Data Analysis Software | MinitabBlog posts and articles with tips for using statistical software to analyze data for quality improvement.
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Sun, 01 Mar 2015 04:22:07 +0000FeedCreator 1.7.3Creating a New Metric with Gage R&R, part 2
http://blog.minitab.com/blog/understanding-statistics/creating-a-new-metric-with-gage-rr-part-2
<p style="line-height: 20.7999992370605px;">In my previous post, I showed you <a href="http://blog.minitab.com/blog/understanding-statistics/creating-a-new-metric-with-gage-rr-part-1">how to set up data collection for a gage R&R analysis</a> using the Assistant in Minitab 17. In this case, the goal of the gage R&R study is to test whether a new tool provides an effective metric for assessing resident supervision in a medical facility. </p>
<p style="line-height: 20.7999992370605px;"><span style="line-height: 20.7999992370605px;">As noted in that post, I'm drawing on one of my favorite bloggers about health care quality, David Kashmer of the Business Model Innovation in Surgery blog, and specifically his</span><span style="line-height: 20.7999992370605px;"> column "</span><a href="http://www.surgicalbusinessmodelinnovation.com/statistical-process-control/how-to-measure-a-process-when-theres-no-metric/" style="line-height: 20.7999992370605px;" target="_blank">How to Measure a Process When There's No Metric</a><span style="line-height: 20.7999992370605px;">." </span></p>
An Effective Measure of Resident Supervision?
<p style="line-height: 20.7999992370605px;">In one scenario Kashmer presents, state regulators and hospital staff disagree about a health system's ability to oversee residents. In the absence of an established way to measure resident <span style="line-height: 20.7999992370605px;">supervision</span><span style="line-height: 20.7999992370605px;">, the staff devises a tool that uses a 0 to 10 scale to rate resident supervision. </span></p>
<p style="line-height: 20.7999992370605px;">Now we're going to analyze the Gage R&R data to test how effectively and reliably the new tool <span style="line-height: 20.7999992370605px;">measures what we want it to measure</span><span style="line-height: 20.7999992370605px;">. The analysis will evaluate whether different people who use the tool </span><span style="line-height: 20.7999992370605px;">(the gauge)</span><span style="line-height: 20.7999992370605px;"> </span><span style="line-height: 20.7999992370605px;">reach the same conclusion (reproducibility) and do it consistently (repeatability). </span></p>
<p style="line-height: 20.7999992370605px;">To get data, three evaluators used the tool to assess each of 20 charts three times each, and recorded their score for each chart in the worksheet we produced earlier. (You can download the completed worksheet <a href="//cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/File/02131d16de689b5864576174e86da023/gage_resident_supervision.MTW">here</a> if you're following along in Minitab.) </p>
<p style="line-height: 20.7999992370605px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/96cfa33c2344135d665b93e2c637b017/data_sheet.gif" style="width: 332px; height: 381px;" /></p>
<p>Now we're ready to analyze the data. </p>
Evaluating the Ability to Measure Accurately
<p style="line-height: 20.7999992370605px;">Once again, we can turn to the Assistant in Minitab Statistical Software to help us. If you're not already using it, your can <a href="http://it.minitab.com/products/minitab/free-trial.aspx">download a 30-day trial version</a> for free so you can follow along. Start by selecting <strong>Assistant > Measurement Systems Analysis...</strong> from the menu: </p>
<p style="line-height: 20.7999992370605px;"><img alt="measurement systems analysis " src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/10b2080fd1ed8b3e1337e7838fd85313/assistant_msa.gif" style="width: 345px; height: 258px;" /></p>
<p style="line-height: 20.7999992370605px;">In my earlier post, we used the Assistant to set up this study and make it easy to collect the data we need. Now that we've gathered the data, we can follow the Assistant's decision tree to the "Analyze Data" option. </p>
<p style="line-height: 20.7999992370605px;"><img alt="measurement systems analysis decision tree for analysis" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/f5fd33e7322a966327a9a5dc659294b6/gage_dialog_analyze.gif" style="width: 600px; height: 450px;" /></p>
<p style="line-height: 20.7999992370605px;">Selecting the right items for the Assistant's Gage R&R dialog box couldn't be easier—when you use the datasheet the Assistant generated, just enter "Operators" for Operators, "Parts" for Parts, and "Score" for Measurements. </p>
<p style="line-height: 20.7999992370605px;"><img alt="gage R&R analysis dialog box" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/b65b3a39837f9875c9748c46d12498f8/grnr_analysis_dialog.png" style="line-height: 20.7999992370605px; width: 600px; height: 397px;" /></p>
<p style="line-height: 20.7999992370605px;"><span style="line-height: 20.7999992370605px;">Before we press OK, though, we need to tell the Assistant how to estimate process variation. When Gage R&R is performed in a manufacturing context, historic data about the amount of variation in the output of the process being studied is usually available. Since this is the first time we're analyzing the performance of the new tool for measuring the quality of resident supervision, we don't have an historical standard deviation</span><span style="line-height: 20.7999992370605px;">, so </span><span style="line-height: 20.7999992370605px;">we will tell the Assistant to estimate the variation from the data we're analyzing. </span></p>
<p style="line-height: 20.7999992370605px;"><span style="line-height: 20.7999992370605px;"><img alt="gage r&r variation calculation options" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/0e55e7fca7d27eeaa22484359e996a8a/grnr_analysis_variation.png" style="width: 546px; height: 110px;" /></span></p>
<p style="line-height: 20.7999992370605px;"><span style="line-height: 20.7999992370605px;">The Assistant also asks for an upper or lower specification limit, or tolerance width</span><span style="line-height: 20.7999992370605px;">, which is the distance from the upper spec limit to the lower spec limit</span><span style="line-height: 20.7999992370605px;">. Minitab uses this to calculate %Tolerance, an optional statistic used to determine whether the measurement system can adequately sort good from bad parts—or in this case, good from bad supervision. For the sake of this example, let's say in designing the instrument you have selected a level of 5.0 as the minimum acceptable score. </span></p>
<p style="line-height: 20.7999992370605px;"><span style="line-height: 20.7999992370605px;"><img alt="gage r and r process tolerance" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/81c00cbc50239082a77d7e3e4c522afe/grnr_analysis_dialog_tolerance.png" style="width: 536px; height: 178px;" /> </span></p>
<p style="line-height: 20.7999992370605px;">When we press OK, the Assistant analyzes the data and presents a Summary Report, a Variation Report, and a Report Card for its analysis. The Summary Report gives us the bottom line about how well the new measurement system works. </p>
<p style="line-height: 20.7999992370605px;">The first item we see is a bar graph that answers the question, "Can you adequately assess process performance?" The Assistant's analysis of the data tells us that the system we're using to measure patient supervision can indeed assess the <span style="line-height: 20.7999992370605px;">resident supervision </span><span style="line-height: 20.7999992370605px;">process. </span></p>
<p style="line-height: 20.7999992370605px;"><img alt="gage R&R summary" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/36cd05b806ba3399849a5a7f562b6892/gage_r_r_summary_report.png" style="width: 600px; height: 471px;" /></p>
<p style="line-height: 20.7999992370605px;">The second bar graph answers the question "Can you sort good parts from bad?" In this case, we're evaluating patient supervision rather than parts, but the Analysis shows that the system is able to distinguish charts that indicate acceptable resident supervision from those that do not. </p>
<p style="line-height: 20.7999992370605px;">For both of these charts, less than 10% of the observed variation in the data could be attributed to the measurement system itself—a very good result.</p>
Measuring the "Unmeasurable"
<p style="line-height: 20.7999992370605px;">I can't count the number of times I've heard people say that they can't gather or analyze data about a situation because "it can't be measured." In most cases, that's just not true. Where a factor of interest—"service quality," say—is tough to measure <em>directly</em>, we can usually find measurable indicator variables that can at least give us some insight into our performance. </p>
<p style="line-height: 20.7999992370605px;">I hope this example, though simplified from what you're likely to encounter in the real world, shows how it's possible to demonstrate the effectiveness of a measurement system when one doesn't already exist. Even for outcomes that seem hard to quantify, we can create measurement systems to give us valuable data, which we can then use to make improvements. </p>
<p style="line-height: 20.7999992370605px;">What kinds of outcomes would you like to be able to measure in your profession? Could you use Gage R&R or another form of measurement system analysis to get started? </p>
<p style="line-height: 20.7999992370605px;"> </p>
Data AnalysisQuality ImprovementStatisticsStatistics HelpThu, 26 Feb 2015 13:00:00 +0000http://blog.minitab.com/blog/understanding-statistics/creating-a-new-metric-with-gage-rr-part-2Eston MartzCreating a New Metric with Gage R&R, part 1
http://blog.minitab.com/blog/understanding-statistics/creating-a-new-metric-with-gage-rr-part-1
<p>One of my favorite bloggers about the application of statistics in health care is David Kashmer, an MD and MBA who runs and writes for the <a href="http://www.surgicalbusinessmodelinnovation.com/" target="_blank">Business Model Innovation in Surgery</a> blog. If you have an interest in how quality improvement methods like Lean and Six Sigma can be applied to healthcare, check it out. </p>
<p>A while back, Dr. Kashmer penned a column called "<a href="http://www.surgicalbusinessmodelinnovation.com/statistical-process-control/how-to-measure-a-process-when-theres-no-metric/" target="_blank">How to Measure a Process When There's No Metric</a>," in which he discusses how you can use the measurement systems analysis method called Gage R&R (or gauge R&R) to create your own measurement tools and validate them as useful metrics. (I select the term “useful” here deliberately: a metric you’ve devised could be very <em>useful </em>in helping you assess your situation, but might not meet requirements set by agencies, auditors, or other concerned parties.) </p>
<p>I thought I would use this post to show you how you can use the Assistant in Minitab Statistical Software to <span style="line-height: 20.7999992370605px;">do this</span><span style="line-height: 1.6;">.</span></p>
How Well Are You Supervising Residents?
<p>Kashmer posits a scenario in which state regulators assert that your health system's ability to oversee residents is poor, but your team believes residents are well supervised. You want to assess the situation with data, but you lack an established way to measure the quality of resident supervision. What to do?</p>
<p>Kashmer says, "You decide to design a tool for your organization. You pull a sample of charts and look for commonalities that seem to display excellent supervision versus poor supervision."</p>
<p>So you work with your team to come up with a tool that uses a 0 to 10 scale to rate resident supervision<span style="line-height: 20.7999992370605px;">, based on various factors appearing on a chart</span>. But how do you know if the tool will actually help you assess the quality of resident supervision? </p>
<p>This is where gage R&R comes in. The gage refers to the tool or instrument you're testing, and the R&R stands for reproducibility and repeatability. The analysis will tell you whether different people who use your tool to assess resident supervision (the gauge) will reach the same conclusion (reproducibility) and do it consistently (repeatability). </p>
Collecting Data to Evaluate the Ability to Measure Accurately
<p>We're going to use the Assistant in Minitab Statistical Software to help us. If you're not already using it, you can <a href="http://it.minitab.com/products/minitab/free-trial.aspx">download a 30-day trial version</a> for free so you can follow along. Start by selecting <strong>Assistant > Measurement Systems Analysis...</strong> from the menu: </p>
<p><img alt="measurement systems analysis " src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/10b2080fd1ed8b3e1337e7838fd85313/assistant_msa.gif" style="width: 345px; height: 258px;" /></p>
<p>Follow the decision tree...</p>
<p><img alt="measurement systems analysis decision tree" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/5f5055a500745c69c183056582dc41a6/msa_decision_tree.gif" style="width: 600px; height: 450px;" /></p>
<p>If you're not sure about what you need to do in a gage R&R, clicking the <strong><em>more...</em></strong> link gives you requirements, assumptions, and guidelines to follow: </p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/62bf793d1ea0e4bbbdad53ffb70783e5/gager_rassumptions.gif" style="width: 600px; height: 454px;" /></p>
<p>After a look at the requirements, you decide you will have three evaluators use your new tool to assess each of 20 charts 3 times, and so you complete the dialog box thus: </p>
<p><img alt="MSA dialog box" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/067ec3e5997c68e845d4061a8251b862/msa_dialog.gif" style="width: 500px; height: 401px;" /></p>
<p style="line-height: 20.7999992370605px;">When you press "OK," the Assistant asks if you'd like to print worksheets you can use to easily gather your data:</p>
<p><img alt="gage R&R data collection form" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/ffb7215f1d123d17cca1eaa3c11211d8/msa_gage_r_r_data_collection_form.gif" style="line-height: 20.7999992370605px; width: 400px; height: 430px;" /></p>
<p>Minitab also creates a datasheet for the analysis. All you need to do is enter the data you collect in the "Measurements" column:</p>
<p><img alt="worksheet" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/cfd21a76a9cab42059705c67e54e5fdc/gage_r_r_worksheet.gif" style="line-height: 20.7999992370605px; width: 355px; height: 357px;" /></p>
<p>Note that the Assistant automatically randomizes the order in which each evaluator will examine the charts in each of their three judging sessions. </p>
<p>Now we're ready to gather the data to verify the effectiveness of our new metric for assessing the quality of patient supervision. Come back for Part 2, where we'll <a href="http://blog.minitab.com/blog/understanding-statistics/creating-a-new-metric-with-gage-rr-part-2">analyze the collected data</a>! </p>
