One-Factor Categoric¶

Enter the Design Parameters¶

Leave the number of factors at its default level of 1. Enter Bowler as the name of the factor. Tab to the Units field and enter Person. Leave Type at its default of Nominal. Set the Levels field to 3. Enter Pat, Mark, and Shari for the names of levels 1-3.

Categoric factor entry – Vertical table orientation selected¶

In the Replicates field type 6 (each bowler rolls six games). Tab to the “Assign one block per replicate” field but leave it unchecked. The program now recalculates the number of runs for this experiment: 18.

Design options entered¶

Press Next. Let’s do the easy things first. Leave the number of Responses at the default of 1. Now click on the Name box and enter Score. Tab to the Units field and enter Pins.

Response name dialog box - completed¶

At this stage you can skip the remainder of the fields and continue on. However, it is good to gain an assessment of the power of your planned experiment. In this case, as shown in the fields below, enter the value 20 for the signal because the bowling captain does not care if averages differ by fewer than 20 pins. Then enter the value 10 for standard deviation (derived from league records as the variability of a typical bowler). The program then computes a signal-to-noise ratio of 2 (20 divided by 10).

Optional power calculator - necessary inputs entered¶

Press Next to view the happy outcome – power that exceeds 80 percent probability of seeing the desired difference.

Results of power calculation¶

Click on Finish to create the design and take you to the design layout window.

Note

Explore the program interface: Before moving on, take a look at the unique branching interface provided by Stat-Ease 360 for the design and analysis of experiments and resulting optimization.

Stat-Ease 360’s easy-to-use branching interface (your run order may differ due to randomization)¶

You will explore some branches in this series of tutorials and others if you progress to more advanced features, such as response surface methods for process optimization.

Save the Design¶

When you complete the design setup, save it to a file by selecting File, Save As…. Type in the name of your choice (for this tutorial, we suggest Bowling) for your data file, which is saved as a *.dxpx type.

Save As dialog box¶

Click on Save.

Examine the Analysis¶

By necessity, the tutorial now turns a bit statistical. If this becomes intimidating, we recommend you attend a basic class on regression, or better yet, a DOE workshop such as Stat-Ease’s computer-intensive Experiment Design Made Easy.

The program now pops up a very specialized plot that highlights factor A—the bowlers—as an emergent effect relative to the statistical error (that is, normal variation) shown by the line of green triangles.

Initial view of the effect of Bowler¶

That is good! It supports what was obvious from the raw results—who bowls does matter.

To get the statistical details, press the ANOVA (Analysis of Variance) tab. Notice to the far right side of the ANOVA table that Stat-Ease 360 verifies that the results are significant.

ANOVA results (annotated), with context-sensitive Help enabled via right-click menu¶

Now, look to the right side of your screen at the Fit Statistics pane to see various summary statistics.

Summary statistics¶

Below the Fit Statistics you will find the Coefficients pane.

Coefficient estimates¶

Here you see statistical details such as coefficient estimates for each model term and their confidence intervals (“CI”). The intercept in this simple one-factor comparative experiment is just the overall mean score of the three bowlers. You may wonder why only two terms, A1 and A2, are provided for a predictive model on three bowlers. It turns out that the last model term, A3, is superfluous because it can be inferred once you know the mean plus the averages of the other two bowlers.

Now let’s move on to the next section within this screen: “Treatment Means.” Click the Treatment Means tab in the ANOVA pane.

Treatment means¶

Here are the averages for each of the three bowlers. Now click the Treatment Contrasts tab and you’ll see these compared via pair-wise t-tests.

Treatment contrasts¶

You can conclude from the treatment comparisons that:

• Pat differs significantly (24.67 pins worse!) when compared with Mark (1 vs 2)

• The 2.5 pins mean difference between Pat and Shari (1 vs 3) is not significant (nor is it considered important by the bowling team’s captain – recall in the design specification for power that a 10-pin difference was the minimum of interest)

• Mark differs significantly (22.17 pins better!) when compared with Shari (2 vs 3).

Analyze Residuals¶

Click the Diagnostics tab to bring up the diagnostic plots. In the layout toolbar, select the single split icon to maximize the normal plot of residuals.

