Minitab | MinitabBlog posts and articles about using Minitab software in quality improvement projects, research, and more.
http://blog.minitab.com/blog/minitab/rss
Sat, 05 Sep 2015 03:39:48 +0000FeedCreator 1.7.3Should Virginia Tech Always Onside Kick Against Ohio State?
http://blog.minitab.com/blog/the-statistics-game/should-virginia-tech-always-onside-kick-against-ohio-state
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/b1f450e1cb8baf91733edabf547941cb/kickoff_tee.jpg" style="float: right; width: 211px; height: 279px;" /></p>
<p>In 2007, the Crayola crayon company encountered a problem. Labels were coming off of their crayons. Up to that point, Crayola had done little to implement data-driven methodology into the process of manufacturing their crayons. But that was about to change. An elementary data analysis showed that the adhesive didn’t consistently set properly when the labels were dry. Misting crayons as they went through the labeling machines solved the problem, and that project’s success prompted Crayola to <a href="http://www.minitab.com/en-us/Case-Studies/Crayola/">expand the use of statistical methods</a>. The following year, the company’s initial wave of Six Sigma projects saved more than $1.5 million, and Crayola now relies on a data-driven culture of continuous improvement to enhance the quality of their crayons.</p>
<p>But statistical success stories don’t have to be confined to the business world. Baseball has already proven that the advancement of statistical analyses can revolutionize a sport. Basketball and hockey teams are also starting to look into how analytics can improve the quality of their team. The only sport that seems to be lagging behind is football. But all it will take is one team to have success implementing statistics into their game plan, and others will surely follow.</p>
<p>Are you listening, Virginia Tech?</p>
<p>The Hokies are playing the defending National Champion Ohio State Buckeyes on Monday night. Ohio State is a double digit favorite in the game, and a good part of the reason is because their offense is great.</p>
<p>Just how great? I’m glad you asked. </p>
Ohio State’s Offense the Previous Two Seasons
<p>Recently, I <a href="http://blog.minitab.com/blog/the-statistics-game/big-ten-4th-down-calculator%3A-creating-a-model-for-expected-points">created a regression model</a> that can calculate the number of points a football team is expected to score based on their field position and whether they are playing at home or on the road. The data comes from every Big Ten conference game the last two seasons. To no surprise, the farther you are from the end zone, the fewer points you’re expected to score. And you’re expected to score fewer points on the road than at home.</p>
<p>Since Ohio State is playing at Virginia Tech, let’s focus on teams playing on the road. I took the data from my previous analysis, removed drives by the home team, and I removed Ohio State. Here is a fitted line plot of the data:</p>
<p><img alt="Fitted Line Plot" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/9bad7db0ccc48c8559e316afd9eeab03/non_ohio_state_fitted_line_plot.jpg" style="width: 576px; height: 384px;" /></p>
<p>This is exactly what we would expect. The farther you are from the end zone, the fewer points you’re expected to score. Now let’s make the same plot for all of Ohio State’s road drives the previous two seasons.</p>
<p><img alt="Fitted line plot" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/02a3cf45867443bc777a5f43ec30e7d2/ohio_state_fitted_line_plot.jpg" style="width: 576px; height: 384px;" /></p>
<p>You can start Ohio State anywhere on the field, and odds are they are going to score you on before you score on them. Start Ohio State on their own 1 yard line, and the model says their expected points are still 2.6 (compared to a value of -1.8 for the other Big Ten teams). But the most impressive part is that the data included 20 drives that Ohio State started inside their own 20 yard line. Here is a bar chart of the next score:</p>
<p><img alt="Bar chart" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/4a988f5e8f698221bdef689e527ea719/bar_chart.jpg" style="width: 576px; height: 384px;" /></p>
<p>Even backed up in their own territory and playing on the road, Ohio State was the next team to score 75% of the time, with almost all of those scores being touchdowns. With an offense that good, it really begs the question.</p>
<p>Why even give them the ball?</p>
The Onside Kick
<p>The onside kick is mainly used at the end of games when a losing team is desperate to get the ball back before the clock runs out. But there is no rule saying you can’t do an onside kick early in the game. Or even do an onside kick every time you have a kickoff.</p>
<p>Would it actually benefit Virginia Tech to attempt an onside kick every time? Let’s calculate the percentage of kicks they would need to recover to make it worth it. If you kick the ball deep, most drives will start from the 25 yard line. So we’ll use that for Ohio State’s starting position on a deep kick. The model above shows that we can expect Ohio State to score about 3.4 points starting from their own 25 yard line.</p>
<p>An onside kick needs to travel at least 10 yards in order for the kicking team to legally recover it. Kickoffs are from the 35 yard line, so if Virginia Tech recovers they will be 55 yards from the end zone, and if Ohio State recovers they will be 45 yards from the end zone. This gives Virginia Tech an expected point value of 2.5 points (<a href="http://blog.minitab.com/blog/the-statistics-game/big-ten-4th-down-calculator%3A-creating-a-model-for-expected-points">calculated using this regression model</a>) and Ohio State would have an expected point value of 4.5 points. Now we can use algebra to calculate the break-even success rate, where p is the probability that Virginia Tech recovers the onside kick.</p>
<p align="center">-3.4 = 2.5*p – 4.5*(1-p)</p>
<p align="center">-3.4 = 2.5*p – 4.5 + 4.5*p</p>
<p align="center">1.1 = 7*p</p>
<p align="center">p = 1.1/7 = .157 = <strong>16%</strong></p>
<p>So if Virginia Tech can recover the onside kick about 16% of the time, their total expected points will be the same as if they were to kick deep. If they can recover a higher percentage, then they should be attempting an onside kick every time.</p>
<p>I couldn’t find any good data on college onside kicks, but in the NFL, non-surprise onside kick recovery rates are approximately 20%. The success rate in college football should be pretty similar. And hey, 20% > 17%, so onside kick every time, right?</p>
<p>Not so fast.</p>
<p>Anytime you perform a data analysis, it’s important to know where your data came from. In this case, our expected points for Virginia Tech came from data from all Big Ten teams. So really, it’s what we would expect an average Big Ten offense to score against an average Big Ten defense. Last year, according to <a href="http://www.footballoutsiders.com/stats/ncaa">Football Outsiders S&P+ ratings</a>, Virginia Tech ranked 85th in offense, and Ohio State ranked 11th in defense. So when Virginia Tech has the ball, it will really be closer to a below average Big Ten offense going up against an above average Big Ten defense. This means our estimate for Virginia Tech’s expected points after a successful onside kick is probably a little too high.</p>
<p>Additionally, Virginia Tech had the #10 ranked defense last year and almost everybody returns from that defense this year. Our model for Ohio State's expected points is based off of an average Big Ten defense. So we should lower Ohio State’s expected points for both a deep kickoff and an unsuccessful onside kick. But how much we should decrease these values by is hard to quantify. So let's look at different values and see how it affects the break-even success rate. In the previous equation, I decreased both Ohio State's expected points and Virginia Tech's expected points by the values in the following table.</p>
Decrease Expected Points By
New Break-Even Success Rate
0.5
18%
1
22%
1.5
27.5%
2
37%
<p>We see that the larger the effect of the better than average defenses and Virginia Tech's poor offense, the more the statistics side with not attempting an onside kick every time.</p>
To Onside Kick or Not to Onside Kick?
<p>With the uncertainty of how much the defenses and Virginia Tech's offense affect the numbers, we can’t definitively say that Virginia Tech should onside kick every single time. However, this data analysis has shown enough that we can definitely say one thing.</p>
<p>Virginia Tech should attempt an onside kick……..<em>at least once</em>.</p>
<p>The 20% value we used for onside kick recoveries was for non-surprise onside kicks. However, in the NFL <em>surprise</em> onside kicks succeed close to 60% of the time. And the first onside kick Virginia Tech attempts will certainly take Ohio State by surprise. And the second one probably will too. Maybe even a third. But eventually Ohio State would adjust their formation of their kick return team and the success rate would drop to that 20% value. </p>
<p>So if Virginia Tech wants to maximize their chances of winning, they really should attempt at least one onside kick. Until Ohio State adjusts their kick return team, anytime Virginia Tech kicks the ball deep they’re just giving away free points.</p>
Ohio State at Home
<p>I’d be remiss if I didn’t share one last thing about the Ohio State offense. The previous data for them only included Big Ten games played on the road. Impressive as it was, their offense gets even better when playing at home. How much better? Well, it doesn’t matter where they start with the football</p>
<p>Like, at all.</p>
<p><img alt="Fitted Line Plot" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/ca45ff40d22b1fa47f923d3a471cf128/ohio_state_at_home.jpg" style="width: 576px; height: 384px;" /></p>
<p>Virginia Tech can be thankful they’re playing Ohio State at home. If it was at Columbus, this data analysis not only would have concluded that they should onside kick every time, but it would have said never to punt either!</p>
<p>Before <a href="http://www.minitab.com/en-us/Case-Studies/Crayola/" target="_blank">Crayola</a> fully embraced the widespread use of statistical analyses, the vice president of manufacturing said he saw people spend more time trying to figure out how to come up with data that supported their thesis rather than letting the data reveal where they needed to go. It’s that kind of thinking that is too prevalent in football today. If you were to never punt against Ohio State, the first time the Buckeyes scored a touchdown after you failed on 4th down, people would point to that as proof that your strategy doesn’t work. But the numbers speak for themselves. Had you punted, Ohio State probably would have scored a touchdown anyway.</p>
<p>So take note Hawaii, Northern Illinois, Western Michigan, Maryland, Penn State, Minnesota, and Michigan State. If you go into Columbus and willingly give Ohio State possession of the football, you’re doing nothing but hurting your football team. Go ahead and ignore the data if you want. Just know that if you do, you might end up with some defective crayons.</p>
Fri, 04 Sep 2015 12:28:00 +0000http://blog.minitab.com/blog/the-statistics-game/should-virginia-tech-always-onside-kick-against-ohio-stateKevin RudyThe Danger of Overfitting Regression Models
http://blog.minitab.com/blog/adventures-in-statistics/the-danger-of-overfitting-regression-models
<p><img alt="Example of an overfit regression model" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/a284ba0ea6c3bf8f6dcec4e7a9d5f1f2/overfitlineplotnoequ.gif" style="float: right; width: 300px; height: 200px;" />In regression analysis, overfitting a model is a real problem. An overfit model can cause the <a href="http://blog.minitab.com/blog/adventures-in-statistics/how-to-interpret-regression-analysis-results-p-values-and-coefficients" target="_blank">regression coefficients, p-values</a>, and <a href="http://blog.minitab.com/blog/adventures-in-statistics/regression-analysis-how-do-i-interpret-r-squared-and-assess-the-goodness-of-fit" target="_blank">R-squared</a> to be misleading. In this post, I explain what an overfit model is and how to detect and avoid this problem.</p>
<p>An overfit model is one that is too complicated for your data set. When this happens, the regression model becomes tailored to fit the quirks and random noise in your specific sample rather than reflecting the overall population. If you drew another sample, it would have its own quirks, and your original overfit model would not likely fit the new data.</p>
<p>Instead, we want our model to approximate the true model for the entire population. Our model should not only fit the current sample, but new samples too.</p>
<p>The fitted line plot illustrates the dangers of overfitting regression models. This model appears to explain a lot of variation in the response variable. However, the model is too complex for the sample data. In the overall population, there is no real relationship between the predictor and the response. You can read about the model <a href="http://blog.minitab.com/blog/adventures-in-statistics/multiple-regession-analysis-use-adjusted-r-squared-and-predicted-r-squared-to-include-the-correct-number-of-variables" target="_blank">here.</a></p>
Fundamentals of Inferential Statistics
<p>To understand how overfitting causes these problems, we need to go back to the basics for inferential statistics.</p>
<p>The overall goal of inferential statistics is to draw conclusions about a larger population from a random sample. Inferential statistics uses the sample data to provide the following:</p>
<ul>
<li>Unbiased estimates of properties and relationships within the population.</li>
<li><a href="http://blog.minitab.com/blog/adventures-in-statistics/understanding-hypothesis-tests%3A-why-we-need-to-use-hypothesis-tests-in-statistics" target="_blank">Hypothesis tests</a> that assess statements about the entire population.</li>