Health Care Quality ImprovementLean Six SigmaSix SigmaWed, 25 Feb 2015 13:00:00 +0000http://blog.minitab.com/blog/understanding-statistics/creating-a-new-metric-with-gage-rr-part-1Eston MartzChoosing Between a Nonparametric Test and a Parametric Test
http://blog.minitab.com/blog/adventures-in-statistics/choosing-between-a-nonparametric-test-and-a-parametric-test
<p>It’s safe to say that most people who use statistics are more familiar with parametric analyses than nonparametric analyses. Nonparametric tests are also called distribution-free tests because they don’t assume that your data follow a specific distribution.</p>
<p>You may have heard that you should use nonparametric tests when your data don’t meet the assumptions of the parametric test, especially the assumption about normally distributed data. That sounds like a nice and straightforward way to choose, but there are additional considerations.</p>
<p>In this post, I’ll help you determine when you should use a:</p>
<ul>
<li>Parametric analysis to test group means.</li>
<li>Nonparametric analysis to test group medians.</li>
</ul>
<p>In particular, I'll focus on an important reason to use nonparametric tests that I don’t think gets mentioned often enough!</p>
Hypothesis Tests of the Mean and Median
<p>Nonparametric tests are like a parallel universe to parametric tests. The table shows related pairs of <a href="http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/hypothesis-tests/basics/hypothesis-tests-in-minitab/" target="_blank">hypothesis tests</a> that <a href="http://www.minitab.com/en-us/products/minitab/features/" target="_blank">Minitab statistical software</a> offers.</p>
<p style="text-align: center;"><strong>Parametric tests (means)</strong></p>
<p style="text-align: center;"><strong>Nonparametric tests (medians)</strong></p>
<p style="text-align: center;">1-sample t test</p>
<p style="text-align: center;">1-sample Sign, 1-sample Wilcoxon</p>
<p style="text-align: center;">2-sample t test</p>
<p style="text-align: center;">Mann-Whitney test</p>
<p style="text-align: center;">One-Way ANOVA</p>
<p style="text-align: center;">Kruskal-Wallis, Mood’s median test</p>
<p style="text-align: center;">Factorial DOE with one factor and one blocking variable</p>
<p style="text-align: center;">Friedman test</p>
Reasons to Use Parametric Tests
<p><strong>Reason 1: Parametric tests can perform well with skewed and nonnormal distributions</strong></p>
<p>This may be a surprise but parametric tests can perform well with continuous data that are nonnormal if you satisfy these sample size guidelines.</p>
<p style="text-align: center;"><strong>Parametric analyses</strong></p>
<p style="text-align: center;"><strong>Sample size guidelines for nonnormal data</strong></p>
<p style="text-align: center;">1-sample t test</p>
<p style="text-align: center;">Greater than 20</p>
<p style="text-align: center;">2-sample t test</p>
<p style="text-align: center;">Each group should be greater than 15</p>
<p style="text-align: center;">One-Way ANOVA</p>
<ul>
<li style="text-align: center;">If you have 2-9 groups, each group should be greater than 15.</li>
<li style="text-align: center;">If you have 10-12 groups, each group should be greater than 20.</li>
</ul>
<p><strong>Reason 2: Parametric tests can perform well when the spread of each group is different</strong></p>
<p>While nonparametric tests don’t assume that your data follow a normal distribution, they do have other assumptions that can be hard to meet. For nonparametric tests that compare groups, a common assumption is that the data for all groups must have the same spread (dispersion). If your groups have a different spread, the nonparametric tests might not provide valid results.</p>
<p>On the other hand, if you use the 2-sample t test or One-Way ANOVA, you can simply go to the <strong>Options</strong> subdialog and uncheck <em>Assume equal variances</em>. Voilà, you’re good to go even when the groups have different spreads!</p>
<p><strong>Reason 3: Statistical power</strong></p>
<p>Parametric tests usually have more <a href="http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/power-and-sample-size/what-is-power/" target="_blank">statistical power</a> than nonparametric tests. Thus, you are more likely to detect a significant effect when one truly exists.</p>
Reasons to Use Nonparametric Tests
<p><strong>Reason 1: Your area of study is better represented by the median</strong></p>
<p><img alt="Comparing two skewed distributions" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/7223b01bc095dbd652bd863be5288cfe/mean_or_median.png" style="float: right; width: 200px; height: 181px; margin: 10px 15px;" />This is my favorite reason to use a nonparametric test and the one that isn’t mentioned often enough! The fact that you <em>can</em> perform a parametric test with nonnormal data doesn’t imply that the mean is the best <a href="http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/summary-statistics/measures-of-central-tendency/" target="_blank">measure of the central tendency</a> for your data.</p>
<p>For example, the center of a skewed distribution, like income, can be better measured by the median where 50% are above the median and 50% are below. If you add a few billionaires to a sample, the mathematical mean increases greatly even though the income for the typical person doesn’t change.</p>
<p>When your distribution is skewed enough, the mean is strongly affected by changes far out in the distribution’s tail whereas the median continues to more closely reflect the center of the distribution. For these two distributions, a random sample of 100 from each distribution produces means that are significantly different, but medians that are not significantly different.</p>
<p>Two of my colleagues have written excellent blog posts that illustrate this point:</p>
<ul>
<li>Michelle Paret: <a href="http://blog.minitab.com/blog/michelle-paret/using-the-mean-its-not-always-a-slam-dunk" target="_blank">Using the Mean in Data Analysis: It’s Not Always a Slam-Dunk</a></li>
<li>Redouane Kouiden: <a href="http://blog.minitab.com/blog/statistics-for-lean-six-sigma/the-non-parametric-economy-what-does-average-actually-mean" target="_blank">The Non-parametric Economy: What Does Average Actually Mean?</a></li>
</ul>
<p><strong>Reason 2: You have a very small sample size</strong></p>
<p>If you don’t meet the sample size guidelines for the parametric tests and you are not confident that you have normally distributed data, you should use a nonparametric test. When you have a really small sample, you might not even be able to ascertain the distribution of your data because the distribution tests will lack sufficient power to provide meaningful results.</p>
<p>In this scenario, you’re in a tough spot with no valid alternative. Nonparametric tests have less power to begin with and it’s a double whammy when you add a small sample size on top of that!</p>
<p><strong>Reason 3: You have ordinal data, ranked data, or outliers that you can’t remove</strong></p>
<p>Typical parametric tests can only assess continuous data and the results can be significantly affected by outliers. Conversely, some nonparametric tests can handle ordinal data, ranked data, and not be seriously affected by outliers. Be sure to check the assumptions for the nonparametric test because each one has its own data requirements.</p>
Closing Thoughts
<p>It’s commonly thought that the need to choose between a parametric and nonparametric test occurs when your data fail to meet an assumption of the parametric test. This can be the case when you have both a small sample size and nonnormal data. However, other considerations often play a role because parametric tests can often handle nonnormal data. Conversely, nonparametric tests have strict assumptions that you can’t disregard.</p>
<p>The decision often depends on whether the mean or median more accurately represents the center of your data’s distribution.</p>
<ul>
<li>If the mean accurately represents the center of your distribution and your sample size is large enough, consider a parametric test because they are more powerful.</li>
<li>If the median better represents the center of your distribution, consider the nonparametric test even when you have a large sample.</li>
</ul>
<p>Finally, if you have a very small sample size, you might be stuck using a nonparametric test. Please, collect more data next time if it is at all possible! As you can see, the sample size guidelines aren’t really that large. Your chance of detecting a significant effect when one exists can be very small when you have both a small sample size and you need to use a less efficient nonparametric test!</p>
Hypothesis TestingStatisticsStatistics HelpThu, 19 Feb 2015 13:00:00 +0000http://blog.minitab.com/blog/adventures-in-statistics/choosing-between-a-nonparametric-test-and-a-parametric-testJim FrostUsing Regression to Evaluate Project Results, part 1
http://blog.minitab.com/blog/statistics-in-the-field/using-regression-to-evaluate-project-results%2C-part-1
<p><em>By Peter Olejnik, guest blogger.</em></p>
<p>Previous posts on the Minitab Blog have discussed the work of the Six Sigma students at <a href="http://rose-hulman.edu/">Rose-Hulman Institute of Technology</a> to reduce the quantities of recyclables that wind up in the trash. Led by Dr. Diane Evans, these students continue to make an important impact on their community.</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/64e636cef00078d6a7b28245f62f8759/recyclables.png" style="border-width: 1px; border-style: solid; margin: 10px 15px; float: right; width: 200px; height: 200px;" />As with any Six Sigma process, the results of the work need to be evaluated. A simple two-sample T test could be performed, but it gives us a very limited amount of information – only whether there is a difference between the before and after improvement data. But what if we want to know if a certain item or factor affects the amount of recyclables disposed of? What if we wanted to know by how much of an effect important factors have? What if we want to create a predictive model that can estimate the weight of the recyclables without the use of a scale?</p>
<p>Sounds like a lot of work, right? But actually, with the use of regression analysis tools in Minitab Statistical Software, it's quite easy!</p>
<p>In this two-part blog post, I’ll share with you how my team used regression analysis to identify and model the factors that are important in making sure recyclables are handled appropriately. </p>
Preparing Your Data for Regression Analysis
<p><span style="line-height: 1.6;">All the teams involved in this project collected a substantial amount of data. But some of this data is somewhat subjective. Also, this data has been recorded in a manner that is geared toward people, and not necessarily for analysis by computers. To start doing analysis in Minitab, all of our data points need to be quantifiable and in long format.</span></p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/9111cc378a9d3a50ad02e8f920d5af6d/data_as_inserted_by_teams_w1024.png" style="width: 800px; height: 485px;" /></p>
<p align="center"><em>The Data as Inserted by the Six Sigma Teams</em></p>
<p align="center"><img alt="data after conversion" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/06c69397a0dc317ee13067c8d9f47ab7/data_after_conversion_w1024.png" style="width: 800px; height: 383px;" /></p>
<p align="center"><em><span style="line-height: 1.6;">The Data, After Conversion into Long Format and Quantifiable Values</span></em></p>
<p><span style="line-height: 1.6;">Now that we have all this data in a computer-friendly format, we need to identify and eliminate any extreme outliers present, since they can distort our final model. First we create a regression model with all of the factors included. As part of this, we generate the residuals from the data vs. the fit. For our analysis, we utilized deleted T-residuals. These are less affected by the skew of an outlier compared to regular T-residuals, making it them better indicator. These can be selected to be displayed by Minitab in the same manner that any other residual can be selected. Looking at these residuals, those with values above 4 were removed. A new fit was then created and the process was repeated until no outliers remain.</span></p>
Satisfying the Assumptions for Regression Analysis
<p>Once the outliers have been eliminated, we need to verify the regression assumptions for our data to ensure that the analysis conducted is valid. We need to satisfy five assumptions:</p>
<ol>
<li style="margin-left: 0.5in;">The mean value of the errors is zero.</li>
<li style="margin-left: 0.5in;">The variance of the errors is even and consistent (or “homoscedastic”) through the data.</li>
<li style="margin-left: 0.5in;">The data is independent and identically distributed (IID).</li>
<li style="margin-left: 0.5in;">The errors are normally distributed.</li>
<li style="margin-left: 0.5in;">There is negligible variance in the predictor values.</li>
</ol>
<p><span style="line-height: 1.6;">For our third assumption, we know that the data points should be IID, because each area’s daily trash collection should have no effect on that of other areas or the next day’s collection. We have no reason to suspect otherwise. The fifth assumption is also believed to have been met, as we have no reason to suspect that there is variance in the predictor value. This means that only three of the five assumptions still need to be checked.</span></p>
<p>Our first and second assumptions can be checked simply by plotting the deleted T-residuals against the individual factors, as well as the fits and visually inspecting them.</p>
<p><br />
<img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/4db6a7f0e802ff26b7641baf76b3326b/scatterplots_1.png" style="width: 800px; height: 537px;" /><br />
<img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/93f6a521a7017db9f74588d7d0a28c98/scatterplots_2.png" style="width: 800px; height: 532px;" /><br />
<img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/79b06f78d420550190a39e49e045765c/scatterplots_3.png" style="width: 800px; height: 532px;" /></p>
<p align="center"><em><img src="file:///C:\Users\emartz\AppData\Local\Temp\msohtmlclip1\01\clip_image007.emz" /> <img src="file:///C:\Users\emartz\AppData\Local\Temp\msohtmlclip1\01\clip_image008.emz" /> <img src="file:///C:\Users\emartz\AppData\Local\Temp\msohtmlclip1\01\clip_image009.emz" /> <span style="line-height: 1.6;"> </span><img src="file:///C:\Users\emartz\AppData\Local\Temp\msohtmlclip1\01\clip_image012.emz" style="line-height: 1.6;" /><span style="line-height: 1.6;">Plots used to verify regression assumptions.</span></em></p>
<p>When looking over the scatter plots, it looks like these two assumptions are met. Checking the fourth assumption is just as easy. All that needs to be done is to run a normality test on the deleted T-residuals.</p>
