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Sun, 19 Apr 2015 01:03:39 +0000FeedCreator 1.7.3Thinking about Predictors in Regression, an Example
http://blog.minitab.com/blog/statistics-and-quality-improvement/thinking-about-predictors-in-regression-an-example
<p>A few times a year, the <a href="http://www.bls.gov/spotlight/2015/long-term-unemployment/home.htm">Bureau of Labor Statistics (BLS)</a> publishes a Spotlight on Statistics Article. The first such article of 2015 recently arrived, providing analysis of trends in long-term unemployment. </p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/048c63ca9af692c560ed58a0105091df/hired.jpg" style="line-height: 18.9090900421143px; margin: 10px 15px; float: right; width: 353px; height: 266px;" /></p>
<p>Certainly an interesting read on its own, but some of the included data gives us a good opportunity to look at how thought can improve your regression analysis. Fortunately, <a href="http://it.minitab.com/en-us/products/minitab/free-trial.aspx">Minitab Statistical Software</a> includes 3-D graphs and Regression Diagnostics that can help you spot opportunities for improvement.</p>
<p>The first chart in the report highlights how high the share of the unemployed who are unemployed for a long time is compared to historical levels. That chart looks a bit like this:</p>
<p><img alt="Percent of total unemployed in each category tend to follow each other." src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/2c134f899638833af4277a85e9816557/time_series_unemployment.png" style="border-width: 0px; border-style: solid; width: 576px; height: 384px;" /></p>
<p>The discussion points out an interesting relationship. The authors note that the record high for those unemployed 27 weeks or longer occurred in the second quarter of 2010. The record high for those unemployed 52 weeks or longer occurred in the second quarter of 2011. The record high for those unemployed 99 weeks or longer occurred in the 4th quarter of 2011. That is, the highest proportion of unemployed in each category happens earlier for shorter terms.</p>
<p>This relationship is where we can see how to put some thought into regression variables. Let’s say that we want to predict the percentage of unemployed who will have been unemployed for 99 weeks or longer, using the other two figures. The most natural setup for the data is for all of the figures to be in the same row by date, like this:</p>
<p><img alt="In this worksheet, each column starts in row 1." src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/e224887ab5cbfa18b531cb83018045db/natural_worksheet.png" style="border-width: 0px; border-style: solid; width: 425px; height: 174px;" /></p>
<p>When your data are set up like this, it’s natural to want to analyze the data this way. The relationship that you get this way is strong. If you looked at the R-squared statistics, you might stop.</p>
<p><span style="font-family: courier new">Model Summary</span></p>
<p><span style="font-family: courier"> S R-sq R-sq(adj) R-sq(pred)<br />
0.963437 94.69% 94.56% 93.96%</span></p>
<p>But if you look a little deeper, you might find that there are some unsatisfactory aspects with the variables this way. Here's what the relationship looks like when you plot all 3 variables on a 3-D graph:</p>
<p><img alt="The relationship between the variables is weaker as the values increase." src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/819be48ca8fb3317ccf269cc1f116bde/3_d_synchronized_variables.png" style="border-width: 0px; border-style: solid; width: 576px; height: 384px;" /></p>
<p>I’ve marked the points on this graph that have unusual predictor values. In the diagnostic report for the model, we can see that these points are followed by large standardized residuals. That is, the lag that the article pointed out in the maximums shows up in the regression relationship as well.</p>
<p><span style="font-family: courier new">Fits and Diagnostics for Unusual Observations</span></p>
<p><span style="font-family: courier new"> 99 weeks<br />
Percent<br />
Obs unemployed Fit Resid Std Resid<br />
63 4.500 3.219 1.281 1.54 X<br />
64 5.800 6.793 -0.993 -1.11 X<br />
65 6.500 8.323 -1.823 -2.03 R X<br />
66 9.500 13.152 -3.652 -3.92 R<br />
67 9.600 12.786 -3.186 -3.40 R<br />
68 10.700 14.019 -3.319 -3.57 R<br />
75 14.300 12.387 1.913 2.04 R</span></p>
<p><span style="font-family: courier new">R Large residual<br />
X Unusual X</span></p>
<p>If you think about the predictor variables, this makes perfect sense. The BLS report notes that finding a job is less likely the longer you are unemployed. People unemployed for more than 27 weeks can become people who are unemployed for longer than 52 weeks. People who are unemployed for more than 52 weeks can become people who are unemployed longer than 99 weeks.</p>
<p>So what are the right predictors to use for the percentage of the unemployed for longer than 99 weeks? The closest we can get with terms provided is probably that people who are unemployed for over 27 weeks can become people who are unemployed for over 99 weeks about 4 quarters later. Similarly, people who are unemployed for over 52 weeks can become people who are unemployed for over 99 weeks about 2 quarters later.</p>
<p>To get these variables in Minitab, use the Time Series menu.</p>
<ol>
<li>Choose<strong> Stat > Time Series > Lag</strong>.</li>
<li>In<strong> Series</strong>, enter '<em>Over 27 Weeks'</em>.</li>
<li>In <strong>Store lags in</strong>, enter<em> ‘Over 27 Lag 4’</em>.</li>
<li>In <strong>Lag</strong>, enter <strong>4</strong>.</li>
<li>Press CTRL + E.</li>
<li>In <strong>Series</strong>, enter <em>'Over 52 Weeks'</em>.</li>
<li>In <strong>Store lags in</strong>, enter<em> ‘Over 52 Lag 2’</em>.</li>
<li>In<strong> Lag</strong>, enter <em>2</em>.</li>
</ol>
<p>The resulting worksheet looks like this:</p>
<p><img alt="New variables are in this worksheet that line up the rows at more logical intervals." src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/fafd01f883d7613f0f100de45b0d66cc/lagged_worksheet.png" style="border-width: 0px; border-style: solid; width: 532px; height: 157px;" /></p>
<p>Now, the value for the percentage unemployed over 27 weeks lines from the first quarter of 1994 lines up with the percentage of unemployed over 52 weeks from the third quarter of 1994 and the percentage unemployed over 99 weeks from the first quarter of 1995. Plot these data and the relationship looks stronger than before:</p>
<p><img alt="The relationship between the response and the lagged predictors looks stronger." src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/22791f44-517c-42aa-9f28-864c95cb4e27/Image/2db1ee6b4f828eda2a556bcb2fa78a01/3_d_lagged_variables.png" style="border-width: 0px; border-style: solid; width: 576px; height: 384px;" /></p>
<p>Highlighting the same 3 points from the first graph in red, the points don’t seem unusual at all. In fact, these points don’t appear in the diagnostic report anymore. One point still has a large standardized residual and it is preceded by an unusual X value. But the regression that compare appropriate time frames explains more variation in the data than the regression that compares simultaneous ones.</p>
<p><span style="font-family: courier new">Model Summary</span></p>
<p><span style="font-family: courier new"> S R-sq R-sq(adj) R-sq(pred)<br />
0.676735 97.50% 97.43% 97.04%</span></p>
<p><span style="font-family: courier new">Fits and Diagnostics for Unusual Observations</span></p>
<p><span style="font-family: courier new"> 99 weeks<br />
Percent<br />
Obs unemployed Fit Resid Std Resid<br />
66 9.500 8.866 0.634 1.01 X<br />