</ul>
<p>An important concept in inferential statistics is that the amount of information you can learn about a population is limited by the sample size. The more you want to learn, the larger your sample size must be.</p>
<p>You probably understand this concept intuitively, but here’s an example. If you have a sample size of 20 and want to estimate a single population mean, you’re probably in good shape. However, if you want to estimate two population means using the same total sample size, it suddenly looks iffier. If you increase it to three population means and more, it starts to look pretty bad.</p>
<p>The quality of the results worsens when you try to learn too much from a sample. As the number of observations per parameter decreases in the example above (20, 10, 6.7, etc), the estimates become more erratic and a new sample is less likely to reproduce them.</p>
Applying These Concepts to Overfitting Regression Models
<p>In a similar fashion, overfitting a regression model occurs when you attempt to estimate too many parameters from a sample that is too small. Regression analysis uses one sample to estimate the values of the coefficients for <em>all</em> of the terms in the equation. The sample size limits the number of terms that you can safely include before you begin to overfit the model. The number of terms in the model includes all of the predictors, interaction effects, and polynomials terms (<a href="http://blog.minitab.com/blog/adventures-in-statistics/curve-fitting-with-linear-and-nonlinear-regression" target="_blank">to model curvature</a>).</p>
<p>Larger sample sizes allow you to specify more complex models. For trustworthy results, your sample size must be large enough to support the level of complexity that is required by your research question. If your sample size isn’t large enough, you won’t be able to fit a model that adequately approximates the true model for your response variable. You won’t be able to trust the results.</p>
<p>Just like the example with multiple means, you must have a sufficient number of observations for each term in a regression model. Simulation studies show that a good rule of thumb is to have 10-15 observations per term in multiple linear regression.</p>
<p>For example, if your model contains two predictors and the interaction term, you’ll need 30-45 observations. However, if the effect size is small or there is high multicollinearity, you may need more observations per term.</p>
How to Detect and Avoid Overfit Models
<p>Cross-validation can detect overfit models by determining how well your model generalizes to other data sets by partitioning your data. This process helps you assess how well the model fits new observations that weren't used in the model estimation process.</p>
<p><a href="http://www.minitab.com/en-us/products/minitab/" target="_blank">Minitab statistical software</a> provides a great cross-validation solution for linear models by calculating <a href="http://blog.minitab.com/blog/adventures-in-statistics/multiple-regession-analysis-use-adjusted-r-squared-and-predicted-r-squared-to-include-the-correct-number-of-variables" target="_blank">predicted R-squared</a>. This statistic is a form of cross-validation that doesn't require you to collect a separate sample. Instead, Minitab calculates predicted R-squared by systematically removing each observation from the data set, estimating the regression equation, and determining how well the model predicts the removed observation.</p>
<p>If the model does a poor job at predicting the removed observations, this indicates that the model is probably tailored to the specific data points that are included in the sample and not generalizable outside the sample.</p>
<p>To avoid overfitting your model in the first place, collect a sample that is large enough so you can safely include all of the predictors, interaction effects, and polynomial terms that your response variable requires. The scientific process involves plenty of research before you even begin to collect data. You should identify the important variables, the model that you are likely to specify, and use that information to estimate a good sample size.</p>
<p>For more about the model selection process, read my blog post, <a href="http://blog.minitab.com/blog/adventures-in-statistics/how-to-choose-the-best-regression-model">How to Choose the Best Regression Model</a>.</p>
Regression AnalysisStatisticsStatistics HelpThu, 03 Sep 2015 12:00:00 +0000http://blog.minitab.com/blog/adventures-in-statistics/the-danger-of-overfitting-regression-modelsJim FrostMonitoring Rare Events with G Charts
http://blog.minitab.com/blog/michelle-paret/monitoring-rare-events-with-g-charts
<p style="line-height: 20.7999992370605px;"><span style="line-height: 1.6;">Rare events inherently occur in all kinds of processes. In hospitals, there are medication errors, infections, patient falls, ventilator-associated pneumonias, and other rare, adverse events that cause prolonged hospital stays and increase healthcare costs. <img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/4caf0e254de4f9a8373137eddabb1b11/hospital_bed.jpg" style="margin: 10px 15px; float: right; width: 200px; height: 200px;" /></span></p>
<p style="line-height: 20.7999992370605px;">But rare events happen in many other contexts, too. Software developers may need to track errors in lines of programming code, or a quality practitioner may need to monitor a low-defect process in a high-yield manufacturing environment. Accidents that occur on the shop floor and aircraft engine failures are also rare events, ideally.</p>
<p style="line-height: 20.7999992370605px;">Whether you’re in healthcare, software development, manufacturing or some other industry, statistical process control is an important component of quality improvement. <span><a href="http://blog.minitab.com/blog/understanding-statistics/control-chart-tutorials-and-examples">Using control charts</a></span>, we can graph these rare events and monitor a process to determine if it’s stable or if it’s out of control and therefore unpredictable and in need of attention.</p>
The G Chart
<p style="line-height: 20.7999992370605px;">There are many different types of control charts available, but in the case of rare events, we can use Minitab <a href="http://www.minitab.com/products/minitab">Statistical Software</a> and the G chart to assess the stability of our processes. The G chart, based on the geometric distribution, is a control chart designed specifically for monitoring rare events.</p>
<p style="line-height: 20.7999992370605px;">G charts are typically used to plot the number of days between rare events. They also can<span style="line-height: 20.7999992370605px;"> be used to plot the number of opportunities between rare events. </span></p>
<p style="line-height: 20.7999992370605px;"><span style="line-height: 20.7999992370605px;">For example, suppose we want to monitor heart surgery complications. We can use a G chart to graph the number of successful surgeries that were performed in between the ones that involved complications.</span></p>
<p style="line-height: 20.7999992370605px;">The G chart is simple to create and use. To produce a G chart, all you need is either the dates on which the rare events occurred or the number of opportunities between occurrences.</p>
Advantages of the G Chart
<p style="line-height: 20.7999992370605px;"><span style="line-height: 20.7999992370605px;"> </span><span style="line-height: 20.7999992370605px;">In addition to its simplicity, this control chart also offers greater statistical sensitivity for monitoring rare events than its traditional counterparts.</span></p>
<p style="line-height: 20.7999992370605px;">Because rare events occur at very low rates, traditional control charts like the <a href="http://blog.minitab.com/blog/the-statistics-of-science/p-and-u-charts-and-limburger-cheese-a-smelly-combination">P chart</a> are typically not as effective at detecting changes in the event rates in a timely manner. Because the probability that a given event will occur is so low, considerably larger subgroup sizes are required to create a P chart and abide by the typical rules of thumb. In addition to the arduous task of collecting more data, this creates the unfortunate circumstance of having to wait longer to detect a shift in the process. Fortunately, G charts do not require large quantities of data to effectively detect a shift in a rare events process.</p>
<p style="line-height: 20.7999992370605px;">Another advantage of using the G chart to monitor your rare events is that it does not require that you collect and record data on the total number of opportunities, while P charts do.</p>
<p style="line-height: 20.7999992370605px;">For example, if you’re monitoring medication errors using a P chart, you must count the total number of medications administered to each and every patient in order to calculate and plot the proportion of medication errors. To create a G chart however, you just need to record the dates on which the medication errors occurred. Note the G chart does assume that the opportunities, or medications administered in this example, are reasonably constant.</p>
<p style="line-height: 20.7999992370605px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/6060c2db-f5d9-449b-abe2-68eade74814a/Image/55f48945cd14690facfb78ca574f8292/g_chart_data.jpg" style="line-height: 20.7999992370605px; margin: 10px 15px; float: right; width: 128px; height: 285px;" /></p>
Creating a G Chart
<p style="line-height: 20.7999992370605px;">Each year, nosocomial (hospital-acquired) infections cause an exorbitant number of additional hospital days nationally, and, <span style="line-height: 1.6;">unfortunately, a considerable number of deaths. Suppose you work for a hospital and want to monitor these infections so you can promptly detect changes in your process and react appropriately if it goes out of control.</span></p>
<p style="line-height: 20.7999992370605px;">In Minitab, you first need to input the dates when each of the nosocomial infections occurred. Then to create a G chart and plot the elapsed time between infections, select <strong>Stat > Control Charts > Rare Event Charts > G</strong>.</p>
<p style="line-height: 20.7999992370605px;"><span style="line-height: 1.6;">In the dialog box, you can input either the 'Dates of events' or the 'Number of opportunities' between adverse events. In this case, we have the date when each infection occurred</span><span style="line-height: 1.6;"> so we can use 'Dates of events' and specify the </span><em style="line-height: 1.6;">Infections</em><span style="line-height: 1.6;"> column.</span></p>
<span style="line-height: 1.6;">Interpreting a G Chart</span>
<p style="line-height: 20.7999992370605px;">Minitab plots the number of days between infections on the G chart. Points above the upper control limit (UCL) are desirable as they indicate an extended period of time between events. Points near or below the lower control limit (LCL) are undesirable and indicative of a shortened time period between events.</p>
<p style="line-height: 20.7999992370605px;">Minitab flags any points that extend beyond the control limits, or fail any other tests for special causes, in red.</p>
<p style="line-height: 20.7999992370605px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/6060c2db-f5d9-449b-abe2-68eade74814a/Image/2264a77810637418e6ea6b0897119aa9/g_chart_of_infections_w1024.jpeg" style="margin: 10px 15px; width: 500px; height: 329px;" /></p>
<p style="line-height: 20.7999992370605px;">The G chart above shows that this hospital went nearly 2 months without an infection. Therefore, you should try to learn from this fortunate circumstance. However, you can also see that the number of days between events has recently started to decrease, meaning the infection rate is increasing, and the process is out of control. You should therefore investigate what is causing the recent series of infections.</p>
Monitoring Rare Events with T Charts
<p style="line-height: 20.7999992370605px;">While G charts are used to monitor the days or opportunities between rare events, you can use a T chart if your data are instead continuous. </p>
<p style="line-height: 20.7999992370605px;">For example, if you have recorded both the dates and time of day when rare events occurred, you can assess process stability using <strong>Stat > Control Charts > Rare Event Charts > T</strong>.</p>
<p style="line-height: 20.7999992370605px;">As more and more organizations embrace and realize the benefits of quality improvement, they will encounter the <em>good</em> problem of increased cases of rare events. As these events present themselves with greater frequency, practitioners across industries can rely on Minitab and the G and T charts to effectively monitor their processes and detect instability when it occurs.</p>
Data AnalysisHealth Care Quality ImprovementLean Six SigmaQuality ImprovementSix SigmaStatisticsThu, 03 Sep 2015 04:00:00 +0000http://blog.minitab.com/blog/michelle-paret/monitoring-rare-events-with-g-chartsMichelle ParetGraphing Wastewater in China
http://blog.minitab.com/blog/statistics-and-quality-improvement/graphing-wastewater-in-china