<p align="center"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/ac5e0f64d22e22a5703b493d4a2e54a3/probability_plot_of_deleted_t_residuals.png" style="width: 577px; height: 385px;" /><br />
<em style="line-height: 1.6;">Normality plot of deleted T-residuals</em></p>
<p><span style="line-height: 1.6;">It appears that our residuals are not normally distributed, as seen by the p-value of our test. This is problematic, as it means any analysis we would conduct would be invalid. Fortunately, all is not lost: we can perform a Box-Cox analysis on our results. This will tell us is if the response variable needs to be raised by a power.</span></p>
<p align="center"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/6539dba1ff25f65482ba8501a0216d3a/box_cox_transformation_analysis.png" style="width: 577px; height: 385px;" /><br />
<em>Box-Cox analysis of the data</em></p>
<p><span style="line-height: 1.6;">The results of this analysis indicate that the response variable should be raised by a constant of 0.75. A new model and residuals can be generated from this modified response variable and the assumptions can be checked.</span></p>
<p align="center"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/c2eac5fd8a48d5476d19510ca69687bd/scatterplots_4.png" style="width: 800px; height: 533px;" /><br />
<img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/42763813beb394fda03a7eaa6ecb53b4/scatterplots_5.png" style="width: 800px; height: 532px;" /><br />
<img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/1eeafd4ef3c00837b425530c40b1a3d3/scatterplots_6.png" style="width: 800px; height: 533px;" /><em>N<span style="line-height: 1.6;">ew plots used in order to verify regression assumptions for revised model.</span></em></p>
<p><span style="line-height: 1.6;">The residuals again appear to be homoscedastic and centered about zero.</span></p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/42df464e7acd7f5b309e6dac0608ba3a/normality_plot.png" style="width: 577px; height: 385px;" /></p>
<p align="center"><em>Normality plot on deleted T-residuals</em></p>
<p>The residuals now are normally distributed. <span style="line-height: 1.6;">Our data is now prepped and ready for analysis!</span></p>
<p>The second part of this post will detail the regression analysis. </p>
<p> </p>
<p><strong>About the Guest Blogger</strong></p>
<p><em>Peter Olejnik is a graduate student at the Rose-Hulman Institute of Technology in Terre Haute, Indiana. He holds a bachelor’s degree in mechanical engineering and his professional interests include controls engineering and data analysis.</em></p>
<p> </p>
Regression AnalysisMon, 16 Feb 2015 13:00:00 +0000http://blog.minitab.com/blog/statistics-in-the-field/using-regression-to-evaluate-project-results%2C-part-1Guest BloggerA Little Trash Talk: Improving Recycling Processes at Rose-Hulman, Part II
http://blog.minitab.com/blog/real-world-quality-improvement/a-little-trash-talk%3A-improving-recycling-processes-at-rose-hulman%2C-part-ii
<p><span style="line-height: 1.6;">I left off last with a </span><a href="http://blog.minitab.com/blog/real-world-quality-improvement/a-little-trash-talk3a-improving-recycling-processes-at-rose-hulman" style="line-height: 1.6;" target="_blank">post</a><span style="line-height: 1.6;"> outlining how the Six Sigma students at Rose-Hulman were working on a project to reduce the amount of recycling thrown in the normal trash cans in all of the academic buildings at the institution.</span></p>
<p>Using the <a href="http://blog.minitab.com/blog/real-world-quality-improvement/dmaic-vs-dmadv-vs-dfss" target="_blank">DMAIC</a> methodology for completing improvement projects, they had already defined the problem at hand: how could the amount of recycling that’s thrown in the normal trash cans be reduced? They collected baseline data for the types of recyclables thrown into the trash, including their weights and frequencies. In order to brainstorm ideas to improve recycling efforts at Rose-Hulman and to determine causes for the lack of recycling in the first place, the students created <a href="http://blog.minitab.com/blog/understanding-statistics/five-types-of-fishbone-diagrams" target="_blank">fishbone diagrams</a>.</p>
Implementing Improvements
<p>The students then entered the ‘Improve’ phase of the project and formed a list of recommended actions based on the variables they could control to motivate recycling practices in a four-week time frame. The short time constraint was fixed due to the length of an academic quarter.</p>
<p>This list of actions included the following:</p>
<ul>
<li>Placing a recycling bin next to each and every trash can throughout the academic buildings, including classrooms.</li>
<li>Constructing and displaying posters next to or on recycling bins indicating what items are recyclable and are not recyclable on campus:</li>
</ul>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/ccb8f6d6-3464-4afb-a432-56c623a7b437/Image/f2c16c8a3b53884cd996ae306074f582/recycle.jpg" style="width: 560px; height: 348px;" /></p>
<ul>
<li>Informing campus about Rose-Hulman recycling policies, as well as the current percentage of recyclables on campus (by weight), determined during the Measure phase. (The information was shared with the entire campus via an email and an article in the school newspaper by Dr. Evans.)</li>
<li>Encouraging good recycling habits through creative posters, contests, incentives, and using concepts related to “<a href="http://www.thefuntheory.com/" target="_blank">The Fun Theory</a>.” Fun theory is used to change people’s behaviors through making activities fun. For example, the class discussed ways to make recycling bins produce amusing sounds when items are placed in it.</li>
</ul>
<p>The students implemented many of these improvements and then gathered post-improvement data at the end of four weeks during four fixed collection periods.</p>
Analyzing Pre-Improvement vs. Post-Improvement Data
<p>There were a total of 15 areas in the academic buildings where recycling data was collected. Fifteen student teams were assigned one of these areas for the entire project, collecting data during the pre- and post-improvement phases. There are a total of 60 data points for both phases.</p>
<p>The teams compared pre-improvement and post-improvement statistics for the percentage of recyclables in the trash with Minitab (using Stat > Basic Statistics > Display Descriptive Statistics in the software):</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/ccb8f6d6-3464-4afb-a432-56c623a7b437/Image/b4217833edb2c6803d89942897093085/descriptivestats.JPG" style="width: 781px; height: 232px;" /></p>
<p><span style="line-height: 1.6;">Some highlights of this analysis:</span></p>
<ul>
<li>The mean percentage of recyclables in trash decreased from 37% to 24%, which is a reduction of 35%.</li>
<li>The median percentage of recyclables in trash decreased from 31% to 17%, which is a reduction of 45%.</li>
<li>The total average weight of recyclables in trash over the baseline period (4 days) decreased from 84.3 pounds with a standard deviation of approximately 7.89 pounds to 45.9 pounds with a standard deviation of approximately 5.19 pounds during the improvement period, which is a reduction in the total average weight of 46%.</li>
<li>The mean recyclable weight for all areas decreased from 1.405 pounds to 0.765 pounds, which is a reduction of 84%.</li>
</ul>
<p>They were also able to view the improvements graphically with boxplots in Minitab:</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/ccb8f6d6-3464-4afb-a432-56c623a7b437/Image/87e11c170f9d5e1e1085bd8b6a3c61bd/boxplots.JPG" style="width: 830px; height: 286px;" /></p>
<p><em><span style="line-height: 1.6;">Boxplots of the percentage of recyclables during the four collection periods in the pre-improvement phase (left plot) and the four collection periods in the post-improvement phase (right plot).</span></em></p>
<p>Although it is not apparent in these boxplots that the mean percentage of recyclables (the circles with the crossbars) has decreased in the improvement phase, it is obvious that the median percentage of recyclables (line within the boxplot) has decreased.</p>
<p>In addition, the students used Minitab plots to track changes in percentage of recyclables in the trash <em>per area</em>, both pre and post-improvement: </p>
<p align="center"><strong><img src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/ccb8f6d6-3464-4afb-a432-56c623a7b437/Image/fceda9f381ab639aae8e8e087cba0b0f/plot.jpg" style="border-width: 0px; border-style: solid; width: 602px; height: 401px;" /></strong></p>
<p><em>Plot of the mean percentage of recyclables in the trash by academic building area for both pre and post-improvement phases. The mean is averaged over the four collection times in each phase.</em></p>
<p><span style="line-height: 1.6;">These plots helped the students to graphically see gaps between the percentages of recyclables collected pre and post-improvement by area. Given the location of each academic area, the changes between pre and post means were justifiable and informative.</span></p>
<p>And in order to statistically determine if the true mean percentage of recyclables post-improvement was significantly less than the true mean percentage of recyclables pre-improvement, the students ran a paired t-test for all 60 data points, pairing by area and day. See below for the Minitab output for this test:</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/ccb8f6d6-3464-4afb-a432-56c623a7b437/Image/661b0f3db3b30fa71345a9d495b2b41e/t_test.JPG" style="width: 780px; height: 140px;" /></p>
<p><span style="line-height: 1.6;">With a t-test statistic of 4.66, it is evident that the recycling improvements made a difference! They ran a paired </span><em style="line-height: 1.6;">t-</em><span style="line-height: 1.6;">test since the pre and post recyclable percentages were linked by area and day. They did not need to check for normality of the paired differences since we had </span><em style="line-height: 1.6;">n</em><span style="line-height: 1.6;"> = 60 data points.</span></p>
<p>After collecting baseline data, the students had created a Pareto Chart to display the type of trash (and recyclables) found in the regular trash cans. They also created a Pareto Chart for the post-improvement data—you can see both below to compare (pre-improvement – left, post-improvement – right):</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/ccb8f6d6-3464-4afb-a432-56c623a7b437/Image/3edfc7284f094d71fa90a356e50ccf26/pareto.JPG" style="width: 820px; height: 284px;" /><span style="line-height: 1.6;">Plastics were the most common recyclable item in the trash both pre- and post-improvement, and overall, besides the Java City coffee cups <em>increasing</em> </span><em style="line-height: 1.6;">post</em><span style="line-height: 1.6;">-improvement, the other categories saw a noticeable decrease post-improvement compared to pre-improvement.</span></p>
<p>To complete their pre- and post-improvement analysis, the students also ran a capability study in Minitab to determine the pre and post-improvement capability of recyclables in the trash. Post-improvement, both their Pp and Ppk values improved.</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/ccb8f6d6-3464-4afb-a432-56c623a7b437/Image/0e554f5242ca484743f2ce779212376e/processcap.JPG" style="width: 841px; height: 319px;" /></p>
<strong style="line-height: 1.2;">R</strong><strong style="line-height: 1.2;">esults</strong><strong style="line-height: 1.2;"> and Future Improvement Efforts</strong>
<p>Of the 15 areas (Spring Quarter 2014) that collected pre-improvement and post-improvement data over the span of two four-day collection periods, only two areas had an increased percentage of recyclables in the trash after the improvements were made. These two areas had “special causes” associated, which can be explained.</p>
<p>One area with increased recyclables <em>after</em> improvements was the Moench Mailroom. <span style="line-height: 1.6;">The Moench Mailroom area is next to the campus mailroom where students pick up their daily mail, graded homework assignments, etc., in their mail slots. It was evident during post-improvement trash collection that a student had emptied an entire quarter’s worth of mail, including junk mail, magazines, and assignments, into the trash can by the mailroom. Since the student’s name was on the mail and assignments, it was clear that that the recyclables discarded in the trash was from this one student. He certainly threw off that area’s post-improvement data!</span></p>
<p>Although the improvement efforts were short-term, the students saw their efforts significantly decrease the percentage of recyclables being discarded in the normal trash cans at the academic buildings. At the beginning of Spring Quarter 2014, 36% of trash cans (by weight) were recyclable items. At the end of Spring Quarter 2014 after the improvement phase, 24% of trash cans (by weight) were recyclable items!</p>
<p>They were not only able to decrease the carbon footprint of their school and aid in their school’s sustainability program, but the increase in recycling also has the potential to create revenue for the school down the road (if they choose to recycle aluminum cans or sell paper, for example).</p>
<p>Dr. Evans and the students have shared their results with the campus community and plan to work with the administration to publish their results, which will hopefully highlight why these improvement efforts should stick around long-term. <span style="line-height: 1.6;">Way to go Dr. Evans and Rose-Hulman Six Sigma Students!</span></p>
<p><em>Many thanks to Dr. Evans for her contributions to this post! </em></p>
Data AnalysisLean Six SigmaStatisticsFri, 13 Feb 2015 13:00:00 +0000http://blog.minitab.com/blog/real-world-quality-improvement/a-little-trash-talk%3A-improving-recycling-processes-at-rose-hulman%2C-part-iiCarly BarryWhatâ€™s the Probability that Your Favorite Football Team Will Win?
http://blog.minitab.com/blog/customized-data-analysis/what%E2%80%99s-the-probability-that-your-favorite-football-team-will-win
<div>
<p>If you wanted to figure out the probability that your favorite football team will win their next game, how would you do it? My colleague <a href="http://blog.minitab.com/blog/understanding-statistics-and-its-application">Eduardo Santiago</a> and I recently looked at this question, and in this post we'll share how we approached the solution. Let’s start by breaking down this problem:<img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/8954fcace8f66a536aca06fad36a4c5a/boy_football_200.png" style="margin: 10px 15px; float: right; width: 200px; height: 200px;" /></p>
<ol>
<li>There are only two possible outcomes: your favorite team wins, or they lose. Ties are a possibility, but they're very rare. So, to simplify things a bit, we’ll assume they are so unlikely that could be disregarded from this analysis.</li>
<li>There are numerous factors to consider.
