The Pro Bowl is the National Football League’s version of an all-star team. In this blog post, I'll look at all the NFL draft picks from 1996 through 2008 and, using Minitab Statistical Software, model the probability of making it to at least one Pro Bowl based on draft order, the NFL team that drafted the player, the NCAA team the player came from, and the position of the player.

I did not include 2009-2013 drafts in this analysis since players drafted in 2009 and beyond haven’t had a reasonable opportunity to make the Pro Bowl. As shown in the graph below, the proportion of drafted players making it to at least one Pro Bowl has already seen a severe drop-off starting in 2008.

As you might expect, analysis with Minitab shows that the first few draft picks have a pretty good chance of going to the Pro Bowl at least once, although there's certainly no guarantee. As the graph below shows, the highest proportion of Pro Bowl players across all draft orders never exceeds 0.70. There are also two interesting outliers: no draft order beats the 44th pick in proportion of times making it to the Pro Bowl, and the 22nd pick never makes it to the Pro Bowl in the timeframe of this dataset (1996-2008). But there's no need to seek out numerology references about the numbers 22 and 44; these outliers are fun to point out, but are likely just extreme ends of random variation.

Position was a highly significant predictor in determining whether or not a player would make it to the Pro Bowl, based on a Binary Logistic Regression Model (p-value = 0.000). In the graph below, we see that Defensive Tackles/Ends and Cornerbacks were less likely to make it to the Pro Bowl for a given draft order than Long Snappers, Kickers, and Punters. 

It makes sense that Long Snappers, Kickers, and Punters are more likely to make the Pro Bowl for a given draft order since the best players in those positions, who typically make the Pro Bowl, are drafted  in later rounds.

The explanation for why fewer than expected Defensive Tackles/Ends and Cornerbacks make it to the Pro Bowl requires a bit more explanation. The draft order adjusted result for player positions above implies that Defensive Tackle/Ends and Cornerbacks tend not to make it to the Pro Bowl as often as they should based on their draft order.

However, the next graph reveals that the proportion of times a Defensive Tackle/End or Cornerback made it to the pro-bowl was similar to, or better than, several other positions. Considering these results together, we might infer that Defensive Tackle/Ends and Cornerbacks were drafted early, resulting in a decent proportion making it to the Pro Bowl, but not as many as would be expected from the draft order.

The graph of average draft order by position shown below confirms that Defensive Tackle/Ends and Cornerbacks were drafted relatively early (the lower the draft order, the earlier a player was drafted).

The bottom line is that it appears to be difficult to determine, at the time of the draft, which Defensive Tackle/Ends and Cornerbacks will make it to the Pro Bowl. 

One interesting result with QB’s is that they had the most variable draft order. They are often drafted very early, but many are drafted late as well, as graph 6 shows.

Here comes the part that makes general managers nervous: Which teams drafted better than expected, after adjusting for draft order and player position?

When they look at the graph below, NFL fans won’t be surprised to see it’s the New England Patriots. However, GM’s can relax, because the difference between teams is not statistically significant (P-value = 0.227).

Now look at the chart of number of draft picks by NFL teams.  It is interesting to note that the Patriots had the third highest number of draft picks (due to getting picks via trades), despite actually losing a 2008 draft pick in the spy-gate scandal. So they probably know they are good at drafting.

This graph shows the overall proportion of players by team who made it to the Pro Bowl. 

Can the data tell us which college teams produce players who make the Pro Bowl most often, relative to their draft order and position? The state of Florida is interesting in that it appears to have the best and worst-case scenarios, which are illustrated in the graphs below. University of Miami (FL) was the most impressive and University of Florida was the least impressive. However, statistical significance was not achieved at a high confidence level (p-value = 0.187).

As the NFL season gets under way, who are you hoping to see make it this year's Pro Bowl?

by Matthew Barsalou, guest blogger

The field of statistics has a long history and many people have made contributions over the years. Many contributors to the field were educated as statisticians, such as Karl Pearson and his son Egon Pearson. Others were people with problems that needed solving, and they developed statistical methods to solve these problems.

One example is Karl Gauss and the standard normal distribution, which is a key element in statistics. The distribution was used by Gauss to analyze astronomical data in the early nineteenth century and is also known as the Gaussian distribution or more simply, the bell curve.

Any normal distribution can easily be converted into the standard normal distribution based on a Z score table. The standard normal distribution is often used when comparing the means of either large samples or populations. For example, an engineer may perform hypothesis testing using the standard normal distribution to compare before-and-after results when attempting to increase the mean of a manufacturing process.

The well-known Student’s t distribution was created by a Guinness brewery employee named William Sealy Gosset, who published in the journal Biometrika under the name Student. Guinness did not permit its employees to publish because of fear of the competition learning about what they were doing, hence Gosset published under a pseudonym.

Gosset created Student's t distribution because previous formulas for estimating the error of samples required a large sample size and Gosset had found that there were often only small samples available. Student’s t distribution is used for small sample sizes and approaches the standard normal distribution as sample size increases.

This aspect permitted Gosset to perform experiments with small sample sizes, and this distribution is just as useful in industry today as it was when Gosset created it. For example, small sample sizes are more economical if a manufacturer wanted to perform experiments on expensive products and the experiments required destructive testing.

In 1924, Walter A. Shewhart presented the management of Western Electric’s Hawthorne plant with his concept of statistical process control (SPC). In his 1931 book Economic Control of Quality of Manufactured Product, Shewhart explained that eliminating assignable causes of variation would lead to a reduced level of inspection and therefore both higher quality and lower costs.

Using control charts such as an Xbar-R chart, a manufacturer can quickly tell when a process is at risk of producing defective parts without needing to individually inspect every item after production. Control charts can also detect a problem before hundreds or thousands of defective parts have been produced.

Four years after Shewhart published Economic Control of Quality of Manufactured Product, Ronald A. Fisher published his classic work The Design of Experiments. Fisher explained the proper methodology for performing Design of Experiments (DOE).

Today, DOE is frequently used in industry for performing experiments and is a key part of the Six Sigma quality improvement methodology. One of the great advantages of DOE is the ability to reduce the number of experimental runs required to get usable results. An experiment performed using DOE can provide the experimenter with information on the main effects of varying the levels of the experimental factors, as well as the interactions between the factors when the levels are varied.

Fisher’s future son-in-law George E. P. Box, with K. B. Wilson, further advanced DOE. They introduced the concept of Response Surface Methodology (RSM) in 1951. This variation on DOE is used to determine the relationship between multiple factors and one or more output variables in order to determine an optimal response. It can be used for process improvement, troubleshooting, and for making a product more robust to outside influences. Response surface methods can be used to produce both surface and contour plots  for analyzing the effects of varying influence factors on a product or process.

The Weibull distribution is named for E. H. Waloddi Weibull and is frequently used in the field of reliability engineering. Weibull was not the first to discover the distribution that bears his name; however, he brought the Weibull distribution to prominence when he introduced in to the American Society of Mechanical Engineers (ASME) in 1951.

This distribution is used to determine the time-to-failure for parts or systems. The Weibull distribution changes shape as parameters change and it can even approach the shape of the standard normal distribution.    

This brief discussion can't cover all statistical methods used in industry, nor all of the discoverers who have contributed to statistical methods. There are, however, commonalities amongst the statistical methods presented here. These breakthroughs in statistics were not discovered by people seeking a breakthrough in statistics; rather, they were found by people who had a problem to solve.

Much of Gauss’ work was done in the field of astronomy, and Gosset was trying to brew good beer at a low cost. Shewhart was at an industrial research laboratory, and Fisher was made his contributions to experimental design when he was attempting to interpret massive quantities of data resulting from years of agricultural experimentation.

Box was a chemist by education, but was confronted with a statistical problem and learned statistics because no other statistician was available to help him. In addition to publishing his namesake distribution, Weibull frequently published on practical engineering-related subjects, such as material strength and material fatigue.

The field of statistics has progressed over the past two centuries and we can expect that it will continue to give us new practical methods to find solutions to real-world problems. Statistics is now an essential part of the modern quality engineer’s body of knowledge.

Perhaps somewhere, right now, an engineer facing a problem on the production floor is creating yet another new statistical method for solving a real-world problem.

Would you like to publish a guest post on the Minitab Blog? Contact publicrelations@minitab.com. 

Failure. Just saying the word makes me cringe. And if you’re human, you’ve probably had at least a couple failures in both your work and home life (that you've hopefully been able to overcome).

But when it comes to Lean Six Sigma projects, there’s really nothing worse than having your entire project fail. Sometimes these projects can last months, involve a large project team, and cost companies a lot of money to carry out, so it can be very upsetting for all involved to know that the project failed (for whatever reason).

At Minitab, we’re always talking to our customers and practitioners in the field to better understand how they’re structuring and completing their projects, what tools they’re using, and the challenges and roadblocks they come across. One common reason practitioners have told us their projects weren’t successful is because the solution recommended at project completion was never even implemented.

When we pried a little further, we got some interesting feedback and heard three different reasons why this outcome occurred. The most common reason was that the process owner was not involved in the project from the start.

If you consider the way that many quality improvement initiatives are structured—with a separate quality improvement team or department responsible for completing and actually “owning” projects taking place all over the company—it’s easy to see how a process owner could be left out of a project from time to time. Maybe the process owner is extremely busy doing his day job, and has little time to devote to the project team. Or maybe for various reasons the process owner is never actually interested in the project. Maybe the process owner wants to take charge of the process and find a solution on his own, or maybe the project team responsible for making the process more efficient could streamline the process so much that the process owner could lose his job? These could all be reasons why the process owner never implemented the solution.

Other feedback suggested that maybe solutions were never implemented because the project team followed the DMAIC methodology, but only completed the define, measure, and analyze phases, and never actually made it to the improve or control phases. In other words, they handed off the project after completing the “DMA” of DMAIC, and expected the process owner to take care of the “I” and “C.”