<p><img alt="Map of China" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/d0259f8718e32fb25d74a071119b4180/hvdc_yunnan_guangdong.PNG" style="width: 330px; height: 179px; float: right;" />Newsweek's recent article, <a href="http://www.newsweek.com/2015/08/21/environmental-crisis-your-closet-362409.html" target="_blank">The Environmental Disaster in Your Closet</a>, led me (through <a href="http://www.greenpeace.org/international/en/campaigns/detox/fashion/detox-catwalk/" target="_blank">Greenpeace's Detox Catwalk</a>) to an interesting new data set on the web. Since I like <a href="http://blog.minitab.com/blog/statistics-and-quality-improvement/3-ways-to-clean-up-data-so-you-can-promote-public-dialog" target="_blank">public data</a>, I thought I'd share some graphs I made from the <a href="http://www.ipe.org.cn/En/about/about.aspx" target="_blank">Chinese Institute for Public and Environmental (IPE)</a> affairs global online platform. The IPE website describes that their goal is "to expand environmental information disclosure to allow communities to fully understand the hazards and risks in the surrounding environment, thus promoting widespread public participation in environmental governance," which sounds great to me. Minitab can make it easy to see patterns in different groups of data over time, including environmental data by region across China.</p>
<strong>Total Wastewater is Highest in Guangdong</strong>
<p>Because the IPE’s aim is to expand environmental information disclosure, it has a lot of environmental data. The data sets exist at a number of different levels, including individual facilities and river basins. I started out looking at <a href="http://www.ipe.org.cn/En/pollution/status.aspx" target="_blank">total wastewater by region</a> (8/31/2015). Here’s what that looks like over time:</p>
<p><img alt="Total wastewater is highest in Guangdong" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/c88b80eed9f85bb2597ac4dd90e4df97/total_wastewater.png" style="border-width: 0px; border-style: solid; width: 576px; height: 384px;" /></p>
<p>Most of the lines are close together on this scale, but the top one stands out. This is the line for Guangdong. Because I hadn’t acknowledged the depth of my ignorance about China, I had to do some research to find out whether there was an explanation for why this region would stand out in terms of their wastewater discharge.</p>
<strong>Looking a Little Deeper</strong>
<p>Turns out that Guangdong was the most <a href="http://www.statista.com/statistics/279013/population-in-china-by-region/" target="_blank">populous region in China in 2013</a> and the region with the highest <a href="http://www.sacu.org/provtable.html" target="_blank">Gross Domestic Product (GDP) in 2009</a>. Either factor could contribute to the amount of wastewater from a region. It turns out that the association between a region’s population in 2013 and the amount of wastewater recorded in the IPE database is fairly strong, which could be one explanation for why Guangdong has so much wastewater. Of course, we'd have to look more closely to establish a causal relationship (See <a href="http://blog.minitab.com/blog/real-world-quality-improvement/common-statistical-mistakes-you-should-avoid" target="_blank">mistake 3</a>). Because the association between population and total wastewater is strong, it’s not surprising that a graph of wastewater per person looks different from the graph of total wastewater. While Guangdong is still one of the higher lines, it’s Shanghai that is the leader in per capita wastewater (using the 2013 population as representative for the years 2004 to 2013).</p>
<p><img alt="Shanghai has the highest amount of wastewater per person" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/fa834affbac6d2a8bd6b680baee81a83/wastewater_per_person.png" style="border-width: 0px; border-style: solid; width: 576px; height: 384px;" /></p>
<p>If you look at the amount of wastewater divided by GDP, it looks like Guangxi will stand out, but a large drop in 2011 puts it closer to the rest of the lines.</p>
<p><img alt="Total wastewater declined more in Guanxi in 2011 than in most other regions." src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/60eb0173559c4df23de94f351766f8fe/wastewater_per_gdp.png" style="width: 577px; height: 385px;" /></p>
<strong>So Many Graphs, So Little Time </strong>
<p>The amount of transparency in society is increasing all the time. The things that you can learn from that data are increasing too. Graphical analysis that Minitab provides can give you quick answers to difficult questions about your data. To see more, take a look at <a href="http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/graphs/basics/graphs-in-minitab/" target="_blank">Which graphs are included in Minitab?</a> for an overview of different ways you can examine your data.</p>
Wed, 02 Sep 2015 16:16:00 +0000http://blog.minitab.com/blog/statistics-and-quality-improvement/graphing-wastewater-in-chinaCody SteeleCalculating the Probability of Converting on 4th Down
http://blog.minitab.com/blog/the-statistics-game/calculating-the-probability-of-converting-on-4th-down
<p><img alt="4th Down" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/388eafa0f8b00f893356dbce728c1677/4th_down.jpg" style="float: right; width: 206px; height: 290px; margin: 10px 15px;" />Imagine a multi-million dollar company that released a product without knowing the probability that it will fail after a certain amount of time. “We offer a 2 year warranty, but we have no idea what percentage of our products fail before 2 years.” Crazy, right? Anybody who wanted to ensure the quality of their product would perform a statistical analysis to look at the <a href="http://blog.minitab.com/blog/statistics-and-quality-data-analysis/reliability-and-survival-the-high-stakes-of-product-performance">reliability and survival</a> of their product. </p>
<p>Now imagine a multimillion-dollar football organization that makes 4th down decisions without knowing the probability that they will convert the 4th down. “We punt on every 4th and 1, but we have no idea what percentage of the time we would keep possession if we went for it.” That's just as crazy, except <em>that seems to be what every football organization does</em>.</p>
<p>But it doesn’t have to be this way. Just like businesses use statistics to improve the quality of their products, football teams should use statistics to improve their chances of winning. So I’m going to use <a href="http://www.minitab.com/products/minitab">Minitab</a>’s binary logistic regression to create a model that will let us know the probability a team has of successfully converting on 4th down.</p>
The Data
<p>We’re continuing our quest to make a <a href="http://blog.minitab.com/blog/the-statistics-game/coming-soon%3A-the-big-ten-4th-down-calculator">Big Ten 4th down calculator</a>, so we’ll start with the same data that we used to create a <a href="http://blog.minitab.com/blog/the-statistics-game/big-ten-4th-down-calculator%3A-creating-a-model-for-expected-points">model for expected points</a>. For every 3rd down in Big Ten conference games the last 2 seasons, I recorded the distance needed to convert, whether the team on offense was at home or away, and whether they converted. I used 3rd down instead of 4th down to increase the sample size. And since the goal on 3rd down is the same as 4th down (convert in one play), the probabilities should be the same.</p>
<p>Speaking of the probabilities, we can use a scatterplot to get an initial look at how distance affects the probability of converting.</p>
<p><img alt="Scatterplot" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/0b03e72f7e28c42fed6b8f4aabe9366e/scatterplot_big_10_prob_vs_dist.jpg" style="width: 576px; height: 384px;" /></p>
<p>The probability of converting decreases pretty consistently as the distance increases. The data does appear to level out a bit between 10 and 15 yards before decreasing again. And there are some outliers at the end of the data, but that is due to small sample sizes.</p>
<p>Now, I do have a different data set with a much larger sample that we can use to eliminate the noise in the data, but first I want to show something with this <a href="//cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/File/5de24a0ccc06069f2b2c02beb2d9e281/expected_points_data.MTW">first data set</a> that we can’t show with the next one.</p>
The Effect of Playing at Home or Away
<p>In the model for expected points, the location of the game affected a team's expected points. Will we see the same effect on the probability of converting on 3rd down? We’ll use binary logistic regression to determine whether Home or Away is a significant term in the model.</p>
<p><img alt="Binary Logistic Regression" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/594dddda4e9f7ee934ccf9061c4f5a84/regression_home_or_away.jpg" style="width: 432px; height: 142px;" /></p>
<p>When it comes to the probability of converting on 3rd down, it doesn’t matter whether the team is home or away. The p-value in the regression analysis is 0.994, which is much greater than the common significance level of 0.05. So why does it matter for expected points, but not here? My best guess is the sample size. Home field advantage has such a small effect on a single play that it doesn’t show up in the 3rd down conversions. But over the course of a multiple play drive (like what we looked at in the expected points model), those small effects add up and the effect of home field advantage becomes noticeable. </p>
<p>So when it comes to a single play, we can ignore home field advantage.</p>
The Data: Part II
<p>To increase our sample size, fellow blogger <a href="http://blog.minitab.com/blog/fun-with-statistics">Joel Smith</a> was kind enough to share data he collected on every college football game from 2006–2012. Because our sample size was so large, we can actually look at 4th downs instead of 3rd downs. Here is a scatterplot of the data:</p>
<p><img alt="Scatterplot" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/9e8dc753b1a9cf0bfd45f93df47b3f3c/scatterplot_of_probability_vs_distance.jpg" style="width: 576px; height: 384px;" /></p>
<p>We see a similar pattern as before. The data decreases until about 10 yards where it levels out a bit before decreasing practically to 0% after 20 yards. And that outlier? Teams were 1 for 3 on 4th and 34. That one success came in the 4th quarter when the team on offense was down by 21 points, so the defense probably no longer had their starters in. That means we should clean up the data to try and remove points like these.</p>
<p>To try and avoid games that were blowouts, I removed any 4th downs where the score differential was greater than 4 touchdowns in the first 3 quarters, and greater than 16 points (3 scores) in the 4th quarter. Finally, I removed any distance greater than 20 yards, since the probability basically drops to 0. This means the decision on anything greater than 4th and 20 should be very easy. Punt or kick a FG unless it’s late in the game and you absolutely need to score a touchdown. So we don't really need to worry about modeling that for our 4th down calculator.</p>
<p>After removing these observations, we still have 11,623 4th downs. <a href="//cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/File/f7360dfae1c5de67594af5d2ced95b55/4th_down_data.MTW">Here's the data I used</a>.</p>
The Final Model
<p>We already saw that it doesn’t matter whether you’re playing at home or on the road, but there is another factor we should take into account. When you get closer to the goal line, the defense has a smaller portion of the field to defend. This might make it harder to convert on 4th down when you have to score a touchdown rather than simply get a first down. So I created a variable to determine whether it was 4th and goal or not to include in the model.</p>
<p>There also appears to be some curvature in the data, so I included the 2nd and 3rd order terms for distance. And lastly, our integers for distance represent the midpoint of the actual distance. For example, on 4th and 4 you could really have to gain anywhere from 3.5 to 4.5 yards. But on 4th and 1, the range is really 0 yards to 1.5 yards. So instead of using the integer 1, I used 0.75.</p>
<p>Now let’s put our data into Minitab and see the results.</p>
<p><img alt="Binary Logistic Regression" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/00dc34aa2041aa23b5f4ce04fa8ec59e/final_model_1.jpg" style="width: 530px; height: 279px;" /></p>
<p><img alt="Binary Logistic Regression" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/7c2dee104803fdbcdcf80df53b544dfb/final_model_2.jpg" style="width: 545px; height: 189px;" /></p>
<p>The p-values for all of our terms are less than 0.05, so we can conclude that they are all significant and keep them in the model. The Deviance R-squared value tells us that 97% of the deviance in the probability of converting on 4th down can be explained by the model. We can now use the model to predict the probability to converting at different distances.</p>
<p align="center"><strong>Distance</strong></p>
<p align="center"><strong>Probability when Goal to go</strong></p>
<p align="center"><strong>Probability when not Goal to go</strong></p>
<p align="center">1*</p>
<p align="center">61%</p>
<p align="center">70%</p>
<p align="center">2</p>
<p align="center">50%</p>
<p align="center">60%</p>
<p align="center">3</p>
<p align="center">43%</p>
<p align="center">53%</p>
<p align="center">4</p>
<p align="center">37%</p>
<p align="center">46%</p>
<p align="center">5</p>
<p align="center">32%</p>
<p align="center">41%</p>
<p align="center">6</p>
<p align="center">29%</p>
<p align="center">37%</p>
<p align="center">7</p>
<p align="center">26%</p>
<p align="center">34%</p>
<p align="center">8</p>
<p align="center">24%</p>
<p align="center">32%</p>
<p align="center">9</p>
<p align="center">22%</p>
<p align="center">30%</p>
<p align="center">10</p>
<p align="center">21%</p>
<p align="center">28%</p>
<p><em>*I used a value of 0.75 for the prediction</em></p>
<p>We see that being at the goal line decreases your chances on 4th down by about 10%. We also see what a drastic effect just a couple of yards makes. Imagine getting a false start penalty and having your 4th and 1 go to 4th and 6. You just cut your odds of converting in half!</p>
<p>So let’s go back to that coach who punts on every 4th and 1. Now that we have our data, we can analyze whether he is making the correct decision. Let’s say he has a 4th and 1 at his own 10 yard line and is playing on the road. We can use our expected points model and our 4th down model to see what the correct decision should be.</p>
