Note

Screenshots may differ slightly depending on software version.

# Multilevel Categoric¶

## Introduction – A Case Study on Battery Life¶

Stat-Ease® software offers a “Multilevel Categoric” option, also known as a “general factorial” on the “Factorial” design tab. If you have completed the General One-Factor Multilevel-Categoric Tutorial (recommended), you’ve seen how this option handles one multilevel, categorical factor. In this two-part tutorial you will learn how to set up a design for multiple categorical factors. Part 2 shows you how to convert truly continuous factors, such as temperature, from categorical to numerical. With this you can generate response surface graphs that provide a better perspective of your system.

If you are in a hurry, skip the boxed bits—these are sidebars for those who want to spend more time and explore things.

The experiment in this case, which comes from Montgomery’s Design and Analysis of Experiments, seeks consistently long life in a battery that will be subjected to extremes in ambient conditions. It evaluates three materials (factor A) at three levels of temperature (factor B). Four batteries are tested at each of the nine two-factor combinations in a completely randomized design. The responses from the resulting 36 runs are shown below.

General factorial design on battery life (data in units of hours)

The following questions must be answered:

• How does material type and temperature affect battery life?

• Do any materials provide uniformly long life regardless of temperature?

The big payoff comes if the battery can be made more tolerant to temperature variations in the field.

This case study provides a good example of applying statistical DOE for robust product design. Let’s get started on it!

To build the design, choose File, New Design as shown below (or click the blank-sheet icon () on the toolbar).

Starting a new design

Then from the default Factorial tab, click Multilevel Categoric. Choose 2 as the number of factors. If you are in Horizontal entry mode, change it to Vertical. (The program will remember this the next time you set up a design.)

Selecting number of factors for multilevel-categoric general-factorial design

Enter Material for factor name A (Categoric). Key in the word Type as your Units. Enter the value 3 for the number of levels. Change the treatment names to A1, A2 and A3. Notice that Type in the far left column defaults to Nominal (named) as opposed to ordinal (ordered). This difference in the nature of factors affects how the program codes the categorical levels, which changes the model coefficients reported under ANOVA in the subsequent response analysis. Your design should now appear as shown below.

Entering material as a nominal factor

Note

Help on Factor Types: Tutorials such as this one on general factorials will quickly get you up to speed on how to use Stat-Ease software, but it does not serve as a statistical primer for design and analysis of experiments. If you crave such details, Help is at your fingertips! Just click the help icon () at the top of your screen.

Now enter factor B data by keying in Temperature for factor name B (Categoric), deg F for units, 3 for the number of levels, and 15, 70 and 125 for the levels. Press Nominal, click the arrow on the drop list, then choose Ordinal as shown below. This change from Nominal to Ordinal indicates that although this factor is being treated categorically (for example, due to controls offering only the three levels), temperature is really a continuous factor.

Entering information for factor B

Enter 4 for replicates. The number of runs (36) won’t be updated until you press the Tab key or move from the cell (or if you use the up/down arrows to change the number of replicates). Leave the blocks option alone because these experiments are completely randomized.

Entering the number of replicates

Click Next to move on to the entry screen for responses. Leave the default responses at 1. Enter name as Life and units as hours.

Now we will walk you through a calculation of power – the ability of your experiment to detect meaningful differences in treatments. If you do too few runs and under-power your experiment, an important change in response (the “signal”) will become obscured by normal system/test variation (the “noise”). That would be a waste of time and materials. The program makes the calculation of power easy and puts it upfront in the design-building process so you have a chance to bolster your experiment, if necessary. Let’s assume that battery life must improve by at least 50 hours to be of any interest and that quality control records produce a standard deviation of 30. Enter these values as shown below, Tab (or click) out of 30, and the program then calculates the signal to noise ratio.

Response entry screen

Press Next to see the power of this design for the difference that the engineers hope to detect, at a minimum. It is calculated to be 94.5 % probability of seeing a difference (delta) as small as 50 hours. This exceeds the rule-of-thumb for power of 80 % at a minimum, thus it can be concluded that the planned design will suffice.

Power calculation

Click Finish to complete the design specification process. The program now displays the 36 runs (in random order) from the replicated 3x3 factorial design.

## Analyze the Results¶

To save time, load the experimental results by clicking Help, Tutorial Data -> Battery Life.

This is a good time to preserve your work: Select File and Save As. Change the file name to Battery.dxpx and Save.

Now go to the Analysis branch of the program and click the node labeled R1:Life. This brings up options for applying response transformations.

First step in the analysis - transformation options

Leave the transformation at the default of “None” and press Start Analysis to bring up the Effects tab displayed next in the toolbar for response analysis. The program now provides an initial effect selection and displays it graphically on a specialized statistical plot called a “half-normal”.

Initial effect selection

Note

How the half-normal plot is constructed for general factorial designs: The program displays the absolute value of all effects (plotted as squares) on the bottom axis. The procedure is detailed in a presentation by Patrick Whitcomb on “Graphical Selection of Effects in General Factorials” (2007 Fall Technical Conference co-sponsored by the American Society for Quality and the American Statistical Association) – contact Stat-Ease for a copy.

The program pre-selected two outstanding effects – the main effects of factors A and B. You can, and in this case should, modify the default effect selection. Move your mouse cursor over the unlabeled square and click it. (Note that this goes both ways, that is, you can deselect chosen effects with a simple mouse click.)

