## Hellbent on Measurement

Any variable that you record in a dataset will have some scale of measurement. Scales of measurement are, simply put, the ways that associated numbers relate to each other. Scales are properties of numbers, not the objects being measured. You could measure the same attribute of an object using more than one scale. For example, say you were doing a study involving cats and wanted to have a measure of each cat’s age. If you knew their actual birth dates, you could calculate their real ages in years, months, and days. If you didn’t know their birth dates, you could have a veterinarian or other knowledgeable individual estimate their ages in years. If you didn’t need even that level of precision, you could simply classify the cats as kittens, adult cats, or mature cats.

Understanding scales of measurement is important for a couple of reasons. Use a scale that has too many divisions and you’ll be fooled by the illusion of precision. Use a scale that has too few divisions and you’ll be dumbing down the data. Most importantly, though, scales of measurement determine, in part, what statistical methods might be applied to a set of measurements. If you want to do a certain type of statistical analysis on a variable, you have to use an appropriate scale for the variable. There are a few intricacies involved with measurement scales, so for now, just know that you have to understand a variable’s scale of measurement in order to analyze those data and interpret what it all means.

In Statistics 101, you’ll learn that there are four types of measurement scales – nominal, ordinal, interval, and ratio. This isn’t entirely true. The four-scale classification, described by Stevens (1946)[1], is just one way that scales are categorized, though it’s mentioned in almost every college-level introduction to statistics. There are actually a variety of other measurement scales, some differing in only obscure details.

The most basic classification of measurement scales involves whether or not the scale defines (1) groups having no mathematical relationship to each other, called grouping scales, or (2) a progression of measurement levels within a group, called progression scales.

## Grouping (Nominal) Scales

Grouping scales define groups, which are finite, usually independent, and non-overlapping (discrete). Nominal scales are grouping scales. They represent categories, names, and other sets of associated attributes. None of the levels within a grouping scale have any sequential relationship to any of the other levels. One level isn’t greater than or less than another level.

Examples of properties that would be measured on a qualitative scale include:

• Names—Kyle, Stan, Eric, Kenny
• Sex—female, male
• Identification—PINs, product serial numbers
• Locations—Wolf Creek, Area 51, undisclosed secure location
• Car styles—sedan, pickup, SUV, limo, station wagon
• Organization—company, office, department, team

Grouping scales are sometimes subdivided by the number of measurement levels. Discrete scales have a finite number of levels. For example, sex has two levels, male and female. Discrete scales with two levels are also called binary or dichotomous scales. Discrete scales with more than two levels are called categorical scales.

Variables measured on grouping scales can be used for counts and statistics based on counts, like percentages. They are also used to subdivide variables measured on progression scales.

## Progression Scales

Progression or continuous scales define some mathematical progression. The number of possible levels may be finite or infinite. They can be limited to integers or use an integer and any number of decimal points after the integer. Ordinal, interval, and ratio scales are all progression scales.

### Ordinal Scales

Ordinal scales have levels that are ordered. The levels denote a ranking or some sequence. One measurement may be greater than or less than another. However, the intervals between the measurements might not be constant.

Examples of properties that would be measured on an ordinal scale include:

• Time—business quarter, geologic period, football quarters
• Rankings—first place, second place, third place …
• Thickness—geologic strata, atmospheric layers

Sometimes the intervals between levels of an ordinal scale are so different they can be treated as if they were grouping scales. Consider geologic time. It’s divided into eon, eras, periods, epochs, and ages, but the divisions aren’t the same lengths. Some periods are four times longer than others and the lengths can change as more is learned about the history of Earth. The units of the scale are also different in different parts of the world. Then there’s Moh’s scale of mineral hardness. It consists of ten levels. However, the interval between levels 1 and 8 is about the same as the interval between levels 8 and 9. The interval between levels 9 and 10 is four times greater than the interval between levels 8 and 9. Geologists must be a bunch of really creative people who aren’t bound by convention.

More frequently, the intervals between levels of an ordinal scale are the same, in theory or reality. Rankings, game segments like innings and periods, business quarters and fiscal years, are all examples.

Counts and statistics based on medians and percentiles can be calculated for ordinal scales. This includes most types of nonparametric statistics. However, there are situations in which averages and standard deviations are used. Surveys present one of those situations because the responses can be considered to be either grouping or progression scales depending on how the levels are defined. Say you have a survey question that has five possible responses:

• Very good
• Good
• No opinion
• Poor
• Very poor

This is a grouping scale because the No Opinion response is not part of a progression. But, if the responses were:

• Very good
• Good
• Fair
• Poor
• Very poor

The scale could be recoded as Very Good=5, Good=4, Fair=3, Poor=2, and Very Poor=1 allowing statistical analyses to be conducted. If it were believed that the intervals between levels were not constant, analyses should be limited to counts and statistics based on medians and percentiles. If the intervals between levels were believed to be fairly constant, calculating averages and standard deviations might be legitimate. This is one of the points of contention with Stevens’s categories of scales. A given measurement’s scale might be perceived differently by different users.

### Ratio Scales

Ratio scales are the top end of progression scales. Their levels consist of integers followed by any number of decimal points. Ratios and arithmetic operations are meaningful. Zero is a constant and a reference to an absence of the attribute the scale measures.

Measurements made by most kinds of meters or other types of measuring device are probably ratio scales. Examples of variables measured on ratio scales include:

• Concentrations, densities, masses, and weights
• Durations in seconds, minutes, hours, or days
• Lengths, areas, and volumes

Any type of statistic can be calculated for variables measured on a ratio scale.

## Other Scales of Measurement

Understanding different types of measurement scales can help you select appropriate techniques for an analysis, especially if you’re a statistical novice. Stevens’s classification of scales works for many applications but it should be viewed as guidance rather than gospel. Interval scales in particular are an exception to the progression of scales form ordinal to ratio scales, and there are other exception scales as well. The following sections describe interval scales and a few scales that don’t quite fit into Stevens’s taxonomy.

### Interval Scales

Interval measurements are ordered like ordinal measurements and the intervals between the measurements are equal. However, there is no natural zero point and ratios have no physical meaning. The classical example of an interval scale is temperature in degrees Fahrenheit or Centigrade. The intervals between each Fahrenheit degree are equal, but the zero point (-32 degrees) is arbitrary. Elevation is sometimes considered to be an interval scale because the choice of sea level as the zero elevation is arbitrary. Time can also be thought of as an interval scale.

Some statisticians consider log-interval scales of measurement, in which the intervals between levels are constant in terms of logarithms, to be a subset of interval scales. Earthquake intensity (Richter and Mercali scales) and pH are examples of log-interval scales.

Statistics for ordinal scales and statistics based on means, variances, and correlations can be calculated for interval scales.

### Counts

Counts are like ratio scales in that they have a zero point, constant intervals and ratios are meaningful, but there are no fractional units. Any statistic that produces a fractional count is meaningless. The classic example of a meaningless count statistic is that the average family includes 2.3 children. Counts are usually treated as ratio scales, but the result of any calculation is rounded off to the nearest whole unit.

### Restricted-Range Scales

A constrained or restricted-range scale is a type of scale that is continuous only within a finite range. Probabilities are examples of constrained scales because any number is valid between the fixed endpoints of 0 and 1. Numbers outside this range are not possible. Percentages can be considered constrained or unconstrained depending on how the ratio is defined. For example, percentages for opinion polls are restricted to the range 0 to 100 percent. Percentages that describe corporate profits can be negative (i.e., losses) or virtually infinite (as in windfall profits). Restricted-range scales must be handled with special statistical techniques, such as logistical regression, that account for fixed scale endpoints.

### Cyclic Scales

Cyclic scales are scales in which sets of units repeat.

#### Repeating Units

Some cyclic scales consist of repeating levels for measuring open-ended quantities. Day of the week, month of the year, and season are examples. Time isn’t the only dimension with repeating scales, either. Musical scales, for instance, repeat yet have very different properties compared to time scales.

Repeating scales can be analyzed either by (1) treating them as an ordinal scale or (2) ignoring the repeating nature of the measure and transforming them into non-repeating linear units, such as day 1, day 2, and so on, or using a specialized statistical technique. The objective of the statistical analysis dictates which approach should be used. The first approach might be used to identify seasonality or determine if some measurement is different on one day or month rather than another. For example, this approach would be used to determine if work done on Fridays had higher numbers of defects than work done on other days. The second approach might be used to examine temporal trends. The third approach is used by statisticians who want to show off.

#### Orientation Scales

Orientation scales are a special type of cyclic scale. Degrees on a compass, for example, are a cyclic scale in which 0 degrees and 360 degrees are the same. Special formulas are required to calculate measures of central tendency and dispersion on circles and spheres.

### Concatenated Numbers and Text

Concatenated numbers and text are not scales in the true sense of variable measurement, but they are part of every data analysis in one way or another. Concatenated numbers contain multiple pieces of information, which must be treated as a nominal scale unless the information can be extracted into separate variables. Examples of concatenated numbers include social security numbers, telephone numbers, sample IDs, date ranges, latitude/longitude, and depth or elevation intervals. Likewise, labels can sometimes be parsed into useful data elements. Names and addresses are good examples.

### Time Scales

Time scales have some very quirky properties. You might think that time is measured on a ratio scale given its ever finer divisions (i.e., hours, minutes, seconds), yet it doesn’t make sense to refer to a ratio of two times any more than the ratio of two location coordinates. The starting point is also arbitrary. This sounds like an interval scale.

Time is like a one-dimensional location coordinate but it can also be linear or cyclic. Year is linear, so it’s at least an ordinal scale. For example, 1953 happened once and will never recur. Some time scales, though, repeat. Day 8 is the same as day 1. Month 13 is the same as month 1. So, time can also be treated as being measured on a nominal scale.