68 10.700 13.357 -2.657 -4.40 R X</span></p>
<p><span style="font-family: courier new">R Large residual<br />
X Unusual X</span></p>
<p>Minitab Statistical Software provides a number of ways for you to evaluate your regression model. If your diagnostics reveal model inadequacies, the you have a lot of easy ways to make improvements. I used lag to create appropriate variables. If you’re ready for more, check out how <a href="http://blog.minitab.com/blog/applying-statistics-in-quality-projects/re-analyzing-wine-tastes-with-minitab-17">Bruno Scibilia</a> uses <a href="http://blog.minitab.com/blog/applying-statistics-in-quality-projects/re-analyzing-wine-tastes-with-minitab-17">includes interactions in his model for wine tasting</a> or <a href="http://blog.minitab.com/blog/applying-statistics-in-quality-projects/how-could-you-benefit-from-a-box-cox-transformation">explains the benefits of a Box-Cox transformation</a>.</p>
Wed, 15 Apr 2015 17:02:00 +0000http://blog.minitab.com/blog/statistics-and-quality-improvement/thinking-about-predictors-in-regression-an-exampleCody SteeleIdentifying the Distribution of Your Data
http://blog.minitab.com/blog/meredith-griffith/identifying-the-distribution-of-your-data
<p><span style="font-size: 13px; line-height: 1.6;">To choose the right statistical analysis, you need to know the distribution of your data. Suppose you want to assess the capability of your process. If you conduct an analysis that assumes the data follow a normal distribution when, in fact, the data are nonnormal, your results will be inaccurate. To avoid this costly error, you must determine the distribution of your data.</span></p>
<p>So, how do you determine the distribution? Minitab’s Individual Distribution Identification is a simple way to find the distribution of your data so you can choose the appropriate statistical analysis. You can use it to:</p>
<ul>
<li>Determine whether a distribution you used previously is still valid for the current data</li>
<li>Choose the right distribution when you’re not sure which distribution to use</li>
<li>Transform your data to follow a normal distribution</li>
</ul>
<p>Let's take a closer look at three ways you can use the Individual Distribution Identification tool in our <a href="http://www.minitab.com/products/minitab">statistical software</a>. </p>
Confirm a Certain Distribution Fits Your Data
<p>In most cases, your process knowledge helps you identify the distribution of your data. In these situations, you can use Minitab’s Individual Distribution Identification to confirm the known distribution fits the current data.</p>
<p>Suppose you want to perform a capability analysis to ensure that the weight of ice cream containers from your production line meets specifications. In the past, ice cream container weights have been normally distributed, but you want to confirm normality. Here’s how you use Individual Distribution Identification to quickly assess the fit.</p>
<ol>
<li>Choose <strong>Stat > Quality Tools > Individual Distribution Identification</strong>.</li>
<li>Specify the column of data to analyze and the distribution to check it against.</li>
<li>Click <strong>OK</strong>.</li>
</ol>
<p><img alt="Probability Plot for Weight" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/809b9e485f550984b027a447a547eb76/distribution_id_plot_for_weight_graph1.jpg" style="width: 450px; height: 300px;" /></p>
<p>A given distribution is a good fit if:</p>
<ul>
<li>The data points roughly follow a straight line</li>
<li>The p-value is greater than 0.05</li>
</ul>
<p>In this case, the ice cream weight data appear to follow a normal distribution, so you can justify using normal capability analysis.</p>
Determine Which Distribution Best Fits Your Data
<p>Perhaps you have successfully used more than one distribution in the past. You can use Individual Distribution Identification to help you decide which distribution best fits your current data. For example, you want to assess whether a particular weld strength meets customers’ requirements, but several distributions have been used to model this data historically. Here’s how you use Individual Distribution Identification to choose the distribution that best fits your current data.</p>
<ol>
<li>Choose <strong>Stat > Quality Tools > Individual Distribution Identification</strong>.</li>
<li>Specify the column of data to analyze and the distributions to check it against.</li>
<li>Click <strong>OK</strong>.</li>
</ol>
<p><img alt="Determine Which Distribution Best Fits Your Data" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/93bb72b716b2b3636510a50921044c55/distribution_id_plot_for_strength_graph2_w1024.jpeg" style="width: 450px; height: 296px;" /></p>
<p>Choose the distribution with data points that roughly follow a straight line and the highest p-value. In this case, the Weibull distribution fits the data best.</p>
<p><strong>Note</strong></p>
<p>When you fit your data with both a 2-parameter distribution and its 3-parameter counterpart, the latter often appears to be a better fit. However, you should use a 3-parameter distribution only if it is significantly better. See Minitab Help for information about <a href="http://support.minitab.com/en-us/minitab/17/topic-library/quality-tools/capability-analyses/distributions-and-transformations-for-nonnormal-data/p-value-for-a-goodness-of-fit-test/#choose-between-a-3-parameter-and-a-2-parameter-distribution">choosing between a 2-parameter distribution and a 3-parameter distribution</a>.</p>
Use a Normal Statistical Analysis on Nonnormal Data
<p>While Minitab offers various options for analysis of nonnormal data, many users prefer to use the broader palette of normal statistical analyses. Minitab’s Individual Distribution Identification can <a href="http://blog.minitab.com/blog/applying-statistics-in-quality-projects/how-could-you-benefit-from-a-box-cox-transformation">transform your nonnormal data using the Box-Cox method</a> so that it follows a normal distribution. You can then use the transformed data with any analysis that assumes the data follow a normal distribution.</p>
<ol>
<li>Choose <strong>Stat > Quality Tools > Individual Distribution Identification</strong>.</li>
<li>Specify the column of data to analyze.</li>
<li>From the Distribution drop-down menu in the main dialog, choose <em>Box-Cox transformation</em>, and select any other distributions to compare it with.</li>
<li>Click <strong>OK</strong> in each dialog box.</li>
</ol>
<p><img alt="USE A NORMAL STATISTICAL ANALYSIS ON NONNORMAL DATA" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/e15daf80da959004a1e44134d2bab016/distribution_id_plot_for_strength_graph3.jpg" style="width: 384px; height: 256px;" /></p>
<p>For the transformed data, check for data points that roughly follow a straight line and a p-value greater than 0.05.</p>
<p>In this case, the probability plot and p-value suggest the transformed data follow a normal distribution. You can now use the transformed data for further analysis.</p>
<p><strong>Note</strong></p>
<p>Data transformations will not always produce normal data. You must check the probability plot and p-value to assess whether the normal distribution fits the transformed data well.</p>
Putting Individual Distribution Identification to Use
<p>It is always good practice to know the distribution of your data before choosing a statistical analysis. Minitab’s Individual Distribution Identification is an easy-to-use tool that can help you identify the distribution of your data as well as eliminate errors and wasted time that result from an inappropriate analysis.</p>
