizer treatments.

  9. Fisher found the data from experiments going on for decades to be basically worthless because of poor experimental design. Fertilizer had been applied to a field one year and not in another in order to compare the yield of grain produced in the two years. BUT It may have rained more, or been sunnier, in different years. The seeds used may have differed between years as well. Or fertilizer was applied to one field and not to a nearby field in the same year. BUT The fields might have different soil, water, drainage, and history of previous use.  Too many factors affecting the results were “uncontrolled.”

  10. Fisher’s solution: Randomization • In the same field and same year, apply fertilizer to randomly spaced plots within the field. • This averages out the effect of variation within the field in drainage and soil composition on yield, as well as controlling for weather, etc.

  11. Randomization prevents any particular treatment from receiving more than its fair share of better units, thereby eliminating potential systematic bias. Some treatments may still get lucky, but if we assign many units to each treatment, then the effects of chance will average out. Replications In addition to guarding against potential systematic biases, randomization also provides a basis for doing statistical inference. (Randomization model)

  12. Start with an initial design Randomly permute (labels of) the experimental units Complete randomization: Pick one of the 72! Permutations randomly

  13. 4 treatments Pick one of the 72! Permutations randomly Completely randomized design

  14. blocking A disadvantage of complete randomization is that when variations among the experimental units are large, the treatment comparisons do not have good precision. Blocking is an effective way to reduce experimental error. The experimental units are divided into more homogeneous groups called blocks. Better precision can be achieved by comparing the treatments within blocks.

  15. After randomization: Randomized complete block design

  16. Wine tasting Four wines are tasted and evaluated by each of eight judges. A unit is one tasting by one judge; judges are blocks. So there are eight blocks and 32 units. Units within each judge are identified by order of tasting.

  17. Block what you can and randomize what you cannot.

  18. Randomization • Blocking • Replication

  19. Incomplete block design 7 treatments

  20. Each of ten housewives does four washloads in an experiment to compare five new detergents. 5 treatments and 10 blocks of size 4.

  21. Incomplete block design 7 treatments

  22. Incomplete block design Balanced incomplete block design Randomize by randomly permuting the block labels and independently permuting the unit labels within each block.

  23. Two simple block (unit) structures • Nesting block/unit • Crossing row ＊ column

  24. Two simple block structures • Nesting block/unit • Crossing row ＊ column Latin square

  25. Wine tasting

  26. Simple block structures Iterated crossing and nesting • cover most, though not all block structures encountered in practice Nelder (1965)

  27. Consumer testing A consumer organization wishes to compare 8 brands of vacuum cleaner. There is one sample for each brand. Each of four housewives tests two cleaners in her home for a week. To allow for housewife effects, each housewife tests each cleaner and therefore takes part in the trial for 4 weeks. 8 treatments Block structure:

  28. Trojan square

  29. Treatment structures • No structure • Treatments vs. control • Factorial structure A fertilizer may be a combination of three factors (variables) N (nitrogen), P (Phosphate), K (Potassium)

  30. Treatment structure Block structure (unit structure) Design Randomization Analysis

  31. Choice of design • Efficiency • Combinatorial considerations • Practical considerations

  32. McLeod and Brewster (2004) Technometrics A company was experiencing problems with one of its chrome-plating processes in that when a particular complex-shaped part was being plated, excessive pitting and cracking, as well as poor adhesion and uneven deposition of chrome across the part, were observed. With the goal being the identification of key factors affecting the quality of the process, a screening experiment was planned. In collaboration with the company’s process engineers, six factors were identified for consideration in the experiment.

  33. Hard-to-vary treatment factors A: chrome concentration B: Chrome to sulfate ratio C: bath temperature Easy-to-vary treatment factors p: etching current density q: plating current density r: part geometry

  34. The responses included the numbers of pits and cracks, in addition to hardness and thickness readings at various locations on the part. Suppose each of the six factors have two levels, then there are 64 treatments. Acomplete factorial design needs 64 experimental runs

  35. Block structure: 4 weeks/4 days/2 runs Treatment structure: A ＊ B ＊ C ＊ p ＊ q ＊ r Each of the six factors has two levels Fractional factorial design

  36. Miller (1997) Technometrics Experimental objective: Investigate methods of reducing the wrinkling of clothes being laundered

  37. Miller (1997) The experiment is run in 2 blocks and employs 4 washers and 4 driers. Sets of cloth samples are run through the washers and the samples are divided into groups such that each group contains exactly one sample from each washer. Each group of samples is then assigned to one of the driers. Once dried, the extent of wrinkling on each sample is evaluated.

  38. Treatment structure: A, B, C, D, E, F: configurations of washersa,b,c,d: configurations of dryers

  39. Block structure:2 blocks/(4 washers＊4 dryers)

  40. Block 1 Block 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 1 0 0 1 1 0 1 1 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 0 0 1 0 0 1 1 0 1 1 0 1 0 0 0 0 1 0 1 0 1 0 0 1 1 0 1 0 1 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 1 0 0 0 0 0 1 1 0 0 0 1 0 1 1 0 1 1 0 1 1 0 0 0 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 1 1 0 0 0 1 1 0 0 1

  41. GenStat code factor [nvalue=32;levels=2] block,A,B,C,D,E,F,a,b,c,d & [levels=4] wash, dryer generate block,wash,dryer blockstructure block/(wash*dryer) treatmentstructure (A+B+C+D+E+F)*(A+B+C+D+E+F) +(a+b+c+d)*(a+b+c+d) +(A+B+C+D+E+F)*(a+b+c+d)

  42. matrix [rows=10; columns=5; values=“ b r1 r2 c1 c2" 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0]Mkey

  43. Akey [blockfactors=block,wash,dryer; Key=Mkey; rowprimes=!(10(2));colprimes=!(5(2)); colmappings=!(1,2,2,3,3)] Pdesign Arandom [blocks=block/(wash*dryer);seed=12345] PDESIGN ANOVA

  44. Outline • Introduction; randomization and blocking • Some mathematical preliminaries • Linear models • Block structures; strata, null ANOVA • Computation of estimates; ANOVA table • Orthogonal designs • Non-orthogonal designs • Factorial designs • Response surface methodology • Other topics as time permits

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