<p style="text-align: center;"><strong>Decision</strong></p>
<p style="text-align: center;"><strong>Expected Points Success</strong></p>
<p style="text-align: center;"><strong>Expected Points Fail</strong></p>
<p style="text-align: center;"><strong>Total Expected Points</strong></p>
<p style="text-align: center;">Go for it</p>
<p style="text-align: center;">-0.64</p>
<p style="text-align: center;">-5.9</p>
<p style="text-align: center;">-2.2</p>
<p style="text-align: center;">Punt*</p>
<p style="text-align: center;">-2.9</p>
<p style="text-align: center;">N/A</p>
<p style="text-align: center;">-2.9</p>
<p><em>* The average net punt in the Big Ten was about 40 yards, so that’s the value I used.</em></p>
<p>By this model, in going for it on 4th down the coach increases his expected points by 0.7 points. That may not sound like much, but imagine making a similar decision 4 or 5 times a game. Those expected points add up to about a field goal. Think there is a coach out there who wouldn’t want an easy way to increase their score by 3 points?</p>
<p>And keep in mind our numbers assume you only gain 1 yard on 4th down. When you account for the fact that you can gain more than 1 yard, the case for going for it only strengthens. As Alabama found out against Ohio State last year, even a <a href="https://youtu.be/M3ZyG_03JlQ?t=2h18m25s">simple running play up the middle has the potential to go the distance</a>.</p>
<p>So now we’re all set to track the 4th down decisions in this upcoming Big Ten season. The first Big Ten conference game is September 19th, when Rutgers takes on Penn State. And the Big Ten 4th down calculator is ready and waiting.</p>
<p>Let the games begin!</p>
Data AnalysisFun StatisticsRegression AnalysisStatistics in the NewsFri, 28 Aug 2015 13:34:00 +0000http://blog.minitab.com/blog/the-statistics-game/calculating-the-probability-of-converting-on-4th-downKevin RudyChi-Square Analysis: Powerful, Versatile, Statistically Objective
http://blog.minitab.com/blog/michelle-paret/chi-square-analysis-powerful-versatile-statistically-objective
<p style="line-height: 20.7999992370605px;">To make objective decisions about the processes that are critical to your organization, you often need to examine categorical data. You may know how to use a t-test or ANOVA when you’re comparing measurement data (like weight, length, <span style="line-height: 1.6;">revenue, </span><span style="line-height: 1.6;">and so on), but do you know how to compare attribute or counts data? It easy to do with <a href="http://www.minitab.com/products/minitab">statistical software</a> like Minitab. </span></p>
<p style="line-height: 20.7999992370605px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/6060c2db-f5d9-449b-abe2-68eade74814a/Image/60bfd1eb8d2c2c3689bce89ea55453ab/chisquare_onevariable_w1024.jpeg" style="line-height: 20.7999992370605px; width: 350px; height: 230px; float: right; margin: 10px 15px;" /></p>
<p style="line-height: 20.7999992370605px;">One person may look at this bar chart and decide that each production line had the same <span style="line-height: 1.6;">proportion of defects. But another person may focus on the small difference between the bars and decide that one of the lines has outperformed the others. Without an appropriate statistical analysis, how can you know which person is right?</span></p>
<p style="line-height: 20.7999992370605px;">When time, money, and quality depend on your answers, you can’t rely on subjective visual assessments alone. To answer questions like these with statistical objectivity, you can use a Chi-Square analysis.</p>
Which Analysis Is Right for Me?
<p style="line-height: 20.7999992370605px;">Minitab offers three Chi-Square tests. The appropriate analysis depends on the number of variables that you want to examine. And for all three options, the data can be formatted either as raw data or summarized counts.</p>
<strong>Chi-Square Goodness-of-Fit Test – 1 Variable</strong>
<p style="line-height: 20.7999992370605px;">Use Minitab’s <strong>Stat > Tables > Chi-Square Goodness-of-Fit Test (One Variable)</strong> when you have just one variable.</p>
<p style="line-height: 20.7999992370605px;">The Chi-Square Goodness-of-Fit Test can test if the proportions for all groups are equal. It can also be used to test if the proportions for groups are equal to specific values. For example:</p>
<ul style="line-height: 20.7999992370605px;">
<li>A bottle cap manufacturer operates three production lines and records the number of defective caps for each line. The manufacturer uses the <strong>Chi-Square Goodness-of-Fit Test</strong> to determine if the proportion of defects is equal across all three lines.</li>
<li>A bottle cap manufacturer operates three production lines and records the number of defective caps for each line. One line runs at high speed and produces twice as many caps as the other two lines that run at a slower speed. The manufacturer uses the <strong>Chi-Square Goodness-of-Fit Test</strong> to determine if the defects for each line is proportional to the volume of caps it produces.</li>
</ul>
<strong>Chi-Square Test for Association – 2 Variables</strong>
<p style="line-height: 20.7999992370605px;">Use Minitab’s <strong>Stat > Tables > Chi-Square Test for Association</strong> when you have two variables.</p>
<p style="line-height: 20.7999992370605px;">The Chi-Square Test for Association can tell you if there’s an association between two variables. In another words, it can test if two variables are independent or not. For example:</p>
<ul style="line-height: 20.7999992370605px;">
<li>A paint manufacturer operates two production lines across three shifts and records the number of defective units per line per shift. The manufacturer uses the <strong>Chi-Square Goodness-of-Fit Test</strong> to determine if the defect rates are similar across all shifts and production lines. Or, are certain lines during certain shifts more prone to defects?</li>
<li>A credit card billing center records the type of billing error that is made, as well as the type of form that is used. The billing center uses a Chi-Square Test to determine whether certain types of errors are related to certain forms.</li>
</ul>
<p style="line-height: 20.7999992370605px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/6060c2db-f5d9-449b-abe2-68eade74814a/Image/7af9e9b2ee624e7d912393d7debe7f1b/chisquare_twovariables_w1024.jpeg" style="width: 500px; height: 329px;" /></p>
<strong>Cross Tabulation and Chi-Square – 2 or more variables</strong>
<p style="line-height: 20.7999992370605px;">Use Minitab’s <strong>Stat > Tables > Cross Tabulation and Chi-Square </strong>when you have two or more variables.</p>
<p style="line-height: 20.7999992370605px;">If you simply want to test for associations between two variables, you can use either <strong>Cross Tabulation and Chi-Square</strong> or <strong>Chi-Square Test for Association</strong>. However, <span><a href="http://blog.minitab.com/blog/understanding-statistics/using-cross-tabulation-and-chi-square-the-survey-says">Cross Tabulation and Chi-Square</a></span> also lets you control for the effect of additional variables. Here’s an example:</p>
<ul style="line-height: 20.7999992370605px;">
<li>A dairy processing plant records information about each defective milk carton that it produces. The plant uses a Cross Tabulation and Chi-Square analysis to look for dependencies between the defect types and the machine that produces the carton, while controlling for any shift effect. Perhaps a particular filling machine is prone to a certain type of defect, but only during the first shift.</li>
</ul>
<p style="line-height: 20.7999992370605px;">This analysis also offers advanced options. For example, if your categories are ordinal (good, better, best or small, medium, large) you can include a special test for concordance.</p>
Conducting a Chi-Square Analysis in Minitab
<p style="line-height: 20.7999992370605px;">Each of these analyses is easy to run in Minitab. For more examples that include step-by-step instructions, just navigate to the Chi-Square menu of your choice and then click Help > example.</p>
<p style="line-height: 20.7999992370605px;">It can be tempting to make subjective assessments about a given set of data, their makeup, and possible interdependencies, but why risk an error in judgment when you can be sure with a Chi-Square test?</p>
<p style="line-height: 20.7999992370605px;">Whether you’re interested in one variable, two variables, or more, a Chi-Square analysis can help you make a clear, statistically sound assessment.</p>
Data AnalysisHypothesis TestingLean Six SigmaQuality ImprovementSix SigmaStatisticsStatistics HelpThu, 27 Aug 2015 12:33:39 +0000http://blog.minitab.com/blog/michelle-paret/chi-square-analysis-powerful-versatile-statistically-objectiveMichelle ParetWho Will Win in 2016? Ask Someone from Connecticut
http://blog.minitab.com/blog/the-statistical-mentor/who-will-win-in-2016-ask-someone-from-connecticut
<p><img alt="Presidential Seal" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/c79fe6c5e9dfe3d3dea8221593afb07a/200px_seal_of_the_president_of_the_united_states_svg_1_.png" style="line-height: 20.7999992370605px; margin: 10px 15px; float: right; width: 200px; height: 200px;" /></p>
<p>There's more data available today than ever before, and with <a href="http://www.minitab.com/products/minitab">statistical software such as Minitab</a> it only takes a couple of seconds to get some significant insights, whether it concerns <a href="http://www.minitab.com/company/case-studies/">how to make your business run better</a> or <span><a href="http://blog.minitab.com/blog/adventures-in-statistics/presidential-politics-political-polls-and-statistics">national politics</a></span>. </p>
<p>For instance, if we look back at the last 9 presidential elections (1980 to 2012), there are some interesting correlations between the percent of state votes for Democrats/Republicans and the percent voting for Democrats/Republicans nationally.</p>
<p>The bar chart below shows how each state's percent Democratic vote count correlates with the percent national vote count. (Each state's votes were taken out of the national count before the correlation was calculated, so that a state like California didn't have a high correlation just because it has a large proportion of the national vote.)</p>
<p><img alt="individual state correlations with national popular vote" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/3bf90520720902adbf46b3a460061d29/win1.png" style="width: 576px; height: 384px;" /></p>
<p>Connecticut had the highest correlation, and the fitted line plot below, which plots Connecticut's percent Democratic voting against the national percentage, shows just how closely correlated the state's percentages have been over the last nine presidential election cycles. </p>
<p><img alt="Connecticut correlation" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/a0d6fa0cb2f1cc3d4b6fc6f67327a0dc/win2.png" style="width: 576px; height: 384px;" /></p>
<p>Other states, including Michigan and Ohio, had similarly high correlations. Keep in mind, however, that no matter how high these correlations are, <a href="http://blog.minitab.com/blog/understanding-statistics/no-matter-how-strong-correlation-still-doesnt-imply-causation">correlation does not imply causation</a>. If a candidate focused on Connecticut thinking their national percent would increase, they would be falling victim to flawed statistical thinking. In fact, that candidate's percent in other states would probably decrease, as opposed to increase, since they would be neglecting the other states.</p>
<p>West Virginia had the lowest correlation with the national voting percentages—so low, in fact, that its correlation was negative. </p>
<p><img alt="West Virginia correlation" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/d8e1b3f225c5caaad89d391bcd938bdb/win3.png" style="line-height: 20.7999992370605px; width: 576px; height: 384px;" /></p>
<p>In years where West Virginia saw a high percentage of voters go Democratic, the national Democratic vote percent was low, and in years where West Virginia had a low Democratic vote percent, the Democratic vote percent was high at the national level. In general, southern states (South Carolina, Mississippi, Louisiana, Georgia, Arkansas, Tennessee, Kentucky, and Alabama) had low correlations with the national percentages. </p>
<p>This data came from The University of California Santa Barbara website, <a href="http://www.presidency.ucsb.edu/elections.php" target="_blank">http://www.presidency.ucsb.edu/elections.php</a>, which has state-by-state historical voting data. As the 2016 election cycle gathers more steam, it will be very interesting to see what the data from a myriad of sources will be able to tell us. </p>
Data AnalysisStatistics in the NewsTue, 25 Aug 2015 12:00:00 +0000http://blog.minitab.com/blog/the-statistical-mentor/who-will-win-in-2016-ask-someone-from-connecticutJim ColtonUsing Probability Distribution Plots to See Data Clearly
http://blog.minitab.com/blog/understanding-statistics/using-probability-distribution-plots-to-see-data-clearly
<p><span style="line-height: 1.6;">When we take pictures with a digital camera or smartphone, what the device <em>really</em> does is capture information in the form of binary code. At the most basic level, our precious photos are really just a bunch of 1s and 0s, but if we were to look at them that way, they'd be pretty unexciting. </span></p>
<p><span style="line-height: 1.6;">In its raw state, all that information the camera records is worthless. T</span><span style="line-height: 1.6;">he 1s and 0s need to be converted into pictures before we can actually see what we've photographed.<img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/8ad3420e0401f47cadb9bd3e9723fc32/camera_with_plot.jpg" style="margin: 10px 15px; float: right; width: 250px; height: 163px;" /></span></p>