<ol style="list-style-type:lower-alpha;">
<li>What will the playing conditions be?</li>
<li>Are key players injured?</li>
<li>Do they match up well with their opponent?</li>
<li>Do they have home-field advantage?</li>
<li>And the list goes on...</li>
</ol>
</li>
</ol>
<p>First, since we assumed the outcome is binary, we can put together a <a href="http://blog.minitab.com/blog/real-world-quality-improvement/using-binary-logistic-regression-to-investigate-high-employee-turnover">Binary Logistic Regression</a> model to predict the probability of a win occurring. Next, we need to find which predictors would be best to include. After <a href="http://www.thepredictiontracker.com/ncaaresults.php" target="_blank">a little research</a>, we found the betting markets seem to take all of this information into account. Basically, we are utilizing the wisdom of the masses to find out what they believe will happen. Since betting markets take this into account, we decided to look at the probability of a win, given the spread of a NCAA football game. </p>
Data Collection
<p>If you are not convinced about how accurate the spreads can be in determining the outcome of a game: win or loss, we collected data for every college football game played <span style="line-height: 20.7999992370605px;">between 2000 and 2014</span><span style="line-height: 1.6;">. The structure of the data is illustrated below. The third column has the spread (or line) provided by casinos at Vegas, and the last column displayed is the actual score differential (vscore – hscore).</span></p>
<p><strong><em>Note</em></strong><em>: In betting lines, a negative spread indicates how many points you are favored over the opponent. In short, you are giving the opponent a certain number of points. </em></p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/52aaa628ea28b55523232a9b2da6b623/table1.png" style="width: 600px; height: 352px;" /></p>
<p><span style="line-height: 1.6;">The original win-or-lose question can be rephrased then as follows: Is the difference between the spreads and actual score differentials statistically significant?</span></p>
<p>Since we have two populations that are dependent we would compare them via a paired t test. In other words, both the <em>Spread</em> and <em>scoreDiffer</em> are observations (a priori and a posteriori) for the same game and they reflect the relative strength of the home team <em>i</em> versus the road team <em>j</em>.</p>
<p>Using <strong>Stat > Basic Statistics > Paired t </strong>in Minitab Statistical Software, we get the output below.</p>
<p style="margin-left: 40px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/5531697c176006a4057f4ab7b6fda7dc/t_test_output.png" style="width: 500px; height: 189px;" /></p>
<p>Since the p-value is larger than 0.05, we can conclude from the 15 years of data that the average difference between Las Vegas spreads and actual score differentials is not significantly different from zero. With this we are saying that the bias that could exist between both measures of relative strength for teams is not different from zero, which in lay terms means that <em>on average</em> the error that exists between Vegas and actual outcomes is negligible.</p>
<p>It is worth noting that the results above were obtained with a sample size of 10,476 games! So we hope you'll excuse our not including <a href="http://blog.minitab.com/blog/understanding-statistics/how-much-data-do-you-really-need-check-power-and-sample-size">power calculations</a> here.</p>
<p>As a final remark on spreads, the histogram of the differences below shows a couple of interesting things:</p>
<ul>
<li>The average difference between the spreads and score differentials seem to be very close to zero. So don’t get too excited yet, as the spreads cannot be used to predict the exact score differential for a game. Nevertheless, with extremely high probability the spread will be very close to the score differential.</li>
<li>The standard deviation, however, is 15.5 points. That means that if a game shows a spread for your favorite team of -3 points, the outcome could be with high confidence within plus or minus 2 standard deviations of the point estimate, which is -3 ± 31 points in this case. So your favorite team could win by 34 points, or lose by 28!</li>
</ul>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/f8f64fe200b85bcd5753a62737735de3/histogram1.png" style="width: 577px; height: 385px;" /></p>
<p align="center"><em>Figure 1 - Distribution of the differences between scores and spreads</em></p>
The Binary Logistic Regression Model
<p>By this point, we hope you are convinced about how good these spread values could be. To make the output more readable we summarized the data as follows:</p>
</div>
<p style="margin-left: 40px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/218aa990c975fa2292d84926ba0002f0/table2.png" style="width: 250px; height: 405px;" /></p>
Creating our Binary Logistic Regression Model
<p>After summarizing the data, we used the Binary Fitted Line Plot (new in Minitab 17) to come up with our model. </p>
<p>If you are following along, here are the steps:</p>
<ol>
<li>Go to <strong>Stat > Regression > Binary Fitted Line Plot</strong></li>
<li>Fill out the dialog box as shown below and click <strong>OK</strong>.</li>
</ol>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/ae06bb10e129b9c527b07700a57a7e2f/dialog1.png" style="width: 600px; height: 457px;" /></p>
<p><span style="line-height: 1.6;">The steps will produce the following graph:</span></p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/e2acef300ee9d1cd605ff089c217c8d5/binary_fitted_line_plot_w1024.png" style="width: 600px; height: 400px;" /></p>
Interpreting the Plot
<p>If your team is favored to win by 25 points or more, you have a very good chance of winning the game, but what if the spread is much closer?</p>
<p>For the 2014 National Championship, Ohio State was an underdog by 6 points to Oregon. Looking at the Binary Fitted Line Plot the probability of a 6-point underdog to win the game is close to 31% in college football. </p>
<p>Ohio State University ended up beating Oregon by 22 points. Given that the differences described in Figure 1 are normally distributed with respect to zero, then if we assume the spread is given (or known), we can compute the probability of the national championship game outcome being as extreme—or more—as it turned out.</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/977aafa9be7a8e2736b1340bca0b3b62/distribution_plot.png" style="width: 576px; height: 384px;" /></p>
<p>With Ohio State 6 point underdogs, and a standard deviation of 15.53, we can run a Probability Distribution Plot to show that Ohio State would win by 22 points or more 3.6% of the time.</p>
<p>Eduardo Santiago and myself will be giving a talk on using statistics to rank college football teams at the upcoming <a href="http://www.amstat.org/meetings/csp/2015/" target="_blank">Conference on Statistical Practice</a> in New Orleans. Our talk is February 21 at 2 p.m. and we would love to have you join. </p>
Fun StatisticsHypothesis TestingRegression AnalysisThu, 12 Feb 2015 13:00:00 +0000http://blog.minitab.com/blog/customized-data-analysis/what%E2%80%99s-the-probability-that-your-favorite-football-team-will-winDaniel GriffithHow to Use Statistical Software to Predict the Exchange Rate
http://blog.minitab.com/blog/voice-of-the-customer/how-to-use-statistical-software-to-predict-the-exchange-rate
<p><span style="font-weight: 600; font-family: 'Open Sans', Helvetica, Arial, sans-serif; line-height: 19px;"><em>The Minitab Fan section of the Minitab blog is your chance to share with our readers! We always love to hear how you are using Minitab products for quality improvement projects, Lean Six Sigma initiatives, research and data analysis, and more. If our software has helped you, please <a href="http://blog.minitab.com/blog/landing-pages/share-your-story-about-minitab/n" style="color: rgb(51, 51, 51); outline: none !important;">share your Minitab story</a>, too!</em></span></p>
<p><img src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/84d328f1-1b81-41d1-aad4-fc00026edd38/Image/a93458e3eaa7a11dd9798d253fd78a7f_w480.jpeg" style="line-height: 20.7999992370605px; width: 200px; height: 228px; border-width: 1px; border-style: solid; margin: 10px 15px; float: right;" /></p>
<p>My LSS coach suggested that I regularly conduct data analysis to refresh my Minitab skills. I'm sure many of you have heard about the devaluation of Russian currency caused by European Union and United States sanctions, and dropping oil prices. </p>
<p>I decided to check this situation with statistical analysis. The question I intended to answer was simple: is there any correlation between <a href="http://en.wikipedia.org/wiki/Brent_Crude" target="_blank">Brent oil price</a> and the exchange rate between the Russian ruble and the U.S. dollar?</p>
<p>I found relevant data from 01-Jan-2014 through 18-Dec-2014 and used the regression tools in Minitab <a href="http://www.minitab.com/products/minitab">statistical software</a> to interpret it.<br />
<br />
First I looked at the model and saw the regression equation was:</p>
<p style="margin-left: 40px;">RR/USD = 78,90 - 0,4071 USD/bbl (Brent), with R-Sq(adj) = 90,2%</p>
<p><img src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/84d328f1-1b81-41d1-aad4-fc00026edd38/Image/798fca93b81fd998cf02590b7c840ec3_w480.png" style="line-height: 20.7999992370605px;" /></p>
<p>This means that the model describe the behavior of RR/USD exchange rate on 90%, which is very good. Another 10% can be assigned to outside USD/bbl (Brent) factors, such as sanctions.<br />
Then I paid attention to Residual Plots.</p>
<p><img src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/84d328f1-1b81-41d1-aad4-fc00026edd38/Image/ca6c11f474b9b2ab837b3ef912075b86_w480.png" style="line-height: 20.7999992370605px;" /><br />
Ideally the residuals should be distributed randomly. In the real world we see that at the end of data observation the residuals are much higher than expected (see graphs in the right column).</p>
<p>This is very much correlated with recent news: many people in Russia are now buying foreign currency to avoid further devaluation. That kind of people behavior significantly increases the currency demand and exchange rate.</p>
<p>If we would build the graph when oil price was moving within $110 and $85 USD/bbl (Brent), then we expect that when the oil price was 60, the RR/USD exchange rate should be about 45-50 rubles per US dollar. But we actually see that it falls within 60-70, which may reflect panic on the Russian exchange market.</p>
<p>The conclusions I draw from my analysis:<br />
<br />
1. There is a strong correlation between Brent oil price and the exchange rate between the Russian ruble and U.S. dollar. About 90% of the ruble's fall can be explained by the oil price.<br />
<br />
2. Be careful when you build a regression model and do not “extend” it for the interval you have not tested it, as you may encounter with another significant factors which are beyond consideration.<br />
<br />
You can see that now the ruble is cheaper that it is expected based on oil prices. So maybe it is a good time for you to visit Russia!<br />
<br />
Alexander Drevalev<br />
Consultant and Master Black Belt<br />
Accenture<br />
Moscow, Russia</p>
<p><br />
</p>
Data AnalysisStatistics in the NewsTue, 10 Feb 2015 13:00:00 +0000http://blog.minitab.com/blog/voice-of-the-customer/how-to-use-statistical-software-to-predict-the-exchange-rateMinitab FanUnderstanding Monte Carlo Simulation with an Example
http://blog.minitab.com/blog/adventures-in-statistics/understanding-monte-carlo-simulation-with-an-example
<p>As someone who has <a href="http://blog.minitab.com/blog/adventures-in-statistics/working-at-the-edge-of-human-knowledge-part-two-data-collection">collected and analyzed real data for a living</a>, the idea of using simulated data for a Monte Carlo simulation sounds a bit odd. How can you improve a real product with simulated data? In this post, I’ll help you understand the methods behind Monte Carlo simulation and walk you through a simulation example using Devize.</p>
<p><img alt="Process capability chart" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/8b31c0befc7c93d3b4ceeea2bc8479e8/main_image.png" style="line-height: 20.7999992370605px; float: right; width: 300px; height: 241px; margin: 10px 15px;" /></p>
<p>What is Devize, you ask? <a href="http://www.minitab.com/products/devize/">Devize</a> is Minitab's exciting new, web-based, Monte Carlo simulation software for manufacturing engineers!</p>
What Is Monte Carlo Simulation?
<p>The Monte Carlo method uses repeated random sampling to generate simulated data to use with a mathematical model. This model often comes from a statistical analysis, such as a <a href="http://support.minitab.com/en-us/minitab/17/topic-library/modeling-statistics/doe/basics/what-is-a-designed-experiment/">designed experiment</a> or a <a href="http://blog.minitab.com/blog/adventures-in-statistics/regression-analysis-tutorial-and-examples">regression analysis</a>.</p>
<p>Suppose you study a process and use statistics to model it like this:</p>
<p style="margin-left: 40px;"><img alt="Regression equation for the process" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/174c81a027515c63241c34903d579ee6/regression_equation.png" style="width: 576px; height: 83px;" /></p>
<p>With this type of linear model, you can enter the process input values into the equation and predict the process output. However, in the real world, the input values won’t be a single value thanks to variability. Unfortunately, this input variability causes variability and defects in the output.</p>
<p>To design a better process, you could collect a mountain of data in order to determine how input variability relates to output variability under a variety of conditions. However, if you understand the typical distribution of the input values and you have an equation that models the process, you can easily generate a vast amount of simulated input values and enter them into the process equation to produce a simulated distribution of the process outputs.</p>
<p>You can also easily change these input distributions to answer "what if" types of questions. That's what Monte Carlo simulation is all about. In the example we are about to work through, we'll change both the mean and standard deviation of the simulated data to improve the quality of a product.</p>
<p>Today, simulated data is routinely used in situations where resources are limited or gathering real data would be too expensive or impractical.</p>
How Can Devize Help You?
<p>Devize helps engineers easily perform a Monte Carlo analysis in order to:</p>
<ul>
<li>Simulate product results while accounting for the variability in the inputs</li>
<li>Optimize process settings</li>
<li>Identify critical-to-quality factors</li>
<li>Find a solution to reduce defects</li>
</ul>
<p>Along the way, Devize interprets simulation results and provides step-by-step guidance to help you find the best possible solution for reducing defects. I'll show you how to accomplish all of this right now!</p>
Step-by-Step Example of Monte Carlo Simulation using Devize