<p>You can use this feature to check the fit of a single distribution, or to compare the fits of several distributions and select the one that best fits your data. If you prefer to work with normal data, you can even use Minitab’s Individual Distribution Identification to transform your nonnormal data to see if they follow a normal distribution.</p>
Data AnalysisStatisticsStatistics HelpTue, 31 Mar 2015 12:00:00 +0000http://blog.minitab.com/blog/meredith-griffith/identifying-the-distribution-of-your-dataMeredith GriffithMaking Better Estimates of Project Duration Using Monte Carlo Analysis
http://blog.minitab.com/blog/statistics-in-the-field/making-better-estimates-of-project-duration-using-monte-carlo-analysis
<p><em style="box-sizing: border-box; font-family: 'Segoe UI', Frutiger, 'Frutiger Linotype', 'Dejavu Sans', 'Helvetica Neue', Tahoma, Arial, sans-serif; line-height: 21px; color: rgb(77, 79, 81); font-size: 14px;">by Lion "Ari" Ondiappan Arivazhagan, guest blogger. </em></p>
<p>Predicting project completion times is one of the major challenges project managers face. Project schedule overruns are quite common due to the high uncertainty in estimating the amount of time activities require, a lack of <span style="line-height: 20.7999992370605px;">historical data about </span><span style="line-height: 1.6;">project completion, organizational culture, inadequate skills, the complex and elaborative nature of projects, and many other factors.</span></p>
<p>PMI’s Pulse of the Profession™ research, which is consistent with other studies, shows that "fewer than two-thirds of projects meet their goals and business intent (success rates have been falling since 2008), and about 17 percent fail outright. Failed projects waste an organization’s money: for every US$1 billion spent on a failed project, US$135 million is lost forever…unrecoverable."</p>
<p>In another report on infrastructure project schedule and cost overruns, released in 2013<span style="line-height: 20.7999992370605px;"> </span><span style="line-height: 20.7999992370605px;">by PMI-KPMG</span><span style="line-height: 1.6;">, 79 percent of the survey respondents agreed that the infrastructure sector in India faces a shortage of skilled project managers with the prerequisite skill set, which results in time/schedule overruns. One of the reasons for inefficient project delivery is the paucity of skilled project managers in the infrastructure sector.</span></p>
<p>Yet predicting an achievable project completion time is more important today than ever before, due to the high liquidated damages (LD) or penalty charges for late completion and growing dissatisfaction among clients and the public.</p>
The Drawbacks of Traditional CPM Technique
<p>Deterministic, single-point estimates of project activities are highly risky as it is impossible to complete all the project activities exactly on the estimated single-point durations. Moreover, most estimators tend to estimate activity durations that are closer to optimistic estimates than to pessimistic ones. The most likely estimates are the modal estimates and the traditional Critical Path Method (CPM), which assumes activities are normally distributed. In a normal distribution, the modal estimates have only a 50% chance of being completed within or below the estimated duration, and hence the critical path duration. In other words, we typically start with estimated project completion time that has a 50% chance of being EXCEEDED from the second the project begins.</p>
Why Probabilistic Method (PERT)
<p>Models that use three-point estimates, such as the PERT model, reduce uncertainty in project completion estimates by taking into account the Optimistic (To), Most-likely (Tml) and Pessimistic (Tp) to some extent. The width of the range (Tp -To) indicates the degree of the risk in each activity duration. While probabilistic estimates can give us three different project completion times based on either To, Tml, or Tp, we generally calculate the project completion time based on an equivalent single-point expected duration by assigning appropriate weights to each of the 3 durations. For example, the PERT model, which assumes a Beta distribution, uses the following formula to calculate the expected duration,Te.</p>
<p style="margin-left: 40px;"><img alt="beta distribution for activity duration estimate" class="center" data-loading-tracked="true" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/b18855de35b95b9147053d80dfb639c2/aaeaaqaaaaaaaagyaaaajgvjowy1ntfhltazzdutndhjni04nju1lwexywewmjmxztrmyw_1_.jpg" style="width: 300px; height: 198px;" /></p>
<p>Expected duration,Te = (To + 4Tml +Tp) / 6</p>
<p><img alt="activity table" class="center" data-loading-tracked="true" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/9cdb231b009720d35f759fbfbb6cc89a/aaeaaqaaaaaaaahsaaaajdk1yziwmzhhltringitndg1oc04zwnllwflmgu0ndy5n2mwza_1_.gif" style="width: 423px; height: 181px;" /></p>
<p>Using the PERT's 3-point estimates of activities whose durations are in weeks, we get the following PERT Network Diagram to calculate the critical path.The expected durations so calculated are then used as single-point durations in the traditional CPM method to arrive at the critical path duration. Please note that the Te values have been used as the fixed length or known activity durations (similar to the CPM) and the critical path is found by the traditional CPM way using forward and backward passes to calculate the total float of each activity.The critical path is shown below in red.</p>
<p><img alt="flowchart" class="center" data-loading-tracked="true" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/72902a470515f19854e9bb7c81516817/aaeaaqaaaaaaaaf5aaaajdc5ntvinmu2ltjmntmtndg2my1inwnhlwjjnjbjymfmotqxmw_1_.png" style="width: 645px; height: 393px;" /></p>
<p>The Critical Path Duration , T = A + E + H + I + J = 6 + 3 + 4 + 2 + 2 = 17 Weeks<br />
<br />
Unfortunately, this PERT project duration, found by adding the critical activities, <em>also </em>enjoys a mere 50% chance of on-time completion.The project completion time, regardless of the distribution shapes of the critical activities, tends to follow an approximately normal distribution if there are a sufficiently large number of activities ( say, >30) in the critical path, according to the <a href="http://blog.minitab.com/blog/understanding-statistics/how-the-central-limit-theorem-works">Central Limit Theorem (CLT)</a>. <span style="line-height: 1.6;">Hence, our problem is still not solved, as PERT-based project completion time is nothing but a glorified-CPM-based completion time. </span></p>
<span style="line-height: 1.6;">Going to Monte Carlo</span>
<p><span style="line-height: 1.6;">This is where simulation techniques, such as Monte Carlo, come in handy. We can use simulation to estimate various project completion times along with their probability of completion so that we can plan contingency reserves (CR) to ensure at least a 90-95% probability of completion (as opposed to 50% by CPM or PERT methods) during the risk management planning stage itself.</span></p>
<p>In <span><a href="http://blog.minitab.com/blog/understanding-statistics/monte-carlo-is-not-as-difficult-as-you-think">Monte Carlo simulation</a></span>, the durations of critical path activities are simulated to take on random values between their Low and High limits, depending on the distributions assumed, using a random number generator until the specified number of simulations—say, 5,000—are exhausted. For each simulation, a set of project completion time and its probability of completion is calculated and stored. When all 5,000 simulations are done, we get 5,000 project completion times and their probability values. </p>