Time units are also used for durations, which are measured on a ratio scale. Durations can be used in ratios, they have a starting point of zero, and they don’t repeat (eight days aren’t the same as one day).

Time formats can be difficult to deal with. Most data analysis software offer a dozen or more different formats for what you see. Behind the spreadsheet format, though, the database has a number, which is the distance the time value is from an arbitrary starting point. Convert a date-time format to a number format, and you’ll see the number that corresponds with the date. The software formatting allows you to recognize values as times while the numbers allow the software to calculate statistics. This quirk of time formatting also presents a potential for disaster if you use more than one piece of software because different programs use different starting dates for their time calculations. Always check that the formatted dates are the same between applications.

### Location Scales

Just as there is time and duration, there is location and distance (or length), but there are a few twists. Time is one-dimensional; at least as we now know it. Distance can be one-, two-, or three-dimensional. Distance can be in a straight line (“as the crow flies”) or along a path (such as driving distance). Distances are usually measured in English inches, feet, yards, and miles or metric centimeters, meters, and kilometers. Locations, though, are another matter. Defining the location of a unique point on a two-dimensional surface (i.e., a plane) requires at least two variables. The variables can represent coordinates (northing/easting, latitude/longitude) or distance and direction from a fixed starting point. Of the coordinate systems, only the northing/easting scheme is a simple, non-concatenated scale that can be used for classical statistical analysis. However, this type of scale is usually not used for published maps, which can be a problem because virtually all environmental data are inherently location-dependent and multidimensional. Thus, coordinate systems usually have to be converted for one to the other. Geostatistical applications, for example, are based on distance and direction measurements but these measurements are calculated from spatial coordinates.

At least three variables are needed to define a unique point location in a three-dimensional volume, so a variable for depth (or height) must be added to the location coordinates. Often, however, a property of an object occurs over a range of depths (or heights or elevations) rather than a finite point. Unfortunately, depth range is a concatenated number (e.g., 2-4 feet). It’s always better to use two variables to represent starting depth and ending depth. Thus, it may take four variables to define an environmental space, such as the sampled interval of a well or soil boring.

## Selecting Scales

In the simplest taxonomy, almost all scales act either to group data or represent the progression in a variable’s attribute, whether simple, ordinal-scale levels or more expansive ratio-scale levels. One way to view these differences is this: nominal (grouping) scales are like stone outcrops, randomly scattered around a garden area. Ordinal scales are like garden steps. You can only be on a step not between steps, and the steps lead progressively upward or downward. There may be many steps or just a few. Ratio scales are like a garden path or ramp. You can be anywhere along the path, at high levels or low. You can move forward or back, in small or large intervals.

Somewhere between those simple, discrete ordinal scales and the finely-divided ratio scales, however, are quite a few types of scales that don’t meet either definition. Just ask yourself these questions to understand the scale you will be dealing with:

• Does the scale represent a progression of values? If not, the scale is a grouping scale.
• Are the scale intervals approximately equal? If not, the scale is may be treated as a grouping scale.
• Is there a constant zero (or other reference point) representing the absence of the attribute being measured? If not, the scale is may be treated like an interval scale.
• Are the limits of the scale limited in any way? Is there a scale minimum or maximum? Are negative numbers prohibited? If so, you may have to use special statistical approaches to analyze data measured on the scale.
• Are the scale values cyclic or repeating? If so, you may have to use special statistical approaches to analyze data measured on the scale.
• Are ratios and other mathematical operations that produce fractional scale levels permissible? If so, you have a ratio scale.

Some people think that an attribute can be measured in only one way. This is untrue more often than it is not. Consider the example of color. To an auto manufacturer, color is measured on a nominal scale. You can buy one of their cars painted red or blue or silver or black. To a gemologist, the color of a diamond is graded on an ordinal scale from D (colorless) to Z (light yellow). To an artist, color is measured on an interval scale because their color wheel contains the sequence: red, red-orange, orange, orange-yellow, yellow, yellow-green, green, green-blue, blue, blue-violet, violet, and violet-red. To a physicist, colors are measured by a continuous spectrum of light frequencies, which employ a ratio scale.

Using a different scale than what might be the convention can provide advantages. Consider this example. Soil texture is usually measured on a nominal scale that defines groups such as loam, sandy loam, clay loam, and silty clay. The information can be made quantitative by recording the percentages of sand, silt, and clay (which define the texture) instead of just the classification. The nominal-scale measure is much easier to collect in the field and is one variable to manage rather than three. On the other hand, the progression-scale measures can be analyzed in more ways. Correlating the clay content of a soil to crop growth, soil moisture, or a pollutant concentration can be done only if soil texture is measured on a progression scale.

If a choice can be made on which type of scale to use, use a ratio scale. Ratio scales are usually best because they provide the most information and can be rescaled easily as ordinal scales. For example, many sports organize contestants using weight class, measured on an ordinal scale, instead of weight, which is measured on a ratio scale. Weight is still measured at weigh-in using a ratio scale but is converted to the ordinal-scale weight classes for simplicity. In contrast, it’s usually not possible to upgrade an ordinal scale to a ratio scale unless the ordinal scale has equal intervals and calculation of percentages or z-scores makes sense. You couldn’t just estimate a contestant’s weight of 178.2 pounds from a weight class of 170-185 pounds.

If you can’t measure an attribute on a ratio or interval scale, think about how an ordinal scale could be applied. You can almost always devise an ordinal scale to characterize an attribute; you just have to be creative. Think of opinion surveys. If you can measure opinions, you can measure anything.

[1] Stevens, S. S. 1946. On the theory of scales of measurement. Science v. 103, No. 2684, p. 677–680.

Read more about using statistics at the Stats with Cats blog. Join other fans at the Stats with Cats Facebook group and the Stats with Cats Facebook page. Order Stats with Cats: The Domesticated Guide to Statistics, Models, Graphs, and Other Breeds of Data analysis at amazon.combarnesandnoble.com, or other online booksellers.

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Need to find something out, just Google it. Now that Google is a verb as well as a noun, it’s easy. But …

## It Hasn’t Always Been Easy

Adults under 30, Millennials, grew up with smartphones, laptop and tablet computers, and the Internet. As a group, they’ve never known a time when technology wasn’t integral to their existence. For those of us who finished school before the 1980s, personal computers were a rarity and the Internet was only then being developed for the military-industrial complex. Browsers didn’t appear until the early 1990s. You couldn’t buy a book from Amazon until 1995.

So, it hasn’t always been easy to find information. For most students, searching for information before 1980 usually involved a trip to the library. There, you would thumb through the 3×5” cards in the drawers of the card catalog looking for information by keywords. You would write down the title of the book referenced on a card along with its location classification (Dewey, Library of Congress). Then you would go to the location in the book stacks and retrieve the book, unless it was already in use, checked out, misplaced, or stolen. Finding enough information to fulfill a need might take hours or days or longer. Then you had to lug the books to a place where you could read them, extract the information you needed, and write it all down on paper. Needless to say, things have changed for the better. Now you can enter your keywords into an Internet search engine, and in a fraction of a second have references to hundreds, if not hundreds of thousands, websites, articles, blogs, books, images, and presentations. You can bookmark sites to read later or just save the relevant information to the cloud. That process might take minutes and will return more relevant information than you could ever access a generation earlier.

## What People Looked For

Not only can people search more information sources faster than ever before but now Big Business and Big Government collects data on all those searches. For example, wordpress.com keeps track of the number of visitors to the Stats with Cats blog site, what country they accessed the blog from, the search terms they used to find the site, and the blogs they visited. This is useful because it reveals what people are looking for, at least those people who ended up at the Stats with Cats blog.

Here are the frequencies for pertinent search terms from May 2010 through June 2016 and the associated word cloud (produced at http://www.wordle.net/; works best in IE).

Perhaps not surprisingly, the most common terms are associated with topics students would search if they were confronted with taking their first statistics class – statistics or stats, school or class, graph or chart, data, variable, and correlation. This may reflect the overpowering anticipation of learning about the some of the fascinating aspects of statistical thinking or, more likely, the fear of number crunching.

People searching for “report” are probably trying to figure out how to convert their statistical results into some meaningful story. How to Write Data Analysis Reports is probably much more than they might have expected.

People searching for the number 30 are looking for the reason they were told that their statistical analysis must have at least 30 samples. They might not like the answer at 30 Samples. Standard, Suggestion, or Superstition? but at least they’ll understand where it started, why they keep hearing it, and why the real answer is so unsatisfying.

## What They Found

There were over 76,000 referrals from 255 sites, of which 97% came from Google. Bing and Facebook each contributed about 1%. Five Things You Should Know Before Taking Statistics 101 was viewed over 100,000 times in five and a half years. Secrets of Good Correlations had nearly 70,000 views in six years.

The following table summarizes the views and the views per year for 56 Stats with Cats blogs.