<p>A materials engineer for a building products manufacturer is developing a new insulation product. The engineer performed an experiment and used statistics to analyze process factors that could impact the insulating effectiveness of the product. (The data for this DOE is just one of the many data set examples that can be found in <a href="http://support.minitab.com/en-us/datasets/">Minitab’s Data Set Library</a>.) For this Monte Carlo simulation example, we’ll use the regression equation shown above, which describes the statistically significant factors involved in the process.</p>
<p><strong>Step 1: Define the Process Inputs and Outputs</strong></p>
<p>The first thing we need to do is to define the inputs and the distribution of their values. The process inputs are listed in the regression output and the engineer is familiar with the typical mean and standard deviation of each variable. For the output, we simply copy and paste the regression equation that describes the process from <a href="http://www.minitab.com/products/minitab/features/">Minitab statistical software</a> right into Devize!</p>
<p>In Devize, we start with these entry fields:</p>
<p><img alt="Setup the process inputs and outputs" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/4dcb307354d2af9c359b7a84d7fd8b3a/setup_input_and_outputs.png" style="width: 935px; height: 345px; border-width: 1px; border-style: solid;" /></p>
<p>And, it's an easy matter to enter the information about the inputs and outputs for the process like this.</p>
<p><img alt="Setup the input values and the output equation" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/fe6bea502da801b31d8cf34bed3acd76/setup_input_and_outputs_with_values.png" style="width: 930px; height: 751px; border-width: 1px; border-style: solid;" /></p>
<p>Verify your model with the above diagram and then click <strong>Simulate</strong>.</p>
<p><em><strong>Initial Simulation Results</strong></em></p>
<p>After you click <strong>Simulate</strong>, Devize very quickly runs 50,000 simulations by default, though you can specify a higher or lower number of simulations. (The <a href="https://licensing.minitab.com/trials?product=devize">free trial of Devize</a> is limited to 75 simulations.)</p>
<p><img alt="Initial simulation results" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/aaf761b1f8844583548acebd146546d7/initial_simulation_results_graph.png" style="width: 944px; height: 412px; border-width: 1px; border-style: solid;" /></p>
<p>Devize interprets the results for you using output that is typical for <a href="http://support.minitab.com/en-us/minitab/17/topic-library/quality-tools/capability-analyses/basics/uses-of-capability-analysis/" target="_blank">capability analysis</a>—a capability histogram, percentage of defects, and the Ppk statistic. Devize correctly points out that our Ppk is below the generally accepted minimum value of Ppk.</p>
<p><em><strong>Step-by-Step Guidance for the Monte Carlo Simulation</strong></em></p>
<p>Devize doesn’t just run the simulation and then let you figure what to do next. Instead, Devize has determined that our process is not satisfactory and presents you with a smart sequence of steps to improve the process capability.</p>
<p>How is it smart? Devize knows that it is generally <a href="http://blog.minitab.com/blog/adventures-in-statistics/quality-improvement-controlling-variability-more-difficult-than-the-mean">easier to control the mean than the variability</a>. Therefore, the next step that Devize presents is <strong>Parameter Optimization</strong>, which finds the mean settings that minimize the number of defects while still accounting for input variability.</p>
<p style="margin-left: 40px;"><img alt="Next steps leading to parameter optimization" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/2066efed65f8943680b3b30a69b8b8fb/simulationresults_next_steps.png" style="width: 949px; height: 88px;" /></p>
<p><strong>Step 2: Define the Objective and Search Range for Parameter Optimization</strong></p>
<p>At this stage, we want Devize to find an optimal combination of mean input settings to minimize defects. After you click <strong>Parameter Optimization</strong>, you'll need to specify your goal and use your process knowledge to define a reasonable search range for the input variables.</p>
<p><img alt="Setup for parameter optimization" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/29d99759d793a6915619ceeb9a541b2c/posetup.png" style="width: 858px; height: 572px;" /></p>
<p>And, here are the simulation results!</p>
<p><img alt="Results of the parameter optimization" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/44c6d95231ea805a09b3c4459e46b37c/po_results_all.png" style="width: 934px; height: 741px; border-width: 1px; border-style: solid;" /></p>
<p>At a glance, we can tell that the percentage of defects is way down. We can also see the optimal input settings in the table. However, our Ppk statistic is still below the generally accepted minimum value. Fortunately, Devize has a recommended next step to further improve the capability of our process.</p>
<p style="margin-left: 40px;"><img alt="Next steps leading to a sensitivity analysis" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/5f4ad804894759bfedeadcbe4b7ee5f1/sa_next_steps.png" style="width: 932px; height: 115px; border-width: 1px; border-style: solid;" /></p>
<p><strong>Step 3: Control the Variability to Perform a Sensitivity Analysis</strong></p>
<p>So far, we've improved the process by optimizing the mean input settings. That reduced defects greatly, but we still have more to do in the Monte Carlo simulation. Now, we need to reduce the variability in the process inputs in order to further reduce defects.</p>
<p>Reducing variability is typically more difficult. Consequently, you don't want to waste resources controlling the standard deviation for inputs that won't reduce the number defects. Fortunately, Devize includes an innovative graph that helps you identify the inputs where controlling the variability will produce the largest reductions in defects.</p>
<p><img alt="Setup for the sensitivity analysis" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/c21d520e44cb813fda17cf0bb99215a3/sa_setup.png" style="width: 861px; height: 627px;" /></p>
<p>In this graph, look for inputs with sloped lines because reducing these standard deviations can reduce the variability in the output. Conversely, you can ease tolerances for inputs with a flat line because they don't affect the variability in the output.</p>
<p>In our graph, the slopes are fairly equal. Consequently, we'll try reducing the standard deviations of several inputs. You'll need to use process knowledge in order to identify realistic reductions. To change a setting, you can either click the points on the lines, or use the pull-down menu in the table.</p>
<p><strong>Final Monte Carlo Simulation Results</strong></p>
<p><img alt="Results of the sensitivity analysis" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/13054bd3a9403f64db72bb9909d44a3a/sa_results.png" style="width: 938px; height: 722px; border-width: 1px; border-style: solid;" /></p>
<p>Success! We've reduced the number of defects in our process and our Ppk statistic is 1.41, which is above the benchmark value. The assumptions table shows us the new settings and standard deviations for the process inputs that we should try. If we ran <strong>Parameter Optimization</strong> again, it would center the process and I'm sure we'd have even fewer defects.</p>
<p>To improve our process, Devize guided us on a smart sequence of steps during our Monte Carlo simulation:</p>
<ol>
<li>Simulate the original process</li>
<li>Optimize the mean settings</li>
<li>Strategically reduce the variability</li>
</ol>
<p>If you want to try Monte Carlo simulation for yourself, sign up for a <a href="http://www.minitab.com/en-us/products/devize/">free trial subscription of Devize</a>!</p>
Monte CarloMonte Carlo SimulationQuality ImprovementStatisticsStatistics HelpThu, 05 Feb 2015 13:00:00 +0000http://blog.minitab.com/blog/adventures-in-statistics/understanding-monte-carlo-simulation-with-an-exampleJim FrostTom Brady Is the Best Super Bowl Quarterback Ever
http://blog.minitab.com/blog/statistics-and-quality-improvement/tom-brady-is-the-best-super-bowl-quarterback-ever
<p><img alt="Tom Brady" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/0733bee29af55af0ec622068e5afaa23/640px_tom_brady.jpg" style="line-height: 20.7999992370605px; float: right; width: 141px; height: 260px; margin: 10px 15px;" /></p>
<p>There’s no shortage of interest this week in whether Tom Brady is the best quarterback to ever play the game of football. As a University of Tennessee alum, I have to recuse myself from that particular debate for lack of objectivity. (Everyone knows Peyton Manning is the best quarterback to ever play <em>the game</em>, right?) But now seems like a good time to look at some numbers that show where Brady fits among the greatest Super Bowl quarterbacks of all time. The passing data are from NFL.com and include only post-merger statistics, with my apologies to Bart Starr, Joe Namath, and Len Dawson. In the graphs that follow, gold indicates a good result for a quarterback and black indicates a bad result.</p>
Narrow the Field
<p>When making the case for someone to be the best Super Bowl quarterback ever, a common starting point is the number of victories. Joe Montana, Tom Brady, and Terry Bradshaw belong in the rarefied group of quarterbacks who have won 4 Super Bowls. Troy Aikman is the only other quarterback to have won at least 3 Super Bowls. If we use the number of victories as a standard for determining the best Super Bowl Quarterback ever, then these 4 make a good list of candidates.</p>
<p><img alt="Tom Brady has played in more Super Bowls than any other quarterback." src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/31514c4167b556dc7e24228c4653ab81/multiple_super_bowls.png" style="width: 576px; height: 384px;" /></p>
A Look at Passing Statistics
<p>We could compare passer ratings among the quarterbacks, but it’s a little unfair across time. While not necessarily a perfect statistic, I’m going to compare each quarterback’s median passer rating in victories to the median passer ratings of other Super-Bowl-winning starting quarterbacks before and after their victories. I’m using the median because Jim Plunkett’s passer rating was so good in Super Bowl XV, and Ben Roethlisberger’s rating was so bad in Super Bowl XL, that I think using the mean would give an unfair advantage to Brady over Bradshaw and Montana. Here’s what the passer rating comparison looks like:</p>
<p><img alt="The median passer ratings for Brady and Montana both exceed the median passer ratings of their competition by 20.4 points." src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/10706963d1b16d114daa32604b52bd39/median_passer_ratings_vs__competition.png" style="width: 576px; height: 384px;" /></p>
<p>In terms of the median, Aikman and Bradshaw have comparable passer ratings to other Super-Bowl-winning quarterbacks near them in time. Brady and Montana are better than their contemporaries. Amazingly, the difference of medians between Brady and his competition is identical to the difference of medians between Montana and his competition: 20.4 points.</p>
<p>The need to try to refine the analysis to compare Brady and Montana more closely leads me to consider an item that comes out in favor of the conclusion that Tom Brady is the best Super Bowl quarterback ever. This last graph shows the same median statistics for the quarterbacks as the previous graph. Each point is labeled with the margin of victory from that Super Bowl. Joe Montana’s best games, while extraordinary athletic accomplishments, came during Super Bowls where his team was much better than the competition. (Sorry Dolphins and Broncos, but you lost by more than 3 touchdowns.) Brady, in contrast, has never played in a Super Bowl where he could pad his stats in an uncompetitive contest.</p>
<p><img alt="Joe Montana's best games were in blowouts. All of Tom Brady's Super Bowls have been competitive matches." src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/fdeaf4d67a320015dd6bd8a84992beab/margin_of_victory.png" style="width: 576px; height: 384px;" /></p>
Wrap Up
<p>Without the number of victories requirement, the field for consideration would get much wider. Roger Staubach, Phil Simms, Doug Williams, and Jim Plunkett are easy names to come up with when you think of extraordinary performances by quarterbacks in Super Bowl victories. You might even consider Russell Wilson who, despite having 1 win and 1 loss, has two performances with higher passer ratings than Tom Brady’s highest passer rating ever. But I think Wilson’s case might be far from stated. For now, with 4 victories, 3 Super Bowl MVP awards, a median passer rating that exceeds his most direct competition by over 20 points, and no victories over clearly inferior competition, Tom Brady is the best Super Bowl quarterback ever.</p>
Bonus
<p>There's lots of color editing in the graphs above. Want to see more about what you can do with graphs in Minitab Statistical Software? Check out <a href="http://support.minitab.com/en-us/minitab/17/getting-started/graphing-data/">Chapter 2 of the Getting Started Guide</a>!</p>
<p><em>The photo of Tom Brady is by <a href="https://www.flickr.com/photos/27003603@N00">Keith Allison</a> and is licensed under this <a href="http://creativecommons.org/licenses/by-sa/2.0/">Creative Commons License</a>.</em></p>
Fun StatisticsWed, 04 Feb 2015 17:20:00 +0000http://blog.minitab.com/blog/statistics-and-quality-improvement/tom-brady-is-the-best-super-bowl-quarterback-everCody SteeleStatistics: Another Weapon in the Galactic Patrolâ€™s Arsenal
http://blog.minitab.com/blog/statistics-in-the-field/statistics-another-weapon-in-the-galactic-patrol%E2%80%99s-arsenal
<p><em><span style="line-height: 1.6;">by Matthew Barsalou, guest blogger. </span></em></p>
<p>E. E. Doc <a href="http://en.wikipedia.org/wiki/E._E._Smith" target="_blank">Smith</a>, one of the greatest authors ever, wrote many classic books such as <a href="http://en.wikipedia.org/wiki/Skylark_%28series%29" target="_blank">The Skylark of </a><a href="http://en.wikipedia.org/wiki/Skylark_%28series%29">Space</a> and his <a href="http://en.wikipedia.org/wiki/Lensman_series" target="_blank">Lensman</a> series. Doc Smith’s imagination knew no limits; his Galactic <a href="http://en.wikipedia.org/wiki/Galactic_Patrol" target="_blank">Patrol</a> had millions of combat fleets under its command and possessed planets turned into movable, armored weapons platforms. Some of the Galactic Patrol’s weapons may be well known. For example, there is the sunbeam, which concentrated the entire output of a sun’s energy into one beam.</p>
<p><span style="line-height: 1.6;"><img alt="amazing stories featuring E. E. "Doc" Smith" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/0d1ef573ea1b75bd2e6364f219ec6a19/docsmithcover.png" style="border-width: 1px; border-style: solid; margin: 10px 15px; float: right; width: 296px; height: 400px;" />The Galactic Patrol also created the negasphere, a planet-sized dark matter/dark energy bomb that could eat through anything. I’ll go out on a limb and assume that they first created a container that could contain such a substance,</span><span style="line-height: 20.7999992370605px;"> </span><span style="line-height: 20.7999992370605px;">at least briefly</span><span style="line-height: 1.6;">.</span></p>
