Homework 02_ANOVA-1

Homework #2: Analysis of Variance Goal: perform an Analysis of Variance on a variety of datasets and interpret the ANOVA statistics. Instructions: Follow the instructions provided, filling in your answers as you go. Make sure the text of your answers is noticeably different from black Times New Roman 12 pt font (that’s what this is), so that your answers stand out. Be sure to include copies of any graphs you are asked to make. Steps that have an R output that you’ll need to copy/paste* are denoted with the  symbol. As you work, you are allowed to discuss the homework with your peers, but note that you must complete all of the steps independently and answer all of the questions in your own words . Submitting work that was completed by another person is a violation of the academic honesty policy of UC Merced. * Note: after you copy/paste, change the font to “ Courier New ”; this is a fixed-width font and will keep the formatting the same as used in R. Part A – Performing an ANOVA by hand - Ecology A plant ecologist collected data regarding the height of plant species X from four different fields. Each field contained a unique type of soil, which the plants grew in. This ecologist measured the height (in inches) of three plants in each of the fields. All four fields were contained in an area of two square miles. The ecologist’s results are tabulated below: 1. What is the null hypothesis that this ecologist can address? - The ecologist can address from the given chart is that there are no differences in growth in different soils. 2. To determine whether there is a significant difference in the height of plants grown in different soils, you will perform an ANOVA analysis. Fill in the following table to help you get started. Note that MY, MF, and FY are just the differences between the values; for example, MY is the difference between the plant height (Y) and the grand mean (M). You can see Chapter 1 in the Grafen & Hails textbook for an example of filling in a table like this. An excel spreadsheet is a convenient format to do these calculations in (a template is provided on CatCourses), but you can also just use a calculator or R to complete this table. Plant Height (Y) Field Grand Mean (M) Treatment Mean (F) MY MF FY 11 A 15.75 11 -.75 -4.75 0 13 A 15.75 11 -2.75 -4.75 2 9 A 15.75 11 -6.75 -4.75 -2 16 B 15.75 20.3 0.25 4.55 -4.3 22 B 15.75 20.3 16.25 4.55 1.7 23 B 15.75 20.3 7.25 4.55 2.7 19 C 15.75 17.6 3.25 1.92 1.33 1 Fields A B C D 11 16 19 13 13 22 13 14 9 23 21 15

13 C 15.75 17.6 -2.75 1.92 -4.67 21 C 15.75 17.6 5.75 1.92 3.33 13 D 15.75 14 -2.75 -1.75 -1 14 D 15.75 14 -1.75 -1.75 0 15 D 15.75 14 -0.75 -1.75 1 Sum of Square (SS) 224.25 150.92 73.77 Degree of Freedom (DF) 11 3 8 3. Calculate the F-ratio. (Be sure to show your calculation setup) F-Ratio = SS DF 150.92 3 = ¿ 50.31 73.77 3 = ¿ 9.17 F-Ratio = 50.31 9.17 F-Ratio = 5.486 4. By using the calculated F-ratio and provided F-table (95% confidence interval), interpret your analysis. We can reject the null hypothesis as a 95% confidence interval since one of the means is different. Part B – Performing an ANOVA in R – Effect of ventilation on blood folate levels Several studies have been published to show that nitrous oxide used in anesthetics can reduce patient blood folate levels. This dataset gives measured levels of folate in red blood cells in patients who received three different methods of ventilation while under anesthesia. The first column in the dataframe, ventilation , is a categorical variable (or “factor”) containing one of the three ventilation methods for each patient: Factor levels Treatment 'N2O+O2, 24h' 50% nitrous oxide and 50% oxygen, continuously for 24 hours 'N2O+O2, op' 50% nitrous oxide and 50% oxygen, only during operation 'O2, 24h' No nitrous oxide, but 35-50% oxygen for 24 hours. The second column in the dataframe, is a numerical variable, folate , which is the folate concentration (μg/l) for each patient. [This dataset is from the “ISwR” library of datasets available in the ISwR package at www.r-project.org .] Procedure 1. First, change the directory in R (under the File menu) to wherever you saved the “red_cell” file. 2. Then read in this data set using the load() command in R . Type the following command: load("red_cell.RData") 2

Alternatively, you can have R open a file browser window as follows: load(file.choose()) 3. Once you’ve read in the data set, you can access the two data vectors in this data frame using the “$” command (i.e. red_cell$ventilation and red_cell$folate) or you can make the individual data vectors available using the command: attach(red_cell) After this command you can access the data vectors just as ventilation and folate  4. Make boxplots and stripcharts of this dataset using the following commands: boxplot(folate~ventilation) stripchart(folate~ventilation) 3