<p>We encounter a similar situation when we try to use <span><a href="http://blog.minitab.com/blog/adventures-in-statistics/how-to-identify-the-distribution-of-your-data-using-minitab">statistical distributions and parameters</a></span> to describe data. There's important information there, but it can seem like a bunch of meaningless numbers without an illustration that makes them easier to interpret.</p>
<p>For instance, if you have data that follows a gamma distribution with a scale of 8 and a shape of 7, what does that really mean? If the distribution shifts to a shape of 10, is that good or bad? And even if <em>you</em> understand it, how easy would it be explain to people who are more interested in outcomes than statistics?</p>
Enter the Probability Distribution Plot
<p>That's where the probability distribution plot comes in. Making a probability distribution plot using Minitab <a href="http://www.minitab.com/products/minitab">Statistical Software</a> will create a picture that helps bring the numbers to life. Even novices can benefit from understanding their data’s distribution.</p>
<p>Let's take a look at a few examples.</p>
Changing Shape
<p><span style="line-height: 20.7999992370605px;">A building materials manufacturer develops a new process to increase the strength of its I-beams. The old process fit a gamma distribution with a scale of 8 and a shape of 7, whereas the new process has a shape of 10.</span><span style="line-height: 20.7999992370605px;"> </span></p>
<p><img alt="estimates" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/3e60c8f30d8ec67ec28c548ef44cbbc2/probability_distribution_plots_1_en_us_1_.gif" style="width: 294px; height: 90px;" /></p>
<p>The manufacturer does not know what this change in the shape parameter means, and the numbers alone don't tell the story. </p>
<p>But if we go in Minitab to <strong>Graph > Probability Distribution Plot</strong>, select the "View Probability" option, and enter the information about these distributions, the impact of the change will be revealed.</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/662165bbd950bcab66ddc18702437d4c/probability_plot_dialog.png" style="width: 369px; height: 207px;" /></p>
<p>Here's the original process, with the shape of 7:</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/e786e9d35416d0eef6b18f4019ef7a18/distributionplot1.png" style="width: 576px; height: 384px;" /></p>
<p>And here is the plot for the new process, with a shape of 10: </p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/ab74192ac9cbacaa4bafb661cf07bb63/distributionplot2.png" style="width: 576px; height: 384px;" /></p>
<p>The probability distribution plots make it easy to see that the shape change increases the number of acceptable beams from 91.4% to 99.5%, an 8.1% improvement. What's more, the right tail appears to be much thicker in the second graph, which indicates the new process creates many more unusually strong units. Hmmm...maybe the new process could ultimately lead to a premium line of products.</p>
Communicating Results
<p>Suppose a chain of department stores is considering a new program to reduce discrepancies between an item’s tagged price and the amount is charged at the register. <span style="line-height: 20.7999992370605px;">Ideally, the system would eliminate any discrepancies, but a </span><span style="line-height: 20.7999992370605px;">± 0.5% </span><span style="line-height: 20.7999992370605px;">difference is considered acceptable. </span><span style="line-height: 1.6;">However, implementing the program will be extremely expensive, so the company runs a pilot test in a single store. </span></p>
<p><span style="line-height: 20.7999992370605px;">In the pilot study, the mean improvement is small, and so is the standard deviation. When the company's board looks at the numbers, they don't see the benefits of approving the program, given its cost. </span></p>
<p><img alt="communicate results data" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/93fc723781cb6d5b48cce1e442c73ecb/probability_distribution_plots_3_en_us_1_.gif" style="width: 266px; height: 92px;" /></p>
<p>The store's quality specialist thinks the numbers aren't telling the story, and decides to show the board the pilot test data in a probability distribution plot instead: </p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/d5707e270065bc777cd9e8eeda66393a/distributionplot3.png" style="width: 576px; height: 384px;" /></p>
<p>By overlaying the before and after distributions, the specialist makes it very easy to see that price differences using the new system are clustered much closer to zero, and most are in the ± 0.5% acceptable range. Now the board can see the impact of adopting the new system. </p>
Comparing Distributions
<p>An electronics manufacturer counts the number of printed circuit boards that are completed per hour. The sample data is best described by a Poisson distribution with a mean of 3.2. However, the company's test lab prefers to use <span><a href="http://blog.minitab.com/blog/quality-data-analysis-and-statistics/assumptions-and-normality">an analysis that requires a normal distribution</a></span> and wants to know if it is appropriate.</p>
<p><span style="line-height: 20.7999992370605px;">The manufacturer can easily compare the known distribution with a normal distribution using the probability distribution plot. </span>If the normal distribution does not approximate the Poisson distribution, then the lab's test results will be invalid.</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/ca8efa0a9ccabe926553c29b7f42d8e6/distributionplot4.png" style="width: 576px; height: 384px;" /></p>
<p><span style="line-height: 1.6;">As the graph indicates, the normal distribution—and the analyses that require it—won’t be a good fit for data that follow a Poisson distribution with a mean of 3.2.</span></p>
Creating Probability D<span style="line-height: 1.6;">istribution Plots in Minitab</span>
<p>It's easy to use Minitab to create plots to visualize and to compare distributions and even to scrutinize an area of interest.</p>
<p>Let's say a market researcher wants to interview customers with satisfaction scores between 115 and 135. Minitab’s Individual Distribution Identification feature shows that these scores are normally distributed with a mean of 100 and a standard deviation of 15. However, the analyst can’t visualize where his subjects fall within the range of scores or their proportion of the entire distribution.</p>
<p>Choose <strong>Graph > Probability Distribution Plot > View Probability</strong>.<br />
Click <strong>OK</strong>.</p>
<p><img alt="dialog box" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/c394e6d6080f4749f93345c431265b6c/probability_plot_dialog_2.png" style="line-height: 20.7999992370605px; width: 431px; height: 386px;" /></p>
<p style="margin-left: 40px;">From Distribution, choose Normal.<br />
In Mean, type 100.<br />
In Standard deviation, type 15.<br />
Click on the "Shaded Area" tab. </p>
<p><img alt="distribution plot dialog box 2" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/1fcc60e71a09408880bf2cff8d8fc545/probability_plot_dialog_3.png" style="width: 431px; height: 386px;" /></p>
<p style="margin-left: 40px;">In Define Shaded Area By, choose X Value.<br />
Click Middle.<br />
In X value 1, type 115.<br />
In X value 2, type 135.<br />
Click OK.</p>
<p>Minitab creates the following plot: </p>
<p><img alt="distribution plot " src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/2f25241390c52c6eb8efbc623bf77e7a/distributionplot5.png" style="width: 576px; height: 384px;" /></p>
<p>About 15% of sampled customers had scores in the region of interest (115-135). This is not a very large percentage, so the researcher may face challenges in finding qualified subjects.</p>
Using Probability Distribution Plots
<p>Just like your camera when it assembles 1s and 0s into pictures, probability distribution plots let you see the deeper meaning of the numbers that describe your distributions. You can use these graphs to highlight the impact of changing distributions and parameter values, to show where target values fall in a distribution, and to view the proportions that are associated with shaded areas. These simple plots also clearly and easily communicate these advanced concepts to a non-statistical audience that might b<span style="line-height: 1.6;">e confused by hard-to-understand concepts and numbers. </span></p>
Data AnalysisStatisticsThu, 20 Aug 2015 12:00:00 +0000http://blog.minitab.com/blog/understanding-statistics/using-probability-distribution-plots-to-see-data-clearlyEston MartzConsidering Defects and Defectives via the Republican Primary
http://blog.minitab.com/blog/statistics-and-quality-improvement/considering-defects-and-defectives-via-the-republican-primary
<p>The difference between <a href="http://support.minitab.com/en-us/minitab/17/topic-library/quality-tools/capability-analyses/data-and-data-assumptions/defects-and-defectives/">defects and defectives</a> lets you answer questions like whether to use a P chart or a U chart in Minitab, so it’s a handy difference to be able to explain. Of course, if you’ve explained it enough times—or if someone’s explained it <em>to</em> you enough times—the whole thing can get a little tired.</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/afae5f883f771199ef33d6078a09e98d/flag_w1024.jpeg" style="margin: 10px 15px; float: right; width: 250px; height: 167px;" />Fortunately, a new explanation of defects and defectives is one more way we can entertain ourselves with the candidates from the <a href="http://blog.minitab.com/blog/adventures-in-statistics/data-driven-analysis-of-the-republican-field-of-presidential-candidates-for-2016">2016 Republican Presidential Primary</a>, even though it’s only 2015. Ready? Here we go!</p>
<strong>Defectives</strong>
<p>A defective item is not acceptable for use, but when we do a statistical analysis we don’t have to be overly literal. In politics, when we do a poll about whether a voter will vote for a certain candidate, we’re using the same math that we do when we talk about defective items. The voter either votes for the candidate or doesn’t vote for the candidate. The voter is either useful to the candidate or not useful to the candidate, in terms of election results.</p>
<p>So when <a href="http://personal.crocodoc.com/KRyE2Wq">a Fox News poll reported on August 3rd</a> that 26% of their poll respondents who answered a question about who they would choose in the Republican primary chose Donald Trump, the other 74% of the respondents were defectives—<em>as far as their usefulness to Donald Trump in that poll</em>.</p>
<p><img alt="26% respond that they would vote for Trump. The other 74% are of no use to him in this poll." src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/40a0422fe7021c02221ea82636ea4984/chart_of_defectives.png" style="border-width: 0px; border-style: solid; width: 576px; height: 384px;" /></p>
<strong>Defects</strong>
<p>A defect is any departure from specifications, but a single defect does not make an item unacceptable for use. In fact, an item can have multiple defects and the defects might not even be noticeable to the person who needs to use the item.</p>
<p>People who do not vote for a candidate are defectives from the perspective of their usefulness to the candidate, but might or might not have defects.</p>
<p>An interesting example of defects comes from <a href="http://www.grammarly.com/blog/2015/republican-primary-candidates-grammar-power-rankings/" target="_blank">grammarly.com’s Grammar Power Rankings</a>, which check the grammar of candidates’ supporters on Facebook. Grammarly determined, for example, that Carly Fiorina’s supporters wrote comments on her Facebook page that contained 6.3 grammatical errors per 100 words. Individual posts can have a higher or lower rate of defects, but the candidate might not care at all. The post, as an item, is still usable.</p>
<p><img alt="Grammar mistakes are good examples of defects. In a Facebook post, they probably can't make a post impossible to use." src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/df11564e85068a4eff26e80d32a4a99d/chart_of_defects.png" style="border-width: 0px; border-style: solid; width: 576px; height: 384px;" /></p>
<strong>Wrap up</strong>
<p>If you need a more traditional explanation of defects and defectives, that information is in the <a href="http://support.minitab.com/en-us/minitab/17/topic-library/quality-tools/capability-analyses/data-and-data-assumptions/defects-and-defectives/">Minitab Support Center</a> (plus a lot more).</p>
<p>If you’re in <a href="http://www.minitab.com/en-us/products/minitab/assistant/">the Assistant</a>, then you can click on “What are you counting” to get an explanation right when you need it. With the support you get with Minitab, you can spend less time looking for answers and more time making decisions.</p>
Fun StatisticsQuality ImprovementStatistics in the NewsWed, 19 Aug 2015 15:27:00 +0000http://blog.minitab.com/blog/statistics-and-quality-improvement/considering-defects-and-defectives-via-the-republican-primaryCody SteeleInterpreting Results from a Split-Plot Design
http://blog.minitab.com/blog/applying-statistics-in-quality-projects/interpreting-results-from-a-split-plot-design
<p>When performing a design of experiments (DOE), some factor levels may be very difficult to change—for example, temperature changes for a furnace. Under these circumstances, completely randomizing the order in which tests are run becomes almost impossible.To minimize the number of factor level changes for a Hard-to-Change (HTC) factor, a <span><a href="http://blog.minitab.com/blog/statistics-and-quality-improvement/what-the-heck-is-a-split-plot-design-and-why-would-i-want-it">split-plot design</a></span> is required.</p>
Why Do We Want to Randomize a Designed Experiment?