 Post Total Views Years Available Views per Year Five Things You Should Know Before Taking Statistics 101 109,329 5.5 19,878 Secrets of Good Correlations 69,212 6.1 11,377 How to Write Data Analysis Reports 32,253 3.5 9,774 How to Tell if Correlation Implies Causation 10,552 1.5 7,035 30 Samples. Standard, Suggestion, or Superstition? 18,151 6.1 2,984 Why Do I Have To Take Statistics? 13,645 6.1 2,243 Ten Fatal Flaws in Data Analysis 13,618 6.1 2,239 Fifty Ways to Fix your Data 11,067 6.1 1,819 Six Misconceptions about Statistics You May Get From Stats 101 8,011 5.5 1,457 Regression Fantasies 7,117 5.5 1,294 The Right Tool for the Job 5,586 6.1 918 The Best Super Power of All 3,511 4.5 780 Why You Don’t Always Get the Correlation You Expect 1,450 2.5 580 Looking for Insight through a Window 224 0.5 448 A Picture Worth 140,000 Words 2,292 5.5 417 The Heart and Soul of Variance Control 2,248 6.1 370 O.U..T…L….I……E……..R………………..S 907 2.5 363 The Five Pursuits You Meet in Statistics 2,005 6.1 330 Ten Ways Statistical Models Can Break Your Heart 144 0.5 288 The Zen of Modeling 1,731 6.1 285 The Foundation of Professional Graphs 1,226 4.5 272 Assuming the Worst 1,550 6.1 255 It’s All Relative 1,303 5.5 237 There’s Something About Variance 1,424 6.1 234 The Measure of a Measure 1,180 6.1 194 Purrfect Resolution 1,167 6.1 192 The Data Scrub 1,145 6.1 188 Limits of Confusion 1,030 5.5 187 Try This At Home 1,133 6.1 186 Grasping at Flaws 1,009 5.5 183 Consumer Guide to Statistics 101 984 5.5 179 It’s All Greek 1,058 6.1 174 It was Professor Plot in the Diagram with a Graph 1,028 6.1 169 Weapons of Math Production 934 6.1 154 Polls Apart 819 5.5 149 You’re Off to Be a Wizard 881 6.1 145 Samples and Potato Chips 866 6.1 142 Time Is On My Side 865 6.1 142 You Can Lead a Boss to Data but You Can’t Make Him Think 833 6.1 137 Types and Patterns of Data Relationships 323 2.5 129 The Santa Claus Strategy 741 6.1 122 It’s All in the Technique 693 6.1 114 The Data Dozen 603 5.5 110 Becoming Part of the Group 589 5.5 107 Reality Statistics 618 6.1 102 Aphorisms for Data Analysts 524 5.5 95 Ten Tactics used in the War on Error 520 5.5 95 The Seeds of a Model 478 6.1 79 Ockham’s Spatula 389 5.5 71 Statistics: a Remedy for Football Withdrawal 384 5.5 70 Many Paths Lead to Models 370 6.1 61 Dealing with Dilemmas 283 5.5 51 Perspectives on Objectives 251 6.1 41 Tales of the Unprojected 241 6.1 40 Getting the Right Answer 197 5.5 36 Resurrecting the Unplanned 202 6.1 33

The message these statistics are sending appears to be that the Stats with Cats blog attracts introductory students who don’t know what to expect from their statistics class or need help in understanding challenging statistical concepts. In contrast, experienced students are acquainted with more statistics professors and students. They own more statistics textbooks and have visited more educational web sites. And as a consequence, they search for more specific statistical terms, like tolerance limits and autocorrelation, that beginners wouldn’t know. It’s ironic, then, that Stats with Cats was written for students who had completed Statistics 101 and were looking for some help in applying what they had learned. Interesting … sometimes statistical analyses reveal things you don’t expect.

Read more about using statistics at the Stats with Cats blog. Read them to your cats. Join other fans at the Stats with Cats Facebook group and the Stats with Cats Facebook page. Order Stats with Cats: The Domesticated Guide to Statistics, Models, Graphs, and Other Breeds of Data Analysis at amazon.com,  barnesandnoble.com, or other online booksellers.

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## Predict the Next President of the United States

The American Statistical Association is sponsoring a new statistics contest for high school and college students. The contest, known as Prediction 2016, challenges students to use statistics to predict the next president of the U.S. The purpose of the contest is to get more students interested in statistics by showing them how it can apply to the real world. It’s part of the larger student education campaign This is Statistics. Here’s more information:

ASA Announces Prediction 2016, a National Student Contest to Predict the Next President of the United States

What:

Sponsored by the American Statistical Association, Prediction 2016 is a contest for high school and undergraduate college students to predict the winner of the U.S. presidential election using statistical methods. Winners will receive a variety of prizes and perks, including exposure to the nation’s leading statisticians and data scientists.

Who:

One winner will be chosen among high school contestants and one among college contestants. Those with the most accurate predictions developed with sound statistical methods will win the contest.

When:

October 24, 2016 at 5pm — Deadline for submitting predictions.

October 27, 2016 — ASA announces which candidate wins in the student predictions.

November 9, 2016 — ASA announces contest winners.

Learn more at ThisisStatistics.org/ElectionPrediction2016. ASA spokespersons are available for interviews about the contest, as well as trends in statistics education and careers that are shaping the economy and workforce.

Media Contact:

• Sarah Litton
• (202) 851-2479
• slitton@stantoncomm.com
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# Common Reasons for Doubting a Regression Model

Finding a model that fits a set of data is one of the most common goals in data analysis. Least squares regression is the most commonly used tool for achieving this goal. It’s a relatively simple concept, it’s easy to do, and there’s a lot of readily available software to do the calculations. It’s even taught in many Statistics 101 courses. Everybody uses it … and therein lies the problem. Even if there is no intention to mislead anyone, it does happen.

Here are eleven of the most common reasons to doubt a regression model.

## Not Enough Samples

Accuracy is a critical component for evaluating a model. The coefficient of determination, also known as R-squared or R2, is the most often cited measure of accuracy. Now obviously, the more accurate a model is the better, so data analysts look large values for R-squared.

R-squared is designed to estimate the maximum relationship between the dependent and independent variables based on a set of samples (cases, observations, records, or whatever). If there aren’t enough samples compared to the number of independent variables in the model, the estimate of R-squared will be especially unstable. The effect is greatest when the R-squared value is small, the number of samples is small, and the number of independent variables is large, as shown in this figure.

The inflation in the value of R-squared can be assesses by calculating the shrunken R-square. The figure shows that for an R-squared value above 0.8 with 30 cases per variable, there isn’t much shrinkage. Lower estimates of R-square, however, experience considerable shrinkage.

You can’t control the magnitude of the relationship between a dependent variable and a set of independent variables, and often, you won’t have total control over the number of samples and variables either. So, you have to be aware that R-squared will be overestimated and treat your regression models with some skepticism.

## No Intercept

Almost all software that performs regression analysis provides an option to not include an intercept term in the model. This sounds convenient, especially for relationships that presume a one-to-one relationship between the dependent and independent variables. But when an intercept is excluded from the model, it’s not omitted from the analysis; it is set to zero. Look at any regression model with “no intercept” and you’ll see that the regression line goes through the origin of the axes.

With the regression line nailed down on one end at the origin, you might expect that the value of R-squared would be diminished because the line wouldn’t necessarily travel through the data in a way that minimizes the differences between the data points and the regression line, called the errors or residuals. Instead, R-squared is artificially inflated because when the correction provided by the intercept is removed, the total variation in the model increases. But, the ratio of the variability attributable to the model compared to the total variability also increases, hence the increase in R-squared.

The solution is simple. Always have an intercept term in the model unless there is a compelling theoretical reason not to include it. In that case, don’t put all your trust in R-square (or the F-tests).

## Stepwise Regression

Stepwise regression is a data analyst’s dream. Throw all the variables into a hopper, grab a cup of coffee, and the silicon chips will tell you which variables yield the best model. That irritates hard-core statisticians who don’t like amateurs messing around with their numbers. You can bet, though, that at least some of them go home at night, throw all the food in their cupboard into a crock pot, and expect to get a meal out of it.

The cause of some statistician’s consternation is that stepwise regression will select the variables that are best for the data set, but not necessarily the population. Model test probabilities are optimistic because they don’t account for the stepwise procedure’s ability to capitalize on chance. Moreover, adding new variables will always increase R-squared, so you have to have some good ways to decide how many variables is too many. There are ways to do this. So using stepwise regression alone isn’t a fatal flaw. Like with guns, drugs, and fast food, you have to be careful how you use it.

If you use stepwise regression, be sure to look at the diagnostic statistics for the model. Also, verify your results using a different data set by splitting the data set before you do any analysis, by randomly extracting observations from the original data set to create new data sets, or by collecting new samples.

## Outliers

Outliers are a special irritant for data analysts. They’re not really that tough to identify but they cause a variety of problems that data analysts have to deal with. The first problem is convincing reviewers not familiar with the data that the outliers are in fact outliers. Second, the data analysts have to convince all reviewers that what they want to do with them, delete or include or whatever, is the appropriate thing to do. One way or another, though, outliers will wreak havoc with R-squared.

Consider this figure, which comes from an analysis of slug tests to estimate the hydraulic conductivity of an aquifer. The red circles show the relationship between rising-head and falling-head slug tests performed on groundwater monitoring wells. The model for this relationship has an R-square of 0.90. The blue diamond is an outlier along the trend (same regression equation) about 60% greater than the next highest value. The R-squared of this equation is 0.95. The green square is an outlier perpendicular to the trend. The R-squared of this equation is 0.42. Those are fairly sizable differences to have been caused by a single data point.

How should you deal with outliers? I usually delete them because I’m usually looking to model trends and other patterns. But outliers are great thought provokers. Sometimes they tell you things the patterns don’t. If you’re not comfortable deciding what to do with an outlier, run the analysis both with and without outliers, a time consuming and expensive approach. The other approach would be to get the reviewer, an interested stakeholder, or an independent expert involved in the decision. That approach is time consuming and expensive too. Pick your poison.

## Non-linear relationships

Linear regression assumes that the relationship between a dependent variable and a set of independent variables are additive, or linear. If the relationship is actually nonlinear, the R-squared for the linear model will be lower than it would be for a better fitting nonlinear model.

This figure shows the relationship between the number of employed individuals and the number of individuals not in the U.S. work force between 1980 and 2009. The linear model has a respectable R-squared value of 0.84, but the polynomial model fits the data much better with an R-squared value of 0.95.

Non-linear relationships are a relatively simple problem to fix, or at least acknowledge, once you know what to look for. Graph your data and go from there.