<p>Monte Carlo Simulation outputs (with 5,000 simulations using the software <a href="http://www.minitab.com/products/devize/video/">Devize</a>® from Minitab) for various target project completion times are given below. These simulated outputs help determine the Contingency Reserves (CR) needed in terms of the project completion time for better planning and completion assurance to clients.</p>
<p><img alt="Devize output" class="center" data-loading-tracked="true" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/50f4f5b024835128212824dde202cd7b/aaeaaqaaaaaaaad7aaaajdi1ytiwmgywlteyn2itndg4ns1iyjzmltu2ytmzzgq1nzqzma_1_.jpg" style="width: 700px; height: 311px;" /><br />
The 5,000-simulation output above predicts that the single-point Critical Path duration of 17 weeks has only a 25.9% chance of completion, or a 74.1% chance of failure (exceeding the estimated duration).</p>
<p>The simulation shown below estimates the probability of completing the project ahead of schedule by 1 month, possibly by fast-tracking. It shows that the chances of completing the project in 16 weeks (as opposed to the baseline duration of 17 weeks from CPM) are only 13.14%. Such predictions are very helpful to project managers in effective planning and deployment of project resources.</p>
<p><img alt="Devize output" class="center" data-loading-tracked="true" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/1596804c3c295c585de2c3e3104c58ae/aaeaaqaaaaaaaahwaaaajge2zmvjzmqyltnhotmtnda0my1iyte0ltkymmzmmza2ztkzmg_1_.jpg" style="width: 700px; height: 311px;" /><br />
If the client wants to know or predict the project completion duration that has at least an 85% chance of success, we can easily do that using <span><a href="http://blog.minitab.com/blog/adventures-in-statistics/understanding-monte-carlo-simulation-with-an-example">simulations performed in Devize</a></span>. In the output below, we can see that the target completion duration of 21 weeks ( USL=21 weeks) has an 86.58% chance of being completed on time.</p>
<p><img alt="monte carlo simulation software output" class="center" data-loading-tracked="true" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/f3a64acc67eb0a963675b3a553060160/aaeaaqaaaaaaaagyaaaajdexzjg0mdhjlwfhmzqtndcyyi04mwq2lwiznmfknte1zwnkna_1_.jpg" style="width: 700px; height: 311px;" /><br />
If the project manager wants to submit a completion time that has at least an 85% chance of completion with all the duration combinations of the critical path activities taken into account, it will be wiser to commit to a completion time of 21 weeks, as opposed to the contractual completion time of 16 weeks, which had only a 13.14% chance of success.</p>
<strong>Monte Carlo Simulation for Project Managers</strong>
<p>Monte Carlo simulation is a boon to project managers in general—and to risk managers in particular—for simulating various possible combinations of the predictor variables within their range of values. <span style="line-height: 20.7999992370605px;">Project managers can use Monte Carlo simulations to make more informed decisions and, as a result, complete more projects within the agreed time. </span><span style="line-height: 1.6;">Software packages such as Devize make the analysis simpler and intuitive, which in turn makes it easier for us to mitigate the overall project schedule risks to an acceptable threshold. </span></p>
<p> </p>
<p class="left"><strong>References </strong></p>
<p class="left">1. <em>An Introduction to Management Science: Quantitative Approaches to Decision Making</em>, by Anderson et al.</p>
<p class="left">2.<em>The PMBOK® Guide </em>- 5th edition, Project Management Institute (PMI).</p>
<p class="left">3. Devize®, Simulation and Optimization software from Minitab® Inc.</p>
<p class="left">4. PMI’s Pulse of the Profession™ -The High Cost of Low Performance. 2013.</p>
<p class="left">5. PMI-KPMG Study on Project Schedule and Cost Overruns - Expedite Infrastructure Projects. 2013.</p>
<p> </p>
<p><strong>About the Guest Blogger: </strong></p>
<p><em>The author, Ondiappan Arivazhagan, "Ari", is an Honors graduate in Civil / Structural Engineering from University of Madras.He is a certified PMP, PMI-SP, PMI-RMP from PMI, USA. He is also a Master Black Belt in Lean Six Sigma and has done Business Analytics from IIM,Bangalore. He has 30 years of professional global project management experience in various countries around the World and has almost 14 years of teaching / training experience in Project management, Analytics, Risk Management and Lean Six Sigma .He is the Founder-CEO of International Institute of Project Management (IIPM), Chennai and can be reached at <a target="_blank">askari@iipmchennai.com.</a></em></p>
<p><em>An earlier version of the article appeared on LinkedIn. </em></p>
Monte CarloMonte Carlo SimulationQuality ImprovementStatisticsThu, 26 Mar 2015 12:00:00 +0000http://blog.minitab.com/blog/statistics-in-the-field/making-better-estimates-of-project-duration-using-monte-carlo-analysisGuest BloggerMy Life as an Outlier
http://blog.minitab.com/blog/statistics-and-quality-data-analysis/my-life-as-an-outlier
<p>I always knew I was different. Even as a kid.</p>
<p>“Is that me? Way out there in left field?” I asked the doc.</p>
<p>“Yes,” he nodded, as he looked at my chart. “I used brushing to identify you on the graph.”</p>
<p>I wasn’t sure I liked getting brushed. It felt like my true identify was being detected and displayed in a window for all to see.</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/ba6a552e-3bc0-4eed-9c9a-eae3ade49498/Image/60cd600d9a51989ba25a081e3a6ae3b5/boxplot_outlier_3.jpg" style="width: 866px; height: 293px; border-width: 1px; border-style: solid;" /></p>
<p>The doctor must have sensed my discomfort.</p>
<p>“It’s not uncommon—even for those from a normal population—to appear as outliers,” he said, doing his best to put a good spin on it.</p>
<p>“For example, based on diagnostic criteria that define an outlier as a value that lies beyond the quartile 1 value minus 1.5 times the inter-quartile range, or beyond the quartile 3 value plus 1.5 the times the inter-quartile range, we’d expect <a href="http://www.amstat.org/publications/jse/v19n2/dawson.pdf" target="_blank">0.8% of observations</a> to appear as outliers, even when they come from a perfectly normal population.”</p>
<p>I wondered where he’d learned to speak <a href="http://omniglot.com/conscripts/vulcan.htm" target="_blank">Golic Vulcan</a> so well. Seeing the blank look on my face, he called in <a href="http://www.minitab.com/products/minitab/assistant">the Assistant</a> to explain in clearer, simpler terms.</p>
<p>“For every 1000 observations,” the Assistant said, “roughly 8 are going to be labelled as funny little stars on this chart—<em>even when they’re perfectly normal</em>. In fact, that’s a very conservative estimate.”</p>
<p>“So maybe I’m just like everybody else?” I asked, hopefully.</p>
<p>“I’d like to do some follow-up tests,” the doctor replied, cautiously.</p>
The Results Come In, and I'm Out
<p>It took about 9 seconds, using Minitab <a href="http://www.minitab.com/products/minitab">Statistical Software</a>, to get my Dixon’s r22 Ratio Test results back. It seemed like forever. </p>
<p style="margin-left: 40px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/ba6a552e-3bc0-4eed-9c9a-eae3ade49498/Image/187b1e83536d4580e3660f55b47ce1e5/outlier_plot_of_c1.jpg" style="width: 576px; height: 384px;" /></p>
<p style="margin-left: 40px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/ba6a552e-3bc0-4eed-9c9a-eae3ade49498/Image/b065073d289bf50d009685ed90c886db/outlier_test_output.jpg" style="width: 582px; height: 324px;" /></p>