<p>When I read about such technology, I always have to wonder “How did they test it?” I can see where Minitab Statistical Software could be very helpful to the Galactic Patrol. How could the Galactic Patrol evaluate smaller, torpedo-sized units of negasphere? Suppose negasphere was created at the time of firing in a space torpedo and needed to be contained for the first 30 seconds after being fired, lest it break containment early and damage the ship that is firing it or rupture the torpedo before it reaches a space pirate.</p>
<p>The table below shows data collected from fifteen samples each of two materials that could be used for negasphere containment. Material 1 has a mean containment time of 33.951 seconds and Material 2 has a mean of 32.018 seconds. But is this difference statically significant? Does it even matter?</p>
<p style="text-align: center;"><strong>Material 1</strong></p>
<p style="text-align: center;"><strong>Material 2</strong></p>
<p style="text-align: center;">34.5207</p>
<p style="text-align: center;">32.1227</p>
<p style="text-align: center;">33.0061</p>
<p style="text-align: center;">31.9836</p>
<p style="text-align: center;">32.9733</p>
<p style="text-align: center;">31.9975</p>
<p style="text-align: center;">32.4381</p>
<p style="text-align: center;">31.9997</p>
<p style="text-align: center;">34.1364</p>
<p style="text-align: center;">31.9414</p>
<p style="text-align: center;">36.1568</p>
<p style="text-align: center;">32.0403</p>
<p style="text-align: center;">34.6487</p>
<p style="text-align: center;">32.1153</p>
<p style="text-align: center;">36.6436</p>
<p style="text-align: center;">31.9661</p>
<p style="text-align: center;">35.3177</p>
<p style="text-align: center;">32.0670</p>
<p style="text-align: center;">32.4043</p>
<p style="text-align: center;">31.9610</p>
<p style="text-align: center;">31.3107</p>
<p style="text-align: center;">32.0303</p>
<p style="text-align: center;">34.0913</p>
<p style="text-align: center;">32.0146</p>
<p style="text-align: center;">33.2040</p>
<p style="text-align: center;">31.9865</p>
<p style="text-align: center;">32.5601</p>
<p style="text-align: center;">32.0079</p>
<p style="text-align: center;">35.8556</p>
<p style="text-align: center;">32.0328</p>
<p><span style="line-height: 1.6;">The questions we're asking and the type and distribution of the data we have should determine the types of statistical test we perform. Many statistical tests for continuous data require an assumption of normality, and this can easily be tested in our <a href="http://www.minitab.com/products/minitab">statistical software</a> by going to <strong>Graphs > Probability Plot…</strong> and entering the columns containing the data.</span></p>
<p><span style="line-height: 1.6;"><img alt="probability plot of material 1" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/ebd5796caf013f0204dbddc33c06df56/probability_plot1.png" style="width: 581px; height: 388px;" /></span></p>
<p><span style="line-height: 1.6;"><img alt="probability plot of material 2" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/a8464e0302753942334c4e11d31482e5/probability_plot2.png" style="width: 580px; height: 388px;" /></span></p>
<p><span style="line-height: 1.6;">The null hypothesis is “the data are normally distributed,” and the resulting P-values are greater 0.05, so we <a href="http://blog.minitab.com/blog/understanding-statistics/things-statisticians-say-failure-to-reject-the-null-hypothesis">fail to reject the null hypothesis</a>. That means we can evaluate the data using tests that require the data to be normally distributed.</span></p>
<p>To determine if the mean of Material 1 is indeed greater than the mean of Material 2, we perform a two sample t-test: go to <strong>Stat > Basic Statistics > 2 Sample t…</strong> and select “Each sample in its own column.” We then choose “Options..” and select “Difference > hypothesized difference.”</p>
<p><img alt="two-sample t-test and ci output" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/3e270a93cceb77f6818345bcb41c9110/2_sample_t_test_output.png" style="width: 546px; height: 226px;" /></p>
<p><span style="line-height: 1.6;">The P-value for the two sample t-test is less than 0.05, so we can conclude there is a statistically significant difference between the materials. But the two sample t-test does not give us a complete picture of the situation, so we should look at the data by going to <strong>Graph > Individual Value Plot...</strong> and selecting a simple graph for multiple Y’s.</span></p>
<p><span style="line-height: 1.6;"><img alt="individual value plot " src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/96d713980aefd4912a402dc156802788/individual_value_plot1.png" style="width: 583px; height: 391px;" /></span></p>
<p><span style="line-height: 1.6;">The mean of Material 1 may be higher, but our biggest concern is identifying a material that does not fail in 30 seconds or less. Material 2 appears to have far less variation and we can assess this by performing an F-test: go to <strong>Stat > Basic Statistics > 2 Variances…</strong> and select “Each sample in its own column.” Then choose “Options..” and select “Ratio > hypothesized ratio.” The data is normally distributed, so put a checkmark next to “Use test and confidence intervals based on normal distribution.”</span></p>
<p><img alt="two variances test output" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/28aa53d2ec2582e3b41e29fb5f55331f/two_variances_test_output.png" style="width: 482px; height: 563px;" /></p>
<p>The P-value is less than 0.05, so we can conclude the evidence does supports the alternative hypothesis that the variance of the first material is greater than the variance of the second material. Having already looked at a graph of the data, this should come as no surprise</p>
<p>No statistical software program can tell us which material to choose, but Minitab can provide us with the information needed to make an informed decision. The objective is to exceed a lower specification limit of 30 seconds and the lower variability of Material 2 will achieve this better than the higher mean value for Material 1. Material 2 looks good, but the penalty for a wrong decision could be lost space ships if the negasphere breaches its containment too soon, so we must be certain.</p>
<p>The Galactic Patrol has millions of ships so a failure rate of even one per million would be unacceptably high so we should perform a capability study by going to<strong> Quality Tools > Capability Analysis > Normal…</strong> Enter the column containing the data for Material 1 and use the same column for the subgroup size and then enter a lower specification of 30. This would then be repeated for Material 2.</p>
<p><img alt="process capability for material 1" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/9c10d14f155707770eb3688aec834ca2/process_capability_report1.png" style="width: 635px; height: 476px;" /></p>
<p><img alt="Process Capability for Material 2" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/eaf12e0874c393730037cf504a90fa8f/process_capability_report2.png" style="width: 638px; height: 476px;" /></p>
<p><span style="line-height: 1.6;">Looking at the Minitab generated capability studies, we can see that Material 1 can be expected to fail thousands of times per million uses, but Material 2 would is not expected to fail at all. In spite of the higher mean, the Galactic Patrol should use Material 2 for the negaspehere torpedoes. </span></p>
<p> </p>
<p> </p>
<div>
<p style="line-height: 20.7999992370605px;"><strong>About the Guest Blogger</strong></p>
<p style="line-height: 20.7999992370605px;"><em><a href="https://www.linkedin.com/pub/matthew-barsalou/5b/539/198" target="_blank">Matthew Barsalou</a> is a statistical problem resolution Master Black Belt at <a href="http://www.3k-warner.de/" target="_blank">BorgWarner</a> Turbo Systems Engineering GmbH. He is a Smarter Solutions certified Lean Six Sigma Master Black Belt, ASQ-certified Six Sigma Black Belt, quality engineer, and quality technician, and a TÜV-certified quality manager, quality management representative, and auditor. He has a bachelor of science in industrial sciences, a master of liberal studies with emphasis in international business, and has a master of science in business administration and engineering from the Wilhelm Büchner Hochschule in Darmstadt, Germany. He is author of the books <a href="http://www.amazon.com/Root-Cause-Analysis-Step---Step/dp/148225879X/ref=sr_1_1?ie=UTF8&qid=1416937278&sr=8-1&keywords=Root+Cause+Analysis%3A+A+Step-By-Step+Guide+to+Using+the+Right+Tool+at+the+Right+Time" target="_blank">Root Cause Analysis: A Step-By-Step Guide to Using the Right Tool at the Right Time</a>, <a href="http://asq.org/quality-press/display-item/index.html?item=H1472" target="_blank">Statistics for Six Sigma Black Belts</a> and <a href="http://asq.org/quality-press/display-item/index.html?item=H1473&xvl=76115763" target="_blank">The ASQ Pocket Guide to Statistics for Six Sigma Black Belts</a>.</em></p>
</div>
Data AnalysisHypothesis TestingStatisticsTue, 03 Feb 2015 13:00:00 +0000http://blog.minitab.com/blog/statistics-in-the-field/statistics-another-weapon-in-the-galactic-patrol%E2%80%99s-arsenalGuest BloggerAnalyzing Qualitative Data, part 2: Chi-Square and Multivariate Analysis
http://blog.minitab.com/blog/applying-statistics-in-quality-projects/analyzing-qualitative-data-part-2-chi-square-and-multivariate-analysis
<p><span style="color: rgb(77, 79, 81); font-family: 'Segoe UI', Frutiger, 'Frutiger Linotype', 'Dejavu Sans', 'Helvetica Neue', Tahoma, Arial, sans-serif; font-size: 14px; line-height: 21px;">In my recent meetings with people from various companies in the service industries, I realized that one of the problems they face is that they were collecting large amounts of "qualitative" data: types of product, customer profiles, different subsidiaries, several customer requirements, etc.</span></p>
<p>As I discussed in my previous post, one way to look at qualitative data is to use different types of charts, including <a href="http://blog.minitab.com/blog/applying-statistics-in-quality-projects/analyzing-qualitative-data-part-1-pareto-pie-and-stacked-bar-charts">pie charts, stacked bar charts, and Pareto charts</a>. In this post, we'll cover how to dig deeper into qualitative data with Chi-square analysis and multivariate analysis. </p>
A Chi-Square Test with Qualitative Data
<p style="line-height: 20.7999992370605px;">The table below shows which statistical methods can be used to analyze data according to the nature of such data (qualitative or numeric/quantitative). Even when the output (Y) is qualitative and the input (predictor : X) is also qualitative, at least one statistical method is relevant and can be used : the Chi-Square test.</p>
<p><strong> X \ Y</strong></p>
<p align="center"><strong>Numeric/quantitative Output</strong></p>
<p align="center"><span style="color: rgb(178, 34, 34);"><strong><u>Qualitative Output</u></strong></span></p>
<p><strong> Numeric/quantitative Input</strong></p>
<p align="center">Regression</p>
<p align="center">Logistic Regression</p>
<p><span style="color: rgb(178, 34, 34);"><strong> <u>Qualitative Input</u></strong></span></p>
<p align="center">ANOVA</p>
<p align="center">T tests</p>
<p align="center"><strong><span style="color: rgb(178, 34, 34);">Chi-Square</span></strong></p>
<p align="center"><span style="color: rgb(178, 34, 34);">Proportion tests</span></p>
<p style="line-height: 20.7999992370605px;">Let's perform the Chi-square test of statistical significance on the same qualitative mistakes data I used in my previous post:</p>
<p style="line-height: 20.7999992370605px;"><img alt="data" src="https://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/31b80fb2-db66-4edf-a753-74d4c9804ab8/Image/545c0823fc7368e795585c38424891d9/quali1.jpg" style="width: 375px; height: 378px;" /></p>
<p style="line-height: 20.7999992370605px;"><span style="line-height: 20.7999992370605px;">In Minitab Statistical Software, go to <strong>Stat > Tables > Cross Tabulation and Chi-square...</strong> In the output below, you can see that for each Employee / Error type combination, observed counts are obtained. Below that, expected counts (based on the assumption that the distribution of types of errors is strictly identical for each employee) are displayed. And below the expected count is displayed that combination's contribution to the overall Chi-Square.</span></p>
<p style="line-height: 20.7999992370605px; margin-left: 40px;"><img alt="" spellcheck="true" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/31b80fb2-db66-4edf-a753-74d4c9804ab8/Image/c34202f08d378b92e7abc1b32d2aab2d/quali10.JPG" style="width: 557px; height: 526px;" /></p>
<p style="line-height: 20.7999992370605px;">A low p-value (p = 0.042 <0.05), shown below the table, indicates a significant difference in the distribution of error types according to the three employees.</p>
<p style="line-height: 20.7999992370605px;">We then need to consider the major contributions to the overall chi-square:</p>
<p style="line-height: 20.7999992370605px;"><strong>Largest contribution: </strong>3.79 for the Mistake type: “Product” & Employee: A combination. Note that in this case, for that particular cell, the number of observed errors for “product” (third row) <u>and</u> employee A (first column of the table) is much larger than the number of expected errors. Due to that difference the contribution for that particular combination is large : 3.79.</p>
<p style="line-height: 20.7999992370605px;"><strong>Second largest contribution:</strong> 2.66 for the Error type: “Address” & employee: C combination. Note that for this particular combination (i.e., this particular cell in the table) the observed number of address errors is much larger than the number of expected errors for Employee C (and therefore the contribution 2.66 is quite large).</p>
Simple Correspondence (Multivariate) Analysis for Qualitative Data
<p><span style="line-height: 20.7999992370605px;">This third approach to analyzing qualitative data is more complex and computationally intensive but this is also a very effective and explicit statistical tool from a graphical point of view. In Minitab, go to <strong>Stat > Multivariate > Simple Correspondence Analysis...</strong></span></p>
<p style="line-height: 20.7999992370605px;">To do this analysis, I rearranged the data in a two way contingency table, with the addition of a column for the employee names :</p>
<p style="line-height: 20.7999992370605px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/31b80fb2-db66-4edf-a753-74d4c9804ab8/Image/f48bee1f16926d5370aa5ca6e1d7d26e/quali8.jpg" style="width: 336px; height: 114px;" /></p>
<p style="line-height: 20.7999992370605px;">The simple correspondence symmetric plot below indicates that “Product” type errors are more likely to be associated with employee A (see on the right part of the graph below the two points are close to one another) whereas "Address" type errors are more likely to be associated with employee C (the two points are visually close on the left part of the graph). This is the same conclusion we found using the Chi-square test.</p>
<p style="line-height: 20.7999992370605px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/31b80fb2-db66-4edf-a753-74d4c9804ab8/Image/7824d6d02af62b303240567beec1081f/quali9.jpg" style="width: 450px; height: 332px;" /></p>
How Can You Use Qualitative Data?