## Overfitting

Overfitting involves building a statistical model solely by optimizing statistical parameters, and usually involves using a large number of variables and transformations of the variables. The resulting model may fit the data almost perfectly but will produce erroneous results when applied to another sample from the population.

The concern about overfitting may be somewhat overstated. Overfitting is like becoming too muscular from weight training. It doesn’t happen suddenly or simply. If you know what overfitting is, you’re not likely to become a victim. It’s not something that happens in a keystroke. It takes a lot of work fine tuning variables and what not. It’s also usually easy to identify overfitting in other people’s models. Simply look for a conglomeration of manual numerical adjustments, mathematical functions, and variable combinations.

## Misspecification

Misspecification involves including terms in a model that make the model look great statistically even though the model is problematical. Often, misspecification involves placing the same or very similar variable on both sides of the equation.

Consider this example from economics. A model for the U.S. Gross Domestic Product (GDP) was developed using data on government spending and unemployment from 1947 to 1997. The model:

GDP = (121*Spending) – (3.5*Spending2) + (136*Time) – (61*Unemployment) – 566

had an R-squared value of 0.9994. Such a high R-squared value is a signal that something is amiss. R-squared values that high are usually only seen in models involving equipment calibration, and certainly not anything involving capricious human behavior. A closer look at the study indicated that the model term involving spending were an index of the government’s outlays relative to the economy. Usually, indexing a variable to a baseline or standard is a good thing to do. In this case, though, the spending index was the proportion of government outlays per the GDP. Thus, the model was:

GDP = (121*Outlays/GDP) – (3.5* (Outlays/GDP)2) + (136*Time) – (61*Unemployment) – 566

GDP appears on both sides of the equation, thus accounting for the near perfect correlation. This is a case in which an index, at least one involving the dependent variable, should not have been used.

Another misspecification involves creating a prediction model having independent variables that are more difficult, time consuming, or expensive to generate than the dependent variable. You might as well just measure the dependent variable when you need to know its value. Similarly with forecasting (prediction of the future) models, if you need to forecast something a year in advance, don’t use predictors that are measured less than a year in advance.

## Multicollinearity

Multicollinearity occurs when a model has two or more independent variables that are highly correlated with each other. The consequences are that the model will look fine, but predictions from the model will be erratic. It’s like a football team. The players perform well together but you can’t necessarily tell how good individual players are. The team wins, yet in some situations, the cornerback or offensive tackle will get beat on most every play.

If you ever tried to use independent variables that add to a constant, you’ve seen multicollinearity in action. In the case of perfect correlations, such as these, statistical software will crash because it won’t be able to perform the matrix mathemagics of regression. Most instances of multicollinearity involve weaker correlations that allow statistical software to function, yet the predictions of the model will still be erratic.

Multicollinearity occurs often in the social sciences and other fields of study in which many variables are measured in the process of model building. Diagnosis of the problem is simple if you have access to the data. Look at correlations between the independent variables. You can also look at the variance inflation factors, reciprocals of one minus the R-squared values for the independent variables and the dependent variable. VIFs are measures of how much the model’s coefficients change because of multicollinearity. The VIF for a variable should be less than 10 and ideally near 1.

If you suspect multicollinearity, don’t worry about the model but don’t believe any of the predictions.

## Heteroscedasticity

Regression, and practically all parametric statistics, requires that the variances in the model residuals be equal at every value of the dependent variable. This assumption is called equal variances, homogeneity of variances, or coolest of all, homoscedasticity. Violate the assumption and you have heteroscedasticity.

Heteroscedasticity is assessed much more commonly in analysis of variance models than in regression models. This is probably because the dependent variable in ANOVA is measured on a categorical scale while the dependent variable in regression is measured on a continuous scale. The solution to this is fairly simple. Break the dependent variable scale into intervals, like in a histogram, and calculate the variance for each interval. The variances don’t have to be precisely equal, but variances different by a factor of five are problematical. Unequal variances will wreak havoc on any tests or confidence limits calculated for model predictions.

## Autocorrelation

Autocorrelation involves a variable being correlated with itself. It is the correlation between data points with the previously listed data points (termed a lag). Usually, autocorrelation involves time-series data or spatial data, but it can also involve the order in which data are collected. The terms autocorrelation and serial correlation are often used interchangeably. If the data points are collected at a constant time interval, the term autocorrelation is more typically used.

If the residuals of a model are autocorrelated, it’s a sure bet that the variances will also be unequal. That means, again, that tests or confidence limits calculated from variances should be suspect.

To check a variable or residuals from a model for autocorrelation, you can conduct a Durban-Watson test. The Durban-Watson test statistic ranges from 0 to 4. If the statistic is close to 2.0, then serial correlation is not a problem. Most statistical software will allow you to conduct this test as part of a regression analysis.

## Weighting

Most software that calculates regression parameters also allows you to weight the data points. You might want to do this for several reasons. Weighting is used to make more reliable or relevant data points more important in model building. It’s also used when each data point represents more than one value. The issue with weighting is that it will change the degrees of freedom, and hence, the results of statistical tests. Usually this is OK, a necessary change to accommodate the realities of the model. However, if you ever come upon a weighted least squares regression model in which the weightings are arbitrary, perhaps done by an analyst who doesn’t understand the consequence, don’t believe the test results.

# Is Your Regression Model Telling the Truth?

There are many technologies we use in our lives without really understanding how they work. Television. Computers. Cell phones. Microwave ovens. Cars. Even many things about the human body are not well understood. But I don’t mean how to use these mechanisms. Everyone knows how to use these things. I mean understanding them well enough to fix them when they break. Regression analysis is like that too. Only with regression analysis, sometimes you can’t even tell if there’s something wrong without consulting an expert.

Here are some tips for troubleshooting regression models.

## Diagnosis

You may know how to use regression analysis, but unless you’re an expert, you may not know about some of the more subtle pitfalls you may encounter. The biggest red flag that something is amiss is the TGTBT, too good to be true. If you encounter an R-squared value above 0.9, especially unexpectedly, there’s probably something wrong. Another red flag is inconsistency. If estimates of the model’s parameters change between data sets, there’s probably something wrong. And if predictions from the model are less accurate or precise than you expected, there’s probably something wrong. Here are some guidelines for troubleshooting a model you developed.

 Your Model Identification Correction Not Enough Samples If you have fewer than 10 observations for each independent variable you want to put in a model, you don’t have enough samples. Collect more samples. 100 observations per variable is a good target to shoot for although more is usually better. No Intercept You’ll know it if you do it. Put in an intercept and see if the model changes. Stepwise Regression You’ll know it if you do it. Don’t abdicate model building decisions to software alone. What’s the fun in that? Outliers Plot the dependent variable against each independent variable. If more than about 5% of the data pairs plot noticeable apart from the rest of the data points, you may have outliers. Conduct a test on the aberrant data points to determine if they are statistical anomalies. Use diagnostic statistics like leverage to evaluate the effects of suspected outliers. Evaluate the metadata of the samples to determine if they are representative of the population being modeled. If so, retain the outlier as an influential observation (AKA leverage point). Non-linear relationships Plot the dependent variable against each independent variable. Look for nonlinear patterns in the data Find an appropriate transformation of the independent variable. Overfitting If you have a large number of independent variables, especially if they use a variety of transformation and don’t contribute much to the accuracy and precision of the model, you may have overfit the model. Keep the model as simple as possible. Make sure the ratio of observations to independent variables is large. Use diagnostic statistics like AIC and BIC to help select an appropriate number of variables. Misspecification Look for any variants of the dependent variable in the independent variables. Assess whether the model meets the objectives of the effort. Remove any elements of the dependent variable from the independent variables. Remove at least one component of variables describing mixtures. Ensure the model meets the objectives of the effort with the desired accuracy and precision.. Multicollinearity Calculate correlation coefficients and plot the relationships between all the independent variables in the model. Look for high correlations. Use diagnostic statistics like VIF to evaluate the effects of suspected multicollinearity. Remove intercorrelated independent variables from the model. Heteroscedasticity Plot the variance at each level of an ordinal-scale dependent variable or appropriate ranges of a continuous-scale dependent variable. Look for any differences in the variances of more than about five times. Try to find an appropriate Box-Cox transformation or consider nonparametric regression or data mining methods. Autocorrelation Plot the data over time, location or the order of sample collection. Calculate a Durbin–Watson statistic for serial correlation. If the autocorrelation is related to time, develop a correlogram and a partial correlogram. If the autocorrelation is spatial, develop a variogram. If the autocorrelation is related to the order of sample collection, examine metadata to try to identify a cause. Weighting You’ll know it if you do it. Compare the weighted model with the corresponding unweighted model to assess the effects of weighting. Consider the validity of weighting; seek expert advice if needed.

Sometimes the model you are skeptical about isn’t one you developed; it is models that are developed by other data analysts. The major difference is that with other analysts’ models, you won’t have access to all their diagnostic statistics and plots, let alone their data. If you have been retained to review another analyst’s work, you can always ask for the information you need. If, however, you’re reading about a model in a journal article, book, or website, you’ve probably got all the information you’re ever going to get. You have to be a statistical detective. Here are some clues you might look for.