Other practitioners told us that project solutions were not implemented because after the project team did all the work and designed the new process, they shared it with upper-level managers who said something along the lines of, “This is not what we expected.” Management might nix a project solution if it’s too complex, expensive, or simply because it’s not the solution they would have come up with themselves.

It might seem pretty simple, but one thing you can do is create a thorough project charter at the outset of each project. What is a project charter? It’s a document, usually completed during the define phase, that answers these basic questions about your project:

• Why is this project important?
• Who is responsible for the success of this project?
• What are the expected benefits of this project?
• When should the expected benefits begin to accrue, and for how long?
• Where are improvement efforts being focused?

The information in a project charter is critical for obtaining leadership commitment to provide the necessary resources for completion of the project. Sometimes it can even serve as your “approval” from management to move forward with the project. This tool is one of the principle communication tools for all project stakeholders, so it’s important that you have it filled out clearly and concisely, as well as updating it with changes that may occur as the project progresses.

Here’s an example of the project charter available in Quality Companion. You’ll notice there’s even a field to name your "process owner" and a field for his sign-off:

Remember, it’s important to include process owners from the start, because they are usually the people responsible for implementing the solution your team recommends. If you do a thorough job on your project charter, you should be able to avoid some of the issues above, and be on your way to completing a successful project!

What are your tips for avoiding a failed project?

In the world of Six Sigma, we’re always looking to improve our process. Whether it’s increasing the strength of building materials or improving the way calls are processed in a call center, it’s always a good idea to use a data-driven analysis to determine the best solution to your process.

The same is true for the NFL. Two years ago, the NFL decided to move kickoffs up from the 30 yard line to the 35. This has resulted in more kicks traveling into the end zone. So NFL coaches have a decision to make on their kick return process:

• Should I have my player take a knee whenever he catches the ball in the end zone?
• Should I have him return it no matter what?
• Is there some middle ground where I have him return it sometimes and take a knee sometimes?

This sounds like a perfect opportunity for data analysis! I’m going to use kickoff data from the last season to determine what I would tell my player if I were an NFL coach! (I know, I’ll never be an NFL coach, but I’m still holding onto my dream of being a clock management coach someday.)

Last year, there were a total of 2,525 kickoffs in the NFL regular season (not including onside kicks). Of those, 2,037 went to the goal line or into the end zone. That’s just over 80% of kickoffs that are going into the end zone. So you can see why making the correct decision about how often to return the kickoff is pretty crucial.  

So when a kick goes into the end zone, how far is it going? Unfortunately, my data contains 710 observations that don’t specify distance. The data (taken from advancednflstats.com) only says “kicked into the end zone for a touchback.” I’m assuming these were either kicked out of the back of the end zone, or the returner decided not to field the kick and it bounced through the end zone. So I’m going to ignore those observations (they are all touchbacks anyway), and focus on the remaining 1,327 where a kick returner was able to catch the kickoff.

The first thing I want to know is how many yards deep the kicks are going into the end zone. The following pie chart breaks it down nicely.

The yards deep are distributed pretty evenly. Five yards and 9 yards are the most common, but none of the yard lines are very different. So as a kick returner, I’d expect to field the ball at a variety of distances inside the end zone.

Now let’s focus on whether or not I should return those kicks!

Of the 1,327 kicks where I know the returner fielded the punt, 893 of them were returned out the end zone (67%). On average, the returner was able to return the kick just past the 23 yard line. That’s better than starting at the 20 (where the ball would be spotted if the returner took a knee in the end zone)! 

Data analysis over! If kick returners are getting past the 20 when they return it from the end zone, always return it!!!

Well, not quite. We previously saw that kicks range from going anywhere from the goal line to 9 yards deep in the end zone. We need to see if the yard line the returner gets to varies based on where they take the kickoff from. So let’s look at another pie chart, this one showing how deep the kick is when it’s returned out of the end zone.

This chart is much different. We see that most of the kicks are being returned between the goal line and 5 yards deep into the end zone. Once you get more than 5 yards into the end zone, players are more likely to take a knee. This seems to make sense, as one would imagine the deeper you are in your end zone, the harder it will be to get past the 20.

Let’s see if the statistics agree. Here is a bar chart that shows the average yard line the kicker returned the ball to for each yard deep they were in the end zone.

In fact, it the data indicate that no matter how deep you are in the end zone, you still make it back to about the same yard line. Even when you return from 9 yards deep in the end zone, you’re still making it past the 20 on average!

But what about being backed up in terrible field position? If you return it from deep inside your own end zone, are you more likely to be tackled inside your own 10 yard line?

This bar chart shows that if you return it from 4 yards deep or shorter, the chance of being tackled inside your own 10 is almost nonexistent. It does increase once you hit the 5 yard mark, but not by much. Nothing is much higher than 5%. So if you return the ball from the end zone, at worst you’re looking at about a 1-in-20 chance that you’ll be tackled inside your own 10. That’s not too high of a risk.

But when you return the ball from your end zone, you’re not looking to minimize your risk--you’re looking for a big play! Or at least some decent field position. So let’s look at one more plot to see the percentage of times the returner makes it past the 30 yard line.

This shows that you’re much more likely to rip off a big gain than end up with terrible field position. And shockingly the highest percentage of returns past the 30 happened when the ball was returned 7 yards deep in the end zone!

No matter how you look at it, there doesn’t seem to be any reason to take a knee when you field the ball cleanly in the end zone. No matter how deep you are, on average you’ll make it back to the 20. Sure, a small percentage of the time you’ll have terrible field position, but that is more than made up for by the times you’ll have much better starting field position, or even a touchdown!

And if you need one more reason to always return the ball out of the end zone, well, it guarantees that this will never happen!

Defects can cause a lot of pain to your customer.

They can also cause a lot of pain inside your body. The picture at right shows my broken right clavicle. Ouch!

You might think of it as the defective output from my bicycling process, which needs improvement.

Sitting around all summer cinched up in a foam orthopedic brace hasn’t exactly been wild and wacky 50s-style fun at the beach.

But the injury has had its perks (a box of mouth-watering dark chocolate ganaches from kind Minitab coworkers, for example!)

It’s also provided me with a rare commodity in the year 2013: Plenty of time to think.

Always on the lookout for a quality improvement opportunity, my brain ponders how this defect could have been prevented.

“If only I had slowed down coming down that hill...”

“If only I hadn’t tried to make that sharp 95-degree turn…”

“If only I had triple somersaulted with a half twist and landed squarely on my feet with a big smile, like a Romanian gymnast ..."

If I could turn back the clock, before I hopped on my bicycle, I'd have carefully examined data from the U.S. National Electronic Injury Surveillance System (NEISS).

(You may have never heard of the NEISS. It's part of powerful, pervasive federal program that secretly records data every time you stub your big toe and say a naughty word.)

Randomly sampling from the nearly 14,000 bicycle-related injuries recorded by the NEISS in 1991, a federal study sought to identify the main causes of bicycling accidents that resulted in rider injury.

If only I'd started my summer by graphically analyzing these data in Minitab...

I'd like to say that I was performing a combination death spiral/12 o'clock wheelie with a switchback handstand when an asteroid fragment suddenly fell from space and lodged in my spokes (see the smallest bars at the bottom, representing 7% and 12% of incidents).

But the simple, boring truth is that I'm aklutz of the most common variety.

The primary causes of my accident are shown by the two longest bars on the chart: uneven road surface and excessive speed. (In my case, it was a perfect storm: the road surface switched from dry, compacted soil to asphalt pavement at the bottom of a hill, which allowed me to maximize my speed just as I hit the uneven surface. Whoo-hoo!)

According to the chart, mechanical malfunction is another frequently cited cause of bicycling accidents that lead to rider injury. But that's a broad category. It couldn't have helped me prevent my mishap.

Unless, of course, I'd taken the time to identify the "vital few" mechanical malfunctions behind most injury-causing bicycling accidents...

Once the types of mechanical malfunctions are displayed in a Pareto chart, the critical malfunctions involved in most bicycling injuries become easy to spot.

Brakes are the No. 1 mechanical malfunction. Look at the associated ascent of the cumulative line. I'd hate to bike down that hill.

And true to the chart, a pair of sneaky, clutching metallic brake pads played a crucial role in my accident:

As I was coming down the hill, I spotted the sharp turn ahead on the road. When I realized I was going too fast to make the turn, I quickly squeezed the brakes. Too little, too late, you might think.  But no: this was too much, too late.

As soon as the bike hit uneven surface, the disc brakes locked up. Kinetic energy kindly did the rest, hurling me over the handlebars and into space -- much like an asteroid fragment.

(I landed somewhere in Russia. To much fanfare. After revealing sensitive NEISS data, I am now seeking permanent asylum).

So there you have it.

My life. My fate. Lying hidden in the vast vaults of NEISS data, just waiting to be teased out in a Minitab bar chart or Pareto chart.

Who needs Tarot cards?

When I first got interested in looking at baseball park factors, I only wanted to know which parks benefited hitters and which benefited pitchers. Once I got started, I got interested in the difference between ESPN's published formula and its results and whether there were obvious reasons for the variation in park factors from year-to-year.

But today I’m returning to the original question: which parks are hitters’ parks, and which are pitchers’ parks?

We already know that the mean and median are inadequate by themselves. For example, consider AT&T Park, where the mean suggests a pitchers’ park but the median suggests a hitters’ park. So we want to use an analysis that also takes into account how variable the park factors are.

Two closely related analyses in Minitab Statistical Software that consider variation are  the analysis of variance (ANOVA) and the analysis of means (ANOM). Let’s take a look at these two analyses to understand the difference in what they do.

One-way ANOVA tests whether all of the group means are equal to each other. One way to go deeper into this kind of analysis is to choose particular comparisons of interest. For example, you might compare all of the parks to the park with the lowest park factor: Petco Park.