<p>Randomization means that the experimental tests are run in a random order specified by Minitab, and that factor level changes occur randomly. Randomization in a DOE is desirable, because it helps ensure that factor estimates are not biased by long-term drifts during the experiments. </p>
<p>Suppose that, due to environmental conditions, a gradual change in temperature takes place during the tests. This gradual change may affect factor effect estimates. If most of the tests are performed at the lower setting for a particular factor at the beginning of the experiment, and then most of the tests at the end use the upper settings of this factor, then temperature effect and the drift will be erroneously attributed to this factor, leading to biases and wrong conclusions.</p>
<p>That's why complete randomization in a DOE is desirable. When you can randomize, a drift in environmental conditions will not have a systematic effect on some factor estimates. There will certainly be a random impact, but not a systematic one. But when HTC factors need to be studied, randomization is not always possible.</p>
<p>Enter the split-plot design.</p>
What Is a Split-Plot design
<p>The first designs of experiments were agricultural experiments at the beginning of the 20th century. Think about a large field in which experiments need to be performed to test different types of plant varieties, fertilizers, soil treatments, etc.</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/31b80fb2-db66-4edf-a753-74d4c9804ab8/Image/65546455a99d3f5dc6dbbcea69602f27/images.jpg" style="line-height: 20.7999992370605px; border-width: 1px; border-style: solid; margin: 10px 15px; float: right; width: 259px; height: 194px;" /></p>
<div>
<p>This experimental field may be divided into plots, and different treatments will be allocated to these plots. If the number of level changes needs to be minimized for a specific HTC factor, large plots of the field will be used. These are referred to as whole plots.</p>
<p>For an Easy-to-Change (ETC) factor, smaller plots may be used. We create these sub-plots by subdividing the whole plots into the smaller subplots. </p>
<p>In the presence of spatial variability in the experimental field, we can expect soil variations to be smaller when subplots are located close to one another, whereas <span style="line-height: 20.7999992370605px;">we would expect soil variations to be much larger between whole plots </span><span style="line-height: 1.6;">since they are located further from one another.</span></p>
<p>In a manufacturing context, whole plots represent long-term variations and sub plots represent short-term variations. To estimate long-term variations, whole plots need to be replicated.</p>
<p><span style="line-height: 20.7999992370605px;">To minimize complex modifications of settings, t</span>he levels of HTC factors are never changed within a whole plot and all the runs within a whole plot are performed together, in the same period of time.</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/31b80fb2-db66-4edf-a753-74d4c9804ab8/Image/240da756baac4b061db3879b3c80ea8c/split_plot_3.JPG" style="width: 634px; height: 267px;" /></p>
<p>The field above has been divided into four whole plots, and the whole plots have then been subdivided into subplots.</p>
<p>A split plot design array as displayed in Minitab <a href="http://www.minitab.com/products/minitab">Statistical Software</a> appears below, with different colors for whole plots and subplots (see below). In the HTC column the 1 or -1 settings are changed much less often than in the ETC column:</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/31b80fb2-db66-4edf-a753-74d4c9804ab8/Image/5d6b37a2213bf30baa390cb77b451d58/split_plot_1.JPG" style="width: 195px; height: 376px;" /></p>
<p>There are two main sources of variations to be considered in a split plot design: short-term variation (between subplots: SP) and long-term variation (between whole plots: WP). Hard-to-Change (WP) factors are affected by long term variability whereas Easy-to-Change (SP) factors are affected by short term variability.</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/31b80fb2-db66-4edf-a753-74d4c9804ab8/Image/449b93a97f9923fda623b8d1a4a67d44/split_plot_4.JPG" style="width: 635px; height: 270px;" /></p>
<ul>
<li>HTC + : treatment at the upper setting for the Hard to change factor</li>
<li>HTC - : treatment at the lower setting for the Hard to change factor</li>
<li>ETC + : treatment at the upper setting for the Easy to change factor</li>
<li>ETC - : treatment at the lower setting for the Easy to change factor</li>
</ul>
<p>To determine whether a factor is significant or not (according to the F test), the effects of ETC factors will be compared to the short-term error term (SP) only, whereas the effects of HTC factors will be compared to the long term error term (WP) only.</p>
<p>Also, two <a href="http://blog.minitab.com/blog/statistics-and-quality-data-analysis/r-squared-sometimes-a-square-is-just-a-square">R²</a> estimates are displayed in a split plot design analysis: a short term R² and a long term R².</p>
<p>To compute the short term R² value (SP), the amount of variability which is explained by short-term, Easy-to-<span style="line-height: 1.6;">Change (SP) factors is compared to the short-term overall (subplot sum of squares) variability.</span></p>
<p>To estimate the long-term R² value (WP), the amount of variability explained by long term Hard-to-Change (WP) factors will be compared to the long term overall (WP sum of squares) variability only.</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/31b80fb2-db66-4edf-a753-74d4c9804ab8/Image/420f2c07b7d2cac3e30636b6b6fabb1e/split_plot_2_w1024.jpeg" style="width: 1024px; height: 439px;" /></p>
Conclusion
<p>In a split plot design, two error terms need to be considered (short term and long term) separately, and two R² values need to be computed (short term and long term). The analysis may look more complex, but that makes the interpretation of the DOE results a lot more realistic.</p>
<p> </p>
</div>
Design of ExperimentsTue, 18 Aug 2015 12:00:00 +0000http://blog.minitab.com/blog/applying-statistics-in-quality-projects/interpreting-results-from-a-split-plot-designBruno ScibiliaBig Ten 4th Down Calculator: Creating a Model for Expected Points
http://blog.minitab.com/blog/the-statistics-game/big-ten-4th-down-calculator%3A-creating-a-model-for-expected-points
<p><img alt="4th Down" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/388eafa0f8b00f893356dbce728c1677/4th_down.jpg" style="margin: 10px 15px; width: 206px; height: 290px; float: right;" />If you want to use data to predict the impact of different variables, whether it's for business or some personal interest, you need to create a model based on the best information you have at your disposal. In this post and subsequent posts throughout the football season, I'm going to share how I've been developing and applying a model for predicting the outcomes of 4th down decisions in Big 10 games. I hope sharing my experiences will help you, whether the questions you want to answer are about football or business logistics. </p>
<p>Here are some questions I was looking to answer when I began thinking about creating a 4th down calculator. If you have a 1st and 10 at your opponent’s 20-yard line, on average you’ll score more points than if you have the ball at your own 20 yard line. But how many more? And how does that number change as you move to different positions on the field. And what if you’re playing on the road as opposed to playing at home?</p>
<p>If you’re trying to use analytics to determine what the best decision is on 4th down, you need to know how many points you (or your opponent) would be expected to score on the ensuing 1st down. So my first step in creating a <a href="http://blog.minitab.com/blog/the-statistics-game/coming-soon%3A-the-big-ten-4th-down-calculator">Big Ten 4th down calculator</a> was to use Minitab Statistical Software to model a team’s expected points on 1st and 10 from anywhere on the field.</p>
The Data
<p>I went through every Big Ten conference game the last two seasons. For each instance a team had 1st and 10, I recorded the field position and the next score. If your opponent was the next team to score, then the value for the next score was negative. If nobody scored before halftime or the end of the game (depending on which half they were in) the value was 0.</p>
<p>I only included conference games because many non-conference games are one-sided (I’m looking at you, Ohio State vs. Kent State in <span style="line-height: 20.79px;">2014</span><span style="line-height: 1.6;">). I also didn’t include the conference championship game, since I want to account for home field advantage and that game is played at a neutral site. Finally, I did my best to exclude drives that ended prematurely because of halftime and drives in the 4th quarter of blowouts. </span></p>
<p><span style="line-height: 1.6;">I ended up with 5,496 drives over the two seasons. You can get both the raw and summarized data </span><a href="//cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/File/5de24a0ccc06069f2b2c02beb2d9e281/expected_points_data.MTW" style="line-height: 1.6;">here</a><span style="line-height: 1.6;">.</span></p>
<p>A bar chart can give us a quick glance at what the most common score is.</p>
<p><img alt="Bar Chart" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/c329e0cb4ca28ed32321e050458fa636/bar_chart_of_next_score.jpg" style="width: 576px; height: 384px;" /></p>
<p>The most common outcome when you have possession of the ball is that you score a touchdown. No revelation there. But surprisingly, it was actually more common for your opponent to get the ball back and score a touchdown than it was for you to kick a field goal. I wouldn’t have expected that.</p>
<p>So now let’s see what happens when we account for the field position and home field advantage.</p>
A Model for Expected Points
<p>I grouped the field position into groups of 5 yards intervals. Then for each group, I took the average of the next score. So first, let’s look at a fitted line plot of the data, <em>without </em>accounting for home field advantage.</p>
<p><img alt="Fitted Line Plot" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/1fc2f2ec3c73acd2c561a0e6fb2c1b92/fitted_line_plot_without_hf_advantage.jpg" style="width: 576px; height: 384px;" /></p>
<p>The regression model fits the data very well. The R-squared value indicates that 96.4% of the variation in Expected Points can be explained by the number of yards to the end zone. That’s fantastic! I added a reference line at the point where the expected value is 0. It crosses our regression line at a distance to the end zone of approximately 85 yards. That suggests you have to be inside your own 15 yard line before the team on defense is more likely to be the next team to score.</p>
<p>Now let’s factor in home field advantage. We’ll start by examining a scatterplot that will show the difference in expected points for home and away teams at each yard line group.</p>
<p><img alt="Scatterplot" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/97951f90653235b48ef016f78eaaf0ec/scatterplot_1.jpg" style="width: 576px; height: 384px;" /></p>
<p>In 17 of the 20 groups, the home team has a higher number of expected points than the away team. And in the 3 cases where the away team is higher, the two values are very close. This gives strong evidence that we need to account for home field advantage. I ran a regression analysis to confirm that we should include that game location in our model.</p>
<p><img alt="Regression" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/8360657dd1ab5ce5f7bf22ad79595849/regression_with_no_interaction.jpg" style="width: 495px; height: 336px;" /></p>
<p>The p-value for location is less than 0.05, and the R-squared value remains very high. I can now use these two equations (one for home games, one for away games) to predict how many points a team with a first down will score from anywhere on the field.</p>
Testing the Interaction Between Home Field Advantage and Yards to the End Zone
<p>There is one last thing I want to look into. Is there an interaction between our two terms? Think about it this way: Say you have 1st and goal inside your opponent’s 10 yard line. You’re so close to the end zone, it seems like it might not matter whether you’re at home or on the road.</p>
<p>Now imagine you have a 1st and 10 inside your own 10 yard line. It seems like a much more daunting task to drive the length of the field on the road with the hostile crowd roaring than it would be with the cheers of a friendly home crowd.</p>
<p>In other words, does the effect of home field advantage increase the further a team is from the end zone? Intuitively, it seems like it <em>should</em>. But we should run a regression analysis to see if the data supports that notion.</p>