 Another Analyst’s Model Identification Not Enough Samples If the analyst reported the number of samples used, look for at least 10 observations for each independent variable in the model. If not, you may be able to estimate the number from a scatterplot. No Intercept If the analyst reported the actual model (some don’t), look for a constant term. Stepwise Regression Unless another approach is reported, assume the analyst used some form of stepwise regression. Outliers Assuming the analyst did not provide plots of the dependent variable versus the independent variables, look for R-squared values that are much higher or lower than expected. Non-linear relationships Assuming the analyst did not provide plots of the dependent variable versus the independent variables, look for a lower-than-expected R-squared value from a linear model. If there are non-linear terms in the model, this is probably not an issue. Overfitting Look for a large number of independent variables in the model, especially if they use different types of transformation Misspecification Look for any variants of the dependent variable in the independent variables. Assess whether the model meets the objectives of the effort. Multicollinearity Assuming relevant plots and diagnostic statistics are not available, there may not be any way to identify multicollinearity. Heteroscedasticity Assuming relevant plots and diagnostic statistics are not available, there may not be any way to identify heteroscedasticity. Autocorrelation Assuming relevant plots and diagnostic statistics are not available, there may not be any way to identify serial correlation. Weighting Compare the reported number of samples to the degrees of freedom. More DF than samples is usually attributable to weighting.

## Follow-up Care

So there are some ways you can identify and evaluate eleven reasons for doubting a regression model. Remember when evaluating other analyst’s models that not everyone is an expert and that even experts make mistakes. Try to be helpful in your critiques, but at a minimum, be professional.

Read more about using statistics at the Stats with Cats blog. Join other fans at the Stats with Cats Facebook group and the Stats with Cats Facebook page. Order Stats with Cats: The Domesticated Guide to Statistics, Models, Graphs, and Other Breeds of Data Analysis at amazon.combarnesandnoble.com, or other online booksellers.

## How to Write Data Analysis Reports in Six Easy Lessons

In every data analysis, putting the analysis and the results into a comprehensible report is the final, and for some, the biggest hurdle. The goal of a technical report is to communicate information. However, the technical information is difficult to understand because it is complicated and not readily known. Add math anxiety and the all too prevalent notion that anything can be proven with statistics and you can understand why reporting on a data analysis is a challenge.

The ability to write effective reports on a data analysis shouldn’t be assumed. It’s not the same as writing a report for a class project that only the instructor will read. It’s not uncommon for data analysts to receive little or no training in this style of technical writing. Some data analysts have never done it, and they fear the process. Some haven’t done it much, and they think every report is pretty much the same. Some learned under different conditions, like writing company newsletters, and figure they know everything there is to know about it. And worst of all, some have done it without guidance and have developed bad habits, but don’t know it.

It’s a pretty safe bet that if you haven’t taken college classes or professional development courses, haven’t been mentored on the job, and haven’t done some independent reading, you have a bit to learn about writing technical reports. Report writing is like any other skill, you get better by learning more about the process and by practicing. Here are four things you can try to improve your skills.

• Educate yourself. Learn what other people think about technical writing. Visit websites on “statistical analysis reports” and “technical writing,” there are millions of them. Take online or local classes. Read books and manuals. Join Internet groups, such as through Yahoo, Google, or LinkedIn. Immerse yourself in the topic as you did when you were in school.
• Understand criticism. Over the course of your career, you’ll give and receive a lot of criticism on technical reports. Not all criticism is created equal. First, consider the source. Some critics have never written a report on a data analysis and some have never even analyzed data. Still, if the critic is the one paying the bills you have to deal with it. For your part, you should learn how to provide constructivecriticism. Unless a report you are reviewing is a complete mess, respect the report writer’s discretion for structure and format. Focus on content. Be nice.
• Download examples. Search the internet for examples of data analysis reports (Hint: adding pdf and download to the search might help). Critique them. Who’s the audience? What’s the message? What’s good and bad about each report? Which reports do you think are good examples? What do they do that you might want to do yourself in the future?
• Find what’s right for you. When you search the Internet for advice on technical writing or take a few classes from knowledgeable instructors, you’ll hear some different opinions. Everyone will talk about audience and content but most will have more limited views of report organization, writing style, and how you work at writing. Ignore what the experts tell you to do if it doesn’t feel right. Just be sure that the path you eventually choose works for you and the audiences who will read your reports.

If you’ve done all that, it’s just a matter of practice. You’ll learn something from each report you write. If you are new to the process of reporting on a data analysis, consider these six easy lessons:

• Lesson 4—Get their attention
• Lesson 5—Get it done
• Lesson 6—Get acceptance.

Start with what you know best. In writing a data analysis report, what you know best would be the statistics, graphing, and modeling you did.

You should be able to describe how you characterized the population, how you generated the data or the sources that provided them, what problems you found in the data during your exploratory analysis, how you scrubbed the data, what you did to treat outliers, what transformations you applied, what you did about dropouts and replicates, and what you did with violations of assumptions and non-significant results.

From that, you’ll need to determine what’s important, and then, what’s important to the reader. Unless you’re writing the report to your Professor in college or your peers in a group of professional data analysts, you can be pretty sure that no one will want to hear about all the issues you had to deal with, the techniques you used, or how hard you worked on the analysis. No one will care if your results came from Excel or an R program you wrote. They’ll just want to hear your conclusions. So, what’s the message you want to deliver? That’s the most important thing you’ll have to keep in mind while writing.

Once you work out your message, write an overview to the report so you’ll know where you’re going. It will help you stay on track. Your summary might take one of three forms:

• Executive Summary. Aimed at decision makers and people with not enough time or patience to read more than 400 words. Limit your summary to less than one-page, do not use any jargon, and provide only the result the decision maker needs to know to take an appropriate action (i.e., the message you want to convey).
• Overview. Aimed at most people, whether they would read the report or not. An overview is an abridged version of what is in the report, with a focus on the message you want to convey. The overview shouldn’t be more than a few pages.
• Abstract.  Aimed at peers and other people who understand data analysis. An abstract summarizes in a page or less everything of importance that you did, from defining the population through assessing effect sizes. Abstracts are most often used in academic articles.

Once you understand who your audience is, you can rewrite the summary to catch the attention of your readers.

# Lesson 2—Know Your Audience

Every self-help article about technical writing starts by telling readers to consider their audience. Even so, probably few report writers do.

In a statistical analysis, you usually start by considering the characteristics of the population about which you want to make inferences. Similarly, when you begin to write a report on an analysis, you usually start by considering the characteristics of the audience with which you want to communicate. You have to think about the who, what, why, where, when, and how of the key people who will be reading your report. Here are some things to consider about your audience.

## Who

Audience is often defined by the role a reader plays relative to the report. Some readers will use the report to make decisions. Some will learn new information from the report. Others will critique the report in terms of what they already know. Thus, the audience for a statistical report is often defined as decision makers, stakeholders, reviewers, or generally interested individuals.

Some reports are read by only a single individual but most are read by many. All kinds of people may read your report. As a consequent, there can be primary, secondary, and even more levels of audience participation. This is problematical; you can’t please everyone. So in defining your audience, focus first on the most important people to receive your message and second on the largest group of people in the audience.

## What

Once you define who you are targeting with your report, you should try to understand their characteristics. Perhaps the most important audience characteristic for a technical report writer is the audience’s understanding of both the subject matter of the report and the statistical techniques being described. You may not be able to do much about their subject matter knowledge but you can adjust how you present statistical information. For example, audiences a data analyst might encounter include:

• Mathphobes. Fear numbers but may listen to concepts. Don’t use any statistical jargon. Don’t show formulas. Use numbers sparingly. For example, substitute “about half” for any percentage around 50%. The extra precision won’t be important to a Mathphobe.
• Bypassers. Understand some but have little interest. Don’t worry about Bypassers, they won’t read past the summary. Be sure to make the summary pithy and highlight the most important finding otherwise they might key on something relatively inconsequential.
• Tourists. Understand some and are interested. Be gentle. Use only essential jargon that you define clearly. Using numbers is fine just don’t use too many in a single table. Round off values so you’re not implying false precision. Stick with nothing more sophisticated than pie charts, bar graphs, and maybe an occasional scatter chart. Don’t use any formulas.
• Hot Dogs. Know less than they think and want to show it. Using jargon is fine so long as you define what you mean. Even a Hot Dog may learn something. In the same vein, using numbers, statistical graphics, and formulas is fine so long as you clearly explain their meanings. Hot Dogs may come to erroneous conclusions if not guided.
• Associates. Other analysts who understand the basic jargon. Anything is fine so long as you clearly explain what you mean.
• Peers. Other data analysts who understand all the jargon. Anything goes.

The audience characteristics provide guidance for report length and writing tone and style

## Why

Are readers likely to be very interested in your report or just curious about it (if they have no interest, they won’t be readers)? Be honest with yourself. Why would anyone be interested in reading your report? What is the objective of the who you defined as your audience? What will they do with your findings? Will they get informed? Will they make a decision or take an action? Is this a big thing for them or just something they have to tune in to?

## Where

Is the report aimed at a finite, confined group, like the organization the analysis was conducted for, or will anyone be able to read it? Is the report aimed at the upper levels of the organization or the rank-and-file (i.e., bottom up or top down)? Are there any concerns for security or confidentiality, either on the individual or organizational levels?

## When

When does the population need to see your report? Who has to review the report and how long might they take before the report is released? How firm are the deadlines? How much time does this leave you to write the report? Will there be enough time to think through what you need to write? Will there be time to conduct additional analyses needed to fill in gaps in the report outline? Will you be outraged when the time taken to review your report is twice as long as the time you took to write it?

Here’s some advice you should take to heart. Never, never, never submit a draft report for review that isn’t your fully complete, edited, masterpiece. I tell myself to follow this rule with every report I write. Unfortunately, like most people, I don’t listen to what I say.