The results show which parks are, statistically speaking, better hitters’ parks than Petco Park. The list of better hitters' parks includes most of the parks with factors as high as AT&T Park's. Some parks that have only a few years of service are hard to distinguish statistically, including Busch Stadium, Target Field, and Marlins Park.

The ANOVA information is very useful. But if we want to classify parks as hitters' parks, neutral parks, and pitchers' parks, ANOVA doesn't give us exactly what we want. We wouldn't really say that AT&T Park is a hitters' park just because it's a better hitters' park than Petco Park.

Instead of testing whether means are equal to each other, ANOM tests whether the means are equal to the overall mean. Minitab makes a graph so that the results are easy to understand.

The points on the graph are the mean park factors. The green line represents the overall mean. The red lines are decision limits that show which parks are different from the overall mean.

These parks are pitchers' parks:

• AT&T Park
• Dodger Stadium
• Petco Park
• Safeco Field

These parks are hitters' parks:

• Chase Field
• Coors Field
• Fenway Park
• Rangers Ballpark

If I think that the requirements to be a hitters’ park or a pitchers’ park are too stringent in the analysis of means, I could redo it with a higher alpha level. This decreases the amount of evidence that I require to classify a park as a hitters’ park or a pitchers’ park.

I think the case of Citi Field is interesting. Citi Field looks like a pitchers’ park, but it’s not classified as a pitchers' park statistically because the park has been in use for only 4 seasons. The variation has been too high to provide statistical evidence that Citi Field will always be a pitchers park.

The uncertainty is true from a more practical standpoint too. The center-right field fence moved in about 6 meters and came down halfway in height for the beginning of the 2012 season. Changes to the park dimensions could make it play very differently from how it was before.

When I looked at the individual value plots earlier, I guessed that Dodger Stadium and AT&T would be neutral parks, conventional wisdom notwithstanding. Statistical analyses like ANOVA and ANOM provide the clarity that we need to make more better decisions from data. Want to see more results for comparing groups? Check out how Riverview Hospital Association identified specific patient groups that gave lower satisfaction scores than other groups so that the association could direct their improvement processes properly.

John Tukey once said, “The best thing about being a statistician is that you get to play in everyone’s backyard.” I enthusiastically agree!

I frequently enjoy reading and watching science-related material. This invariably raises questions, involving other "backyards," that I can better understand using statistics. For instance, see my post about the statistical analysis of dolphin sounds.

The latest topic that grabbed my attention was an apparent error in the BBC program Wonders of Life. In the episode “Size Matters,” Professor Brian Cox presents a graph with a linear regression line that illustrates the relationship between the size of mammals and their metabolic rate.

Brian Cox, a theoretical physicist, is a really smart guy and one of my favorite science presenters. So, I was surprised when his interpretation of the linear regression model seemed incorrect. Below is a closer look at the graph he presents, and his claim.

Cox points out the straight line and states, “That implies, gram-for-gram, large animals use less energy than small animals . . . because the slope is less than one.”

For linear regression, the slope being less than 1 is irrelevant. Instead, the fact that it is a straight line indicates that the same relationship applies for both small and large mammals. If you increase mass by 1 unit for a small mammal and for a large mammal, metabolism increases by the same average amount for both sizes. In other words, gram-for-gram, size doesn’t seem to matter!

However, it’s unlikely that Cox would make such a fundamental mistake, so I conducted some research. It turns out that biologists use a log-log plot to model the relationship between the mass of mammals and their basal metabolic rate.

Perhaps Cox actually presented a log-log plot but glossed over the details?

If so, this dramatically changes the interpretation of this graph, because log-log plots transform both axes in order to model curvature while using linear regression. If the slope on a log-log plot of metabolic rate by mass is between 0 and 1, it indicates that the nonlinear effect of mass on metabolic rate lessens as mass increases.

This description fits Cox’s statements about the slope and how mass effects metabolic rate.

Let’s test Cox’s hypothesis ourselves! Thanks to the PanTHERIA database*, we can fit the same type of log-log plot using similar data.

I’ll use the Fitted Line Plot in Minitab statistical software to fit a regression line to 572 mammals, ranging from the masked shrew (4.2 grams) to the common eland (562,000 grams). You can find the data here.

In Minitab, go to Stat > Regression > Fitted Line Plot. Metabolic rate is our response variable and adult mass is our predictor. Go to Options and check all four boxes under Transformations to produce the log-log plot.

The slope (0.7063) is significant (p = 0.000) and its value is consistent with recently published estimates. Because the slope is between 0 and 1, it confirms Cox’s interpretation. Gram-for-gram, larger animals use less energy than smaller animals. In order to function, a cell in a larger animal requires less energy than a cell in a smaller animal.

I’m quite amazed that the R-squared is 94.3%! Usually you only see R-squared values this high for a low-noise physical process. Instead, these data were collected by a variety of researchers in different settings and cover a huge range of mammals that live in completely different habitats.

So Cox presented the correct interpretation after all: for mammals, size matters. However, he presented an oversimplified version of the underlying analysis by not mentioning the double-log transformations. This is television, after all!

There are important implications based on the fact that this model is curved rather than linear. If the increase in metabolic rate remained constant as mass increased (the linear model), we’d have to eat 16,000 calories a day to sustain ourselves. Further, mammals wouldn’t be able to grow larger than a goat due to overheating!

Cox also presented the idea that mammals with a slower metabolism live longer than those with a faster metabolism. However, he didn’t present data or graphs to support this contention. Fortunately, we can test this hypothesis as well.

In Minitab, I used Calc > Calculator to divide metabolic rate by grams. This division allows us to make the gram-for-gram comparison. Time for another fitted line plot with a double-log transformation!

The negative slope is significant (0.000) and tells us that as metabolic rate per gram increases, longevity decreases. The R-squared is 45.8%, which is pretty good considering that we’re looking at just one of many factors than can impact maximum lifespan!

However, it is not a linear relationship because this is a log-log plot. Maximum longevity asymptotically approaches a minimum value around 13 months as metabolism increases. The graph below shows the curvilinear relationship using the natural scale.

A one-unit increase in the slower metabolic rates (left side of x-axis) produces much larger drops in longevity than a on-unit increase in the faster metabolic rates (right side of x-axis).

Once again, we’ve shown that size does matter! Larger mammals tend to have a slower metabolism. And animals that have a slower metabolism tend to live longer. That’s fortunate for us because without our slower metabolism, we’d only live about a year!

________________________________

*Kate E. Jones, Jon Bielby, Marcel Cardillo, Susanne A. Fritz, Justin O'Dell, C. David L. Orme, Kamran Safi, Wes Sechrest, Elizabeth H. Boakes, Chris Carbone, Christina Connolly, Michael J. Cutts, Janine K. Foster, Richard Grenyer, Michael Habib, Christopher A. Plaster, Samantha A. Price, Elizabeth A. Rigby, Janna Rist, Amber Teacher, Olaf R. P. Bininda-Emonds, John L. Gittleman, Georgina M. Mace, and Andy Purvis. 2009. PanTHERIA: a species-level database of life history, ecology, and geography of extant and recently extinct mammals. Ecology 90:2648.

Tired of squinting at your computer screen?  Want to make an impact with the trainees in the last row of your class?  Try zooming in! 

Here are some helpful tips on how you can blow up the size of Minitab Statistical Software screen elements for easier viewing.  Who knew Minitab looked so great close up?

Select any cell in the worksheet.  Hold down the CTRL key on your keyboard while simultaneously rolling your mouse wheel up and down.  This will zoom the worksheet in and out so that you see more or fewer rows and columns in the window.

From the Tools menu, select Customize.  Click the Options tab in the Customize window then select the “Large Icons” check box. Voila!  Your toolbar icons will instantly double in size.

Again, from the Tools menu, select Options.  In the menu tree on the left, click the plus sign next to Session Window to expand the options.  Click I/O Font to see your font options in the right of the dialog box.  Increase the font size to 18 or 20 or more!  The same can be done with the Session Window Title Font option.  Be sure to make the title font a little bit bigger than your I/O font.  

Even veteran Minitab users are often surprised by some of the lesser-known features in our statistical software. Sign up for one of our upcoming freewebinars to learn more tips and tricks in Minitab!

In a previous post, I discussed how to avoid a Lean Six Sigma project failure, specifically if the reason behind the failure is that the project solution never gets implemented.

In this post, let's discuss a few other project roadblocks that our customers cited when we asked them about the challenges they come across in completing projects. I’ll also go into detail about suggestions our industry-seasoned trainers at Minitab offer to avoid these failures.

One common reason quality improvement projects get started on the wrong foot is that their scope is too large.

In fact, one of our customers provided us with a great example.

In this particular case, the customer's project goal was to “improve the profitability of the company’s South American Division.” In theory, this sounds like a pretty good project goal, but when they actually began work on the project, the project team needed to hone in on that broad goal in order to reach something to work on that was less vague and more specific (not to mention measurable).

To help the team drill down to a more concrete project goal, they created a CTQ (Critical To Quality) tree:

This CTQ tree above was created in Quality Companion, but you could certainly employ this tool using pen and paper, a whiteboard and a marker, etc.

The CTQ tree starts with a broad project goal, such as “increasing the profitability of the South American Division,” and works downward to identify factors “critical to” achieving this goal. In this case, the team identified that improving sales by 20% and improving the sales margin by 20% was “critical to” achieving this goal (see the green boxes).

Now they drill down even further. In order to improve sales by 20%, the team identified that it was critical to improve Internet sales by 25%. To improve Internet sales, the team determined that they needed to reduce complaints about order-to-fill time, and to do that they needed to decrease order-to-fill time by 45%.

The team could have gone on (and on …), but they decided to end here, since they now had a much more measurable project goal than they originally started with. This smaller project—decreasing the Internet order-to-fill time—supports the overall goal to improve the profitability of the entire division, but now the team has a specific goal to focus on.