Ideally this will be a straight line, indicating no outlying abnormalities.

Notice that the points are coded by color to the level of response they represent – going from cool blue for lowest values to hot red for the highest. In this example, the red point is Mark’s outstanding 195 game. Pat and Shari think Mark’s 195 game should be thrown out because it’s too high. Is this fair? Click this point so it will be selected on this and all the other residual graphs on the Diagnostics Tool (choose how many graphs are displayed at once via the blue layout icons above the Diagnostics tab).

Normal probability plot of residuals (195 game highlighted)¶

Note

The Diagnostics tool dropdown: Notice on the Diagnostics Tool that they are “studentized” by default. This converts raw residuals, reported in original units (‘pins’ of bowling in this example), to dimensionless numbers based on standard deviations, which come out in plus or minus scale. More details on studentization reside in the Help. Raw residuals can be displayed by choosing it off the down-list on the Diagnostics Tool shown below. Check it out!

Other ways to display residuals¶

In any case, when runs have greater leverage (another statistical term to look up in the Help), only the Studentized form of residuals produces valid diagnostic graphs. For example, if Pat and Shari succeed in getting Mark’s high game thrown out (don’t worry – they won’t!), then each of Mark’s remaining five games will exhibit a leverage of 0.2 (1/5) versus 0.167 (1/6) for each of the others’ six games. Due to potential imbalances of this sort, we advise that you always leave the Studentized feature checked (as done by default). So if you are on Residuals now, go back to the original choice that came up by default (externally* studentized).

* P.S. Another aspect of how Stat-Ease 360 displays residuals by default is them being done “externally”. This is explored in the Two-Level Factorial Tutorial For now, suffice it to say that the program chooses this form of residual to provide greater sensitivity to statistical outliers. This makes it even more compelling not to throw out Mark’s high game.

Now select the Resid. vs. Pred. tab to view a plot of residuals for each individual game versus what is predicted by the response model.

Let’s make it easier to see which residual goes with which bowler by pressing the down-list arrow for the Color by option in the Diagnostics Tool and selecting A:Bowler.

Residuals versus predicted values, colored by bowler¶

The size of the studentized residual should be independent of its predicted value. In other words, the vertical spread of the studentized residuals should be approximately the same for each bowler. In this case the plot looks OK. Don’t be alarmed that Mark’s games stand out as a whole. The spread from bottom-to-top is not out of line with his competitors, despite their protestations about the highest score (still highlighted).

Bring up the next graph on the list – Resid. vs Run (residuals versus run number).

Residuals versus run chart¶

Here you might see trends due to changing alley conditions (the lane re-oiling, for example), bowler fatigue, or other time-related lurking variables.

More importantly in this case, all points fall within the limits (calculated at the 95 percent confidence level). In other words, Mark’s high game does not exhibit anything more than common-cause variability, so it should not be disqualified.

View the Means and Data Plot¶

Select the Model Graphs tab to continue the analysis and display a plot containing all the response data and the average value at each level of the treatment (factor). This plot gives an excellent overview of the data and the effect of the factor levels on the mean and spread of the response. Note how conveniently the program scaled the Y axis from 140 to 200 pins in increments of 10.

One-factor effects graph with Mark’s scores in the middle¶

The squares in this effects plot represent predicted responses for each factor level (bowler). Vertical ‘I-beam-shaped’ bars represent the 95% least significant difference (LSD) intervals for each treatment. Mark’s LSD bars don’t overlap horizontally with Pat’s or Shari’s, so with at least 95% confidence, Mark’s mean is significantly higher than the means of the other two bowlers.

Pat and Shari’s LSD bars overlap horizontally, so we can’t say which of them bowls better. It seems they must spend a year in a minor bowling league and see if a year’s worth of games reveals a significant difference in ability. Meanwhile, Mark will be trying to live up to the high average he exhibited in the tryouts and thus justify being chosen for the Stat-Ease bowling team.

That’s it for now. Save your results by going to File, Save (or by clicking the icon). You can now Exit if you like, or keep it open and go on to the next tutorial – part two for general one-factor design and analysis. It delves into advanced features via further adventures in bowling.