Chapter 2 Planning Experiments

Transcription

1 Some Standard Designs Page 1 of 9 Chapter 2 Planning Experiments Careful planning of an experiment is crucial for good analysis with required precision. 2.2 A Checklist for Planning Experiments A. Define the objectives of the experiment. B. Identify all sources of variation, including ( i ) treatment factors and their levels ( ii ) experiment units ( iii ) blocking factors, noise factors and covariates. C. Choose a rule for assigning experimental units to the treatments. D. Specify the measurements to be made, the experimental procedure, and the anticipated difficulties. E. Run a pilot experiment. F. Specify the model. G. Outline the analysis H. Calculate the number of observations needed. I. Review the above decisions. Revise if necessary.

2 2.3 A Real Experiment ---- Cotton-Spinning Experiment (Robert Peake, 1953, Journal of Applied Statistics) Checklist A. Define the objective of the experiment 1. to investigate how the degrees of twist (measured in turns per inch) affected the breakage rate of the roving, and 2. to compare the ordinary flyer with the newly devised special flyer. A flyer is the rotary guide that produces the twist. B. Identifying all sources of variation.

3 Some Standard Designs Page 3 of 9 (i) Treatment factors and their levels. There are two treatment factors: type of flyer (ordinary and special, coded as 1 and 2), and degrees of twist. Based on a pilot study, 4 unequally spaced levels were selected for the degree of twist, 1.63, 1.69, 1.78, Coding these levels as 1, 2, 3, and 4. The 8 treatment combinations are 11, 12, 13, 14, 21, 22, 23, 24 as shown in the following table: Twist Ordinary Flyer Special From the pilot experiment, treatments 11 and 24 resulted in higher breakage rates. They were not used in the formal experiment. (ii) Experimental units. A unit is the thread on the set of full bobbins in a machine on a given day. (iii) Blocking factors:

4 Apart from the treatment factors, other sources of variation: the different machines, the different operators. One blocking factor was used, i.e., the combination of machine and operator. C. Choose a rule to assign the experimental units to the treatments. A randomized complete block design was selected. The 6 experimental units within each block were randomly assigned to the 6 treatments. D. Specify the measurements to be made, the experimental procedure, and the anticipated difficulties The response variable is the number of breaks per 100 pounds of material. Part of the job of the machine operator was mending every break in the roving so it was easy for the operator to keep a record of every break that occurred. The experiment was to take place during the normal routine. The major difficulties include the length of time involved for each observation, the loss of production time caused by changing the flyers, and the fact that it was not known in advance how many machines would be available for the experiment. E. Run a pilot experiment. F. Specify the model. Breakage rage = constant + effect of treatment combination+ effect of block + error.

5 Some Standard Designs Page 5 of 9 This kind of model will be discussed in Chapter 10. G. Outline the analysis The analysis was planned to compare differences in the breakage rates caused by the 6 treatments. Further, the trend in breakage rates as the degree of twist was increased was of interest for each flyer separately. H. Calculate that number of observations that need to be taken. In order to detect a true difference in breakage rates of at least 2 breaks per 100 pounds with high probability, 56 blocks were needed. The calculation of the required number of blocks will be discussed in section I. Review the above decisions. Revise, if necessary. Since each block could take about a week to observe, it was decided that 56 blocks was impossible. The experimenters decided to analysis the data after 13 blocks had been run. Example. Design an experiment to compare the effects of different colors of exam paper on students performance in an examination. Complete the steps A-D.

6 2.4 Some Standard Designs Completely Randomized Designs It is a design in which the experimental units are assigned to the treatments completely at random, subject only to the number of observations to be taken on each treatment. It is used when an experiment involves no blocks. Model: Response=constant+effect of treatment+error. Factorial experiments often have a large number of treatments. This number may exceed the number of available experimental units, so that only a subset of the treatment combinations can be observed. This is called a fractional factorial design and will be discussed in Chapter 15.

7 Some Standard Designs Page 7 of Block Designs In a block design, the experimenters partition the experimental units into blocks, determine the allocation of treatments to blocks, and assigns the experimental units within each block to the treatments completely at random. Model Response=constant+effect of block+effect of treatment+error. The blocks are treated as a levels of a single blocking factor, even though there are more than one blocking factors. Complete block designs: each treatment is observed the same number of times. If each treatment is observed only once in each block, the design is a randomized complete block design or, simply, randomly block design. Incomplete block design: when the block size

8 (i.e., the number of experimental units in the block) is smaller than the number of treatments, not all treatments can be observed in a block. The design is an incomplete design Designs with Two or More Blocking Factors Crossed factors: all combinations of levels of the factors are used. Nested factors: a particular level of one of the blocking factors occurs at only one level of the other factor. Row-Column Design: a design involving two crossed blocking factors. Nested (or hierarchical) blocking factors: For example, the experiment units may be samples of some experimental material(e.g., cotton) taken from several different batches that are from different suppliers. The samples, which are to be assigned to the treatments, are nested within batches, and the batches are nested within suppliers Split-Plot Designs For example, the yields of three different varieties of soybeans are to be compared under two different levels of

9 Some Standard Designs Page 9 of 9 a fertilizer. Suppose we want 2 observations for each treatment combination. Then 12 plots would be needed. Logistically, it would be easier to have 4 whole plots and randomly assign them to the levels of the fertilizer, then split the whole plot into 3 sub-plots. The 3 sub-plot within each whole plot are randomly assigned to the 3 varieties. This is a split-plot design. Note there is a two-stage of randomization: First, levels of fertilizer are randomly assigned to the whole plots; second, the levels of variety are randomly assigned to the sub-plot within each whole plot.