<p>“The p-value for the outlier test is less than the significance level of 0.05,” the doctor began. “So we must reject the null hypothesis that you come from the same normal population as others.“</p>
<p>He paused to take a deep drag on a Lucky Strike that he held between the two thumbs of his left hand. Then he droned on in measured tones, summarizing each and every analysis that seemed to confirm my diagnosis as a delinquent datum. </p>
<p>But I didn’t hear a word he said. I was already a million miles away, wondering how my parents would react.</p>
Mom and Dad Try to Interpret Their Outlier
<p>When my dad saw the individuals chart, he hit the roof.</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/ba6a552e-3bc0-4eed-9c9a-eae3ade49498/Image/ff0ccd4d1d3f7c00378c6a739535c60a/i_chart_of_c1.jpg" style="width: 576px; height: 384px;" /></p>
<p>“He’s out of control!!” Dad exploded.</p>
<p>“I’m sure there must be some special cause for it,” my mother reasoned.</p>
<p>“He’s never learned to respect limits,” he said.</p>
<p>“Let’s not overreact, dear. This might be a false alarm.”</p>
<p>“False alarm, huh?” my dad sneered. “Then what about this stem-and-leaf I found in his bedroom?”</p>
<p style="margin-left: 40px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/ba6a552e-3bc0-4eed-9c9a-eae3ade49498/Image/d79134a457caf04f34782a5037fd564b/stem_leaf_outlier.jpg" style="line-height: 20.7999992370605px; width: 250px; height: 302px; border-width: 1px; border-style: solid;" /></p>
<p>“What were you doing in my bedroom?!” I protested."Did you brush me?"</p>
<p>“Maybe it belongs to one of his friends…” my mother said, with the same vague, speculative tone you’d use to say, “Maybe there’s life on other planets…"</p>
Rebel without a Special Cause
<p>When you treat someone a certain way, they begin living up (or down) to your expectations.</p>
<p>Once the world pegged me as an outlier, my attitude quickly changed. If I was going to be treated as an outcast, by god, I’d be an extreme one. <em>Then</em> they’d find out just how problematic a single, aberrant datum could be!</p>
<p>At first, I started messing with simple parametric statistics, like the mean. They were so sensitive and easy to push around, especially when they weren’t part of a large crowd.</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/ba6a552e-3bc0-4eed-9c9a-eae3ade49498/Image/a871fd11ff8ca8f5211363cc7ebde099/summary_report_for_without_me.jpg" style="width: 600px; height: 450px;" /></p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/ba6a552e-3bc0-4eed-9c9a-eae3ade49498/Image/8c69ccda6bb11a6fe8f866eca5043059/summary_report_for_with_me.jpg" style="width: 600px; height: 450px;" /></p>
<p>Man, what a power trip! Single-handedly I could drag down an arithmetic average. Or blow a variance sky-high, until it reached over 50 times its original magnitude. Sweet! </p>
<p>See the data huddle fearfully inside a single histogram bin when I’m around? Heh heh heh...they' so afraid they even begin to question their <em>own</em> normality!</p>
<p>As time went on, my insatiable craving for deviation made me move on to bigger things. That's when I started going out at night to wreck models.</p>
<p>I loved to ruin a clean, shiny model and instantly make it a disjointed, insignificant mess.</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/ba6a552e-3bc0-4eed-9c9a-eae3ade49498/Image/182788196ed789fea4a40208886b9f75/wreck_model_1.jpg" style="width: 661px; height: 425px;" /><br />
When I was feeling even more insidious, I'd use sleight of hand to make an insignificant relationship <em>appear</em> significant, to the unsuspecting. Little did they know, as soon as I walked away their perfect little model would crumble into a million little unrelated pieces. Ha ha ha!</p>
<p><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/ba6a552e-3bc0-4eed-9c9a-eae3ade49498/Image/df349145fc41c1567c57efe49ff832d9/wreck_model_2.jpg" style="width: 645px; height: 415px;" /></p>
<p>Ah, those were the days. The grand vicissitudes of youth! My pointy, pixelated head was either soaring high in the clouds, or spiraling down to the bottom of a subterranean sinkhole.</p>
<p>Then I got busted.</p>
<p>Remember that Assistant in the doctor's office? The one who could cogently explain the maximum likelihood function to a group of rodeo clowns? Turns out he's also a part-time policeman who conducts routine data checks.</p>
<p>One day he flagged me running a red light. Then the gig was up. My deviance was exposed for all to see.</p>
What To Do with Me?
<p>Once I'd been apprehended and booked, the debate began. How should the world deal with the error of my ways?<img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/ba6a552e-3bc0-4eed-9c9a-eae3ade49498/Image/e3c1174814b71c8d5dbc2cda160b43f1/debate_1.jpg" style="float: right; width: 300px; height: 225px;" /></p>
<p>Some wished I'd never existed in the first place. They believed I wasn't fit to live with other normal data. I upset the natural balance.</p>
<p>"How simple and peaceful and wonderful the results would be," they argued, "If we could just delete this errant value."</p>
<p>Others argued it was ethically wrong to expunge me. They believed that, with the right transformation, I could be successfully reformed.</p>
<p>To curb my extremist tendencies, some statistical shrinks recommended that I undergo the rigors of a square root or logarithmic transformation. A few even advised shipping me off to the Box-Cox Boarding School for the Delinquent Datum.</p>
<p>"It can work wonders on reforming outliers like your son," the headmaster told my parents.</p>
<p>Yet others felt the reformist approach was just a charade. A sneaky scaling maneuver with smoke and mirrors--one that really didn't change the true nature of my underlying character. They argued against treating me as an aberration.</p>
<p>"There's nothing really wrong with him," they said. "He doesn't need changing. He's just crying out for attention."</p>
<p>These people recognized a simple, basic truth about me. </p>
<p>All I'd ever really wanted, was to be understood.</p>
Data AnalysisFun StatisticsLearningTue, 24 Mar 2015 12:31:00 +0000http://blog.minitab.com/blog/statistics-and-quality-data-analysis/my-life-as-an-outlierPatrick RunkelPlanning a Trip to Disney World: Using Statistics to Keep It in the Green
http://blog.minitab.com/blog/cpammer/planning-a-trip-to-disney-world%3A-using-statistics-to-keep-it-in-the-green
<p>Our vacation planning has begun. My daughter has requested a trip to <a href="http://disneyworld.disney.go.com/" target="_blank">Disney World</a> as her high school graduation present. For most people, trip planning might mean a simple phone call to the local travel agent or an even simpler do-it-yourself online booking.</p>
<p>Not for me.</p>
<p>As a statistician, a request like this means I’ve got a lot of data analysis ahead. So many travel questions require (in my world, anyway) data-driven decisions. What is the best time to book tickets? What is the best flight/airline to use given the probability of cancellation and/or missed connections traveling from our small airport? How do we schedule our in-park time so we aren’t waiting in line most of the day?</p>
<p style="font-face:Arial,Tahoma,Sans-serif; font-style:italic; font-size:12px; line-height:14px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/883b3a04d5b6300523fd74b68de36f75/castle.JPG" style="width: 250px;" /><br />
The statistician and the graduate-to-be during a previous visit to Disneyland Paris. </p>
