Dataplot Analysis of a 2**3 Full factorial Design: Defective Springs We recommend the following as a bare minimum Dataplot analysis of a 2**3 full factorial design. The analysis consists of the following 7 steps: 1. pre-view the data file 2. read in the data 3. generate an effects plot for main factors 4. generate an effects plot for main factors & 2-term interactions 5. generate a matrix of effects plots 6. do a Yates analysis to estimate effects 7. generate a contour plot of the most important 2 factors Experimental Background You are a manufacturer of metal springs and you wish to reduce the number of defective springs per batch. You suspect that 3 factors are affecting the number of defectives: the temperature of the oven for making the steels, the carbon concentration of the steel, and the temperature of the quenching temperature after the steel is removed from the oven. You run a 2**3 full factorial design (8 runs) in which Y = response variable = % acceptable per batch X1 = oven temperature (1450 and 1600 degress F) X2 = carbon concentration (.5 and .7%) X3 = quenching temperature (70 and 120 degrees F) Your objective of course to determine which of these factors (and interactions) effect the % acceptable springs, and ultimately to determine the settings of the factors which maximize the percent acceptable springs. Dataplot has several hundred auxiliary data files (ASCII) that have been provided for user perusal and which contain a variety of interesting data sets. The names of the available Dataplot auxiliary data files are in the file DATASETS.TEX You can thus scan Dataplot's available auxiliary data file names by entering any of the following 3 equivalent commands: VIEW DATASETS.TEX PREVIEW DATASETS.TEX LIST DATASETS.TEX Hypothetical data from the above 2**3 defective springs experiment has been placed in the Dataplot auxiliary file BOXSPRIN.DAT. This example and data were drawn from the manuscript [Box & Bisgaard, 19xx] 1. Preview the data file You can view the contents of the ASCII data file BOXSPRIN.DAT by entering: VIEW BOXSPRIN.DAT The following will appear on your screen: Note that this ASCII file has 25 lines of non-data header information (most Dataplot auxiliary files have 25 lines of header information) plus 8 lines of data. The 25 header lines describe the contents of the file and how to read ths file into Dataplot. The 8 lines of data consist of 4 data columns: column 1 = response = % acceptable springs per batch column 2 = factor 1 = coded oven temperature column 3 = factor 2 = coded carbon concentration column 4 = factor 3 = coded quench temperature 2. Read in the data You can read the 4 data columns in the file BOXSPRIN.DAT into Dataplot by entering: SKIP 25 ignore the 25 header lines READ BOXSPRIN.DAT Y X1 X2 X3 read the data in This read will be free-format. The default separator of adjacent numbers on a file line is one (or more) blanks. Dataplot will read all data in until down to the end-of-file. The 4 data columns in the file will be read into 4 data columns (= variables = vectors) internal to Dataplot. We have chosen to name these 4 internal variables as Y, X1, X2, and X3. You can name them anything you like by changing the READ command. For example, READ BOXSPRIN.DAT PERCACC OVEN CARB QUENCH would name the 4 internal variables PERCACC, OVEN, CARB, and QUENCH. In Dataplot, characters beyond the first 8 in a name are ignored, thus choose whatever names you like but make them unique within characters 1 to 8. Upon entering the original READ command (READ BOXSPRIN.DAT Y X1 X2 X3) the following feedback message appears on your screen: 3. Generate an effects plot for main factors You can generate an effects plot (main factors only) by entering DEX MEAN PLOT Y X1 X2 X3 The following will appear on your screen: The vertical axis is the mean response. The horizontal axis (1, 2, and 3) is the coded 3 factors: 1 = factor 1 = coded oven temperature 2 = factor 2 = coded carbon concentration 3 = factor 3 = coded quench temperaTURE The line on the plot above "1" connects the mean response (xx) for factor 1 (oven temperature) at the low setting (1450 degrees F), and the mean response (xx) for factor 1 (oven temperature) at the high setting (1600 degrees F). The difference between these 2 mean values is the least squares estimate of the factor 1 effect. In this case, the estimated factor 1 effect is xx-yy = zz. You thus see from the plot that mean response for oven temperature at 1450 degrees = xx mean response for oven temperature at 1600 degrees = xx estimated oven temperature effect = xx mean response for carbon concentration at .5% = xx mean response for carbon concentration at .7% = xx estimated carbon concentration = xx mean response for quench temperature at 70 degrees = xx mean response for quench temperature at 120 degrees = xx estimate quench temperature effect = xx Main effects plots compares the relative importance of the 3 factors and gives you a procedure for constructing a ranked list of the 3 factors. If a line on a main effects plot is steep, the estimated factor effect is large, and the factor is important. If a line on a main effects is shallow, the estimated factor effect is small (near zero), and the factor is unimportant. In this case, you see that 1. factor 1 has the steepest line; 2. factor 2 has the next steepest line; 3. factor 3 has the least steep line; thus 1. factor 1 (oven temperature) is most important; 2. factor 2 (carbon concentration) is next in importance; 3. factor 3 (quench temperature) is least important. The main effects plot also gives you a good method of determining best settings for each of the 3 factors. From the plot, for each factor, you choose the setting which (of course) maximizes the % acceptable springs Your best settings are therefore: 1. factor 1 (oven temperature): 1600 degrees 2. factor 2 (carbon concentration): .5% 3. factor 3 (quench temperature): 120 degrees Other considerations (to be discussed later) may temper our choose for these best settings, but for now these settings are clearly superior in terms of improving our % acceptable springs. 4. Generate an effects plot for main factors & interactions Main effects tell only part of the story in many physical phenomenona; interactions are often also important. Since this experiment involves 3 factors, there are three 2-term interactions: 1. oven temperature x carbon concentration 2. oven temperature x quench temperature 3. carbon concentration x quench temperature and one 3-term interaction: oven temperature x carbon concentration x quench temperature For 2-level experiments such as you ran above, you can extend the above effects plot in a very simple fashion to tell if interactions are important. Since the 2 settings of each of the 3 factors X1, X2 and X3 have been coded as -1 and +1, note that you may form the 2-term interaction variables X1*X2, X1*X3, and X2*X3