Factorial Design Worksheet
This worksheet will introduce you to a couple exploratory data analysis (EDA) features in Minitab. The EDA methods use graphics and interaction with graphics to uncover structure in a factorial experiment. After using these methods, you should have a good idea as to whether or not main effects or interaction effects are present in your data. You should also have some insight into the nature of these effects. The data we will be using is cited in the XLISPSTAT reference and manual and is taken from Devore and Peck. The response is yield of tomato plants, factor A is the planting density (4 levels) and factor B is the tomato variety (3 levels). Enter the data using the set command:
set c1 3(1:4)3 set c2 (1:3)12 set c3 7.9 9.2 10.5 11.2 12.8 13.3 12.1 12.6 14.0 9.1 10.8 12.5 8.1 8.6 10.1 11.5 12.7 13.7 13.7 14.4 15.5 11.3 12.5 14.5 15.3 16.1 17.5 16.6 18.5 19.2 18.0 20.8 21 17.2 18.4 18.9 end
Be sure to name the columns. This experiment has 3 replications for each treatment. In order to produce an interaction plot, we need to find the means for each treatment. The simplest way to do this in Minitab is to pull down the Statistics menu, choose ANOVA and then Interactions Plot. Select C1 and C2 as factors in the Factors dialog box, click the button next to Raw Response Data in: and select C3 as the response. Summarize the effect of planting density. Does interaction seem to be present? Summarize the effect of each factor. We'll discuss this data in class.
The next method gives a little more insight into the data, though we use it for essentially the same purpose-to gain an understanding of the magnitude and direction of main effects and interaction effects. Pull down the Graph menu and select Matrix Plot. Enter C1-C3 as the graph variables and press OK. A hard-to-read matrix plot will appear on screen-maximize it. The three plots in the lower diagonal are just the transposes of the three plots in the upper diagonal; we will work with the plots in the upper diagonal. The plot of C1 vs. C2 doesn't look very interesting since you're only plotting factor levels against factor levels, but this is actually the window we will use when brushing. Notice in the plot of C1 vs. C2 that there are four rows corresponding to the 4 planting densities of C1 and three columns corresponding to the three varieties of C2. We will use a brush to highlight rows (levels of C1) or columns (levels of C2) of this plot.
Choose Brush under the Editor menu or click on the brush icon on the menu bar. A window that displays the rows of the data set being brushed will open in the upper left-hand corner of the screen. The cursor should appear as a pointing hand when in the graphics window. Position the pointer so that you can highlight the first column (level 1 of C2) of the plot with a narrow vertical box. Draw the box by depressing and holding down the right-hand button of the mouse while tracing around the four data points. Now look at the plot of C1 vs. C3; the points in the box will be highlighted in color. If they are not highlighted, then the highlight color has been set to black. Under the Editor menu, choose Set Brushing Color and select a color of your choice. What can you say about the plot of C1 vs. C3 for the observations at level 1 of C2? When a box has been traced, an open hand replaces the pointing hand. Use the open hand to drag the box over to levels 2 and 3 of C2. Does the pattern of highlighted points maintain its shape? Its position? What are the implications in terms of interaction between planting variety and yield?
Repeat this exercise exchanging the roles of C1 and C2-you can trace a new box (and make the old one disappear) simply by depressing the right-hand button of the mouse. The box highlighting the levels of C1 will be short and wide instead of tall and narrow.