<p>My list of questions goes on and on, and will keep me very busy in the weeks to come. But to keep this at a reasonable length for a blog post, let’s just focus on the last one. Specifically, how do we minimize queue time and maximize fun? There are many valid approaches to looking at a question like this, but to keep things simple and use available data, I’m going to take advantage of some <a href="http://www.minitab.com/en-us/products/minitab/whats-new/#">new Data window features</a> available in Minitab Statistical Software 17.2.</p>
<p>Disney queue time data is available on several websites with varying levels of sophistication, but I chose to use <a href="http://www.easywdw.com/cheatsheets/mk_cheatsheet.pdf" target="_blank">a very simple set of average wait times</a>. It’s well known that park attendance is highly seasonal, so I chose to only look at data that matches the predicted crowd level for the days we will be there. We’re also going to focus this particular analysis on my family’s seven must-see attractions at The Magic Kingdom.</p>
<div>
<div id="_com_3" uage="JavaScript">
<p>If you want to follow along in Minitab 17.2, please download my <a href="//cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/File/ba72cfe9eb69927859bedf6d1a023674/disneywaittimes.mtw">data sheet</a>. </p>
<p><span style="line-height: 20.7999992370605px;">My primary variable of interest is wait time in minutes. I want to investigate wait time by time of day using the specific ride as a grouping variable. </span>To get a quick overview of the data, I started with a <a href="http://blog.minitab.com/blog/statistics-and-quality-data-analysis/time-series-plots-theres-gold-in-them-thar-hills">time series plot</a> (<strong>Graph > Times Series Plot > Simple</strong>.) You can overlay multiple graphs (in this case, our seven must-see rides) on a single plot like this by selecting <strong>Overlaid on the same graph</strong> under the <strong>Multiple Graphs</strong> button. </p>
<div>
<div id="_com_2" uage="JavaScript">
<p><img alt="time series plot of Magic Kingdom rides" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/818aa748ebfad35c2c7b0c64a981b398/magic_kingdom_time_series_plot.png" style="width: 577px; height: 385px;" /></p>
<p>From this graph, I see that the new Seven Dwarfs Mine Train—the yellow line at or near the top for every hour—is going to be a tough one to ride without a substantial wait, but with the right timing, we can get through It’s a Small World pretty quickly. Everything else falls somewhere in between.</p>
<p>This is certainly good to know. What would be more useful, though, is to look at the actual data in our Minitab spreadsheet and set up some rules based on what we believe are acceptable wait times. My personal wait time tolerance can be roughly described as:</p>
<ul>
<li>Under 20 minutes: totally Happy.</li>
<li>20 to 35 minutes: may get a little Sleepy.</li>
<li>More than 35 minutes: this better be the best ride ever or I’ll become very Grumpy.</li>
</ul>
<p>Wouldn’t it be great if I could use information to visualize my data in the Data window? Fortunately, with Minitab 17.2, I can use <a href="http://support.minitab.com/minitab/17/topic-library/minitab-environment/data-and-data-manipulation/conditional-formatting/conditional-formatting-overview/">conditional formatting</a> to do this. Simply click in the Data window, and either right-click or choose <strong>Editor > Conditional Formatting</strong>. I used the options under <strong>Highlight Cell </strong> to set three rules: Less than 20, Between 20 and 35, and Greater than 35. I can now use the resulting Green, Yellow, and Red formatting to plan my day at The Magic Kingdom.</p>
<p><img alt="conditional formatting of data for magic kingdom rides" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/f88d04106f2ba1fa85d32e2abf2f0dca/conditional_formatting_w1024.png" style="width: 800px; height: 342px;" /></p>
<p>Although Seven Dwarfs Mine Train never makes it into the green zone, putting the wait at the very end of our day—when our feet will be tired from walking anyway—may be reasonable. To avoid red as much as possible, we might want to hit either Space Mountain or Big Thunder Mountain first and move around from there.</p>
<p>I still need to collect information about the distance between rides to complete our plan, but I think we’re off to a promising start!</p>
</div>
</div>
</div>
</div>
Data AnalysisFun StatisticsStatisticsStatsTue, 17 Mar 2015 14:00:00 +0000http://blog.minitab.com/blog/cpammer/planning-a-trip-to-disney-world%3A-using-statistics-to-keep-it-in-the-greenCheryl PammerP-value Roulette: Making Hypothesis Testing a Winner’s Game
http://blog.minitab.com/blog/rkelly/p-value-roulette-making-hypothesis-testing-a-winner%E2%80%99s-game
<p>Welcome to the Hypothesis Test Casino! The featured game of the house is roulette. But this is no <em>ordinary</em> game of roulette. This is p-value roulette!</p>
<p>Here’s how it works: We have two roulette wheels, the Null wheel and the Alternative wheel. Each wheel has 20 slots (instead of the usual 37 or 38). You get to bet on one slot.</p>
<p><img alt="http://upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Edvard_Munch_-_At_the_Roulette_Table_in_Monte_Carlo_-_Google_Art_Project.jpg/256px-Edvard_Munch_-_At_the_Roulette_Table_in_Monte_Carlo_-_Google_Art_Project.jpg" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/8647ae2930d63e128d09f0b2cc5cdb87/p_value_roulette.jpg" style="line-height: 20.7999992370605px; border-width: 1px; border-style: solid; margin: 10px 15px; width: 256px; height: 166px; float: right;" /></p>
<p>What happens if the ball lands in the slot you bet on? Well, that depends on which wheel we spin. If we spin the Null wheel, you lose your bet. But if we spin the Alternative wheel, you win!</p>
<p>I’m sorry, but we can’t tell you which wheel we’re spinning.</p>
<p>Doesn’t that sound like a good game?</p>
<p>Not convinced yet? I assure you the odds are in your favor <em>if </em>you choose your slot wisely. Look, I’ll show you a graph of some data from the Null wheel. We spun it 10,000 times and counted how many times the ball landed in each slot. As you can see each slot is just as likely as any other, with a probability of about 0.05 each. That means there’s a 95% probability the ball won’t land on your slot, so you have only a 5% chance of losing—no matter what—<em>if</em> we happen to spin the Null wheel.</p>
<p><img alt="histogram of p values for null hypothesis" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/dc5efcd7001f33a77bea1c635af837e5/histogram_of_p_values_null_hypothesis.png" style="width: 576px; height: 384px;" /></p>
<p>What about that Alternative wheel, you ask? Well, we’ve had quite a few different Alternative wheels over the years. Here’s a graph of some data from one we were spinning last year:</p>
<p><img alt="histogram of p values from alternative hypothesis" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/dd0cafe3375f3202adaf3542d15eb9ab/histogram_of_p_values_alternative_hypothesis.png" style="width: 576px; height: 384px;" /></p>
<p>And just a few months ago, we had a different one. Check out the data from this one. It was very, very popular.</p>
<p><img alt=" histogram of p-values from popular alternative hypothesis" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/fc6f0ff641e7eb4d3f7750c8163ac968/histogram_of_p_values_alternative_hypothesis_2.png" style="width: 576px; height: 384px;" /></p>
<p>Now that’s what I call an Alternative! People in the know always picked the first slot. You can see why.</p>
<p>I’m not allowed to show you data from the current game. But I assure you the Alternatives all follow this same pattern. They tend to favor those smaller numbers.</p>
<p>So, you’d like to play? Great! Which slot would you like to bet on?</p>
Is this on the level?