## How

Finally, consider how the report should be presented so that the audience will get the most out of it. Here are five considerations:

• Package. How will your writing be packaged (i.e., assembled into product for distribution)? Will it be a short letter report, a  comprehensive report, a blog or an Internet article, a professional journal article, a white paper, or will your writing be included as part of another document?
• Format. Will your report be distributed as an electronic file of as a paper document? If it will be an electronic document, will it be available on the Internet? Will it be editable? Will it be restricted somehow, such as with a password?
• Appearance. Will the report be limited to black-and-white or will color be included? What will be the ratio of graphics to text? Will the report be conventional or glitzy, like a marketing brochure? Will there be 11”x17” foldout pages or oversized inserts like maps.
• Specialty items. Will you need to provide some items apart from the report, such as electronic data files, analysis scripts or program codes, and outputs? Will you have to create a presentation from the contents of the report? Will your graphics be used for courtroom or public presentations?
• Accessibility. Do you need to follow the guidelines of Section 508 of the Rehabilitation Act of 1973, which may affect your use of headings, tables, graphic objects, and special characters? Should you account for common forms of color blindness in your color graphics?

## Take a Few Moments

You won’t have to address all of these details in evaluating your audience and many will only require a few moments of thought. But, if you think through these considerations, you’ll have a much better idea of who you are writing the report for and how you should write it.

# Lesson 3—Know Your Route

You’ve been taught since high school to start with an outline. Nothing has  changed with that. However, there are many possible outlines you can follow depending on your audience and what they expect. The first thing you have to decide is what the packaged report will look like.

Will your report be an executive brief (not to be confused with a legal brief), a letter report, a summary report, a comprehensive report, an Internet article or blog, a professional journal article, or a white paper to name a few. Each has its own types of audience, content, and whiting style. Here’s a summary of the differences.

Writing a report is like taking a trip. The message is the asset you want to deliver to the ultimate destination, the audience. The package is the vehicle that holds the message. Now you need a map for how to reach your destination. That’s the outline.

Just as there are several possible routes you could take with a map, there are several possible outline strategies you could use to write your report. Here are six.

• The Whatever-Feels-Right Approach. This is what inexperienced report writers do when they have no guidelines. They do what they might have done in college or just make it up as they go along. This might work out just fine or be as confusing as The Maury Show on Father’s Day. Considering that the report involves statistics, you can guess which it would be.
• The Historical Approach. This is another approach that inexperienced report writers use. They do what was done the last time a similar report was produced. This also might work out fine. Then again, the last report may have been a failure, ineffective in communicating its message.
• The “Standard” Approach. Sometimes companies or organizations have standard guidelines for all their reports, even requiring the completion of a formal review process before the report is released. Many academic and professional journals use such a prescriptive approach. The results may or may not be good, but at least they look like all the other reports.
• The Military Approach. You tell ‘em what you’re going to tell ‘em, you tell ‘em, and then you tell ‘em what you told ‘em. The military approach may be redundant and boring, but some professions live by it. It works well if you have a critical message that can get lost in details.
• The Follow-the-Data Approach. If you have a very structured data analysis it can be advantageous to report on each piece of data in sequence. Surveys often fall into this category. This approach makes it easy to write the report because sections can be segregated and doled out to other people to write, before being reassembled in the original order. The disadvantage is that there usually is no overall synthesis of the results. Readers are left on their own to figure out what it all means.
• The Tell-a-Story Approach. This approach assumes that reading a statistical report shouldn’t be as monotonous as mowing the lawn. Instead, you should pique the reader’s curiosity by exposing the findings like a murder mystery, piece by piece, so that everything fits together when you announce the conclusion. This is almost the opposite of the follow-the-data approach. In the tell-a-story approach, the report starts with the simplest data analyses and builds, section by section, to the great climax—the message of the analysis. Analyses that are not relevant to the message are omitted. There are usually arcs, in which a previously introduced analytical result is reiterated in subsequent sections to show how it supports the story line. Graphics are critical in this approach; outlines are more like storyboards. There may be the equivalent of one page of graphics for every page of text. Telling a story usually takes longer to write than the other approaches but the results are more memorable if your audience has the patience to read everything (i.e., don’t try to tell a story to a Bypasser.)

So, be sure that you have an appropriate outline but don’t let it constrain you. Having a map doesn’t mean you can’t change your route along the way, you just need to get to the destination. In building the outline, try to balance sections so the reader has periodic resting points. Within each section, though, make the lengths of subsections correspond to their importance.

# Lesson 4—Get Their Attention

If you’re writing a report about statistics, you have to expect that many readers will lose interest after a while, if they even had it to begin with. So, in writing the report, think about how you might engage your audience. Here are five ideas.

• Find Common Ground.  Every relationship begins with having something in common. Fighting a common foe or solving a common problem can form the strongest and longest lasting of bonds. So the first thing you should try to establish in your report is that common ground. This isn’t so difficult if you are working on an analysis at the behest of a client. The client is already immersed in the data and has invested in you to help solve the problem. Establishing common ground is not so easy if you are proffering an uninvited message. Some people, perhaps subconsciously, don’t really want the message you are offering, especially when you’re analyzing data in their area of expertise. Try to establish common ground in other areas. Perhaps your analysis touches on a similar or analogous issue the reader might have. Maybe the analysis procedure could be used on a different problem the reader might have.
• Clear the Decks. Get rid of everything that doesn’t add to the progression of the report. That doesn’t necessarily mean you have to omit the content. You can relegate it to an appendix, which is pretty much the same thing. Unless required to be in the body of the report, things like the data, data collection surveys and forms, and scrubbing and analysis procedures should all be put in an appendix.
• Set the Tone. Your writing style can either add to or detract from the readability of your report. A formal tone, with strict adherence to grammar rules, complex sentence structures, use of third-person point-of-view and passive voice, and plentiful jargon, is appropriate for most data analysis reports. Formal tones are good for describing details, specifications, and step-by-step instructions. However, formal tones can be more difficult to understand, especially for individuals not accustomed to reading technical reports. An informal tone, with simple grammar and vocabulary, colloquialisms, contractions, analogies, and humor, works well for blogs. Informal tones are good for discussing ideas and concepts, and for inspiring readers or communicating a vision. They are more engaging and tend to be easier for most individuals to understand. If you’re being paid to write the report, a formal tone is usually more appropriate. This is problematical, of course, because formal writing is usually harder to read and maintain an interest in.
• Add Mind Candy. A Harry Potter novel consisting of page-after-page of text will keep readers, young and old, transfixed for hours. A data analysis report consisting of page-after-page of text will put readers into a coma faster than a handful of barbiturates taken with a glass of warm milk in a tub of hot water while meditating. The difference is that the novel engages readers with mental images. Data analysis reports need to use visual imagery, which for the most part means good graphics. Granted, most readers won’t understand anything more complicated than a pie chart or a bar chart, but don’t add to the confusion. Three-dimensions are a no-no. Avoid graphing data in more than a few categories to avoid making the slices and bars uninterpretable. And most importantly, make sure they add to the analysis. You can do more, too. Break up the text with subheadings and bullets. Reiterate information nuggets in boxes instead of just letting them get lost in the text. Use tables for explaining differences in data groups and not just for number buckets. Add footnotes or hyperlinks to explain collateral concepts.

• Make it Better. Just when you think you’re done writing, you’re not. That’s the time when you have to do even more to make the report better. First, take some time off if you can. Then, read it through again making improvements along the way. Read it aloud if you need to, even record it when you read it aloud and then play it back so you can engage both your vision and hearing. Consider getting a second opinion, especially if you can’t distance yourself from the report by setting it aside for a few days. A second opinion may come from a data analysis peer, but don’t ignore nontechnical editors. A good editor can help with spelling, grammar, punctuation, word choice, style and tone, formatting, references, and accessibility. It’s usually worth the effort. This is the time to go for purrfection.

# Lesson 5—Get It Done

Perhaps the hardest part of writing a data analysis report is just getting it completed. It takes discipline and persistence to stay on track. Even so, it’s easy to get distracted. Sometimes the problem is that the story of the analysis hasn’t been thought all the way through. Sometimes there are gaps in the analysis that necessitate stopping to complete more calculations. Sometimes there are too many interruptions and distractions to maintain focus. Sometimes, the process of writing becomes boring and requires a great effort to continue.

Writer’s block is an impediment experienced by all writers. Writer’s block might be attributable to not knowing what to write next, trying to write text that is perfect, or fear of failure. Any of these reasons may be applicable to the report writer. Here are ten ways to fight off writer’s block.

1. Stick with a routine. Keep writing even if you are dissatisfied with what you’ve written. You can, and should, edit your draft after you’re done. Try to identify your productivity tipping point. For some people, accomplishing a specific goal by a certain time in a day helps ensure the rest of your day is productive. For example, my productivity tipping point is beginning to write by 8AM. If I do, I’ll be writing productively all day.

2. Visualize. If you’ve never used visualization techniques before, now is a good time to develop the skill. The idea is to close your eyes, get relaxed, and think about what you want to do or see. Start by visualizing what the next few sentences you have to write might look and sound like. Eventually, you’ll be able to visualize what paragraphs, sections, and even the entire final product will look like.

3. Eschew perfection. If it’s not perfect the first time you write it, leave it alone. Let it age while you write the rest of the report. You can reevaluate and rewrite it later when you know more about the rest of the report.

4. Write in parallel. Some parts of reports, like introductions and summaries, and descriptions of variables and other details, are almost formulaic. Write all the similar parts at the same time. Set up a second file in your word processing software to serve as a staging area for the repeated parts. Then, copy and paste the standardized parts to your report and edit the text as appropriate.

5. Grow the outline. Instead of trying to write the report section by section, try using the outline as a template rather than a map. Add key phrases, instructions, notes, sentences, and even paragraphs to the template-outline. You can skip around the template-outline as you come up with ideas for what to write. Eventually, you can consolidate these ideas into paragraphs and then sections. Continue to expand the template-outline until it ultimately becomes the complete report.

6. Tiptoe through the tables. Create all or most of your graphics (i.e., tables and figures) before starting to write. Lay the graphics out in your word processing software and write the text that would go with each graphic. Then, go back and fill in the gaps between graphics. Continue joining the pieces until the report is complete.