<p><img alt="Regression" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/290b2ad01ddba5340b3604b1834fa0d8/regression_with_interaction.jpg" style="width: 534px; height: 158px;" /></p>
<p>The data does not support my intuition. The p-value for the interaction term is much higher than 0.05, indicating that it is not a significant term, and thus that we should not include it in our model. To visualize why, let’s revisit the previous scatterplot, but this time I'll add regression lines to each group.</p>
<p><img alt="Scatterplot" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/98828cf26b5dd282a08d2d7aaeeff6d7/scatterplot_2.jpg" style="width: 576px; height: 384px;" /></p>
<p>If there were an interaction between our two terms, we would expect the two lines to be close together at small distances to the end zone. Then they should move farther apart as the yards to the end zone increase. But you can see here that the lines are pretty parallel to each other. So we can safely remove the interaction term from our model.</p>
The Final Model
<p>Let’s take a final look at the model created by this regression analysis.</p>
<p><img alt="Model Equations" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/362b968414eaa5a21f46de82b8b88f57/model_equations.jpg" style="width: 492px; height: 120px;" /></p>
<p>The equations indicate that if you start a drive on the road, you’ll be expected to score approximately 0.6 fewer points than you would if you were playing at home. Because there is no interaction term, the slopes are the same for both equations. The value of -0.075 means that for every yard you move away from the end zone, your expected points decrease by 0.075. So if you decide to punt the football away and get a net of 40 yards (the average in the Big Ten last year), this model indicates you’ll have saved yourself about 3 points on average.</p>
<p>Of course, that 3 points assumes that you turned the ball over on downs. <span style="line-height: 1.6;">But a third option exists: successfully converting on 4</span>th<span style="line-height: 1.6;"> down. </span></p>
<p><span style="line-height: 1.6;">Will the reward of a successful conversion outweigh the risk of losing those 3 points you would gain by punting? That all depends on the probability of successfully converting on 4</span>th<span style="line-height: 1.6;"> down. And that’s exactly what I'll look at in my next post. Once we can determine the probability of converting on 4</span>th<span style="line-height: 1.6;"> down, we’ll be able to get some data-driven insights into what the correct decision is on 4</span>th<span style="line-height: 1.6;"> down. Stay tuned!</span></p>
<p> </p>
Data AnalysisFun StatisticsRegression AnalysisStatisticsFri, 14 Aug 2015 14:21:00 +0000http://blog.minitab.com/blog/the-statistics-game/big-ten-4th-down-calculator%3A-creating-a-model-for-expected-pointsKevin RudyThe Null Hypothesis: Always “Busy Doing Nothing”
http://blog.minitab.com/blog/using-data-and-statistics/the-null-hypothesis-always-busy-doing-nothing
<p>The 1949 film <a href="http://www.imdb.com/title/tt0041259/" target="_blank"><em>A Connecticut Yankee in King Arthur's Court</em></a> includes the song “Busy Doing Nothing,” and this could be written about the <a href="http://blog.minitab.com/blog/understanding-statistics/things-statisticians-say-failure-to-reject-the-null-hypothesis">Null Hypothesis</a> as it is used in statistical analyses. </p>
<p></p>
<p>The words to the song go:</p>
<p style="margin-left: 40px;"><em>We're busy doin' nothin'<br />
<span style="line-height: 1.6;">Workin' the whole day through<br />
Tryin' to find lots of things not to do </span></em></p>
<p><span style="line-height: 1.6;">And that summarises the role of the Null Hypothesis perfectly. Let me explain why.</span></p>
<span style="line-height: 1.6;">What's the Question?</span>
<p>Before doing any statistical analysis—in fact even before we collect any data—we need to define what problem and/or question we need to answer. Once we have this, we can then work on defining our Null and Alternative Hypotheses.</p>
<p>The null hypothesis is always the option that maintains the status quo and results in the least amount of disruption, hence it is “Busy Doin’ Nothin'”. </p>
<p>When the probability of the Null Hypothesis is very low and we reject the Null Hypothesis, then we will have to take some action and we will no longer be “Doin Nothin'”.</p>
<p>Let’s have a look at how this works in practice with some common examples.</p>
<p><strong>Question</strong></p>
<p><strong>Null Hypothesis</strong></p>
Do the chocolate bars I am selling weigh 100g?
Chocolate Weight = 100g<br />
<br />
If I am giving my customers the right size chocolate bars I don’t need to make changes to my chocolate packing process.<br />
Are the diameters of my bolts normally distributed?
<p>Bolt diameters are n<span style="line-height: 1.6;">ormally distributed.</span></p>
<p>If my bolt diameters are normally distributed I can use any statistical techniques that use the standard normal approach.<br />
</p>
Does the weather affect how my strawberries grow?
Number of hours sunshine has no effect on strawberry yield<br />
<br />
Amount of rain has no effect on strawberry yield<br />
<br />
Temperature has no effect on strawberry yield<br />
<p>Note that the last instance in the table, investigating if weather affects the growth of my strawberries, is a bit more complicated. That's because I needed to define some metrics to measure the weather. Once I decided that the weather was a combination of sunshine, rain and temperature, I established my null hypotheses. These all assume that none of these factors impact the strawberry yield. I only need to control the sunshine, temperature and rain if the probability that they have no effect is very small.</p>
Is Your Null Hypothesis Suitably Inactive?
<p><span style="line-height: 1.6;">So in conclusion, in order to be “Busy Doin’ Nothin’”, your Null Hypothesis has to be as follows:</span></p>
<ul>
<li>A logical question.</li>
<li>Focused on one objective.</li>
<li>Requires action only if <a href="http://blog.minitab.com/blog/michelle-paret/alphas-p-values-confidence-intervals-oh-my">its probability of being true</a> is low (typically 5%).</li>
</ul>
Hypothesis TestingStatisticsWed, 12 Aug 2015 12:00:00 +0000http://blog.minitab.com/blog/using-data-and-statistics/the-null-hypothesis-always-busy-doing-nothingGillian GroomThe Bubble Plot: It's A Beautiful Display
http://blog.minitab.com/blog/data-analysis-and-quality-improvement-and-stuff/the-bubble-plot%3A-its-a-beautiful-display
<p>As you may know, <a href="http://blog.minitab.com/blog/starting-out-with-statistical-software/introducing-the-bubble-plot">we added Bubble Plots to Minitab's menu of meaningful graphs</a> in Release 17. If you are familiar, I think you'll agree that Bubble Plots make a perfect addition to the pantheon of impressive and powerful plots that you can produce in Minitab. They’re great. Of course, they would have been even greater if they used my idea...but that’s spilt milk under the bridge now.</p>
<p>If you haven’t met the Bubble Plot yet, it’s a lot like a scatterplot, only the dots on the plot (a.k.a. the aforementioned “bubbles”) are different sizes so you can visualize the value of a 3rd variable in addition to the x-variable and the y-variable. For example, the bubble plot below shows gross sales (in thousands of dollars) on the y-axis and quarter of the year (1 through 4) on the x-axis. The size of each bubble indicates the number of orders that were received during the quarter.</p>
<p>The first bubble (far left) shows that the company earned approximately $350,000 in revenue during Quarter 1. The second bubble is smaller and lower than the first bubble, which indicates that both the number of orders and total sales revenues were down in Q2 as compared to Q1. Things rebounded a bit in Q3. Q4 was in progress when the graph was made, but the preliminary data look promising. The red bubble was added to show the projected orders and sales for Q4.</p>
<p>It’s a great graph, and it really speaks to you. But it doesn’t quite <em>sing</em>.</p>
<p><img alt="Boring Bubble Plot" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/8de770ba-a50a-4f6b-9144-9713c3b99f66/Image/e01d2ea5962f31640a01dceea71e941d/boring_bubble_plot.jpg" style="width: 576px; height: 384px;" /></p>
<p>You can’t say that I didn’t try. I traveled endlessly up and down the hollowed corridors of Minitab and shared my idea with all who would listen (and several who would not).</p>
<p>I started with a visit to the Software Development department. The developers seemed generally impressed with my idea. At least they smiled a lot while I was explaining it. But they said that I should talk to Research and Development first, so I ventured over there.</p>
<p>The R&D folks were inquisitive and asked thoughtful questions like, “You want to do <em>what</em>?” and “Are you serious?” and “Is that even <em>legal</em>?” In order to address that last question, I took a trip to the Legal Department.</p>
<p>Initially, I was concerned that the folks in Legal would talk over my head. I imagined that they would use Latin words and legal jargon and cite obscure precedents from volumes of landmark court cases. In the end, however, I found them to be quite plain-spoken. I think the exact words were, “Ain’t nobody got time for that.” Legal then sent me to Human Resources. As a reward for my brilliance, HR added an extensive psychotherapy rider to my existing health insurance policy and encouraged me to use it. Which I did. (It’s going very well, by the way. I’m learning a lot about my mother.)</p>
<p>You get the idea. I basically got the run around. Frankly, I think that everyone is simply jealous or embarrassed that they didn’t think of this themselves. Especially since it’s so <em>obvious </em>when you think about it. I mean, instead of settling for a mere <strong>bubble plot</strong>, who wouldn’t want to showcase their data in a fabulous <strong>Bublé Plot!</strong></p>
Introducing the <u><em>Bublé Plot</em></u>
<p>Just think of the extra attention that you’ll garner at your next meeting when your data are brought to life...not by boring old bubbles, but by the viral and vivacious visage of the one and only Mr. Michael Bublé!</p>
<p><img alt="The Bublé Plot" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/8de770ba-a50a-4f6b-9144-9713c3b99f66/Image/e5b3a301ff451cbc9f60d39d1804c70d/bubl___plot.jpg" style="width: 576px; height: 384px;" /></p>
<p>Now <em>there’s</em> a graph that just sings out to you. Looking at that graph, how can you possibly doubt that things are looking up? (Or at least looking left?)</p>
<p>And when that happy day comes and you finally <em><strong>do</strong></em> meet those fourth-quarter projections, how do you want to receive the good news? Would you rather stare blankly at expressionless bubbles? Or crack a smile with the chart that smiles back with a look that says, “You did it, Kid! You’re the greatest. That therapist doesn’t know what he’s talking about.”</p>
<p><img alt="You did it Kid!" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/8de770ba-a50a-4f6b-9144-9713c3b99f66/Image/153d4bb81797aabb941e59befa86f9e5/bubl___plot__projections_met.jpg" style="width: 576px; height: 384px;" /></p>
<p>I know which graph <em><strong>I’d</strong></em> rather use. Reminds me of a song ...</p>
<p style="margin-left: 40px;"><em>I’m not surprised</em><br />
<em>There’s been slump</em><br />
<em>I’m not gonna let that get me,</em><br />
<em>down in the dumps</em><br />
<em>Revenues they come in,</em><br />
<em>and expenditures out</em><br />
<em>We get all worked up</em><br />
<em>then we let our guard down</em></p>
<p style="margin-left: 40px;"><em>We’ve tried so very hard to improve it<br />
Now is not the time for excuses<br />
Let’s think of every source of variability</em></p>
<p style="margin-left: 40px;"><em>And I know that Q4 it’ll all turn out<br />
They’ll make us work so we can work to work it out<br />
And we promised, yes we did, and will, but we haven’t quite met<br />
Fourth quarter projections yet</em></p>
<p> </p>
<p> </p>
<p style="font-size:10px;"><em>Credit for the</em><em> <a href="http://commons.wikimedia.org/wiki/File:Michael_Buble_and_Meredith_Vieira.jpg">original image</a> </em><em>of a smiling Mr. Bublé goes to www.vancityallie.com. </em><em> Credit for the <a href="http://commons.wikimedia.org/wiki/File:Michael_Buble_by_Dallas_Bittle.jpg">original image</a> of a smoldering Mr. Bublé goes to Dallas Bittle<span>. Both are available under Creative Commons License 2.0. </span></em></p>
<p style="font-size:10px;"><em>Credit for the bubbles in the first plot go to the colors Blue and Red, and to the letter Q. All are creative, common, and available in Minitab Statistical Software.</em></p>
Data AnalysisFun StatisticsStatisticsTue, 11 Aug 2015 12:00:00 +0000http://blog.minitab.com/blog/data-analysis-and-quality-improvement-and-stuff/the-bubble-plot%3A-its-a-beautiful-displayGreg FoxHigh School Researchers: What Do We Do with All of this Data?
