Visualizations versus Infographics

Visualizations and infographics are both visual representations of data that are often confused. In fact, there is not a clear line of demarcation between the two. Both are informative. Both can be static or animated. Both require a knowledgeable person to create them.

VizInfo5-13-2017

Visualizations Explore

Data visualizations are created to make sense of data visually and to explore data interactively. Visualization is mostly automatic, generated through the use of data analysis software, to create graphs, plots, and charts. The visualizations can use the default settings of the software or involve Data Artistry and labeling (i.e., these Enhanced Visualizations fall in the intersection of the two circles in the figure). The processes used to create visualizations can be applied efficiently to almost any dataset. Visualizations tend to be more objective than infographics and better for allowing audiences to draw their own conclusions, although the audience needs to have some skills in data analysis. Data visualizations do not contain infographics.

Infographics Explain

Infographics are artistic displays intended to make a point using information. They are specific, elaborate, explanatory, and self-contained. Every infographic is unique and must be designed from scratch for visual appeal and overall reader comprehension. There is no software for automatically producing infographics the way there is for visualizations. Infographics are combinations of illustrations, images, text, and even visualizations designed for general audiences. Infographics are better than visualizations for guiding the conclusions of an audience but can be more subjective than visualizations.

Visualization Infographic
Objective Analyze Communicate
Audience Some data analysis skills General audience
Components Points, lines, bars, and other data representations Graphic design elements, text, visualizations
Source of Information Raw data Analyzed data and findings
Creation Tool Data analysis software Desktop publishing software
Replication Easily reproducible with new data Unique
Interactive or Static Either Static
Aesthetic Treatment Not necessary Essential
Interpretation Left to the audience Provided to the audience

REFERENCES

http://jacobjwalker.effectiveeducation.org/blog/2017/05/12/data-artistry-using-and-sharing-the-knowledge-in-an-effective-manner/

http://killerinfographics.com/blog/data-visualization-versus-infographics.html

http://killerinfographics.com/infographic-design-start-finish.html

http://www.arena-media.co.uk/blog/2012/09/whats-the-difference-between-an-infographic-and-data-visualisation/

http://www.dummies.com/programming/big-data/big-data-visualization/understanding-the-difference-between-data-visualization-and-infographics/

http://www.thefunctionalart.com/2014/03/infographics-to-reveal-visualizations.html

https://eagereyes.org/blog/2010/the-difference-between-infographics-and-visualization

https://visage.co/throwdown-data-visualization-vs-infographics/

img_8475c (1)

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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How to Analyze Text

Statisticians love to analyze numbers, but what do they do when what they want to explore is unformatted text? It happens all the time. The text may come from opencat-diary-ended responses on surveys, social networking sites, email, online reviews, public comments, notations (e.g., medical, customer relations), documents and text files, or even recorded and transcribed interactions. But before anything can happen, you have to accomplish three tasks:

  • Get the text into a spreadsheet or other software that you can use to manipulate it.
  • Break the text into analyzable fragments – letters, words, phrases, sentences, paragraphs, or whatever.
  • Assign properties to the text fragments

How you might complete these tasks depends on what you want to do and the software you have. Nonetheless, you’ll be surprised by how much you can do with just a spreadsheet and an internet connection if you have the time and focus. This article will show you how.

Approaches

Ther0402a6_fd87fbc829ec41faaf10aa7aa1cbed88-mv2_d_2000_1333_s_2e are several ways that you can analyze text. You can:

  • Count the occurrence of specific letters, words, or phrases, often summarized as Word Clouds. There are quite a few free web sites that will help you construct word clouds.
  • Categorize text by key themes, topics, or commonalities, called Text Mining.
  • Classify attitudes, emotions, and opinions of a source toward some topic, called Sentiment Analysis or opinion mining. There are many applications of sentiment analysis in business, marketing, customer management, political science, law, sociology, psychology, and communications.
  • Explore relationships between words using a Word Net. The relationships can reflect definitions or other commonalities.

Some of these analyses can be performed using free web apps, others, require special software.

Specialized Software

Some text analytics can be performed manually, but it is a time consuming process so having software can be crucial. Unfortunately, the biggest and best software is proprietary, like SAS and SPSS, and costs a lot. There are also free and low-cost alternatives, as well as free web sites that preform less sophisticated analyses. There are a lot of software options so there are probably a lot of people analyzing text. Let Google be your guide.

Manual Analyses

Even if you don’t have access to specialized software for text analyses, you can also still perform two types of analyses with nothing more than a spreadsheet program and an internet connection. You can count the number of times that a letter, word, or phrase appears in a text passage. Word frequency turns out to be relatively easy to produce but once you have the counts, the analysis and interpretation may be a bit more challenging. You can also do simple topic analyses or sentiment analyses. Parsing the sentences or sentence fragments and analyzing them is straightforward but time consuming, though the interpretation is usually easier.

Word Counts

If you are just looking for keywords or counting words for some diagnostic purpose, you’ll find that it’s not that difficult. Here’s how to do word counts.

Step 1 – Find the text you want to analyze.