Another tip for ensuring your project scope is appropriate is to limit the project to one geographical location, one measurable product/defect, and clearly define the customer up front. Another tool that can help define the project scope is a SIPOC.  Basically, a SIPOC is a high-level process map that defines the scope of a process. Check out this post for more on how to use this tool.
 

Another reason customers say quality improvement projects have been unsuccessful is a failure to link the project to finances. One of our trainers gave me the following example from his years working as Six Sigma project leader.

One quality improvement team from a manufacturing company had a goal to reduce the number of press shutdowns caused by mold in equipment from 37 to less than or equal to 14 per month. Even though everyone in the facility understood the importance of reducing the number of shutdowns, when it came time to implement the team's solution, questions began to arise about the large costs involved.

The team now faced a challenge—they needed to prove that the expensive solution was actually a good idea to implement over the long term.

Here’s what they did: They calculated that just one shutdown would cost the company $8,000. They also calculated that once implemented, the solution would reduce shutdowns from 37 to 14 per month (23 fewer per month), thus saving them over $2 million dollars per year. So was the solution worth implementing?  Yes!

The company decided that the high cost to implement the solution wasn’t really so bad after all, and the project team was given the OK to make it happen.

This example project illustrates the importance of having a demonstrated financial “link” throughout your project, and being able to answer the following questions: 

• How much money do you expect to save overall?
• How much money will it cost to implement your project solution?
• What is the time frame for realizing the expected financial benefits?

How can these questions be answered? Carry out a financial analysis of your project. There are many ways to do this. If you're using Quality Companion, the software includes a form (in both transactional and manufacturing versions) you can utilize to help guide your project financial analysis:

To help you provide the “linkage” between your project and bottom-line impacts on the organization—which is an important component in obtaining full support from management and project stakeholders—a project financial analysis is the way to go.

What other tips have helped you to carry out successful projects?

Recently, a customer called our Technical Support team about a Design of Experiment he was performing in Minitab Statistical Software. After they helped to answer his question, the researcher pointed our team to an interesting DOE he and his colleagues conducted that involved using nasal casts to predict the drug delivery of nasal spray.

The study has already been published, and you can read more about it here, but I wanted to highlight this use of the DOE tools in Minitab in this blog post.

The nose is a convenient route of administration for delivering drugs that treat respiratory ailments. For example, nasal sprays target and deliver drugs to regions of the nasal cavity that become irritated by allergens such as pollen and pet dander. But evaluating the ability of new nasal delivery technologies to accurately target specific regions in the nasal cavity is complicated, and clinical trials that do so can be expensive, time-consuming, and offer only qualitative findings.

Alternatively, researchers can predict regional deposition using nasal casts. The casts mimic the anatomy of human nasal passages and can be tested under controlled conditions. Nasal casts provide a cost-effective and simpler way to test and optimize device and formulation factors that affect drug delivery. Nasal casts also have the potential to let researchers precisely measure how much of a nasal spray reaches a given region of the nose, which could allow for more efficient treatment of diseases in the nasal cavity and fewer side effects.

Here’s an image of the various sections of a nasal cast:

For a nasal cast to be effective at predicting deposition, and for use in optimizing new nasal spray devices, it needs to be set up to mimic human deposition. As an “in-vitro,” or simulation, tool, there were many settings to consider.

To evaluate different settings on the nasal cast that would be most predictive of human deposition, the research team developed success criteria from several published nasal spray deposition studies conducted on humans. These studies gave them an understanding of the different regions of the cast where they could expect to see drug deposition, as well as insight into the total percentage of the drug that was being deposited into the various nasal regions after the spray was used. Their goal was to adjust the settings on the cast to get a deposition pattern of the drug that was similar to what you would see in humans. They wanted the settings to be repeatable, in order to prove that nasal casts could be used as a cost-effective way to compare new device and formulation technologies for nasal drug delivery.

The nasal cast the team used, which was made of nylon, was created based on 3D computer images of the nasal cavities of healthy humans. The cast was made up of five regions that mimic the human nasal cavity, which were coated with a special solution to simulate the mucus in human nasal passages. After assembling and orienting the cast, a model nasal spray formulation was sprayed into it. To analyze the spray’s deposition, the cast was disassembled and the amount of drug that reached each region was measured using a technique known as an HPLC, or high-pressure liquid chromatography.

To identify which factors were significant, and to reduce the number of factors they needed to optimize, the team used Minitab’s DOE tools to collect information on the ranges of factors influencing the deposition of spray droplets throughout the nasal cast.

With a designed experiment, researchers can change more than one factor at a time, and then use statistics to determine which factors have significant effects on an outcome. Using a DOE reduces the number of experimental runs needed to gather reliable data, making studies less expensive and more efficient.

In situations where many factors or settings need to be considered, an exploratory or screening experiment can help researchers determine which to focus on. The team began with an exploratory DOE—a full factorial with five sets of spray angles, three airflow rates, and two speeds, but no repeats. Once the range was determined, they used a half-factorial design to screen the important factors identified in the exploratory experiments.

Main effects plots of the half factorial DOE showed how different levels of each factor affected the various nasal cast components.

The team conducted additional optimization experiments to examine the effect of different levels of the significant factors on deposition. They chose two factors—airflow time and tilt angle—for follow-up experiments because they appeared to affect deposition the most.

After collecting the additional data about these two factors, the team used Minitab’s Response Optimizer to identify the optimal settings for the airflow time and tilt angle. Using the optimized settings, two operators tested two nasal casts five times, and the team performed Gage R&R analysis in Minitab to determine if either the operator or cast were causing any variation in the process.

A Minitab Bar Chart of mean percent deposition in different regions of the nasal cast showed the team that deposition patterns from their experiments were meeting success criteria for predicting human deposition.

The studies showed that neither nasal cast nor operator had a significant effect on nasal deposition in any of the cast regions, and the Gage R&R analysis revealed that with optimal settings, the nasal cast provided a robust measurement system for deposition within the nasal cast.

The optimized settings for the nasal cast produced repeatable results, and proved that the cast could be used to cost-effectively compare new nasal spray devices and formulations that target specific regions of the nasal cavity. Although a nasal cast cannot be used to predict clinical response, nasal casts could be used in combination with clinical studies to relate biological drug response to the drug deposition pattern in accord with nasal spray use.

Research discussed here was originally published in the Journal of Aerosol Medicine and Pulmonary Drug Delivery, March 2013: “Design of Experiments to Optimize an In Vitro Case to Predict Human Nasal Drug Deposition.”

Do you have an interesting DOE you’d like to share with us? Tells us at http://blog.minitab.com/blog/landing-pages/share-your-story-about-minitab.

by Arun Kumar, guest blogger

One of the most commonly used statistical methods is ANOVA, short for “Analysis of Variance.” Whether you’re analysing data for Six-Sigma styled quality improvement projects, or perhaps just taking your first statistics course, a good understanding of how this technique works is important.

A lot of concepts are involved in any analysis using ANOVA and its subsequent interpretation. You’re going to have to grapple with terms such as Sources of Variation, Sum of Squares, Mean Squares, Degrees of Freedom, and F-ratio—and you’ll need to understand what statistical significance means.

The table below shows a typical ANOVA table, with call-outs to label each of the columns. This is the kind of output you’ll see when you perform ANOVA in statistical software such as Minitab.

I’ve found an analogy from the field of Management Accounting to be very helpful in understanding ANOVA. And while we don’t all do accounting for businesses, we do all have personal budgets we need to keep track of. Therefore, rather than thinking about an organisation’s expenses in this example, let’s instead use your household’s monthly expenses as the basis of discussion.

Remember that everything has variation—whether it’s the parts coming off an assembly line, a chemical process, the weather each day...or the amount of your expenses from month to month.

The whole purpose of Analysis of Variance is to break up the variation into component parts, and then look at their significance. But there's a catch: in statistics, Variance (the square of Standard Deviation) is not an “additive” quantity—in other words, you can’t just add the variance of two subsets and use the total as the variance of the combination.

That’s why we need to look at Sums of Squares (SS), which are additive in nature. What ANOVA therefore involves is looking at the Sum of Squares for the total data, and then breaking that up into a number of component parts, including Sum of Squares due to “error.”

This is akin to breaking your total monthly household expenses into categories such as clothing, entertainment, education, travel, etc.

Typically, our budgets also include a “miscellaneous” for expenses that happen, but aren’t readily identifiable or easily classified. This is the same thing as “Error” in the ANOVA table: it’s a place to capture variation that isn’t explicitly accounted for in our data.

In terms of our budgets, doing this helps us look at two things:

1. How big is a given expense in relation to Total expenses?
2. How does any particular expense compare with the Miscellaneous category?

In ANOVA, we call the collection of factors we’re using to assess variation a “model.” A good model ensures that the Sum of Squares due to error (a la “Miscellaneous” household expenses) is relatively small compared to that due to the factors in our model. This is why the Sum of Squares attributed to the model factors is called Explained Variation and the Sum of Squares due to error is called the Unexplained Variation.

Good accounting in households requires the explained to be more than the unexplained. In ANOVA, the F-ratio is the tool we use to compare each of the sources of variation. However, getting the F-ratio requires a bit of adjustment to the Sum of Squares...we need to consider the Degrees of Freedom for each factor, which is akin to assigning weights to your household expense items. The Sum of Squares adjusted for—that is, divided by—the Degrees of Freedom is the statistical term we call Mean Square (MS).

The F-ratio is the MS for a factor divided by the MS for error. So, in the example above where the MS for Factor 1 is 959.25 and the MS for Error is 103.667, the F-ratio is:

F-ratio = MS (Factor 1) / MS (Error) = 959.25/103.667 = 9.25

In general, the higher the F-ratio—the ratio of the variation due to any component to the variation due to error—the more significant the factor is. In our budget analogy, these factors would be our more significant the household expense categories.

Ultimately, the value of budgeting is to give you an overall understanding of where your money is going. The value of ANOVA is to give you an overall understanding of where the variation you observe is coming from. 

