Testing associations between two categorical variables

 

Over the coming months I will be adapting various bits from a work in progress, a new book on building predictive models using regression, aimed at people working in nonprofit and higher education advancement.

 

(If you’re interested, I suggest subscribing via email — see the box to the right — to have the inside track on this project.)

 

In my previous post, I wrote about “Exploring associations between variables.” Today I go deeper to describe exactly how to test for an association or relationship between two variables that are categorical.

 

These directions are specific to the statistics software Data Desk, but apply equally to any stats package. I make frequent reference to contingency tables, which you may know better as cross-tabs or R x C tables (row by column tables).

 

As well, I mention a data file from “Data University.” This is a file of anonymized sample data provided to me by Peter Wylie and John Sammis, which readers of my book will be able to download and use to learn the concepts.

 

In your data file for Data University, you’ve got a variable for “Business phone present” and “Job title present”. You’d like to know if these two factors are related to each other. If they are completely unrelated, they are said to be independent of each other. We can analyze this in Data Desk using contingency tables. (If you don’t know why we care about dependence or independence among variables, read my previous post.)

 

Click once on the icon for ‘BusPhone Present’ to select it as the first variable, then Shift-click on ‘JobTitle Present’ to select it as the second variable. Under the Calc menu at the top of the screen, select Contingency Tables. The displayed result will depend on what default settings are in effect, but it might look like the table below. (Just as with frequency tables, you can play with display options from a menu available by clicking on the little triangle in the top left corner of the window.)

 

 

A contingency table gives us counts of how many cases fall into the categories that result from the intersection of two categorical variables. In this example, we have two binary variables, therefore there are four possibilities, arranged in a two-by-two matrix. The two “levels” of ‘BusPhone Present’ are labeled vertically (down the rows), and the two ‘levels’ of ‘Job Title Present’ are labeled horizontally (across the columns).

 

The labeling of rows and columns in contingency tables can be a bit confusing when the category names are just ones and zeroes, so here’s the same table with more descriptive labels.

 

 

If these variables are properly coded, with no missing values, each and every case in our data falls into one of the four cells of the matrix. We see for example that 2,240 cases are ‘0’ for ‘BusPhone Present’ and ‘0’ for ‘JobTitle Present’ – a large number of cases have neither piece of information coded in the database.

 

Looking at counts of cases in a contingency table can sometimes show cells with unusually many or unusually few cases. In this example, it seems to be fairly common that whenever Data University has a business phone number, it tends to have a job title as well, and it’s also common for both to be missing. It’s much less common for one to be present and the other not. This suggests there’s a relationship.

 

We’ve made a judgment call, just looking at the numbers. Fortunately the field of statistics provides some tests for making decisions more consistently and easily. If you have some training in stats you will readily grasp how we can apply these tests. For our purposes, we will rely on rules of thumb and simplified explanations rather than delve into a deep understanding of the tests themselves.

 

Click on the options menu for the contingency table, and add “Expected value” to the table, then click OK. This adds a new set of numbers to the table. The section at the bottom of the table, “table contents,” lists these elements in the order they appear in each cell. The first element is the count of cases that we’ve already seen. The other element is Expected Values.

 

 

“Expected values” are the counts of cases that would be expected to result if ‘BusPhone Present’ and ‘JobTltle Present’ were statistically independent of each other. In other words, if the two variables were completely unrelated, the probability that any one case would end up in a particular cell depends only on the probability that the case falls in a specified row and the probability that it falls in a specified column. In other words, we would expect the number of people with a business phone number AND who have a job title in the database to depend solely on the number of each in the sample – one doesn’t depend on the other. By this logic, the probability of a person having a business phone in the database is about the same whether or not they have a job title in the database, if it is true that one is not related to the other.

 

Have a quick look to compare each pair of numbers. You’ll notice that two actual counts are higher than the expected value, and two counts are lower than the expected value. If the variables were not associated with each other (independent), the actual counts would be a lot closer to the expected counts. It appears that if one variable is a zero or one, the other variable is more likely than not to have the same value – that is, we are likely to have either both or neither. Therefore, the variables do seem to be related.

 

Being able to compare the expected values to the observed values is helpful in identifying a pattern or relationship. But again, we are still just eyeballing the numbers and making a guess. Let’s take another step forward.

 

An assertion that two variables are not related is based on a concept in statistics called the null hypothesis. The null hypothesis in this case would state, “There is no relationship between Business Phone Present and Job Title Present.” In a test for independence between Business Phone and Job Title, if there is no relationship, then the null hypothesis is true. The alternative hypothesis is that the variables are in fact related – they are dependent, rather than independent.

 

The test for determining whether the null hypothesis is true in this case is called the chi-square test for independence. For our purposes, it is not necessary to formally state any kind of hypothesis, nor is it necessary to understand how chi-square is calculated. (Chi is pronounced “kai”.) Data Desk takes care of the calculations, and a rule of thumb for interpreting the result will suffice. (1)

 

Go back to the options for the contingency table. Deselect “Expected value”, and choose “Chi-square value” instead. Data Desk will then calculate the chi-square statistic. If it’s large, that is enough to reject the null hypothesis. This number is not interpreted on its own. Rather, the important statistic to watch for appears directly underneath it: p. The p-value in the table below is less than or equal to 0.0001 – a very small value.

 

 

The lowercase p stands for “probability.” A p-value of 0.0001 is the same as a 0.01% chance of something happening. We ask this question: “If there were no relationship between ‘BusPhone Present’ and “JobTitle Present’, what is the probability that you would get a result for chi-square at least as high as 1,141?” That probability is your p-value.