Another effect chosen

Interaction AB is now identified. Notice that the program adjusts the line to exclude the chosen effects. You will gain more practice on the use of half-normal plots for picking effects in the Two-Level Factorial Tutorial. It’s best to now press ahead in this case.

Note

Numeric effects list: For statistical details, view the Numeric tab.

Numeric Selection showing Effects

Notice the designation “” for the selected model terms A, B and AB and the “” next to the pure error line in this statistical spreadsheet. You may be wondering why there are so many estimates of pure error. (If not, skip ahead!) Each subgroup of 4 replications provides 3 degrees of freedom (“df”) of pure error. This was done for all 9 factor combinations (3x3) which yields 27 df (= 3*9) in total for estimating pure error.

This screen also provides many features for model selection, which will be covered in tutorials on response surface methods (RSM).

Click the ANOVA tab to see the analysis of variance for this chosen model. If you do not see annotations in blue text as shown below, select View, Annotated ANOVA.

Annotated ANOVA Report

Note

Other details provided under the ANOVA tab: look at the different panes such like Fit Statistics to see post-ANOVA statistics such as R-Squared. As you can conclude for yourself by reading the annotations, the results look good. Further down the report are details of the model based on nominal contrasts. To keep this tutorial moving, it’s best not to get bogged down in the mathematics of modeling categorical factors, so press ahead.

Open the Diagnostics tab and examine the residual graphs. By default you see the normal plot of residuals, which ideally fall more-or-less in line. The pattern here is a bit askew but not badly abnormal, so do not worry.

Normal plot of residuals - looks OK

The tedious, but necessary, model-fitting and statistical validation is now completed, so you are free and clear to finally assess the outcome of the experiment and decide whether any materials provide uniformly long battery-life regardless of temperature.

## Present the Experimental Findings¶

Click the Model Graphs to view the long-awaited results. The program automatically presents the AB interaction plot – identified by the Term window on the floating Factors Tool.

Default model graph - interaction plot with A on bottom (X1) axis

Note

An over-simplified one-factor plot: Choose the One Factor plot view via the Graphs Toolbar. Another way to bring up a one-factor plot (the main effect of A or B, in this case) is by clicking the drop-down menu for the Term selection on the Factors Tool. You will be warned about presenting main effects of factors that interact. This can be very misleading. In this case it will be a mistake to look at either material or temperature effects alone, because the effect of one factor depends on the other. However, while you are at this, explore options on the Factors Tool for B: Temperature. In the Term drop-down menu, select each level and notice how much the effects change due to the interaction. Then select average over from the dropdown, as shown below.

Viewing each Temperature and then its average for the effect plot of Material

Notice how the least significant difference (LSD) bars contract after the averaging. But nevertheless, this is not helpful because it obscures the interaction. On the Graphs Tool press the Interaction plot to bring back the true picture.

Graphs tool - interaction plot selected

Right-click the box next to B:Temperature factor on the Factors Tool and change it to the X1 Axis, thus producing an interaction graph with the ordinal factor displayed in a continuous manner and the nominal factor (material) laid out discretely as separate lines. This makes it easier to interpret your results.

Effect graph with temperature on bottom axis

Note

How the software identifies points: Click the highest point (green) at the upper left of the graph.

Point highlighted for identification

Note how to the left of the plot the software identifies the point by:

• Standard order number (2) and run number (randomized)

• Factor levels “X” (temperature of 15 with material A2).

• Actual response “Y” (188)

The actual results are represented by various-colored circles. If there are multiples, the program displays a number; in this case quite a few labeled “2”. Click these points multiple times to see details on each and every one of them. You can also click on the non-circular symbols (square, triangle or diamond) to display the predicted outcome pairwise comparisons. Try this!

To produce a cleaner looking plot, go to View and deselect Show Legend. Now let’s do some more clean-up for report purposes: Right-click over the graph and select Graph Preferences.

Right-click menu selection for graph preferences

Now under All Graphs turn off (uncheck) the Show design points on graph option, as shown below.

Turning off design points

Press OK

Note

Copy/paste to Microsoft Word or Powerpoint: This is an optional sidetrack on this tutorial: To have your graph look like that shown below for reporting purposes, do the following: Edit, Copy from Stat-Ease, then Paste into Microsoft Word, or right-click and select Export Graph to Word or Export Graph to Powerpoint.

Clean-looking interaction graph

From this graph you can see that all three materials work very well at the low temperature (15 degrees). Based on the overlapping LSD bars, it would be fair to say that no material stands out at this low temperature end of the scale. However, the A1 material clearly falls off at the 70 degree temperature, which would be encountered most often, so it must be rejected. None of the materials perform very well at the highest temperature (125 degrees), but the upper end of the LSD bar for A2 barely overlaps the bottom end of the LSD bar for A3. Therefore, with respect to temperature sensitivity, material A3 may be the most robust material for making batteries.

Finally, if you do have an opportunity to present graphics in color, here’s a dazzling new and easy way to display general factorial effects with Stat-Ease: Click 3D Surface on the Graphs Toolbar. Next, right-click the graph, select Graph Preferences, and turn Show design points back on.

Now place your mouse cursor on the graph – notice that it changes to a hand (). While pressing the left mouse button, spin the graph so the temperature axis is at the bottom.

3D surface plot - rotated slightly for a better view

The 3D view presents a different perspective of the general factorial effects – more on a macro level of the overall experimental landscape. Now the inferiority of material A1 (red bars) becomes obvious: The other two materials tower over it at the mid-temperature of 70 degrees F. Clearly the next step is to eliminate material A1 from contention and perhaps do some further investigations on A2 and A3.