First, we need to decide how "expensive" a category must be in order to be considered importance. In statistics, we call this "alpha." The value of alpha (level of significance of the analysis/ test – typically 5%, or 0.05) represents the concept of materiality in our household example.

Then we need to figure out if each factor meets that standard of importance, or "significance." For that, we look at the p-value. The p-value which is the probability of getting the observed F-value if the variance between factors is equal.  To determine significance, compare the p-value for each factor with the alpha value (0.05). If the p-value is less than alpha, that factor is significant, because there's less than a 5% chance we'd see that factor's F value otherwise.  If the p-value is greater than alpha, we reject the factor's significance.

A final component of the ANOVA output to consider is R-squared. The R-squared statistic is simply the percentage of variation that can be explained by the factors included in our model. The greater the significance of the factors we can find through analysis, the greater the R-squared value (which is really a reflection of a good model). 

And what good household manager would not want to account for as much budget variation as possible?

About the Guest Blogger:

Arun Kumar is based in the United Kingdom and has been a Master Black Belt at Xchanging plc for 9 years. He can be contacted at arunkumar1976@gmail.com. 

If you would like to be a guest blogger on the Minitab Blog, contact publicrelations@minitab.com.

Two weeks ago Penn State and Michigan played in a quadruple-overtime thriller that almost went into a 5th overtime. Had Penn State coach Bill O’Brien kicked a field goal in the 4th overtime instead of going for it on 4th and 1, the game would have continued. But the Nittany Lions converted the 4th down (which, by the way, wasn’t a gamble) and went on to score the game winning touchdown in the 4th overtime.

Watching this game got me asking a bunch of questions. How many college football overtime games go into 4 overtimes? Did Penn State still have home-field advantage since they were playing at home? How often is there an overtime period where neither team scores? (This happened twice in the Penn State-Michigan game.) If a team scores to tie the game in the closing minute (Penn State tied the game with 27 seconds left) do they have momentum on their side, and are thus more likely to win in overtime?

What better way to answer these questions than to conduct a data analysis using Minitab Statistical Software? Let's get to it!

With all the detail I wanted to get into, I wasn’t able to find any satisfactory overtime data online. So I had to collect all the data I wanted manually. It was slow going, but I was able to record every overtime game since 2008 (including bowl games). This gives us a sample of 156 overtime games. Obviously I would have liked to include every overtime game since 1996 (when college football started having overtime), but there are only so many hours in a day. So for now, our 156 games will have to suffice. If you want to follow along, you can get the data here.

Now that the boring part is over, let’s get to the statistics!

Penn State and Michigan played 4 overtimes. A few weeks earlier Buffalo and Stony Brook played a 5-overtime game. Are these results typical when a game goes to overtime, or does it usually take only one extra session?

Over the last 5 years, over 71% of games took only 1 overtime to produce a winner. And there are only a handful of games that make it past 2 overtimes. So if you hear somebody complaining that the college football overtimes rules are bad because it makes games last forever, know that most of the time, that’s not actually the case. 

Of course, there will always be outliers, as the Penn State-Michigan game showed us. But don't worry, Nittany Lion and Wolverine fans—odds are you won't have to go through 4 extra periods of heart-attack city again anytime soon.

For each game, I recorded whether the home team or away team won. There were also 13 bowl games, so I simply marked those games as “Neutral.”

Of the 143 games played at one of the teams' home field, the home team won 80 of them, which comes out to 56%. After playing 4 quarters of football and having a tie score, I'm going to assume the teams are about equal. So if there was no home-field advantage, we would expect the home team to win about 50% of the time. We can perform a 1 proportion test to see if our sample proportion of 56% is significantly greater than 50%. If it is, we can conclude that home-field advantage does still exist even when a game goes to overtime.

At the α = 0.10 level, we can conclude that our sample proportion is significantly greater than 50%. The p-value is 0.09, which means if the true proportion of the home team winning was actually 50%, the probability that we would have seen a sample proportion of .56 or higher is only 9%.

I think this is enough evidence to conclude that home field advantage still exists in overtime.

Typically, an α of 0.05 is used, and we would reach a different conclusion if we use 0.05 for α. But I believe the reason the test is not significant at the 0.05 level is because we do not have enough power. In hypothesis testing, power is the likelihood that you will find a significant difference when one truly exists. We can use Minitab’s Power and Sample Size analysis to determine the power of our test when α is 0.05.

This tells us that even if home teams did in fact have a 56% chance of winning in overtime (as opposed to 50%), we would only have a 41.6% chance of detecting a difference with our sample size. That's not very high. To increase the power of the test, we would need to collect a larger sample. But in the meantime, concluding home field advantage exists at the α = 0.10 level works for me.

When Penn State scored against Michigan with 27 seconds left, Bill O’Brien thought about going for a 2-point conversion instead of tying the game with an extra point. However, he decided on the latter. Seeing as Penn State was playing at home, this was likely the correct decision, as the odds were slightly in their favor to win in overtime (whereas, on average, teams convert 2-point conversions less than 50% of the time).

We just saw that home teams have a slight advantage in overtime. The consensus thought is that teams that start overtime on defense also have an advantage. That way you get to play offense 2nd, and you know exactly what you need to score to either win or tie the game. But will the statistics show that the team that gets the ball 2nd actually wins more often?

When a team was victorious in overtime, they started on offense 2nd 61.5% of the time. This appears to be a huge advantage. But is it significantly different form 50% (the percentage we would expect if there were no advantage)?

With a p-value of 0.002, we can conclude that a team that starts on offense 2nd has a greater than 50% chance of winning the game. Coaches are clearly making the correct decision in overtime periods when they choose to start on offense 2nd.

And this brings us back to Bill O’Brien’s 4th and 1 call in the 4th overtime. Had Penn State kicked the field goal, they would have gone to a 5th overtime where Michigan would have been on offense 2nd. Tying the game would have actually given the advantage back to the Wolverines. By trying to win the game right there, O’Brien was making the correct decision. And the outcome worked out pretty well for him too, as Penn State went on to score the game winning touchdown on that drive.

Notice how I separated the decision from the outcome in the previous paragraph. It's always important to consider coaching decisions independently from the outcome. Even if Penn State had failed on the 4th-and-1 play, it was still the correct decision. The only difference is that outcome would have been poor. Coaches can't tell the future, so they have to make decisions without knowing what the outcome will be. Keep that in mind the next time somebody is using hindsight to judge a coaching decision.

What would a blog post on college football overtime be if it didn't extend into part 2?!?! There is so much more I want to get to! What is the most common combined score by both teams in overtime? How often is there an overtime period where neither team scores? Do teams that score to tie the game carry that momentum into the extra session for an overtime victory? It’s all going to have to wait for another day. Check back next week where I’ll (hopefully) finish off this blog post in double overtime!

Design of experiments (DOE) is an extremely practical and cost-effective way to study the effects of different factors and their interactions on a response.

But finding your way through DOE-land can be daunting when you're just getting started. So I've enlisted the support of a friendly golden retriever as a guide dog to walk us through a simple DOE screening experiment.

Nala, the golden retriever, is shown at right. Notice how patiently she sits as her picture is being taken. She's a  true virtuoso with the "Sit" command.

But "Lay Down" is another story...

Although Nala knows the "Lay Down" command, she doesn't perform it quickly or consistently.

Why not?  I decided to design a simple experiment to find out. The goal of this experiment is to screen potential factors and determine which ones, separately or in combination with each other, may be affecting her response to "Lay Down."

Note: The main point of this example is to illustrate the basic process of a DOE  screening experiment—we'll talk about its shortcomings as an actual DOE application after the experiment.

First determine a list of candidate factors, based on process knowledge, that you think are likely to affect the response of interest.

In this case, my dog is the process. Based on my knowledge of this "process," here are some factors that I think might influence the response.

There are 7 factors in this screening experiment. Each factor has only 2 levels. All the factors here are categorical, but in a real experiment you'd typically have numeric factors like Temperature (Low, High) or Speed (Slow, Fast) and so on.

The response is the number of seconds it takes the dog to lay down after being given the command, as measured on a stopwatch.

Because there only 2 levels for each factor, this experiment is called a 2-level factorial design.

A run of an experiment refers to each combination of factor settings at which the experiment is performed. For example, one run of this experiment is to give the "Lay Down" command using these settings.

How many different unique runs are possible with 7 factors and 2 levels of each factor? If you remember your junior high school math, there are  2*2*2*2*2*2*2 = 27 = 128 possible combinations. So to perform the full experiment using all possible settings, I’d have to perform a total of 128 trials (runs).

My dog would certainly love that much attention. But I can’t invest that much time in the experiment (or that much money on honey-cured ham). Luckily, Minitab Statistical Software can help me pare down this experiment to make it more feasible.

To choose a design for the experiment, in Minitab choose Stat > DOE > Factorial > Create Factorial Design. Recall that this is a 2-level factorial with 7 factors, so the dialog box is completed as shown below:

To see what designs are possible with the experiment, click Display Available Designs.

The table at the top of the dialog box summarizes possible designs for a 2-level factorial study. This experiment has 7 factors, so the available designs are listed under the 7 column.

There are 5 designs shown in the column. The Roman numeral for each design refers to its resolution. Roughly speaking, the resolution tells you how clearly you'll be able to "see" the effects of the experiment. Designs shaded in green provide the best resolution (from Resolution V to Full Resolution). Designs shaded in red provide the poorest resolution (Resolution III). Yellow indicates a mid-range resolution.

Full resolution allows you to see all the effects in your experiment. For lower resolutions, the Roman numeral (III, IV, or V) describes the extent that effects you're investigating will be "blurred together"—that is, confounded—with each other.

For this experiment, here's how the main effects (the effects from each single factor) are confounded with interactions between the factors at each resolution:

See the pattern in the table? For each resolution, the size of the interaction that's confounded with the main effects is one less than the resolution number. There are other confounded interactions as well. We'll talk more about these confounding patterns when we create the design in Part II.