<p>No, I don’t really have a casino with two roulette wheels. My graphs are simulated p-values for a <a href="http://blog.minitab.com/blog/statistics-and-quality-data-analysis/what-is-a-t-test-and-why-is-it-like-telling-a-kid-to-clean-up-that-mess-in-the-kitchen">1-sample t-test</a>. The null hypothesis is that the mean of a process or population is 5. The two-sided alternative is that the mean is different from 5. In my first graph, the null hypothesis was true: I used Minitab to generate random samples of size 20 from a normal distribution with mean 5 and standard deviation of 1. For the other two graphs, the only thing I changed was the mean of the normal distribution I sampled from. For the second graph, the mean was 5.3. For the final graph, the mean was 5.75.</p>
<p>For just about any hypothesis test you do in Minitab <a href="http://www.minitab.com/products/minitab">Statistical Software</a>, you will see a p-value. Once you understand how p-values work, you will have greater insight into what they are telling you. Let’s see what we can learn about p-values from playing p-value roulette.</p>
<ol>
<li>Just as you didn’t know whether you are spinning the Null or Alternative wheel, you don’t know for sure whether the null hypothesis is true or not. But basing your decision to reject the null hypothesis on the p-value favors your chance of making a good decision.<br />
</li>
<li>If the null hypothesis is true, then any p-value is just as likely as any other. You control the probability of making a Type I error by rejecting only when the p-value falls within a narrow range, typically 0.05 or smaller. A <a href="http://blog.minitab.com/blog/the-stats-cat/understanding-type-1-and-type-2-errors-from-the-feline-perspective-all-mistakes-are-not-equal">Type I error</a> occurs if you incorrectly reject a true null hypothesis.<br />
</li>
<li>If the alternative hypothesis is true, then smaller p-values become more likely and larger p-values become less likely. That’s why you can think of a small p-value as evidence in favor of the alternative hypothesis.<br />
</li>
<li>It is tempting to try to interpret the p-value as the probability that the null hypothesis is true. But that’s not what it is. The null hypothesis is either true, or it’s not. Each time you “spin the wheel” the ball will land in a different slot, giving you a different p-value. But the truth of the null hypothesis—or lack thereof—remains unchanged.<br />
</li>
<li>In the roulette analogy there were different alternative wheels, because there is not usually just a single alternative condition. There are infinitely many mean values that are not equal to 5; my graphs looked at just two of these.<br />
</li>
<li>The probability of rejecting the null hypothesis when the alternative hypothesis is true is called the power of the test. In the 1-sample t-test, the power depends on how different the mean is from the null hypothesis value, relative to the standard error. While you don’t control the true mean, you can reduce the standard error by taking a larger sample. This will give the test greater power.<br />
</li>
</ol>
You Too Can Be a Winner!
<p>To be a winner at p-value roulette, you need to make sure you are performing the right hypothesis test, and that your data fit the assumptions of that test. Minitab’s <a href="http://www.minitab.com/en-us/products/minitab/assistant/">Assistant menu</a> can help you with that. The Assistant helps you choose the right statistical analysis, provides easy-to-understand guidelines to walk you through data collection and analysis. Then it gives you clear graphical output to let you know how to interpret your p-value, while helping you evaluate whether your data are appropriate, so you can trust your results.</p>
<p> </p>
Hypothesis TestingStatisticsStatistics HelpStatsThu, 12 Mar 2015 11:00:00 +0000http://blog.minitab.com/blog/rkelly/p-value-roulette-making-hypothesis-testing-a-winner%E2%80%99s-gameRob KellyPredicting the Barclay's Premier League with Regression Analysis
http://blog.minitab.com/blog/starting-out-with-statistical-software/predicting-the-barclays-premier-league-with-regression-analysis
<p style="line-height: 20.7999992370605px;">In England, with only a few months left, the Barclay’s Premier League is about to enter the final run in to finish up the season. While the top two spots seem pretty locked up with Chelsea and Manchester City showing their class, the fight for the other two spots in the coveted top 4 promises to entertain to the very last weekend. This is key, because only the top 4 finishers qualify for next season's UEFA Champions Leagues.</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/889bfd6a0e2195d4c8a34d7b2d6f7623/600px_soccer_ball_svg_1_.png" style="margin: 10px 15px; float: right; width: 200px; height: 200px;" />Right now, there are five teams who have a realistic chance at qualifying for the last two Champions League spots: Manchester United, Southampton, Arsenal, Tottenham, and Liverpool.</p>
<p style="line-height: 20.7999992370605px;">We’re going to use Minitab’s Prediction dialog to forecast, based on some statistics, who will finish top 4 and qualify for next season’s UEFA Champions League. Using our <a href="http://www.minitab.com/products/minitab">statistical software</a>, we ran a <a href="http://blog.minitab.com/blog/adventures-in-statistics/regression-analysis-tutorial-and-examples">regression</a> using data from the past five seasons, with Total Points as our response variable (in the Premier League, you receive 3 points for every win, and 1 point for every draw). Our predictors included a few different team-based statistics, namely Shots per game; Possession, which tracks the percentage of time a team controls the ball, pass completion percentage; and goal difference.</p>
<p style="line-height: 20.7999992370605px;">After running the data through the <strong>Stat > Regression > Regression > Fit Regression Model</strong> command in Minitab, we arrived at the following final model:</p>
<p style="line-height: 20.7999992370605px; margin-left: 40px;">Points = 45.39 - 0.157 Shots per game + 0.115 Possession + 0.040 Pass %<span style="line-height: 20.7999992370605px;"> + 0.5945 Goal Difference</span></p>
<p style="line-height: 20.7999992370605px;">Now, using the Predict dialog in the regression menu, we can forecast and see which of the five teams competing for a Champions League spot will come out on top<span style="line-height: 20.7999992370605px;">, based on our model</span>.</p>
<p style="line-height: 20.7999992370605px;">To do this, after we have fit a regression model like we did above, we go back to <strong>Stat > Regression > Regression > Predict</strong>. </p>
<p style="line-height: 20.7999992370605px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/732ead34-1005-4470-b034-d7f8b87fabcf/Image/e701b582bf368b722547b6eedbab4021/blog1.png" style="width: 688px; height: 393px;" /></p>
<p style="line-height: 20.7999992370605px;">Here we are presented with a straightforward dialog that allows us to enter <span style="line-height: 20.7999992370605px;">either </span><span style="line-height: 20.7999992370605px;">individual values to predict on, or we can enter a column of values if we are interested in multiple predictions. For this analysis, we’re going to enter a column. Our worksheet contains the following table, which includes the statistics for each of our five teams, as well as a prorated goal differential, which will be used to forecast each team’s point total at the end of the year.</span></p>
<p style="line-height: 20.7999992370605px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/732ead34-1005-4470-b034-d7f8b87fabcf/Image/459fc2a2b94a2c0f7d2d6f07a8a9cb3e/blog2.png" style="width: 494px; height: 161px;" /></p>