<p style="line-height: 20.7999992370605px;">Counts of qualitative data may obviously be used to provide relevant information to decision takers, process owners, quality professionals etc., and several graphical or statistical tools are available for that in Minitab. Our <a href="http://www.minitab.com/products/minitab">statistical software</a> includes statistical tools that are useful to analyze qualitative values, but that I didn't have space to present in this short blog (for example, Kappa studies, Attribute sampling inspection, Nominal Logistic regression...). </p>
<p style="line-height: 20.7999992370605px;">Quantitative analysis and statistics might still be used more extensively in the service sector to improve quality and customer satisfaction. Of course, analyses of qualitative data are also often performed in the manufacturing industry. If you're not already using it, please download our <a href="http://it.minitab.com/products/minitab/free-trial.aspx">free 30-day trial</a> and see what you can learn from your data!</p>
Data AnalysisQuality ImprovementStatisticsStatsWed, 28 Jan 2015 13:00:00 +0000http://blog.minitab.com/blog/applying-statistics-in-quality-projects/analyzing-qualitative-data-part-2-chi-square-and-multivariate-analysisBruno ScibiliaHow to Choose the Best Regression Model
http://blog.minitab.com/blog/adventures-in-statistics/how-to-choose-the-best-regression-model
<p><img alt="Rodin's statue, The Thinker" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/742d7708-efd3-492c-abff-6044d78e3bbd/Image/381a4964475703a0136b974f98c6c47f/rodin_the_thinker2.jpg" style="float: right; width: 275px; height: 367px; margin: 10px 15px;" />Choosing the correct linear regression model can be difficult. After all, the world and how it works is complex. Trying to model it with only a sample doesn’t make it any easier. In this post, I'll review some common statistical methods for selecting models, complications you may face, and provide some practical advice for choosing the best regression model.</p>
<p>It starts when a researcher wants to mathematically describe the relationship between some <a href="http://support.minitab.com/en-us/minitab/17/topic-library/modeling-statistics/regression-and-correlation/regression-models/what-are-response-and-predictor-variables/" target="_blank">predictors</a> and the <a href="http://support.minitab.com/en-us/minitab/17/topic-library/modeling-statistics/regression-and-correlation/regression-models/what-are-response-and-predictor-variables/" target="_blank">response variable</a>. The research team tasked to investigate typically measures many variables but includes only some of them in the model. The analysts try to eliminate the variables that are not related and include only those with a true relationship. Along the way, the analysts consider many possible models.</p>
<p>They strive to achieve a Goldilocks balance with the number of predictors they include. </p>
<ul>
<li><strong>Too few</strong>: An underspecified model tends to produce biased estimates.</li>
<li><strong>Too many</strong>: An overspecified model tends to have less precise estimates.</li>
<li><strong>Just right</strong>: A model with the correct terms has no bias and the most precise estimates.</li>
</ul>
Statistical Methods for Finding the Best Regression Model
<p>For a good regression model, you want to include the variables that you are specifically testing along with other variables that affect the response in order to avoid biased results. <a href="http://www.minitab.com/en-us/products/minitab/features/" target="_blank">Minitab statistical software </a>offers statistical measures and procedures that help you specify your regression model. I’ll review the common methods, but please do follow the links to read my more detailed posts about each.</p>
<p><strong><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">Adjusted R-squared and Predicted R-squared</a></strong>: Generally, you choose the models that have higher adjusted and predicted R-squared values. These statistics are designed to avoid a key problem with <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">regular R-squared</a>—it increases <em>every</em> time you add a predictor and can trick you into specifying an overly complex model.</p>
<ul>
<li>The adjusted R squared increases only if the new term improves the model more than would be expected by chance and it can also decrease with poor quality predictors.</li>
<li>The predicted R-squared is a form of cross-validation and it can also decrease. Cross-validation determines how well your model generalizes to other data sets by partitioning your data.</li>
</ul>
<p><strong><a href="http://blog.minitab.com/blog/adventures-in-statistics/how-to-interpret-regression-analysis-results-p-values-and-coefficients" target="_blank">P-values for the predictors</a></strong>: In regression, low p-values indicate terms that are statistically significant. “Reducing the model” refers to the practice of including all candidate predictors in the model, and then systematically removing the term with the highest p-value one-by-one until you are left with only significant predictors.</p>
<p><strong><a href="http://blog.minitab.com/blog/adventures-in-statistics/regression-smackdown-stepwise-versus-best-subsets" target="_blank">Stepwise regression and Best subsets regression</a></strong>: These are two automated procedures that can identify useful predictors during the exploratory stages of model building. With best subsets regression, Minitab provides Mallows’ Cp, which is a statistic specifically designed to help you manage the tradeoff between precision and bias.</p>
Real World Complications
<p>Great, there are a variety of statistical methods to help us choose the best model. Unfortunately, there also are a number of potential complications. Don’t worry, I’ll provide some practical advice!</p>
<ul>
<li>The best model can be only as good as the variables measured by the study. The results for the variables you include in the analysis can be biased by the significant variables that you don’t include. <a href="http://blog.minitab.com/blog/adventures-in-statistics/collecting-good-data-its-a-messy-world-confound-it" target="_blank">Read about an example of omitted variable bias</a>.</li>
<li>Your sample might be unusual, either by chance or by data collection methodology. <a href="http://blog.minitab.com/blog/adventures-in-statistics/not-all-p-values-are-created-equal" target="_blank">False positives</a> and false negatives are part of the game when working with samples.</li>
<li>P-values can change based on the specific terms in the model. In particular, <a href="http://blog.minitab.com/blog/adventures-in-statistics/what-are-the-effects-of-multicollinearity-and-when-can-i-ignore-them" target="_blank">multicollinearity</a> can sap significance and make it difficult to determine the role of each predictor.</li>
<li>If you assess enough models, you <em>will</em> find variables that appear to be significant but are only correlated by chance. <a href="http://blog.minitab.com/blog/adventures-in-statistics/four-tips-on-how-to-perform-a-regression-analysis-that-avoids-common-problems" target="_blank">This form of data mining can make random data appear significant</a>. A low predicted R-squared is a good way to check for this problem.</li>
<li>P-values, predicted and adjusted R-squared, and Mallows’ Cp can suggest different models.</li>
<li>Stepwise regression and best subsets regression are great tools and can get you close to the correct model. However, studies have found that <a href="http://blog.minitab.com/blog/adventures-in-statistics/which-is-better%2C-stepwise-regression-or-best-subsets-regression" target="_blank">they generally don’t pick the correct model</a>.</li>
</ul>
Recommendations for Finding the Best Regression Model
<p>Choosing the correct regression model is as much a science as it is an art. Statistical methods can help point you in the right direction but ultimately you’ll need to incorporate other considerations.</p>
<p><strong>Theory</strong></p>
<p>Research what others have done and incorporate those findings into constructing your model. Before beginning the regression analysis, develop an idea of what the important variables are along with their relationships, coefficient signs, and effect magnitudes. Building on the results of others makes it easier both to collect the correct data and to specify the best regression model without the need for data mining.</p>
<p>Theoretical considerations should not be discarded based solely on statistical measures. After you fit your model, determine whether it aligns with theory and possibly make adjustments. For example, based on theory, you might include a predictor in the model even if its p-value is not significant. If any of the coefficient signs contradict theory, investigate and either change your model or explain the inconsistency.</p>
<p><strong>Complexity</strong></p>
<p>You might think that complex problems require complex models, but many studies show that <a href="http://blog.minitab.com/blog/adventures-in-statistics/four-tips-on-how-to-perform-a-regression-analysis-that-avoids-common-problems" target="_blank">simpler models generally produce more precise predictions</a>. Given several models with similar explanatory ability, the simplest is most likely to be the best choice. Start simple, and only make the model more complex as needed. The more complex you make your model, the more likely it is that you are tailoring the model to your dataset specifically, and generalizability suffers.</p>
<p>Verify that added complexity actually produces narrower <a href="http://blog.minitab.com/blog/adventures-in-statistics/applied-regression-analysis-how-to-present-and-use-the-results-to-avoid-costly-mistakes-part-2" target="_blank">prediction intervals</a>. Check the predicted R-squared and don’t mindlessly chase a high regular R-squared!</p>
<p><strong>Residual Plots</strong></p>
<p>As you evaluate models, <a href="http://blog.minitab.com/blog/adventures-in-statistics/why-you-need-to-check-your-residual-plots-for-regression-analysis" target="_blank">check the residual plots</a> because they can help you avoid inadequate models and help you adjust your model for better results. For example, the bias in underspecified models can show up as patterns in the residuals, such as the need to <a href="http://blog.minitab.com/blog/adventures-in-statistics/curve-fitting-with-linear-and-nonlinear-regression" target="_blank">model curvature</a>. The simplest model that produces random residuals is a good candidate for being a relatively precise and unbiased model.</p>
<p>In the end, no single measure can tell you which model is the best. Statistical methods don't understand the underlying process or subject-area. Your knowledge is a crucial part of the process!</p>
<p>If you're learning about regression, read my <a href="http://blog.minitab.com/blog/adventures-in-statistics/regression-analysis-tutorial-and-examples">regression tutorial</a>!</p>
<p><em>* The image of Rodin's </em>The Thinker <em>was taken by flickr user innoxius and licensed under <a href="https://creativecommons.org/licenses/by/2.0/" target="_blank"><span style="font-size: 12px; display: inline;">CC BY 2.0</span></a>.</em></p>
Regression AnalysisThu, 22 Jan 2015 13:00:00 +0000http://blog.minitab.com/blog/adventures-in-statistics/how-to-choose-the-best-regression-modelJim FrostTom Brady and the Danger of Selective Endpoints
http://blog.minitab.com/blog/the-statistics-game/tom-brady-and-the-danger-of-selective-endpoints
<p>Last Friday I had an interesting tweet come across my Twitter feed.</p>
<p><img alt="Tweet" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/e9b3b34597f15a072ce760cdb7f90a1c/pats_tweet.jpg" style="width: 600px; height: 194px;" /></p>
<p>And that was <em>before</em> the Patriots failed to cover their first playoff game of 2015 against the Ravens. When you include that, the record becomes 3-11, good for a winning percentage of only 21%! With the Patriots set to play another playoff game against the Colts, it seems like the smart thing to do is to bet the Colts to cover. But wait, 14 games is a pretty small sample. We should do a <a href="http://blog.minitab.com/blog/understanding-statistics/what-statistical-hypothesis-test-should-i-use">hypothesis test</a> to determine whether this percentage is significantly less than 50%.</p>
<p><img alt="1 Proportion Test" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/e6e6f63ddde4905115625fc149a8e2fb/1_proportion.jpg" style="width: 516px; height: 172px;" /></p>
<p><a href="http://www.minitab.com/products/minitab">Minitab Statistical Software</a> returns a p-value of 0.029, which is less than the alpha value of 0.05, so we can be 95% confident that the true percentage of games that the Patriots cover during the playoffs is less than 50%. Great! Now it’s time to get my ATM card and bet a mortgage payment on the Colts. Thank you, statistics!</p>
<p>But wait, there is one more question I should probably ask pertaining to that tweet.</p>
<p>Why only the last 13 games?</p>
<p style="text-align: center;"><strong>Date</strong></p>
<p style="text-align: center;"><strong>Patriots Opponent</strong></p>
<p style="text-align: center;"><strong>Spread</strong></p>
<p style="text-align: center;"><strong>Score</strong></p>
<p style="text-align: center;"><strong>Cover the Spread?</strong></p>
<p style="text-align: center;">1/19/2014</p>
<p style="text-align: center;">@ Denver</p>
<p style="text-align: center;">+5</p>
<p style="text-align: center;">L 16-26</p>
<p style="text-align: center;">L</p>
<p style="text-align: center;">1/11/2014</p>
<p style="text-align: center;">Indianapolis</p>
<p style="text-align: center;">-7.5</p>
<p style="text-align: center;">W 43-22</p>
<p style="text-align: center;">W</p>
<p style="text-align: center;">1/20/2013</p>
<p style="text-align: center;">Baltimore</p>
<p style="text-align: center;">-8</p>
<p style="text-align: center;">L 13-28</p>
<p style="text-align: center;">L</p>
<p style="text-align: center;">1/13/2013</p>
<p style="text-align: center;">Houston</p>
<p style="text-align: center;">-9.5</p>
<p style="text-align: center;">W 41-28</p>
<p style="text-align: center;">W</p>
<p style="text-align: center;">2/5/2012</p>
<p style="text-align: center;">New York Giants</p>
<p style="text-align: center;">-3</p>
<p style="text-align: center;">L 17-21</p>
<p style="text-align: center;">L</p>
<p style="text-align: center;">1/22/2012</p>
<p style="text-align: center;">Baltimore</p>
<p style="text-align: center;">-7</p>
<p style="text-align: center;">W 23-20</p>
<p style="text-align: center;">L</p>
<p style="text-align: center;">1/14/2012</p>
<p style="text-align: center;">Denver</p>
<p style="text-align: center;">-14</p>
<p style="text-align: center;">W 45-10</p>
<p style="text-align: center;">W</p>
<p style="text-align: center;">1/16/2011</p>
<p style="text-align: center;">New York Jets</p>
<p style="text-align: center;">-9.5</p>
<p style="text-align: center;">L 21-28</p>
<p style="text-align: center;">L</p>
<p style="text-align: center;">1/10/2010</p>
<p style="text-align: center;">Baltimore</p>
<p style="text-align: center;">-3.5</p>
<p style="text-align: center;">L 14-33</p>
<p style="text-align: center;">L</p>
<p style="text-align: center;">2/3/2008</p>
<p style="text-align: center;">New York Giants</p>
<p style="text-align: center;">-12.5</p>
<p style="text-align: center;">L 14-17</p>
<p style="text-align: center;">L</p>
<p style="text-align: center;">1/20/2008</p>
<p style="text-align: center;">San Diego</p>
<p style="text-align: center;">-14</p>
<p style="text-align: center;">W 21-12</p>
<p style="text-align: center;">L</p>
<p style="text-align: center;">1/12/2008</p>
<p style="text-align: center;">Jacksonville</p>
<p style="text-align: center;">-13.5</p>
<p style="text-align: center;">W 31-20</p>
<p style="text-align: center;">L</p>
<p style="text-align: center;"><span style="color:#FF0000;">1/21/2007</span></p>
<p style="text-align: center;"><span style="color:#FF0000;">Indianapolis</span></p>
<p style="text-align: center;"><span style="color:#FF0000;">+3.5</span></p>
<p style="text-align: center;"><span style="color:#FF0000;">L 34-38</span></p>
<p style="text-align: center;"><span style="color:#FF0000;">L</span></p>
<p style="text-align: center;"><span style="color:#FF0000;">1/14/2007</span></p>
<p style="text-align: center;"><span style="color:#FF0000;">San Diego</span></p>
<p style="text-align: center;"><span style="color:#FF0000;">+5</span></p>
<p style="text-align: center;"><span style="color:#FF0000;">W 24-21</span></p>
<p style="text-align: center;"><span style="color:#FF0000;">W</span></p>
<p style="text-align: center;"><span style="color:#FF0000;">1/7/2007</span></p>
<p style="text-align: center;"><span style="color:#FF0000;">New York Jets</span></p>
<p style="text-align: center;"><span style="color:#FF0000;">-9.5</span></p>
<p style="text-align: center;"><span style="color:#FF0000;">W 37-16</span></p>
<p style="text-align: center;"><span style="color:#FF0000;">W</span></p>
<p>Here are the last <em>fifteen</em> games the Patriots played prior to the tweet (so, not including the most recent Baltimore game). I’ve highlighted the 13th, 14th, and 15th games in red. All three of these games were played 1 week apart, but the 13th game was included in the tweet, while the 14th and 15th games were conveniently left off.</p>
<p>Why? Because 3-10 against the spread sounds more impressive than 5-10.</p>
<p>This is using selective endpoints to manipulate statistics to help prove your point. It’s these kind of things that lead people to say “There are three kinds of lies: lies, damned lies, and statistics.” The conclusions you can make from your statistical analysis are only as good as the data behind it is. That’s why you should always make sure you collect a random, unbiased, sample. And before you believe the conclusions made by others, ensure they collected the data correctly too!</p>