For now, just be aware that the lower the resolution, the lower the size of the interaction that's confounded with the main effects, and the less clear your results will be. The red shading for Resolution III warns you that confounding main effects with 2-factor interactions is not something you want to do if you can avoid it.

For each design, the number of runs required is shown in the far left column. For this experiment, the Full design requires 128 runs, just as we calculated above.The four other designs require fewer runs (64, 32, 16, and 8), but have lower resolutions.

Choose a design based on your available resources and the goal of your experiment. Using this dialog is a bit like comparison shopping to evaluate potential costs vs potential benefits/drawbacks.

For a given number of factors, the more runs you perform, the higher resolution you’ll get, but the more "expensive" the experiment becomes.

For example, although performing all 128 runs gives me Full Resolution, which is the best, I just can't afford to perform that many runs. Using half as many runs (64), I can get Resolution VII, which is still in the green range. But even that design is too costly for me.

Using 16 or 32 runs, I can get Resolution IV design. At this point, my goal is primarily to screen for significant main effects and simple 2-way interactions, so I'm going to save my pennies and go with a 16-run Resolution IV design. I've knocked down the "price" of my experiment from 128 runs to 16 runs--what a bargain!

Because my chosen design requires only 1/8 the number of runs as the full factorial design, it's called a 1/8 fractional factorial design.

Next time we'll create this 1/8 fractional factorial design in Minitab. We'll see how Minitab sets up the data collection worksheet and indicates confounding patterns in the design.

Unlike Nala, it does it all automatically—and without requiring a hunk of honey-cured ham.

Last week I took a look at some different aspects of overtime in college football. I found that most games only last one overtime period, home field advantage still exists, and that the team that gets the ball 2nd has an advantage.

Now I want to continue the data analysis and ask some more questions. Specifically, how many points do teams combine for in overtime, and does the team that ties the game in regulation carry that momentum to overtime? You can follow along by getting the data here.

The first thing I did was make a bar chart of all the different scoring combinations in overtime periods the last 5 years.

Three points is the most likely outcome, occurring 27.2% of the time. This is because if the first team doesn’t score, the second team only needs a field goal to win. If a game needs another overtime, it’s most likely because both teams scored a touchdown (16.1% of the time). And we see that neither team scoring is very rare, happening only 4.5% of the time.

Penn State and Michigan actually had two scoreless overtimes. So according to these statistics, what were the odds? In this case, we have a binomial distribution (there were only 2 outcomes, neither team scored, or at least one team scored) with 4 trials (since there were 4 overtimes) and an event probability of 0.045 (the odds of neither team scoring). What we want to know is the probability that 2 out of 4 of those overtimes would be scoreless.

So given the fact that there were 4 overtimes (which, as we saw in part 1, is rare to begin with), there was only a 1.1% chance that 2 of those overtimes were scoreless.

But an even more interesting question is, what were Michigan’s chances of winning in the 2 overtimes where Penn State got the ball first and failed to score?

There were 67 cases where the team that started the ball did not score. In 57 of those cases (85%), the team that had the ball 2nd went on to win in that overtime. And of those 57 winning teams, 42 of them (74%) won by kicking a field goal, as opposed to scoring a touchdown.

So after Penn State missed a field goal in the 1st overtime, and fumbled in the 3rd overtime, Michigan had about an 85% chance of winning in each overtime. But they didn’t, and Brady Hoke was crushed for playing conservative both times. However, the numbers show most coaches do the exact same thing and it works out for them a majority of the time. Perhaps you could criticize the play calling in those 2 overtimes (not a single rush attempt for Devin Gardner other than centering the ball?), but playing for a field goal does work most of the time.

On the flip side, the odds that Penn State failed to score in two different overtimes and forced another overtime in both were a slim 0.152 = 2.25%. Nittany Lion fans can consider themselves very lucky.

When it comes to sports commentators, momentum has to be one of their favorite words! From quarter to quarter, drive to drive, or even play to play, announcers are always saying teams are "trying to gain momentum" or keep the other team from getting any. Surely if momentum were real, we would see its effect in college overtime games. The team that scores to tie the game in regulation would have all the momentum, right? Therefore, they should be more likely to win in overtime!

For each game, I recorded whether the team that tied the game in regulation won or lost. Let’s see if the team that scores last in regulation wins more than 50% of the time.

The team that scored last in regulation did win more often, but only by a very slim margin. It doesn’t take a Six Sigma black belt to tell you that the proportion of teams that won is not significantly greater than 50%.

But our quest to find momentum is not lost. The data above does not account for when the team tied the game. Consider the 2008 game between West Virginia and Colorado. West Virginia tied the game with 4:49 left in the 3rd quarter! Nobody scored again until Colorado kicked the game-winning field goal in overtime. Surely any momentum West Virginia gained when they tied the game was gone when overtime started.

So to account for this, I also recorded how many seconds were left when the final score of regulation occurred. Just to let you know, the median value was 54 seconds. So in the 156 overtime games since 2008, over half of them were tied with less than a minute to go in the game.

But to get a larger sample size, I first want to look at teams that tied the game with less than two minutes to go. They had to have gained some momentum for overtime, so let’s see if they won more often!

This table shows us that when a team scored with less than 2 minutes left in the game, they only went on to win 49% of the time. In fact, they were more likely to win if they tied the game with more than 2 minutes to go! That doesn’t say much for momentum.

But I’m not giving up. Maybe the game-tying score has to be really dramatic. Like, less-than-30-seconds-left dramatic. Just when the other team thinks they’re about to win, you completely deflate them and force them to play more football. With them shaking their heads at how close they were to winning, and you fired up that you’re still alive, surely that momentum will carry over to overtime!

Wait, what? Teams that scored in the last 30 seconds went on to win only 41% of the time! It turns out that it doesn’t matter how we slice it, there just doesn’t seem to be any momentum when it comes to college football overtime.

Penn State drove 80 yards in 23 seconds, capping the drive off with the game tying touchdown with 27 seconds left in the game. Nittany Lion receiver Allen Robinson had an amazing catch on that drive, which became Sportscenter’s #1 highlight of the day. The sold-out homecoming crowd of over 100,000 people roared as loud as any stadium in the country. And Penn State used all that momentum to gain 2 yards in 3 plays and miss a field goal on their first possession in overtime.

Momentum? Nope. As Bill Barnwell would say, it looks more like Nomentum.

And that’ll wrap things up! So when it comes to college football overtime, remember that if you want to gain an advantage, you’re better off playing at home and starting off on offense 2nd. And if somebody says a team just gained all the momentum by tying the game in the closing seconds, slap them on the head and tell them they're wrong. Wait, on second thought, don't do that—just show them the statistics instead. After all, you can't argue with facts!

As Halloween is almost here, I'm ready to check out some Halloween statistics. You can have a lot of fun with Minitab on Halloween.

The National Retail Foundation (NRF) released the results of their Halloween Consumer Spending Survey last month. The basics are easy to summarize:

Because we have Minitab, we can dig a little deeper into the data. The NRF gives some information about the proportion of respondents who participate and the proportion of participators who will celebrate with different activities. The proportions for participators are broken down by different age groups. There’s some rounding error, but I used the proportions in the data to estimate the number of celebrators in each age group, the number of celebrants planning to buy candy, and the number of celebrants who plan to hand out candy.

I’m going to use binary logistic regression in Minitab to compare the age groups to each other.

Age group

Celebrants

Buy Candy

Give Candy

18-24

385

353

213

25-34

575

543

350

35-44

959

927

669

45-54

400

390

316

55-64

463

450

373

65+

676

643

558

First, notice that the 45-54 age group is not printed in the table. This group had the highest proportion of celebrants who said they would buy candy, so I made them the reference level that Minitab compares the other groups to. The p-values show strong evidence that adults younger than 35 are less likely to buy candy than the 45-54 year olds. The statistical evidence is a bit weaker that people over 65 are less likely to buy candy than 45-54 year olds.

The odds ratios are a convenient way to quantify how much less likely an age group is to buy candy than the 45-54 year olds. The smallest odds ratio belongs to the 18-24 year old group, who are only 0.28 times as likely to buy candy as the 45-54 year olds.

For celebrants planning to hand out candy, I kept the same reference level, 45-54 year olds. This time, the p-values show that the group of younger people who are statistically different from the 45-54 year olds extends all the way to 44. The older groups are statistically similar to the 45-54 year old group.

The biggest difference in the odds ratios is still the 18-24 year old group, who are only 0.33 times as likely to pass out candy as the 45-54 year olds.

It’s fairly clear that there are statistical differences between the different age groups. In general, being older makes you more likely to buy candy and more likely to pass out candy. But the biggest mystery Halloween mystery of all becomes apparent when you look across categories of activities. If 97.4% of 45-54 year olds are going to buy candy, and only 78.9% are going to hand that candy out, what are the other 18.5% planning to do with it? These data won’t answer that question for us, believe it, or not.

Candy is always a big part statistics, and Halloween, for me. If your preference is for the macabre, Check out Eston's chi-square analysis of how gender relates to mode of death in the Halloween and Friday the 13th horror franchises!

In my previous post, I highlighted recent academic research that shows how the presentation style of regression results affects the number of interpretation mistakes. In this post, I present four tips that will help you avoid the more common mistakes of applied regression analysis that I identified in the research literature.

I’ll focus on applied regression analysis, which is used to make decisions rather than just determining the statistical significance of the predictors. Applied regression analysis emphasizes both being able to influence the outcome and the precision of the predictions.

Before beginning the regression analysis, you should already have an idea of what the important variables are along with their relationships, coefficient signs, and effect magnitudes based on previous research. Unfortunately, recent trends have moved away from this approach thanks to large, readily available databases and automated procedures that build regression models.

If you want see the problem with data mining in action, simply create a worksheet in Minitab Statistical Software that has 101 columns, each of which contains 30 rows of random data, or use this worksheet. Then, perform stepwise regression using one column as the response variable and all of the others as the potential predictor variables. This simulates dredging through a data set to see what sticks.