This is usually easy except for there being so many choices. You have to start with an electronic file. If you have hard copy, you’ll have to sc
an it and correct the errors. If you have text from separate sources, you’1399360333213ll want to aggregate them to make things easier. If you have text on a website, you can usually highlight it and copy it using <ctrl-C>. If the passage is long, you can use <ctrl-A> to select everything before copying it, but you’ll have to edit out the extraneous material. You can do these operations in most word processors.

Step 2 – Scrub the data

You should scrub the text to be sure you’ll be counting the correct things. Take out entries that aren’t part of the flow of the text, like footnotes and section numbers. Correct misspellings. Take out punctuation that might become associated with words, like em dashes.

Step 3 – Count the words.

The quickest way to count words is to go to an Internet site for that purpose. Just copy your scrubbed text, paste it into the box on the site, and press submit. You’ll get a column of words and their frequencies. Parse the numbers from the text and you’re ready to analyze the data. It’s a good idea to review the results of the counting to be sure no errors have crept into the process.

Another way to do this solely in a spreadsheet is to replace all the punctuation with blanks and then replace the blanks with paragraph marks. This will give you a column of words. Copy it and remove the duplicates then you can use a formula to count each word.

Once you have the counts, the analysis is up to you. You can compare word statistics from different sources or analyze word frequencies within a single source. The possibilities are endless. Interpretation is another matter. Here are some examples.

table

One thing you can do with word counts is to produce a word cloud. There are many web sites that will generate these graphics. My favorite is Wordle, but be advised, you have to use Internet Explorer for it to work. Here’s an example of a word cloud produced with Wordle.

wordle1

Text Mining

Topic or Sentiment Analyses are straightforward but more time consuming than word counts. Unless you are analyzing text for work or school, relax and turn on Netflix. This isn’t very sophisticated, but it’ll take a while and you’ll need frequent breaks to maintain your focus.

There are six steps.

Step 1 – Get the Data into a Spreadsheet

As with word counts, you have to get the text file into a text manager, preferably a spreadsheet. Highlight your text or use <ctrl A> and then <ctrl C> and <ctrl V>. You’ll need to parse any block text into sentences or whatever length fragment you want to analyze. You can usually do this by replacing periods with paragraph marks. Start with a small dataset, perhaps fewer than fifty fragments, until you get used to the process.

Step 2 – Scrub the Responses

Format the fragments into a single column with one fragment per row. Delete extraneous fragments. Don’t worry about misspellings and punctuation. If you make a mistake, <ctrl Z> will undo it.

Step 3 – Assign Descriptors

In a column next to the column with the fragments, enter your first descriptor. It can be a keyword, theme, sentiment, length, or whatever you want to analyze. Unless you have predetermined descriptors you are looking for, don’t worry too much about the descriptors you use. You’ll review and edit them in the next step.

cat-writingStep 4 – Count the Fragments Assigned to Each Descriptor

When you count the fragments assigned to each descriptor, you’ll probably find a few descriptors with only a few fragments. Consider combining them with other descriptors. When you’re satisfied with the assignments, you might want to subdivide the descriptor groups with another set of descriptors.

Step 5 – Repeat Steps 3 and 4

You can repeat the last two steps as many times as you feel is necessary. You can use these hierarchical descriptor groups to characterize subsets of the text so don’t have too many or too few fragments in each descriptor group. When you’re done, your data set would look something like this.

spreadsheet

If you have a predetermined set of descriptors, you can assign each one to a column of the spreadsheet and code them as 0 or 1 for presence or absence.

Step 6 – Analyze

Once you have built your data set, you can analyze it statistically by counts and percentages, or graphically using word clouds. Consider this example. On December 29, 2016, Tanya Lynn Dee asked the question on her Facebook page, “Without revealing your actual age, what [is] something you remember that if you told a younger person they wouldn’t understand?” There were over 1,000 responses (at the time I saw the post), which I copied and classified into common themes. The results are here.

So, try analyzing some text (and other things) at home. You won’t need parental supervision.

cat-news3

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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Top 50 Statistics Blogs And Websites on the Web

Number 28

Reading Stats with  Cats

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Hellbent on Measurement

tape-measure-on-a-catAny 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.

sound-27302bac00000578-3021300-image-a-35_1427885938930In 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, Kennycup-beaker-vv9areh
  • 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.

ordinal-progression

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
  • Survey responses—very good, somewhat good, average, somewhat bad, very bad

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.

cup-70694-002132826jpg-vf2a

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:scale-twitch_scale

  • 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:five

  • 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 temperature-should-hospital_e2d565717fa09970because 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

count-the-cats

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.cat-bicycle

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 temcompass-20130531-182857poral 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-daylight-savings-time-cat-checks-for-accuracy

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-d71_2271

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 multidimefly-c2bac65889b946dec4996a0a248e2ba0nsional. 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 othe_cat_stairsr 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 ontape-itskhnz 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.

scale-fat-cat-on-a-scale-30630

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 hobook-92e6f20565c2a9472dda6410939a44a6w 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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Searching for Answers

scaleNeed 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.

student-using-the-card-catalog-1971So, 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.

dt951116dhc0

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 bcat-using-iphonelog 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).

keywords2Perhaps 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.

search-terms

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.

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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

Cat for presidentThe 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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Regression Fantasies

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.

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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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