<p style="line-height: 20.7999992370605px;">If we then go to <strong>Stat > Regression > Regression > Predict</strong>, we can fill out the dialog as follows, with our new columns. :</p>
<p style="line-height: 20.7999992370605px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/732ead34-1005-4470-b034-d7f8b87fabcf/Image/14aaf98af6882bf033b7262c3ff77e03/blog3.png" style="width: 688px; height: 393px;" /></p>
<p style="line-height: 20.7999992370605px;">Before pressing "OK," click "Results" and make sure Prediction Table is checked. We can check our Session Window output to see predicted values for Total Points:</p>
<p style="line-height: 20.7999992370605px; margin-left: 40px;"><img alt="" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/732ead34-1005-4470-b034-d7f8b87fabcf/Image/b5f7072bd3c7868a4ff108aca933e9fd/blog4.png" style="width: 451px; height: 855px;" /></p>
<p style="line-height: 20.7999992370605px;">So what do our results tell us? Which of these five teams will finish in the Top 4? We can look at the raw point totals for each of the teams, which is listed under "Fit." Judging by these, we can rank the teams as follows, by point total:</p>
<p style="line-height: 20.7999992370605px; margin-left: 40px;">Arsenal - 72 <br />
<span style="line-height: 20.7999992370605px;">Manchester United - 71</span><br />
<span style="line-height: 20.7999992370605px;">Southampton - 70</span><br />
<span style="line-height: 20.7999992370605px;">Liverpool - 59</span><br />
<span style="line-height: 20.7999992370605px;">Tottenham - 57</span></p>
<p style="line-height: 20.7999992370605px;">According to our prediction, both Arsenal and Manchester United will qualify, with Southampton just on the outside looking in. Liverpool and Tottenham seem well behind according to our prediction. This makes sense, as the most important predictor in our model is goal difference, and those two teams are well behind the other three. Only time will tell if our predictions are correct, but for now, we'll pick Arsenal and Manchester United. </p>
<p style="line-height: 20.7999992370605px;"> </p>
Tue, 10 Mar 2015 10:00:00 +0000http://blog.minitab.com/blog/starting-out-with-statistical-software/predicting-the-barclays-premier-league-with-regression-analysisEric HeckmanThe Falling Child Project : Using Binary Logistic Regression to Predict Borewell Rescue Success
http://blog.minitab.com/blog/statistics-in-the-field/the-falling-child-project-%3A-using-binary-logistic-regression-to-predict-borewell-rescue-success
<p><em>by Lion "Ari" Ondiappan Arivazhagan, guest blogger. </em></p>
<p>An alarming number of borewell accidents, especially involving little children, have occurred across India in the recent past. This is the second of a series of articles on Borewell accidents in India. In the first installment of the series, I used the <a href="http://blog.minitab.com/blog/statistics-in-the-field/using-the-g-chart-control-chart-for-rare-events-to-predict-borewell-accidents">G-chart in Minitab</a> Statistical Software to predict the probabilities of innocent children falling into open borewells, which are sunk by farmers for agricultural and drinking water, while playing in the fields.</p>
<p>In this article, I will use the power of predictive analytics to predict the probability of successfully rescuing a trapped child based on the inputs of the child's age and gender using <a href="http://blog.minitab.com/blog/fun-with-statistics/analyzing-titanic-survival-rates-part-ii-v1">Binary Logistic Regression</a>.</p>
<p>In Minitab, we can use <strong>Stat > Regression > Binary Logistic Regression</strong> to create models when the response of interest (Rescue, in this case) is <em>binary</em> and only takes two values: successful or unsuccessful. </p>
<p>Borewell accidents data collected and provided by The Falling Child Project (www.fallingchild.org), a non-governmental organization (NGO) based in the United States, has been used for this predictive analysis.</p>
<p>Part of the raw data provided by the NGO is shown Table 1 below. A total of 62 borewell accident cases in India have been documented from 2001 to January 2015.</p>
<p><img alt="data" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/07bbd31b1482ae1d91c814a04c9ffe2d/170c6d6_1_.jpg" style="width: 461px; height: 285px;" /></p>
<p>As part of the analysis, Minitab will predict probabilities for the events you are interested in, based on your model. The predicted probabilities for unsuccessful events versus the Predicted Age and Predicted Gender are shown in the scatterplot below.</p>
<p><img alt="scatterplot of events" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/836e2dfc2fc0061df235fc2bfd594b41/30987c8_1_.png" style="width: 577px; height: 385px;" /></p>
<p>We can predict, with 70% confidence, that the probability of unsuccessful rescue is 15% higher for a male child of age 2 than that for a female child of same age. However, it is surprising to note that above age 5, girls have about a 10 % higher chance of an unsuccessful rescue attempt than boys.</p>
<p><img alt="output" src="http://cdn.app.compendium.com/uploads/user/458939f4-fe08-4dbc-b271-efca0f5a2682/479b4fbd-f8c0-4011-9409-f4109cc4c745/Image/34a90bfdffb4d41a10d71a9770c87622/37a7c0c_1_.jpg" style="width: 512px; height: 288px;" /></p>
<p>I should note that one outlier, a male of age 60, was replaced with a male of age 6 to reduce the unnecessary effect of outlier on the whole analysis / output.</p>
<strong>Inferences</strong>
<p>From the Binary Logistic Regression analysis above, we can predict that boys of age 5 and above have a greater chance of being successfully rescued than do girls of the same age. Although the analysis indicates a P-value of 0.736,hinting that there is not much of interaction between the age of the child and its gender in predicted probabilities, the over all model's P-Value is reasonable at 0.291, hinting at a moderate 70% confidence level in the model.</p>
<p>However, the scatter plot of predicted probabilities shown above paints a different picture. The age 5 seems to be critical age beyond which girls have lesser chances of being rescued alive than boys do.</p>
<p>My goal in performing this analysis and sharing my findings is to be helpful to the rescue teams that plan these rescue efforts, so that they can increase the chances of successfully rescuing every trapped child, boy or girl.</p>
<p> </p>
<p><strong>About the Guest Blogger:</strong></p>
<p><em>Ondiappan "Ari" Arivazhagan is an honors graduate in civil / structural engineering from the University of Madras. He is a certified PMP, PMI-SP, and PMI-RMP from the Project Management Institute. He is also a Master Black Belt in Lean Six Sigma and has done Business Analytics from IIM, Bangalore. He has 30 years of professional global project management experience in various countries and has almost 14 years of teaching / training experience in project management, analytics, risk management, and Lean Six Sigma. He is the Founder-CEO of International Institute of Project Management (IIPM), Chennai, and can be reached at <a href="mailto:askari@iipmchennai.com?subject=Minitab%20Blog%20Reader" style="box-sizing: border-box; color: rgb(66, 139, 202); text-decoration: none; background: 0px 0px;" target="_blank">askari@iipmchennai.com</a>.</em></p>
<p><em>An earlier version of this article was published on LinkedIn.</em></p>
<p> </p>
Regression AnalysisStatistics in the NewsMon, 02 Mar 2015 13:00:00 +0000http://blog.minitab.com/blog/statistics-in-the-field/the-falling-child-project-%3A-using-binary-logistic-regression-to-predict-borewell-rescue-successGuest BloggerCreating 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 BarryWhat’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 Fan