<p><img alt="Tom Brady -- Keith Allison. Used under Creative Commons Attribution-ShareAlike 2.0" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/c5c413bce99fe8f74543b520463a28e8/tombrady.jpg" style="border-width: 1px; border-style: solid; margin: 10px 15px; float: right; width: 165px; height: 240px;" />In our Patriots situation, we could go back and look at every playoff game the Patriots have played in. But I don’t think their games in 1963 have any effect on their games this season. So instead, the best thing to do is to associate this Patriots team with Tom Brady. So we should sample <em>all</em> the playoff games that Tom Brady has played in. That includes the 16 previous games (in which he went 5-11 against the spread) and 11 games he played before 2007 (in which he went 6-4-1). This gives us a final record of 11-15-1, which is a winning percentage of 42%.</p>
<p>Once we obtained a legitimate sample of data, we see that Tom Brady and the Patriots record against the spread in the playoffs isn’t nearly as bad as we were originally led to believe. While 42% is still less than 50%, it is no longer significantly different.</p>
<p><img alt="1 Proportion Test" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/fe2c58f6-2410-4b6f-b687-d378929b1f9b/Image/207a813845e6ff925d8fac8c124d0a77/1_proportion_2.jpg" style="width: 529px; height: 172px;" /></p>
<p>So could the Patriots still fail to cover against the Colts this weekend? Of course. But I'm not going to go bet a mortgage payment on it. </p>
<p> </p>
<p><em>Photo of Tom Brady by Keith Allison, used under Creative Commons 2.0.</em></p>
Fri, 16 Jan 2015 13:00:00 +0000http://blog.minitab.com/blog/the-statistics-game/tom-brady-and-the-danger-of-selective-endpointsKevin RudyBirds Versus Statisticians: Testing the Gambler's Fallacy
http://blog.minitab.com/blog/statistics-in-the-field/birds-versus-statisticians%3A-testing-the-gamblers-fallacy
<p><em><span style="line-height: 1.6;">by Matthew Barsalou, guest blogger</span></em></p>
<p>Recently Minitab’s Joel Smith posted a blog about an incident in which he was pooped on by a bird. <a href="http://blog.minitab.com/blog/fun-with-statistics/poisson-processes-and-probability-of-poop">Twice</a>. I suspect many people would assume the odds of it happening twice are very low, so they would incorrectly assume they are safer after such a rare event happens.</p>
<p>I don’t have data on how often birds poop on one person, and I assume Joel is unwilling to stand under a flock of berry-fed birds waiting to collect data for me, so I’ll simply make up some numbers for illustration purposes only.</p>
<div><img alt="Joel, that bird's got that look in his eye again...." src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/21770465cb23a69e5b7d99c4cc3351b9/bird1.jpg" style="line-height: 20.7999992370605px; border-width: 1px; border-style: solid; margin: 10px 15px; float: right; width: 278px; height: 200px;" />
<p>Suppose there is a 5% chance of being pooped on by a bird during a vacation. That means the probability of being pooped on is 0.05. The probability of being pooped on twice during the vacation is 0.0025 (0.05 x 0.05) or 0.25%, and the probability of being pooped on three times is 0.000125 (0.05. x 0.05 x 0.05).</p>
<p>Joel has already been pooped on twice. So what is the probability of our intrepid statistician being pooped on a <em>third </em>time?</p>
<p>The probability is 0.05. If you said 0.000125, then you may have made a mistake known as the <a href="http://en.wikipedia.org/wiki/Gambler%27s_fallacy" target="_blank">Gambler’s Fallacy</a> or the Monte Carlo Fallacy. This fallacy is named after the mistaken belief that things will average out in the short-term. A gambler who has suffered repeated losses may incorrectly assume that the recent losses mean a win is due soon. Things <em>will </em>balance out in the long term, but the odds do not reset after each event. Joel could correctly conclude the probability of a bird pooping on him during his vacation are low and the odds of being pooped on twice are much lower. But being pooped on one time does not affect the probability of it happening a second time.</p>
<p>There is a caveat here. The probabilities only apply if the meeting of poop and Joel are random events. Perhaps birds, for reasons understood only by birds, have an inordinate fondness for Joel. Our probability calculations would no longer apply in such a situation. This would be like calculating the probabilities of a coin toss when there is some characteristic that causes the coin to land more on one side than on the other.</p>
<p>We can perform an experiment to determine if Joel is just a victim of the odds or if there is something that makes the birds target him. The generally low occurrence rate would make it difficult to collect data in a reasonable amount of time so we should perform an experiment to collect data. We could send Joel to a bird sanctuary for two weeks and record the number of times he is pooped on. Somebody of approximately the same size and appearance as Joel could be used as a control. Both Joel and the control should be dressed the same to ensure that birds are not targeting a particular color or clothing brand. The table below shows the hypothetical results of our little experiment.</p>
<p align="center"><img src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/0f8accc5b70a622475b15c5e70c34aa0/table1.png" style="border-width: 0px; border-style: solid; width: 187px; height: 322px;" /></p>
<p>We can see that Joel was hit 99 times, while the control was only hit 80 times. But does this difference mean anything? To find out, we can use <a href="http://www.minitab.com/products/minitab">Minitab Statistical Software</a> to determine if there is a statistically significant difference between the number of times Joel was hit and the number of times the control was hit.</p>
<p>Enter the data into Minitab and then go to <strong>Stat > Basic Statistics > 2-Sample Poisson Rate</strong> and select “Each sample is in its own column.” Go to Options and select “Difference > hypothesized difference” as the alternative hypothesis for a one-tailed upper tailed test. The resulting P-value shown in the output below is 0.078. That's greater than the alpha of 0.05 so we <a href="http://blog.minitab.com/blog/understanding-statistics/things-statisticians-say-failure-to-reject-the-null-hypothesis">fail to reject the null hypothesis</a>. Although there was a higher occurrence rate for Joel, we have no reason to think that birds are especially attracted to him.</p>
<p align="center"><img alt="Test and CI for two-sample Poisson rates output" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/565188c12e19c8bb56aed22dd0e48e9e/output.png" style="border-width: 0px; border-style: solid; width: 564px; height: 333px;" /></p>
<p>Joel is well aware of the Gambler’s Fallacy, so we can be assured that he is not under a false sense of security. He must know the probability of him getting struck a third time has not changed. But has he considered that these may not be random events? The experiment described here was only hypothetical. Perhaps Joel should consider wearing a<a href="http://www.merriam-webster.com/dictionary/sou'wester" target="_blank"> sou’wester</a> and rain coat the next time he takes a vacation in the sun.</p>
<p> </p>
<p style="line-height: 20.7999992370605px;"><strong>About the Guest Blogger</strong></p>
<p style="line-height: 20.7999992370605px;"><em><a href="https://www.linkedin.com/pub/matthew-barsalou/5b/539/198" target="_blank">Matthew Barsalou</a> is a statistical problem resolution Master Black Belt at <a href="http://www.3k-warner.de/" target="_blank">BorgWarner</a> Turbo Systems Engineering GmbH. He is a Smarter Solutions certified Lean Six Sigma Master Black Belt, ASQ-certified Six Sigma Black Belt, quality engineer, and quality technician, and a TÜV-certified quality manager, quality management representative, and auditor. He has a bachelor of science in industrial sciences, a master of liberal studies with emphasis in international business, and has a master of science in business administration and engineering from the Wilhelm Büchner Hochschule in Darmstadt, Germany. He is author of the books <a href="http://www.amazon.com/Root-Cause-Analysis-Step---Step/dp/148225879X/ref=sr_1_1?ie=UTF8&qid=1416937278&sr=8-1&keywords=Root+Cause+Analysis%3A+A+Step-By-Step+Guide+to+Using+the+Right+Tool+at+the+Right+Time" target="_blank">Root Cause Analysis: A Step-By-Step Guide to Using the Right Tool at the Right Time</a>, <a href="http://asq.org/quality-press/display-item/index.html?item=H1472" target="_blank">Statistics for Six Sigma Black Belts</a> and <a href="http://asq.org/quality-press/display-item/index.html?item=H1473&xvl=76115763" target="_blank">The ASQ Pocket Guide to Statistics for Six Sigma Black Belts</a>.</em></p>
</div>
Data AnalysisFun StatisticsTue, 13 Jan 2015 13:29:00 +0000http://blog.minitab.com/blog/statistics-in-the-field/birds-versus-statisticians%3A-testing-the-gamblers-fallacyGuest BloggerUnderstanding Qualitative, Quantitative, Attribute, Discrete, and Continuous Data Types
http://blog.minitab.com/blog/understanding-statistics/understanding-qualitative-quantitative-attribute-discrete-and-continuous-data-types
<p>"Data! Data! Data! I can't make bricks without clay."<br />
— Sherlock Holmes, in Arthur Conan Doyle's <em>The Adventure of the Copper Beeches</em></p>
<p>Whether you're the world's greatest detective trying to crack a case or a person trying to solve a problem at work, you're going to need information. Facts. <em>Data</em>, as Sherlock Holmes says. </p>
<p><img alt="jujubes" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/96d7c87addccc11b6072d6dfa38d0039/jujubes.jpg" style="line-height: 20.7999992370605px; margin: 10px 15px; float: right; width: 200px; height: 200px;" /></p>
<p>But not all data is created equal, especially if you plan to analyze as part of a quality improvement project.</p>
<p>If you're using Minitab Statistical Software, you can access the Assistant to <a href="http://www.minitab.com/products/minitab/assistant">guide you through your analysis step-by-step</a>, and help identify the type of data you have.</p>
<p>But it's still important to have at least a basic understanding of the different types of data, and the kinds of questions you can use them to answer. </p>
<p>In this post, I'll provide a basic overview of the types of data you're likely to encounter, and we'll use a box of my favorite candy—<a href="http://en.wikipedia.org/wiki/Jujube_(confectionery)" target="_blank">Jujubes</a>—to illustrate how we can gather these different kinds of data, and what types of analysis we might use it for. </p>
The Two Main Flavors of Data: Qualitative and Quantitative
<p>At the highest level, two kinds of data exist: <em><strong>quantitative</strong></em> and <em><strong>qualitative</strong></em>.</p>
<p><strong><em>Quantitative</em> </strong>data deals with numbers and things you can measure objectively: dimensions such as height, width, and length. Temperature and humidity. Prices. Area and volume.</p>
<p><strong><em>Qualitative </em></strong>data deals with characteristics and descriptors that can't be easily measured, but can be observed subjectively—such as smells, tastes, textures, attractiveness, and color. </p>
<p>Broadly speaking, when you measure something and give it a number value, you create quantitative data. When you classify or judge something, you create qualitative data. So far, so good. But this is just the highest level of data: there are also different types of quantitative and qualitative data.</p>
Quantitative Flavors: Continuous Data and Discrete Data
<p>There are two types of quantitative data, which is also referred to as numeric data: <em><strong>continuous </strong></em>and <em><strong>discrete</strong>. </em><span style="line-height: 20.7999992370605px;">As a general rule, </span><em style="line-height: 20.7999992370605px;">counts </em><span style="line-height: 20.7999992370605px;">are discrete and </span><em style="line-height: 20.7999992370605px;">measurements </em><span style="line-height: 20.7999992370605px;">are continuous.</span></p>
<p><strong><em>Discrete </em></strong>data is a count that can't be made more precise. Typically it involves integers. For instance, the number of children (or adults, or pets) in your family is discrete data, because you are counting whole, indivisible entities: you can't have 2.5 kids, or 1.3 pets.</p>
<p><strong><em>Continuous</em> </strong>data, on the other hand, could be divided and reduced to finer and finer levels. For example, you can measure the height of your kids at progressively more precise scales—meters, centimeters, millimeters, and beyond—so height is continuous data.</p>
<p>If I tally<span style="line-height: 1.6;"> the number of individual Jujubes in a box, that number is a piece of discrete data. </span></p>
<p style="margin-left: 40px;"><img alt="a count of jujubes is discrete data" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/f5e3c44269356903cf156c065b10746a/jujubes_count_tally.jpg" style="width: 200px; height: 200px;" /></p>
<p><span style="line-height: 1.6;">If I use a scale to measure the weight of each Jujube, or the weight of the entire box, that's continuous data. </span></p>
<p style="margin-left: 40px;"><span style="line-height: 1.6;"><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/d11051162c9e2375e531ac589fd5a20e/jujube_weight_continuous_data.jpg" style="width: 200px; height: 200px;" /></span></p>
<p>Continuous data can be used in many different kinds of <a href="http://blog.minitab.com/blog/understanding-statistics/what-statistical-hypothesis-test-should-i-use">hypothesis tests</a>. For example, to assess the accuracy of the weight printed on the Jujubes box, we could measure 30 boxes and perform a 1-sample t-test. </p>
<p>Some analyses use continuous and discrete quantitative data at the same time. For instance, we could perform a <a href="http://blog.minitab.com/blog/adventures-in-statistics/regression-analysis-tutorial-and-examples">regression analysis</a> to see if the weight of Jujube boxes (continuous data) is correlated with the number of Jujubes inside (discrete data). </p>
Qualitative Flavors: Binomial Data, Nominal Data, and Ordinal Data
<p>When you classify or categorize something, you create <em>Qualitative</em> or attribute<em> </em>data. There are three main kinds of qualitative data.</p>
<p><em><strong>Binary </strong></em>data place things in one of two mutually exclusive categories: right/wrong, true/false, or accept/reject. </p>
<p>Occasionally, I'll get a box of Jujubes that contains a couple of individual pieces that are either too hard or too dry. If I went through the box and classified each piece as "Good" or "Bad," that would be binary data. I could use this kind of data to develop a statistical model to predict how frequently I can expect to get a bad Jujube.</p>
<p>When collecting <em><strong>unordered </strong></em>or <em><strong>nominal </strong></em>data, we assign individual items to named categories that do not have an implicit or natural value or rank. If I went through a box of Jujubes and recorded the color of each in my worksheet, that would be nominal data. </p>
<p style="margin-left: 40px;"><img alt="" src="http://cdn2.content.compendiumblog.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/ce64d648ac395d5c8098985caabc754f/jujubes_sorted_nominal_data.jpg" style="width: 200px; height: 97px;" /></p>
<p>This kind of data can be used in many different ways—for instance, I could use <a href="http://blog.minitab.com/blog/understanding-statistics/chi-square-analysis-of-halloween-and-friday-the-13th-is-there-a-slasher-movie-gender-gap">chi-square analysis</a> to see if there are statistically significant differences in the amounts of each color in a box. </p>
<p>We also can have <strong><em>ordered </em></strong>or <em><strong>ordinal </strong></em>data, in which items are assigned to categories that do have some kind of implicit or natural order, such as "Short, Medium, or Tall." <span style="line-height: 1.6;">Another example is a survey question that asks us to rate an item on a 1 to 10 scale, with 10 being the best. This implies that 10 is better than 9, which is better than 8, and so on. </span></p>
<p>The uses for ordered data is a matter of some debate among statisticians. Everyone agrees its appropriate for creating bar charts, but beyond that the answer to the question "What should I do with my ordinal data?" is "It depends." Here's a post from another blog that offers an excellent summary of the <a href="http://learnandteachstatistics.wordpress.com/2013/07/08/ordinal/" target="_blank">considerations involved</a>. </p>
Additional Resources about Data and Distributions
<p>For more fun statistics you can do with candy, check out this article (PDF format): <a href="http://www.minitab.com/uploadedFiles/Content/Academic/sweetening_statistics.pdf">Statistical Concepts: What M&M's Can Teach Us.</a> </p>
<p>For a deeper exploration of the probability distributions that apply to different types of data, check out my colleague Jim Frost's posts about <a href="http://blog.minitab.com/blog/adventures-in-statistics/understanding-and-using-discrete-distributions">understanding and using discrete distributions</a> and <a href="http://blog.minitab.com/blog/adventures-in-statistics/how-to-identify-the-distribution-of-your-data-using-minitab">how to identify the distribution of your data</a>.</p>
Fun StatisticsLearningStatistics HelpFri, 19 Dec 2014 13:00:00 +0000http://blog.minitab.com/blog/understanding-statistics/understanding-qualitative-quantitative-attribute-discrete-and-continuous-data-typesEston Martz