http://blog.minitab.com/blog/statistics-in-the-field/high-school-researchers-what-do-we-do-with-all-of-this-data
<p><em>by Colin Courchesne, guest blogger, representing his Governor's School research team. </em></p>
<p>High-level research opportunities for high school students are rare; however, that was just what the New Jersey Governor’s School of Engineering and Technology provided. </p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/64c1ec6f5eb944a2866e5a1c9b2e80e9/cup_149682_640.png" style="line-height: 20.7999992370605px; margin: 10px 15px; float: right; width: 200px; height: 212px;" /></p>
<p>Bringing together the best and brightest rising seniors from across the state, the Governor’s School, or GSET for short, tasks teams of students with completing a research project chosen from a myriad of engineering fields, ranging from biomedical engineering to, in our team's case, industrial engineering.</p>
<p>Tasked with analyzing, comparing, and simulating queue processes at Dunkin’ Donuts and Starbucks, our team of GSET scholars spent five days tirelessly collecting roughly 250 data points on each restaurant. Our data included how much time people spent waiting in line, what type of drinks customers ordered, and how much time they spent waiting for their drinks after ordering.</p>
<p><img alt="data collection interface" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/ec87ea6790ddc0e8ddf576c7efd1e6e9/gov_school_1.png" style="line-height: 20.7999992370605px; width: 600px;" /><br />
<em>The students used a computerized interface to collect data about customers in two different coffee shops.</em></p>
<p>But once the data collection was over, we reached a sort of brick wall. What do we <em>do </em>with all this data? <span style="line-height: 1.6;">As research debutantes not well versed in the realm of statistics and data analysis, we had no idea how to proceed. </span></p>
<p><span style="line-height: 1.6;">Thankfully, the helping hand of our project mentor, engineer Brandon Theiss, guided us towards <a href="http://www.minitab.com/products/minitab">Minitab</a>.</span></p>
Getting Meaning Out of Our Data
<p>Our original, raw data told us nothing. In order to compare data between stores and create accurate process simulations, we needed a way to sort the data, determine descriptive statistics, and assign distributions; it is these very tools that Minitab offered. Getting started was both easy and intuitive.</p>
<p>First, we all managed to download Minitab 17 (thanks to <a href="http://it.minitab.com/products/minitab/free-trial.aspx">the 30-day trial</a>). Our team then went on to learn the ins and outs of Minitab, both through <a href="http://www.minitab.com/support/videos/">instructional videos on YouTube</a> as well as <a href="http://support.minitab.com/minitab/17/">helpful written guides</a>, all of which are provided by Minitab. Less than an hour later, we were able to navigate the program with ease.</p>
<p>The nature of the simulations our team intended to create called for us to identify the arrival process for each store, the distributions for the wait time of a customer in line at each restaurant, as well as the distributions for the drink preparation time, sectioned off by both restaurant as well as drink type. In order to input this information into our simulation, we also needed certain parameters that were dependent on the distribution. Such parameters ranged from alpha and beta values for Gamma distributions to means and standard deviations for Normal distributions.</p>
<p>Thankfully, running the necessary hypothesis tests and calculating each of these parameters was simple. We first used the “<a href="http://blog.minitab.com/blog/statistics-and-quality-data-analysis/overfit-those-skintight-jeans-fit-perfect-when-you-bought-them-but">Goodness of fit for Poisson</a>” test in order to analyze our arrival rates.</p>
All Necessary Information
<p>Rather than having to fiddle with equations and arrange cells like in Excel, Minitab quickly provided us with all necessary information, including our P-value to determine whether the distribution fit the data as well as parameters for shape and scale.</p>
<p>As for distributions for individual drink preparation times, the process was similarly simple. Using the “<a href="http://blog.minitab.com/blog/meredith-griffith/identifying-the-distribution-of-your-data">Individual Distribution Identification</a>” tool, Minitab ran a series of hypothesis tests, comparing our data against a total of 16 possible distributions. The software output graphs along with P-values and Anderson-Darling values for each distribution, allowing us to graphically and empirically determine the appropriateness of fit. </p>
<p><img alt="Probability Plot for Latte S" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/621113f0e0af63fcb51a67e84498e0a9/gov_school_2.png" style="width: 576px;" /></p>
<p>Within 3 hours, we had sorted and analyzed all of our data.</p>
<p>Not only was Minitab a fantastic tool for our analysis purposes, but the software also provided us with a graphical platform, a means by which to produce most of the graphs used in our research paper and presentation. Once we determined which distribution to use with what data, we used Minitab to output histograms with fitted data distributions for each set of data points. The ease of use for this feature served to save us time, as a series of simple clicks allowed us to output all 10 of our required histograms at the same time.</p>
<p><img alt="Histogram of Line Time S" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/d696fa8296af19f5efec2ef9815249fa/gov_school_3.png" style="width: 580px;" /></p>
<p>The same tools first used to analyze our data were then finally used to analyze the success of our simulations; we ran a Kolmogorov-Smirnov test to determine whether two sets of data—in this case, our observed data and the data output by our simulation—share a common distribution. Like most other features in Minitab, it was extremely easy to use and provided clear and immediate feedback as to the results of the test, both graphically and through the requisite critical and KS values</p>
<img alt="Empirical CDF of Iced" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/687a0c85c34e919d3f793ffb9d278b2c/gov_school_4.png" style="float: left; width: 400px; height: 267px;" />
<img alt="Simulated vs Actual " src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/4f7ad45d38e9646b3d585b3907be83df/gov_school_5.png" style="width: 400px; height: 267px;" />
<p><span style="line-height: 1.6;">Research isn’t always fun. It’s often long, tedious, and amounts to nothing. Thankfully, that wasn’t our case. Using Minitab, our entire analysis process was simple and painless. The software was easy to learn and was able to run any test quickly and efficiently, providing us with both empirical and graphical evidence of the results as well as high-quality graphs which were used throughout our project. It really was a pleasure to work with.</span></p>
<p><img alt="GSET Coff(IE) Team" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/13f3ba56ecd0b8f0c6db49e5f8f08a5c/gov_school_6_w1024.png" style="width: 500px; " /></p>
<p><em>—The GSET COFF[IE] Team, whose members were Kenneth Acquah, Colin Courchesne, Sheela Hanagal, Kenneth Li, and Caroline Potts. The team was mentored by Juilee Malavade and Brandon Theiss, </em>PE<em>. Photo courtesy Colin Courchesne. </em></p>
<p> </p>
<p> </p>
<p><strong>About the Guest Blogger:</strong></p>
<p><i>Colin Courchesne was a scholar in the 2015 New Jersey Governor's School of Engineering and Technology, </i><em>a summer program for high-achieving high school students. Students in the program complete a set of challenging courses while working in small groups on real-world research and design projects that relate to the field of engineering. Governor’s School students are mentored by professional engineers as well as Rutgers University honors students and professors, and they often work with companies and organizations to solve real engineering problems.</em></p>
<p> </p>
<p><strong>Would you like to publish a guest post on the Minitab Blog? Contact <a href="mailto:publicrelations@minitab.com?subject=Guest%20Blogger">publicrelations@minitab.com</a>.</strong></p>
<p> </p>
<p> </p>
Data AnalysisFun StatisticsStatistics in the NewsWed, 05 Aug 2015 12:00:00 +0000http://blog.minitab.com/blog/statistics-in-the-field/high-school-researchers-what-do-we-do-with-all-of-this-dataGuest BloggerKappa Studies : What Is the Meaning of a Kappa Value of 0?
http://blog.minitab.com/blog/applying-statistics-in-quality-projects/kappa-studies-%3A-what-is-the-meaning-of-a-kappa-value-of-0
<p>Kappa statistics are commonly used to indicate the degree of agreement of nominal assessments made by multiple appraisers. They are typically used for visual inspection to identify defects. Another example might be inspectors <em>rating </em>defects on TV sets: Do they consistently agree on their classifications of scratches, low picture quality, poor sound? Another application could be patients examined by different doctors for a particular disease: How often will the doctors' diagnoses of the condition agree?</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/86f90d39a93f794eaaf067d8ca432954/kappa.png" style="margin: 10px 15px; width: 170px; height: 170px; float: right;" />A <a href="http://support.minitab.com/en-us/minitab/17/topic-library/quality-tools/measurement-system-analysis/attribute-agreement-analysis/kappa-statistics-and-kendall-s-coefficients/">Kappa study</a> will enable you to understand whether an appraiser is consistent with himself (within-appraiser agreement), coherent with his colleagues (inter-appraiser agreement) or with a reference value (standard) provided by an expert. If the kappa value is poor, it probably means that some additional training is required.</p>
<p>The higher the kappa value, the stronger the degree of agreement.</p>
<p>When:</p>
<ul>
<li>Kappa = 1, perfect agreement exists.</li>
<li>Kappa < 0, agreement is weaker than expected by chance; this rarely happens.</li>
<li>Kappa close to 0, the degree of agreement is the same as would be expected by chance.</li>
</ul>
<p>But what is exactly the meaning of a Kappa value of 0?</p>
<p>Remember your years at school? Suppose that you are expected to go through a very difficult examination (with multiple choice questions) and that for this very particular subject, you had, unfortunately, no time at all to review any course material due to a lack of time, family constraints and other very understandable and valid reasons. Suppose that this exam gave you five possible choices and only one correct answer for every question.</p>
<p>If you tick randomly to select one choice per question, you might end up having 20% correct answers, by chance only. Not bad after all, considering the minimal amount of effort involved, but in this case, a 20% agreement with the correct answers would result in a…kappa score of 0.</p>
Kappa Measure in Attribute Agreement Analysis
<p>In an <a href="http://blog.minitab.com/blog/understanding-statistics/got-good-judgment-prove-it-with-attribute-agreement-analysis">attribute agreement analysis</a>, the kappa measure takes into account the agreement occurring by chance only.</p>
<p>The table below shows the odds of by-chance agreement between correct answer and appraiser assessment:</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/31b80fb2-db66-4edf-a753-74d4c9804ab8/Image/f651d523ff46057b658756e76b5af30e/kappa.JPG" style="width: 578px; height: 181px;" /></p>
<p>To estimate the Kappa value, we need to compare the <em>observed </em>proportion of correct answers to the <em>expected </em>proportion of correct answers (based on chance only):</p>
<p style="margin-left: 40px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/31b80fb2-db66-4edf-a753-74d4c9804ab8/Image/872e4ed3524e0b94d73e4bc6af9a84c1/kappa_stat.JPG" style="width: 445px; height: 119px;" /> </p>
<p>Kappas can be used only with <a href="http://blog.minitab.com/blog/marilyn-wheatleys-blog/coffee-or-tea-analyzing-categorical-data-with-minitab-v2">binary or nominal-scale ratings</a>, they are not really relevant for ordered-categorical ratings (for example "good," "fair," "poor").</p>
<p>Kappas are not only restricted to visual inspection in a manufacturing environment. A call center might use this approach to rate the way incoming calls are dealt with, or a tech support service might use it to rate the answers provided by employees. In an hospital this approach could be used to rate the adequacy of health procedures implemented for different types of situations or different symptoms.</p>
<p>Where could you use Kappa studies? </p>
<p> </p>
Quality ImprovementStatisticsTue, 04 Aug 2015 04:00:00 +0000http://blog.minitab.com/blog/applying-statistics-in-quality-projects/kappa-studies-%3A-what-is-the-meaning-of-a-kappa-value-of-0Bruno Scibilia