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Unformatted text preview: Stat 511 Midterm II 7 April 2011 1. A large group of ISU chemists and biologists is studying a chemical produced by Echinacea purpurea plants. This chemical is one of the main active chemicals in Echinacea herbal medicines. Measuring the concentraion of the active chemical requires 3 separate steps: ex traction, clarification, and measurment. The data analyzed here was collected by growing 25 plants then separating each plant into three tissues: leaves, flowers, and roots. Each of the 75 biological samples was divided into two parts; each biological part was extracted separately. Each extract was divided into two parts; each extract part was clarified separately. Each clarified sample was measured 2 times. In summary: 25 plants, 75 biological samples, 150 extracts, 300 clarified samples, and 600 measurements. A potential model is: y ijklm = + i + j + ij + ijk + ijkl + ijklm , i { 1 ,..., 25 } identifies plants j { L,F,R } identifies tissue k { 1 , 2 } identifies extract within biological sample l { 1 , 2 } identifies clarified sample within extract m { 1 , 2 } identifies measurement within clarified sample i N (0 , 2 plants ) ij N (0 , 2 biol.samp ) ijk N (0 , 2 extract ) ijkl N (0 , 2 clar.samp ) ijklm N (0 , 2 measurement ) (a) 10 pts. Write out the sources of variation and corresponding degrees of freedom for the ANOVA table corresponding to this model The investigators are not sure about the appropriate model for the random effects. For example, is 2 biol.samp 0 or > 0? One way to evaluate this is to compare a model with the ij term to a model without that term. Similarly, 2 extract may be 0 or > 0 and 2 clar.samp may be 0 or > 0. There are 8 possible models representing all possible combinations of these three variance components. The investigators also consider two models where plant effects ( i ) are considered fixed effects. Each model was fit using REML. AIC was calculated using k = number of random effect parameters. AIC statistics for each of the 10 models are: 1 Stat 511 Midterm II 7 April 2011 Model Plants 2 biol.samp 2 extract 2 clar.samp AIC 1 Random > > > 105.1 2 Random > > 103.2 3 Random > > 110.7 4 Random > 108.9 5 Random > > 115.4 6 Random > 113.5 7 Random > 118.1 8 Random 116.2 9 Fixed > > 98.7 10 Fixed 113.0 (b) 5 pts. Calculate the REML log likelihood for model 1. (c) 10 pts. Based on the data, which of the 10 models is the most reasonable? Explain your choice. (d) 5 pts. Are there any other models that might also be reasonable to consider? If so, which model(s)? Explain your choice(s). 2. The following data are based on a study of the effect of video games on reaction time, defined as the time required for an individual to respond to a stimulus. The two treatments are to play a video game (treated group) or to listen to music from that video game (control group). A subject does one of those two treatments for 10 minutes, then their reaction timegroup)....
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 Spring '08
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