Experimental Designs in Statistics: A Systematic Approach with Examples - Prof. Jiwei Zhao, Assignments of Data Analysis & Statistical Methods

Download Experimental Designs in Statistics: A Systematic Approach with Examples - Prof. Jiwei Zhao and more Assignments Data Analysis & Statistical Methods in PDF only on Docsity! STAT572: EXPERIMENTAL DESIGNS Prepared by YoungDeok Hwang Department of Statistics, UW-Madison March 29, 2009 1 Experiment: The tea tasting lady The lady of the Rothamsted Experiment Station staff claimed that she could discriminate between a cup of tea made with milk and one with tea added first. Fishers experiment design consisted of making 8 cups of tea with 4 made in one way and 4 in the other. The lady was told of this structure. The 8 cups are presented to the lady in random order and she has to partition the 8 cups into two sets of 4. The interpretation will be made on the basis that there are 70 partitions. So, if the assignment was random, the probability of the lady obtaining the correct partition is 1/70, if she can not discriminate. Because 1/70 is a small probability, it is rational to conclude that if she obtains the correct partition she has given evidence in favor of her claim. (R. A. Fisher, The Design of Experiments, 1935.) 2 Definitions  Experiment : an operation or procedure carried out under controlled conditions in order to discover an unknown effect or law, to test or establish a hypothesis, or to illustrate a known law (Merriam-Webster).  Factor : variable whose influence upon a response variable is being studied in the experiment.  Factor Level : numerical values or settings for a factor.  Trial (or run ) : application of a treatment to an experimental unit.  Treatment or level combination : set of values for all factors in a trial.  Experimental unit : object to which a treatment is applied.  Design : a set of carefully chosen factor values used for conducting the experiment. 3 A Systematic Approach to Experimentation  State the objective of the study.  Choose the response variable . . . should correspond to the purpose of the study.  Nominal-the-best, larger-the-better or smaller-the-better.  Choose factors and levels.  Use flow chart or cause-and-effect diagram (see Figure 1).  Choose experimental design (i.e., plan).  Perform the experiment (use a planning matrix to determine the set of treatments and the order to be run).  Analyze data (design should be selected to meet objective so that the analysis is efficient and easy).  Draw conclusions. 4 Fundamental Principles:Replication  Each treatment is applied to units that are representative of the population (example : measurements of 3 units vs. 3 repeated measurements of 1 unit).  Replication vs Repetition (i.e., repeated measurements).  Enable the estimation of experimental error. Use sample standard deviation. 5 Fundamental Principles:Randomization  Use of a chance mechanism (e.g., random number generators) to assign treatments to units or to run order. It has the following advantages.  Protect against latent variables or lurking variables.  Reduce influence of subjective bias in treatment assignments (e.g., clinical trials). 6 Fundamental Principles:Blocking  A block refers to a collection of homogeneous units. Effective blocking : larger between-block variations than within-block variations. (Examples: hours, batches, lots, street blocks, pairs of twins.)  Run and compare treatments within the same blocks. (Use randomization within blocks.) It can eliminate block-block variation and reduce variability of treatment effects estimates.  Block what you can and randomize what you cannot. 7 Full Factorial Design  Goal: Explore the possible combinations of settings effectively. Successfully investigate the effects of two or more factors simultaneously.  2k Design 1. k factors: Total k factors are involved in the experiment. 2. 2 levels: We assume two levels of treatment for each factor. 3. 2k Design: So we will observe at least 2 × 2 × · · · × 2 = 2k observations. 8 23 Design 9 Epitaxial Layer Growth Experiment  An AT&T experiment based on 24 design; four factors each at two levels. There are 16 (=24) level combinations. Table: Factors and Levels, Layer Growth Experiment Level Factor − + A. susceptor-rotation method continuous oscillating B. code of wafers 668G4 678D4 C . deposition temperature(◦C) 1210 1220 D. deposition time short long 10 Main Effects plot • • • • • • • • 13 .8 14 .0 14 .2 14 .4 - A + - B + - C + - D + 11 Interaction Effects plot B 13 .6 13 .8 14 .0 14 .2 14 .4 14 .6 - + A-- A+ AB C 13 .6 13 .8 14 .0 14 .2 14 .4 14 .6 - + A-- A+ AC D 13 .6 13 .8 14 .0 14 .2 14 .4 14 .6 - + A-- A+ AD C 13 .6 13 .8 14 .0 14 .2 14 .4 14 .6 - + B-- B+ BC D 13 .6 13 .8 14 .0 14 .2 14 .4 14 .6 - + B-- B+ BD D 13 .6 13 .8 14 .0 14 .2 14 .4 14 .6 - + C-- C+ CD 12