Wuchap_03a_09fall - Unit 3: Experiments with More Than One...

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Unformatted text preview: Unit 3: Experiments with More Than One Factor Sources : Chapter 3. • Paired comparison design (Section 3.1). • Randomized block design (Section 3.2). • Two-way and multi-way layout with fixed effects (Sections 3.3 and 3.5). • Latin and Graeco-Latin square design (Sections 3.6 and 3.7). • Balanced incomplete block design (Section 3.8). • Split-plot design (Section 3.9). • Analysis of covariance (ANCOVA) (Section 3.10). • Transformation of response (Section 3.11). 1 Sewage Experiment • Objective : To compare two methods MSI and SIB for determining chlorine content in sewage effluents; y = residual chlorine reading. Table 1: Residual Chlorine Readings, Sewage Experiment Method Sample MSI SIB d i 1 0.39 0.36- 0.03 2 0.84 1.35 0.51 3 1.76 2.56 0.80 4 3.35 3.92 0.57 5 4.69 5.35 0.66 6 7.70 8.33 0.63 7 10.52 10.70 0.18 8 10.92 10.91- 0.01 • Experimental Design : Eight samples were collected at different doses and contact times. Two methods were applied to each of the eight samples. It is a paired comparison design because the pair of treatments are applied to the same samples (or units). 2 Paired Comparison Design vs. Unpaired Design • Paired Comparison Design : Two treatments are randomly assigned to each block of two units. Can eliminate block-to-block variation and is effective if such variation is large. Examples : pairs of twins, eyes, kidneys, left and right feet. (Subject-to-subject variation much larger than within-subject variation). • Unpaired Design : Each treatment is applied to a separate set of units, or called the two-sample problem. Useful if pairing is unnecessary; also it has more degrees of freedom for error estimation (see page 5). 3 Paired t tests • Paired t test : Let y i 1 ,y i 2 be the responses of treatments 1 and 2 for unit i,i = 1 ,...N . Let d i = y i 2- y i 1 , ¯ d and s 2 d the sample mean and variance of d i . t paired = √ N ¯ d/s d The two treatments are declared significantly different at level α if | t paired | > t N- 1 ,α/ 2 . (1) 4 Unpaired t tests • Unpaired t test : The unpaired t test is appropriate if we randomly choose N of the 2 N units to receive one treatment and assign the remaining N units to the second treatment. Let ¯ y i and s 2 i be the sample mean and sample variance for the i th treatment, i = 1 and 2. Define t unpaired = ( ¯ y 2- ¯ y 1 ) / radicalBig ( s 2 2 /N ) + ( s 2 1 /N ) . The two treatments are declared significantly different at level α if | t unpaired | > t 2 N- 2 ,α/ 2 . (2) Note that the degrees of freedom in (1) and (2) are N- 1 and 2 N- 2 respectively. The unpaired t test has more df’s but make sure that the unit-to-unit variation is under control (if this method is to be used). 5 Analysis Results : t tests t paired = . 4138 . 321 / √ 8 = . 4138 . 1135 = 3 . 645 , t unpaired = 5 . 435- 5 . 0212 radicalbig (17 . 811 + 17 . 012) / 8 = . 4138 2 . 0863 = 0 . 198 . The p values are Prob ( | t 7 | > 3 . 645) = . 008 , Prob ( | t 14 | > . 198) = . 848 .....
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This note was uploaded on 12/12/2010 for the course STAT 425 taught by Professor Ma,p during the Fall '08 term at University of Illinois, Urbana Champaign.

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Wuchap_03a_09fall - Unit 3: Experiments with More Than One...

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