7. Chunk it up. Don’t try to write the entire report by yourself. Break it up into pieces and get help.

8. Set deadlines. Sometimes it helps to be able to work towards an interim goal. Set deadlines for sections or other tasks you have to accomplish. Make them challenging but achievable.

9. Give it a rest. Absence makes the mind grow sharper. Consider taking some time off from report writing, but make sure you use the time productively. Schedule that colonoscopy you’ve been putting off. Clean the garage and paint the house. Visit your in-laws. Don’t just play video games or watch Netflix.

10. Do something different. If your routine isn’t working, try doing something different. If you can’t get anywhere because you’re pressing, work on something else or take some time off. If you can’t get anywhere because you’re slacking, try researching. If you can’t get anywhere because you’re stuck on writing, pull together graphics or the appendices. If you can’t get anywhere because you’re procrastinating, ask yourself why.

# Lesson 6—Get Acceptance

Data analysis reports have to go through one more hurdle after they are completely written. They have to be approved for acceptance by a gatekeeper. The approval for acceptance may involve allowing report distribution, starting the publishing process, issuing payment for your services, or just acknowledging that your work is done. The gatekeeper may be your client, your supervisor, your publisher, or for blog writers, you. To get that approval, formal reports usually have to be reviewed by reviewers. Reviewers are usually individuals the gatekeeper chooses based on their technical background or role in the gatekeeper’s organization. Sometimes, reviewers are individuals the gatekeeper is forced to listen to, like regulatory reviewers. In academic publishing, you may not even know who the peer reviewers are.

Logically, the acceptance review shouldn’t take too long compared to the time you took to analyze the data and write the report. After all, the reviewers only have to read it. In practice, though, reviews take far longer than report preparation. The report you wrote in a month may take six months to be reviewed. Don’t panic. It’s just the way things seem to happen.

The number of comments you get from the reviewers is inconsequential. Great reports can get dozens of highly critical comments. Again, don’t panic. The only review you should be concerned about is the one that provides no comments. That usually signals a lack of interest by the reviewers and the gatekeeper.

When the review is complete, be sure to get the comments in writing. If you don’t, some comments may be forgotten or misunderstood. If there is more than one reviewer, compile all the comments together. This is essential because sometimes reviewers provide conflicting comments. The gatekeeper may compile the comments for you if he or she wants to control the process. The comments should be placed in the order they correspond to in the report. Be sure to identify the source of each comment. If a single comment has many parts, break the comment apart so you can respond to each part individually.

Then comes the challenging part—you must respond to each comment separately. Create a new document listing all the compiled comments. For each comment in this document, either describe what you’ll do in response or explain why you won’t make any changes. Start with the easy comments, such as those involving grammar and spelling. As you describe your response to a comment in the document, make the associated change in the report. Proceed through increasingly more difficult comments until you are done. For very complex comments, try to parse the ideas and respond to each separately. If a particular comment is very difficult to address, you may have to conduct additional analyses or information research. Cite information sources if appropriate.

When you’re done, reread both the response document and the changes in the report. Be sure all the changes were made in the report and that they are consistent with the rest of the report. Also, make sure the tone of your response is even; be stoic.

If you’ve written an informal piece, like a blog, you don’t have to go through the grueling process of responding to formal comments from an acceptance review. Since you are the gatekeeper, you can release your blog whenever you feel it is done. But after you release the blog, you may well get comments. That’s good because it shows that people are reading your blog. Furthermore, there’s no pressure to compile these comments and document your responses. Unfortunately, at least some of the comments will come from spammers, trolls, 13-year-olds, head cases, angry arguers, and other individuals who won’t be providing constructive criticism. Therefore, first consider the source of each comment. In some cases, you won’t have to respond to any of them. Your blogging software will allow you to delete unwelcome comments. Beware of the overly gracious comments, too. Sometimes malicious commenters use addresses that link to spam or malware. If you don’t trust your instincts, just delete the comment.

Don’t get upset by reviewers pointing out flaws in your report. That’s what they’re supposed to do. Having been on both sides of the writer/reviewer divide, I can tell you that creating a report takes a hundred times more knowledge, creativity, effort, and time than reviewing a report. Providing constructive criticism on a report requires a hundred times more experience, situational awareness, and interpersonal sensitivity than creating a report. Good writing combined with constructive reviewing makes a data analysis report the best it can be.

Read more about using statistics at the Stats with Cats blog. Join other fans at the Stats with Cats Facebook group and the Stats with Cats Facebook page. Order Stats with Cats: The Domesticated Guide to Statistics, Models, Graphs, and Other Breeds of Data Analysis at amazon.com,  barnesandnoble.com, or other online booksellers.

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## Ten Ways Statistical Models Can Break Your Heart

Models are beautiful. The ways their features are combined sets them apart from each other.  Each has its own personality, sometimes pleasant, sometimes not, and often not what you would expect.

Here are ten ways your love affair with statistical models can end up on the rocks.

## Relationship-Building Disasters

Modeling is more than just meeting a dataset on the internet and jumping into some R code together. You have to develop relationships with the data and everyone associated with them. For example:

• Miscommunications. There are often quite a few people who have some stake in the model. They usually have different experiences and levels of understanding of modeling and, of course, different agendas for how the model will be treated. They won’t necessarily trust you. You have to try to keep them all happy and on the same page.
• Interference. You may be doing all the heavy lifting with the data and the modeling but there are often individuals, like a boss, the client, or independent reviewers, who poke their fingers into your efforts.
• Delays. You may feel under the gun to complete a modeling project but that doesn’t mean everyone associated with the project will share your constraints. You may be asked to redo the model every time new data become available, attend meetings, make presentations, and wait for decisions from upper management.
• Skepticism. Not everyone is driven to make decisions after a careful analysis of relevant data. Some people prefer to rely on their gut feel. They may look at your model but then ignore those results and use their own intuition.
• Indifference. On occasion, you might create a model, even what you consider a groundbreaking model, but nobody pays attention to it. Your model may be ignored for an inferior model, like an undrafted football player being benched in favor of a million-dollar bust. Or, people just don’t appreciate the importance of the model like you do. You’ll still need to get their acceptance.

## Unrequited Models

You put your heart and soul into modeling the dataset but you get … NOTHING. No love in return. No matter how much you’ve planned, you can’t find a collection of independent variables that will adequately model your dependent variable. It happens to data analysts everywhere, all the time, for a variety of reasons. There may be non-linear relationships, outliers, or excessive uncontrolled variance. The variables may be inappropriate or inefficient.

What can you do?

First, you should reexamine the theory behind your model. Are your hypothesis and assumptions valid? Are your data suspect? Are the metrics you’re using as variables problematical? Are there latent concepts you could explore in a Factor Analysis? Do your samples need to be categorized in some way? Might conducting a Cluster Analysis provide insight?

Second, examine your correlations thoroughly. See if there are any transformations that might be helpful.

Third, if you have appropriate software, consider looking into nonlinear statistical regression, neural networks, and data mining solutions. Finally, there may be ways to construct probabilistic models, or models based on optimization procedures, or relative solutions from experts using a Delphi Method.

In the end:

Some models were not meant to be. If you can’t fit the model to the data, you have to be prepared to call it quits. In a way, this is equivalent to a Do Not Resuscitate order in medicine, and likewise, it can be a sensitive subject. It’s usually easier to create new variables or try some other statistical manipulation than it is to give the bad news, and the bill, to the client.

## Muddled Models

Sometimes models go wrong right out of the box because they are improperly specified. You may not be pursuing the relationship for the right reasons or in the right ways. For example:

• The dependent or independent variables may be too expensive to collect. The model may even cost more to run than addressing the problem is worth.The dependent variable may not be actionable, at last not within the limits set by the client.
• An independent variable might incorporate part of the dependent variable, if one or the other is a ratio.
• The structure of the model may be wrong, for example, the model might be better as a multiplicative or other non-linear form instead of linear.

## Wandering Eye Models

There are many different types of models, like fish in the sea. Some people are always looking for something better, even if what they have is pretty good.

For example, you might have a good model but it’s not what the client expected. Perhaps the results are not what the client wanted to hear or the model may look good for general trends but not be an adequate representation of the phenomenon for extreme or special cases. He wants you to try over and bring him something better.

One concept that often confuses novice model builders are the differences between models aimed at prediction vs explanation. Explanatory models are based on theory. They need to incorporate independent variables that make theoretical or logical sense to be associated with the dependent variable. Prediction models don’t rely on theory. They need independent variables that produce large values of the Coefficient of Determination (r2) but low values of the Standard Error of Estimate (sxy or SEE). Explanatory models assume (or hope) that there are cause-effect relationships between the dependent variable and the independent variables; prediction models do not.

That’s where some clients balk if the model doesn’t have the variables they feel should be in a prediction model. It usually doesn’t matter if the model produces excellent predictions, they feel it would be better if their favorite variables were there … even though it wouldn’t.

It’s not just clients, though. There are times when model builders, especially young professionals, want to try out some new analytical breakthrough. The tried-and-true regression approach may produce results that are nearly as good, but the cutting edge model looks and sounds so much sexier. It’s seductive, and for some, hard to resist.

## Deceptive Profile Models

Don’t you just hate it when you see something that isn’t at all the way it was described? “Hey, you should try analyzing this dataset. It’s a perfect match for you.” But then when you meet up, it’s nothing like you expected.

Maybe the expected population from which the data are drawn doesn’t really exist. Maybe the quality of the data is questionable or needs a lot of cleanup. Maybe the samples are biased or misleading.

And it’s not just what goes into a model that might be disappointing but also what comes out of modeling activities. The regression model itself might be improperly specified or misleading. Sometimes correctly specified models are poorly calibrated. Fortunately, there are also a variety of statistical diagnostics and plots that can be used to identify the problems.