The results below are for the entirely random data. Each column in the output shows the model fit statistics for the first 5 steps of the stepwise procedure. For five predictors, we got an R-squared of 84.23% and an adjusted R-squared of 80.12%! The p-values (not shown) are all very low, often less than 0.01!

While stepwise regression and best subsets regression have their place in the early stages, you need more reason to include a predictor variable in a final regression model than just being able to reject the null hypothesis.

This model is too complex. Read why here.

While it may seem reasonable that complex problems require complex models, many studies show that simpler models generally produce more precise predictions. How simple? In many cases, three predictor variables are sufficient.

Start simple, and only make the model more complex as needed. Be sure to confirm that the added complexity truly improves the precision. While complexity tends to increase the model fit (r-squared), it also tends to lower the precision of the predictions (wider prediction intervals).

I write more about this tradeoff and how to include the correct number of variables in my post about adjusted and predicted r-squared.

This statistical truth seems simple enough. However, in regression analysis, people often forget this rule. You can have a well-specified model with significant predictors, a high r-squared, and yet you might only be uncovering correlation rather than causation!

Regression analysis outside of an experimental design is not a good way to identify causal relationships between variables.

In some cases, this is just fine. Prediction doesn’t always require a causal relationship between predictor and response. Instead, a proxy variable that is simply correlated to the response, and is easier to obtain than a causally connected variable, might produce adequate predictions.

However, if you want to affect the outcome by setting predictor values, you need to identify the truly causal relationships.

To illustrate this point, it has been hard for studies that don’t use randomized controlled trials to determine whether vitamins improve health, or if vitamin consumption is simply correlated to healthy habits that actually improve health (read my post). Put simply, if vitamin consumption doesn’t cause good health, then consuming more vitamins won’t improve your health.

Confidence intervals and statistical significance provide consistent information. For example, if a statistic is significantly different from zero at the 0.05 alpha level, you can be sure that the 95% confidence interval does not contain zero.

While the information is consistent, it changes how people use the information. This issue is similar to that raised in my previous post, where presentation style affects interpretation accuracy. A study by Cumming found that reporting significance levels produced correct conclusions only 40% of the time, while including confidence intervals yielded correct interpretations 95% of the time.

For more on this, read my post where I examine when you should use confidence intervals, prediction intervals, and tolerance intervals.

For a good regression analysis, the analyst:

• Uses large amounts of trustworthy data and a small number of predictors that have well established causal relationships.
• Uses sound reasoning for including variables in the model.
• Brings together different lines of research as needed.
• Effectively presents the results using graphs, confidence intervals, and prediction intervals in a clear manner that ensures proper interpretation by others.

Conversely, in a less rigorous regression analysis, the analyst:

• Uses regression outside of an experiment to search for causal relationships.
• Falls into the trap of data-mining because databases provide a lot of convenient data.
• Includes a variable in the model simply because he can reject the null hypothesis.
• Uses a complex model to increase the r-squared value.
• Reports only the standard statistics of coefficients, p-values, and r-squared values, even though this approach tends to produce inaccurate interpretations even among experts.

________________________________

Armstrong J., Illusions in Regression Analysis, International Journal of Forecasting, 2012 (3), 689-694.

Cumming, G. (2012), Understanding the New Statistics: Effect Sizes, Confidence Intervals, and Meta Analysis. New York: Routledge.

Ord, K. (2012), The Illusion of Predictability: A call to action, International Journal of Forecasting, March 5, 2012.

Zellner, A. (2001), Keep it sophisticatedly simple. In Keuzenkamp, H. & McAleer, M. Eds. Simplicity, Inference, and Modelling: Keeping it Sophisticatedly Simple. Cambridge University Press, Cambridge.

It’s back! World Quality Month will be celebrated this month, and the American Society for Quality is again heading up this year’s festivities.

Quality is definitely our “thing” here at Minitab, and we’re excited to celebrate World Quality Month with quality professionals worldwide. It’s also a good time to remind our customers that we’re as dedicated as ever to providing the best tools for improving quality.

In honor of World Quality Month, be sure to check out the resources and learning opportunities available on Minitab.com:

Case Studies
We’re very proud to recognize the accomplishments of our customers. The companies that use our products and services come in all sizes, represent all industries, and are located all over the world. Get ideas and read about how others are improving quality.

Tutorials
Learn how to analyze your data faster and more easily with Minitab Statistical Software. All of our "Accessing the Power of Minitab" tutorials include examples and illustrated instructions.

Webinars
We offer several free, live webinars each month on how our software can help you improve quality. Topics include product overviews, introduction to key features, and more.

Videos
View our recorded webcasts on all sorts of topics—from how to get started using Minitab, to how to create control charts and value stream maps.

So I’ve done a lot of talking about what Minitab is doing to celebrate World Quality Month—but what am I personally doing to celebrate? For someone who talks a big game about using quality tools in real-life, I’ve been slacking lately. My goal this month is to use quality tools to sort out a possible major upcoming financial decision my husband and I are facing—whether or not now is the time to buy a new car.

First stop: Quality Companion’s brainstorming tools to draw a fishbone diagram. We’re going to lay out all of the factors—the condition of each of our current cars, mileage, trade-in value, and what kind of monthly payments we can comfortably afford. Of course we both want a new car, but I’m hoping that laying out the facts will make the decision easier and cut down on any potential arguments. Depending on how it goes, I may blog about it!

Are you doing anything special to commemorate World Quality Month? Tell us in the comments section!

For more information about World Quality Month and to view a calendar of events, as well as quality resources and success stories, visit WorldQualityMonth.org. 

by Matthew Barsalou, guest blogger

I recently moved, and right after finishing the less-than-joyous task of unpacking I decided to take and break and relax by playing with Minitab Statistical Software.  

As a data source I used the many quotes I received from moving companies. I'd invited many companies to look around my previous home, and then they would provide me an estimate with the price in Euros as well as an estimate on the amount of goods that would need to be transported. The "amount of goods" estimate was given in boxes. I don’t know what size boxes where referred to, but all the moving companies used boxes as a standard estimate of cubic area.

I had planned on using 35 boxes; most companies told me it would be 110-120 boxes. Since I was not even finished packing books when I had used up the first 50 boxes, I think I can safely assume the movers proved to be generally better at estimating shipping volume than I am.

Let’s suppose I wanted to determine the regression line for the cost of moving and the number of boxes that need to be moved. I rounded the estimates to the nearest 25 and changed the moving company names. Below is a data table with the estimates I received for cost and amount of goods:

Moving Company

Cost Estimate

(in Euro)

Material Estimate

(in Boxes)

Company A

1700

115

Company B

1850

120

Company C

3400

145

Company D

1650

80

Company E

1675

90

Company F

2000

110

Company G

1950

115

 I was a bit suspicious of the estimate from Company C. The young man who gave me that estimate may not have even been born at a time when many of the other estimators where already working in the moving industry, so I wondered about his experience. Had the estimate been different, it may not have stood out, but his estimates were far higher than the others. Part of the reason this estimate was so high may be because he included extra costs for using conveyor outside my window as a labor-saving device.

I would be happy to pay for a labor-saving device that lowers my overall costs, but I was not so happy with extra costs for an expensive labor saving device that actually raised the overall expense.

I suspected Company C was an outlier, so to get a quick look at the situation I entered my data into a Minitab worksheet and created a scatterplot:

In the dialog box, I then selected worksheet column C1 Euros as the Y variable and worksheet column C2 Boxes as the X variable.

The resulting scatter plot is shown below. The red dot in the upper right hand corner is the result for Company C.

I am generally hesitant to discard potential outliers because I may be inadvertently throwing away valuable data, but in this case I decided that the estimates from Company C were just wrong and could throw off my regression model. Therefore, I removed them from the data set.

I then went to Stat > Regression > Regression… as depicted:

In the dialog box, I selected worksheet column C1 euros as the Response and worksheet column C2 Boxes as the predictor.

 Minitab produced the following output:

The regression equation is

Euros = 1,240 + 5.19 Boxes

This means the cost in Euros is equal to 1,240 plus 5.19 times the number of boxes. Using the resulting regression equation I can calculate the cost in Euro for any given number of boxes. For example, 100 boxes should cost: 1,240 + 5.19 x (100) = 1,759 Euro.

There are some things to keep in mind when performing regression. This is a statistical calculation based on the available data. If my data set (the moving companies) is not be as inclusive as I think it is, the next moving company I contact may not match my sample. For example, two movers with an almost-falling-apart truck would generally charge much less than a luxury moving company that offers far more than just a transportation service.

We also need to be aware of the hazards of extrapolating beyond the data set. Suppose I bought an entire library full of books on statistics. I now have 400 boxes to transport and may be able to get a discount from a moving company that is happy to have such a large, but easy contract. The move may take a few trucks, but pre-packed books are faster to move than boxes full of fine china or large furniture items that need to be disassembled, and the price estimate would reflect this.

I am rather certain that this regression model will fall apart on the low side. The cost should go down as the number of boxes to transport is decreased; however, contrary to what the regression model may indicate, I find it improbable that a moving company would give the same proportional rate to transport just one box. According to the regression model the cost for just one box is: 1,240 + 5.19 x (1) = 1,245.19 Euro. There is far less labor involved in the transport of only one box. The moving company does not need to supply a driver and four people for carrying boxes, so the estimate may actually be much lower.

Unfortunately, Minitab can’t tell us that the biggest expense in the transportation of one box would be the moving trucks’ fuel, so a moving company is not the type of company to use when transporting only one box!  This is an example of why process knowledge is so important: if you didn't know alternative types of transport were available, you wouldn't know the moving company was a poor choice for shipping one box! 

