Effects of Fertilizers on Tomato Plant Height: One-Way ANOVA Model - Prof. Spencer Muse, Lab Reports of Data Analysis & Statistical Methods

Download Effects of Fertilizers on Tomato Plant Height: One-Way ANOVA Model - Prof. Spencer Muse and more Lab Reports Data Analysis & Statistical Methods in PDF only on Docsity! Today, we will talk about the one-way ANOVA model. This model is used to look at the effects of different treatments on a dependent variable. For example, in the data set we will look at today, we will look at the effects of different fertilizers on the height of tomato plants one month after planting. Since we are looking at plants grown five different types of fertilizers, we cannot use our simple linear regression. Consider if we tried: our parameter estimates would be an intercept (sounds reasonable enough…) and a slope term, describing the change in plant height per change in unit of fertilizer type. But this doesn’t really make sense, as there is no “unit” with which to measure Fertilizer A or Fertilizer B; they’re simply different types of fertilizer. Rather, they are class variables. Therefore, we use what is called the one-way ANOVA model: (Note that this model looks slightly different from the one presented in the text. The simple conversion: will give you the model from the text.) In this model, there are three terms. Here, “ ” represents the overall population mean, while the ‘s represent the differences between the sub- population means (the means for the different treatment groups) and the overall population mean, while the ‘s are the error terms. There are different treatment groups and observations in the treatment group. Thus, the overall sample size is: For this dataset, for all treatment groups, so : A 39.6 34.3 38.5 42 B 37.3 42.6 40.8 44.8 C 15.8 16.2 21 18.3 D 31.6 27.4 28.3 29 E 39.2 42.8 35 34.2 Here, we see our five treatment groups, A, B, C, D, and E arranged in a column with four observations for each group in a row. You’ll note that this data is formatted differently from what we are accustomed to. Therefore, we can either re-arrange the data (tedious and often times impossible for very large data sets) or we can use a different method for inputing the data. Below is SAS code for a DO-loop which will correctly read in our data: options formdlim="_"; DATA tomato; INPUT fertilizer $ @; DO rep=1 TO 4; INPUT y @; OUTPUT; END; DATALINES; A 39.6 34.3 38.5 42 B 37.3 42.6 40.8 44.8 C 15.8 16.2 21 18.3 D 31.6 27.4 28.3 29 E 39.2 42.8 35 34.2 ; RUN; TITLE 'fertilizer data'; proc print;run; The way the DO-loop works is, after the INPUT statement, SAS reads in the “fertilizer” column, and then stops reading (because of the “@” symbol). Then for “rep” (repetition) 1 through 4, it reads the value for “y” and stops the cursor. The OUTPUT statement tells SAS to go ahead and put the data row (with three values, one for “fertilizer”, one for “rep” and one for “y”) into the data file we’re creating. You can see the results with the proc print statement. As always, we’ll start by plotting our data. Use the following code to create a scatter plot for the data. title 'scatter plot of fertilizers'; proc gplot; plot y*fertilizer; run; While useful, the scatter plot may not be the clearest way to display this type of data, especially when dealing with larger data sets. Below is the code for a side-by-side box plot: title 'side-by-side box plot of fertilizers'; proc boxplot; plot y*fertilizer/ boxstyle=skeletal boxconnect=mean cboxes=green cboxfill=brown boxwidth=10; run; Despite the hideous color scheme, (you can specify your own, more aestetically pleasing choices if you wish) this box plot gives us a clear picture of where each sample lies. Now that we have plotted our data, it’s time to perform our ANOVA analysis. Some of the things we may be looking for when performing a one-way ANOVA include estimates for the effects of the different treatments (parameter estimates), a test for whether or not there are statistically significant effects from different treatments, or perhaps estimates for the differences in dependent variable between different treatment groups. While you can use PROC ANOVA to do most of the analysis we’re performing today, we will be using PROC GLM, which is a more powerful and versatile procedure. (The actual coding for the two is mostly the same, but with GLM, you have more options and will be able to do more analysis in the future.) Using the following code, perform a one-way ANOVA test on our data: proc glm data=tomato;