Note

Screenshots may differ slightly depending on software version.

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.

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).

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.)

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.

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.

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.

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.

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.

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

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.

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”.

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.)

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.

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**.

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.

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.

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.

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.

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.

Note

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

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**.

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

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**.

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.

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.