Who knows, maybe some day Minitab will be able to do the all of the thinking for us!  For now, whether calculating a regression model for costs/boxes or sales price/units, some knowledge of statistics and its limitations is still needed. Regression is an excellent way to make predictions and Minitab makes this easier; but it does not remove the need to have an understanding of the statistics being used.

Would you like to publish a guest post on the Minitab Blog? Contact publicrelations@minitab.com.

Moving box image by Hsing Wei, used under Creative Commons 2.0 license.

A colleague of mine at Minitab, Cheryl Pammer, was recently featured in "A Statistician's Journey," a monthly feature that appears in the print and online versions of the American Statistical Association's AMSTAT News magazine.  

Each month, the magazine asks ASA members to talk about the paths they took to get to where they are today. Cheryl is a "user experience designer" at Minitab. In other words, she's one of the people who help determine how our statistical software does what it does, and tries to make it as helpful, useful, and beneficial as possible. Cheryl is always looking for ways to make our software better so that the people who use it can get more out of their data. 

It's exciting that one of Minitab's statisticians was selected to be profiled, and it's always great when someone whom you know does great work receives some public recognition. But I was particularly interested to see what Cheryl had to say about her work at Minitab -- you know, what it is that motivates her to come into work every day.  Here's how she answered that question: 

A tremendous amount of data exists out there, most of it being analyzed without the help of a degreed or accredited statistician. As a designer of statistical software, my main goal is to promote good statistical practices by presenting appropriate choices to the software user and displaying results in a meaningful way. It is exciting to know that the work I do makes a difference in how thousands of people will use and interpret the data they have.

This really struck home with me. Before I joined Minitab, I worked in higher education as an editor, and I oversaw a magazine that covered the work of scientists from a wide variety of fields. I needed to keep track of circulation and many other metrics, and I did not have a clue how to do it properly. I muddled through it using spreadsheets, and even pencil and paper, but I never had confidence in my conclusions and always had a nagging suspicion that I'd probably missed something critical...something that would either invalidate any good news I'd seemed to find, or would make data that already didn't look so good even worse. 

Since then, I've come a long way in terms of analyzing data, even completing a graduate degree in applied statistics.  But I remember vividly how it felt to look at a collection of numbers and not have the vaguest idea how to start making sense of it. And I remember seeing research results and analyses and wishing they'd been expressed in some way that was easier to understand if, like me back then, you didn't have a good background in statistics.  

And that's where the Assistant comes in. People like Cheryl designed the Assistant in Minitab Statistical Software to help people like me understand data analysis.  When you select the type of analysis you want to do in the Assistant -- like graphical analysis, hypothesis testing, or regression -- the Assistant guides you through it by asking you questions.  

For example, if you're doing a hypothesis test, the Assistant will ask you whether you have continuous or categorical data. Don't know the difference?  Click on a button and the Assistant will give you a crystal-clear explanation so you can make the right choice. Back when I was trying to figure out how our science magazine was performing, this would have saved me a lot of wasted time.  It also would have made me a lot more sure about the conclusions I reached.  

Describing it doesn't really do it justice, though.  Here's a video that provides a quick overview of how the Assistant works:  

Even though I've learned a lot about analyzing data since my magazine days, I still find the Assistant tremendously helpful because:

A.  I'm usually sharing the results of an analysis with people who don't know statistics, and
B. The Assistant explains those results in very clear language that anyone can understand. 

For ANOVA, capability analysis, measurement systems analysis, and control charts, the Assistant's output includes not only graphs and bottom-line results, but also report cards and summaries that tell you how well your data meet the the statistical assumptions for the analysis, and whether there are trouble spots or specific data points you should take a look at.  So if you're explaining the results to your boss, your colleagues, or a group of potential clients, you can present the information and provide assurance that the analysis has followed good statistical practice.  

We've heard the same thing from consultants, Six Sigma black belts, researchers, and other people who know how to wrangle a data set:  these experts certainly can do their analysis without the Assistant, but the Assistant makes it easier to communicate what the analysis means and how reliable it is, both of which are critical. 

Which brings us back to Cheryl -- and her colleagues in Minitab's software development teams -- who work so hard to make data analysis accessible to more people. H. G. Wells famously said "Statistical thinking will one day be as necessary a qualification for efficient citizenship as the ability to read and write."  In a world where so much data is so readily available to all of us, it's an honor to be part of a team working to make statistical thinking and the ability to make better use of that data more available. 

Nala, our 6-year-old golden retriever, loves her dogma. That's her sitting in front of church on Sunday morning.

But she's not crazy about her catechism. For example, she doesn't always dutifully follow the "Lay Down" commandment.  

What factors may be influencing her response? We're performing a DOE screening experiment to find out.

In this post, we'll use Minitab Statistical Software to

• Create the design for the experiment
• Determine the confounding pattern for this design
• Set up the data collection worksheet

In the previous post, we used the Display Design dialog box in Minitab to compare  2-level factorial designs for an experiment with 7 factors. Based on available resources, we decided to use a 1/8 fractional factorial design.

To create this design in Minitab, choose Stat > DOE > Factorial > Create Factorial Design. In Number of factors, enter 7. Click Designs and choose the 1/8 fraction design as shown below.

Click OK, then click Factors. Enter the name of each factor in the experiment, its type (numeric or text), and its low and high levels. Notice each separate factor is designated with a letter of the alphabet: A, B, C, D, and so on.

Click OK in each dialog box. Minitab will automatically do two things: Summarize the alias structure of your design and set up a randomized data collection worksheet for the experiment.

To see the summary of the design you just created, including its alias structure, look in the Minitab Session window (Tip: to make the Session window the active window, press Ctrl+M).

At the top, you'll see a summary of the number of factors, the number of runs, the type of factorial, its resolution, and other details of your design.

Fractional Factorial Design

Factors:   7   Base Design:         7, 16   Resolution:   IV
Runs:     16   Replicates:              1   Fraction:    1/8
Blocks:    1   Center pts (total):      0

At the bottom, you'll see clumps of letters separated by plus signs.

Alias Structure

I + ABCE + ABFG + ACDG + ADEF + BCDF + BDEG + CEFG

A + BCE + BFG + CDG + DEF + ABCDF + ABDEG + ACEFG
B + ACE + AFG + CDF + DEG + ABCDG + ABDEF + BCEFG
C + ABE + ADG + BDF + EFG + ABCFG + ACDEF + BCDEG
D + ACG + AEF + BCF + BEG + ABCDE + ABDFG + CDEFG
E + ABC + ADF + BDG + CFG + ABEFG + ACDEG + BCDEF
F + ABG + ADE + BCD + CEG + ABCEF + ACDFG + BDEFG
G + ABF + ACD + BDE + CEF + ABCEG + ADEFG + BCDFG
AB + CE + FG + ACDF + ADEG + BCDG + BDEF + ABCEFG
AC + BE + DG + ABDF + AEFG + BCFG + CDEF + ABCDEG
AD + CG + EF + ABCF + ABEG + BCDE + BDFG + ACDEFG
AE + BC + DF + ABDG + ACFG + BEFG + CDEG + ABCDEF
AF + BG + DE + ABCD + ACEG + BCEF + CDFG + ABDEFG
AG + BF + CD + ABDE + ACEF + BCEG + DEFG + ABCDFG
BD + CF + EG + ABCG + ABEF + ACDE + ADFG + BCDEFG
ABD + ACF + AEG + BCG + BEF + CDE + DFG + ABCDEFG

The Alias Structure table can look a bit frightening at first glance. But this quasi "Scrabble board" in your output provides important information about your design.

Each row of the table shows you which effects and interactions in your experiment will be confounded with each other (that is, which effects you won't be able to tell apart) based on the design you've created.

For example, consider the first row of the main table body:

A + BCE + BFG + CDG + DEF + ABCDF + ABDEG + ACEFG

"A" represents the main effect for Factor A, which for this experiment is the tone of voice used to give the Lay Down command. Everything in this row will be confounded with main effect of A:

• The 3-way interactions BCE, BFG, CDG, DEF
• The 5-way interactions ABCDF, ABDEG, ACEFG

The interactions in this row will also be confounded with each other.

What does this mean, in concrete terms? A is in the same row as DEF. Therefore, I won't be able to discriminate between the effect of the tone of voice (A) with the interaction between the hand signal (D), the location (E), and the reward used (F).

This confounding is the "sacrifice" I made when I chose the 1/8 fractional design, with its lower resolution, to reduce the number of runs in my experiment.

If I had created a full factorial design with these 7 factors, there wouldn't even be an aliasing table in the design summary, because all the effects could be estimated separately:

Full Factorial Design

Factors:    7   Base Design:         7, 128
Runs:     128   Replicates:               1
Blocks:     1   Center pts (total):       0

All terms are free from aliasing.

If the alias table shows that potentially important interactions in your experiment are confounded, you may want to create a different design with a higher resolution before you start collecting data.

For this rough screening experiment, my main concern was that the 2-way interactions would not be confounded with the main effects. You can verify that this is the case by looking at the rows in the alias table: None of the rows that contain main effects (A B C D E F G) contain any 2-way interactions (AB AC AD and so on). Therefore, main effects and 2-way interactions can be estimated separately.

Note: Many of the 2-way interactions are confounded with each other--that could affect the interpretation of the results.

To see the data collection worksheet for the experiment, look in the Data window. (Tip: to make the Data window the active window, press Ctrl+D).

Minitab automatically randomizes the run order to prevent experimental bias, such as possible time-dependent effects. The random order is given by the RunOrder column (C2). The StdOrder column indicates the conventional order of the runs if they are not randomized.

Use the next available empty column for the response data. In this example, I've labeled column c12 Response.

As you perform the experiment, just follow the run order in the worksheet and record the measured response for each run.

In this experiment, I'll record the number of seconds (to the nearest hundredth of a second) that it takes Nala to lay down after being given the command.

Will the sweet aroma of honey-cured ham inspire Nala to bring her belly to the floor?  And will adding the "cush factor" (carpeting) make her lay down even more quickly?

Find out in the next post....