## Mercurial Models

Every measurement of a phenomenon includes characteristics of the population and natural variability as well as unwanted sampling variability, measurement variability, and environmental variability. You can’t understand your data unless you control extraneous variance attributable to the way you select samples, the way you measure variable values, and any influences of the environment in which you are working. If you plan to conduct a statistical analysis, you need to understand the three fundamental Rs of variance control — Reference, Replication, and Randomization. Using the concepts of reference, replication and randomization, you can control, minimize, or at least be able to assess the effects of extraneous variability using: procedural controls; quality samples and measurements; sampling controls; experimental controls; and statistical controls.

Even after spending considerable effort trying to control extraneous variance in data collection, though, sometimes the models produced from them don’t share the precision. The models may have good accuracy, shown by large values of the Coefficient of Determination (r2) but low precision, shown by the large Standard Error of Estimate (sxy or SEE). You might have an accurate predictive model but it lacks enough precision to be useful. This is a surprisingly common occurrence. Some data analysts don’t seem to look past the r2. The sxy is ignored.

Look at any studies you can that involve predictive modeling. Do they discuss the uncertainty in the predictions? What do you think?

## Run-away Models

Sometimes you spend months and even longer getting to know your data and building a relationship only to have the model taken away. Maybe it’s a boss or more senior co-worker. Maybe it’s the client. You can chase after your model, keep up to speed with what’s happening in the model’s life, but that’s about it. There’s not much else you can do. It’s somebody else’s responsibility now.

## Irreconcilable Difference Models

You and your model may reach a point where you might want to go to the next level in your relationship only to find there are differences you did not expect and can’t overcome. When you try to extend the relationship to new situations, everything fails. There are several possible reasons. Maybe you have a multi-level model. What worked for the samples you used doesn’t work when they are aggregated into higher level associations. Maybe you’re a victim of Simpson’s Paradox. What worked for the samples you used doesn’t work when they are separated into component groups. Then again, maybe it’s something you did. Maybe your model is overfit. Perhaps you capitalized on chance and found associations that weren’t pervasive and lasting. The only thing you can do is reexamine the relationship and either start over or move on.

## Marry or Break Up

There comes a time when you have to decide whether to commit to the effort to build a relationship or back out of the commitment. Maybe you don’t have enough samples. Maybe your goals don’t fit what the model needs. Perhaps the model is being asked to do something it wasn’t designed for. What works for describing a population may not be suited to describing individuals in the population. Then there might also be ethical issues to consider. But statisticians rarely get to make these decisions. If they accepted the assignment, the product belongs to the client.

## Happily Never After

Deploying a model can sometimes change the behaviors of the population the model is based on. This is especially true when humans are involved; humans just love to game the rules. For example, if you develop a model for allocating resources, you can be assured that the potential recipients will do whatever it takes to increase their advantage. Once they do that, the model is no longer useful. That’s why models are often kept secret.

Read more about using statistics at the Stats with Cats blog. Join other fans at the Stats with Cats Facebook group and the Stats with Cats Facebook page. Order Stats with Cats: The Domesticated Guide to Statistics, Models, Graphs, and Other Breeds of Data Analysis at amazon.com,  barnesandnoble.com, or other online booksellers.

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## Looking for Insight through a Window

At a press briefing on February 12, 2002, then Secretary of Defense Donald Rumsfeld addressed the absence of evidence linking the government of Iraq with weapons of mass destruction:

There are known knowns. There are things we know that we know. There are known unknowns. That is to say, there are things that we now know we don’t know. But there are also unknown unknowns. There are things we do not know we don’t know.

Now, despite the statement being a transparently irresponsible attempt to cover up a monumental failure in the collection and analysis of information or just a REALLY BIG LIE, the statement actually makes some sense. Similar words have been attributed to Confucius and others. But whether he realized it or not, Mr. Rumsfeld was describing a type of data analysis window.

Analytical windows are a type of matrix plot. Matrix plots are just grids for organizing information. The cells of a matrix plot can contain data, tables, graphs, or text. Windows consist of two criteria, or dimensions, defined by rows and columns. Each dimension usually has two categories, or levels, resulting in four cells, or panes. Rumsfeld’s window would look like this:

 Things that We Know Don’t Know We Know Things that we know we know. Things that we don’t know we know. We Don’t Know Things that we know we don’t know. Things that we don’t know we don’t know.

So, for example, a Rumsfeld Window could be used for planning a statistical study.

• Things that we know we know would be things like background information on the study environment, the underlying theory on the phenomenon being explored, and the statistical characteristics of the population.
• Things that we don’t know we know would be things like the statistical assumptions we make to perform the analysis — independence of observations, normality and homoscedasticity of errors.
• Things that we know we don’t know would be things like the results of the research questions and test hypotheses we plan to focus on.
• Things that we don’t know we don’t know would be things like the causes of outliers and other data and analysis anomalies.

The beauty of a window is the way it can organize sometimes complex information into simple binary categories. As a consequence, windows are used in many ways to analyze data.

### Johari Windows

A Johari Window is a tool used by psychologists to help individuals and groups evaluate interpersonal communications. Its name comes from the first names of Joseph Luft and Harry Ingham, who created it in 1955. To use the window, subjects are told to pick five or six adjectives they feel describe their own personality from a standard list of 56 adjectives. Peers of the subject are then given the same standard list of 56 adjectives, and each pick five or six adjectives that describe the subject. These adjectives are then paced in the appropriate pane of the Johari Window.

 Known to Self Not Known to Self Known to Others Open Blind Not Known to Others Hidden Unknown

Johari windows were featured on a 2010 episode of the television series Fringe, which was seen by six million viewers, most of whom probably had no idea what they are.

### Variance Windows

Windows can also be applied to planning how to control extraneous variance in the process of collecting data. If you plan to conduct a statistical analysis, you’ll need to understand the three fundamental Rs of variance control — Reference, Replication, and Randomization. Every measurement of a phenomenon includes characteristics of the population and natural variability as well as unwanted sampling variability, measurement variability, and environmental variability. You can’t understand your data unless you control extraneous variance attributable to the way you select samples, the way you measure variable values, and any influences of the environment in which you are working. Using the concepts of reference, replication and randomization, you can control, minimize, or at least be able to assess the effects of extraneous variability using: procedural controls; quality samples and measurements; sampling controls; experimental controls; and statistical controls.

 Sources of Variance that we Understand Don’t Understand Control Sampling and measurement variance Sampling and measurement variance, environmental variance Don’t Control Natural variance Sampling and measurement variance, environmental variance

To use a window to plan a variance control program, fill the panes of the window with all the sources of variability you can think of, categorized by how well you understand the source and think you can control it. Then identify a control measure for each source of variation.

### Pick Charts

A Pick Chart is a Lean Six Sigma tool for comparing difficulty of implementation (in terms of costs, effort, complexity, or time) to possible results (paybacks, returns, impacts, or improvements) for actions being considered. These two concepts serve as the axes of a data analysis window having four quadrants:

• Possible“ideas that are considered “low hanging fruit”. The effort to implement is low, but the impact is also low. These should only be implemented after everything in the “Implement” quadrant.”
• Implement“ideas that should be implemented as they will have a high impact and require low effort.”
• Challenge“ideas that should be considered for implementation after everything in the “Implement” column. The impact is high, but the effort is also high.”
• Kill“ideas that should be “killed” or not implemented. The effort to do so is high and the impact is low.”

Here’s an example involving the federal Employee Viewpoint Survey. In this pick chart, eighteen EVS question areas are compared according to:

• Payoff from the actions being considered to improve EVS scores
• Difficulty anticipated in successfully undertaking the actions.

Payoff was calculated (after scale adjustments) as the product of the score for a question and the decline in the scores from 2012 to 2014. Difficulty was based on: (1) who would have to be involved in implementing the change (i.e., many or few staff; in the main office or satellite offices; at staff, supervisor, or senior leader levels); (2) if existing programs or policies would be used or if they would have to be created; and (3) the funding required to implement the change. Payoff is based on actual EVS data so there is not much uncertainty. Difficulty is based on judgments concerning what generic actions might be taken to improve job satisfaction, so there is considerable uncertainty. Thus, the positions of the icons representing the EVS question areas are likely to shift horizontally, depending on the nature of specific projects being considered, but not vertically.

### Performance Windows

A performance window is a way to convey the results of a statistical test or classification. It is a table with two rows and two columns that summarize the number of correct classifications (true positives and true negatives), and the number of misclassifications (false positives and false negatives). This type of window is also called a confusion matrix, an error matrix, or a matching matrix.

Here are performance windows for classifications and statistical tests.

 Predicted Classification A B Actual Classification A Correct Classification Misclassification B Misclassification Correct Classification
 Statistical Test Null hypothesis is not rejected Null hypothesis is rejected Actual Condition True Correct Inference False Positive – Type I Error False False Negative -Type II Error Correct Inference

A contingency table is a type of matrix plot, frequently for more than two levels on the dimensions or even more than two dimensions, which summarizes the occurrence of data. They are also called cross tabulation ‎tables.

### Windows on Scatter Plots

The concept of dividing areas of information into more understandable parts can be extended to scatter plots. Plots can be divided into quadrants, for example, using the means (or medians) of the data points for each axis. In essence, the window is overlain on the scatter plot. The window can be subdivided further by standard deviations (or quartiles).

The performance window for this scatter plot would be:

 Math Grade Below Average Above Average English Grade Above Average 9 18 27 Below Average 16 8 24 25 26 51

Read more about using statistics at the Stats with Cats blog. Join other fans at the Stats with Cats Facebook group and the Stats with Cats Facebook page. Order Stats with Cats: The Domesticated Guide to Statistics, Models, Graphs, and Other Breeds of Data Analysis at amazon.com, barnesandnoble.com, or other online booksellers.