 

By convention, you need a p-value of 0.05 (i.e. 5 percent) or less to consider the probability low enough to conclude that there must, in fact, be a relationship between ‘BusPhone Present’ and ‘JobTitle Present’. (In other words, you can reject the null hypothesis.) The probability of obtaining a value of chi-square this large if the variables were independent is extremely low: less than 0.01%. The p-value in this case is very small, therefore the variables are strongly related.

 

We will encounter p-values again when we do regression. (2)

 

Given the foregoing discussion, you might be thinking that exploring relationships among variables is a complicated and subtle business. Not really. You don’t have to study expected values or formulate a null hypothesis – I introduced those things to help you understand where chi-square comes from. You only need these steps to test for presence of a relationship:

 

1. Select the icons of the two variables.
2. Create a contingency table from the Calc menu.
3. In the table options menu, select “Chi-square value.”
4. Regardless of the chi-square value, if the p-value is less than 0.05, the variables are not independent – they are related somehow.

 

You can create a new contingency table whenever you want to test two new variables, or you can simply drag a new category variable into an existing table. Just drag a new categorical variable icon on top of the name of the variable you want to replace. These names are in bold face text near the top of the table window, and will highlight when touched by the cursor during the drag. You can replace both factors at once by selecting two new variables and dragging them both into the centre of the table.

 

Before we move on, try dragging other binary variables into the contingency table to quickly see which combinations yield some interesting associations.

 

A more relevant example

 

Dragging variables one by one into a contingency table is a useful exploration step while evaluating potential predictors for use in a model. This implies that one of the variables must be the target (outcome) variable.

 

Let’s say you plan to build a predictive model to identify which alumni are more likely to make a donation at a higher level than usual, and you want to make a preliminary assessment of potential predictors so you can toss out any that don’t look promising, while keeping the others for use in the regression analysis. You can create a target variable using a lifetime giving variable that will allow you to do this analysis using contingency tables.

 

Working with the Data University file, create a new derived variable called ‘Big donor’ with the expression:

 

‘Lifetime HC’ > 999

 

This will create a binary variable that evaluates to ‘1’ for alumni with lifetime hard-credit giving greater than $999, and ‘0’ for everyone else.

 

You can use whatever dollar value you like, but $1,000 and up will work here. According to a frequency table of ‘Big donor’, 602 people have lifetime giving of $1,000 or more.

 

Click on the icon for ‘Big donor’ to select it and create a contingency table from the Calc menu, and change the table options to include chi-square. Now the window is ready for testing variables one by one, by dragging variables into the space that is initially labeled “Drag Variable Here.”

 

Chi-square is great for indicating that two variables are related, but a little more information will help you understand the nature of the relationship. When you drag in “BusPhone Present,” for example, it seems apparent that the relationship is significant, to judge from the chi-square value. One tweak will tell you something more concrete: Go back to the table options, deselect “Count” and select “Percent of column total.”

 

 

This replaces cell counts with percent values – the percent breakdown for each column. The circled values in the table above are the ones to pay attention to – they are the cases that have a ‘1’ for ‘Big donor’. Here is what we can read from this:

 

• The first column of figures includes all cases for which there is no business phone in the database (0). Only 8.39% of people with no business phone are also top donors.
• The second column includes all cases with a business phone (1) – 18.4% of people with a business phone are top donors.

 

People with a business phone are more than twice as likely to be top donors, and the chi-square value indicates that the relationship is significant. It might be hasty to conclude that ‘BusPhone Present’ is truly a “predictor” of high-level giving, but the association is clearly there in the data.

 

(These percentages total on the columns. If ‘Big donor’ were your second variable instead of your first, the matrix would be ordered in the other direction, and you would choose “Percent of row total” from the table options instead.)

 

Try dragging other binary variables from the Data University set into the table, replacing ‘BusPhone Present’, and observe how the percent breakdowns differ from group to group, and whether that difference is significant.

 

In my book-in-progress, I go on to talk about exploring categorical variables with multiple categories – not just binary variables – but that’s all for today. Again, if you’re interested in knowing more as this project progress, please subscribe to this blog via email from the box in the right sidebar.

 

End notes

 

1) If you must know, the value of chi-square is the sum of the squared standardized residuals across all cells in the table. The standardized residual describes the extent to which the observed count differs from the expected count. You can display this statistic in Data Desk along with the others, but there is no need. Another abbreviation in the table is “df”, which stands for degrees of freedom. Probability values of chi-square depend upon the degrees of freedom, which in turn is related to the number of rows and columns in the table.

 

2) Use and misuse of p-values is a hot topic in statistics, especially in connection with academic publishing, where putting too much stock in p-values and the rather arbitrary 0.05 threshold causes a great deal of angst. As predictive modelers we need to keep these controversies in perspective: We are not testing the effects of new medical treatments nor are we seeking to publish in a scientific journal. We are making use of “good-enough” tools that will produce a useful result. That said, applying the 0.05 threshold without judgment and common sense means that we will detect “relationships” that are not real – potentially about 5% of the time!

 

 

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Logistic regression vs. multiple regression

by Peter Wylie, John Sammis and Kevin MacDonell

(Click to download printer-friendly PDF: Logistic vs MR-Wylie Sammis MacDonell)

The three of us talk about this issue a lot because we encounter a number of situations in our work where we need to choose between these two techniques. Many of our late night/early morning phone/internet discussions have been gobbled up by talking about which technique seems to be better under what circumstances. More than a few times, I’ve suggested we write something up about our experience with both techniques. In the end we’ve always decided to put off doing that because … well, because we’ve thought it might put a lot of people to sleep. Disagree as we might about lots of things, we’re of one mind on the dictum: “Don’t bore people.” They have enough tedious stuff in their lives; we don’t need to add to their burden.

On the other hand, as analytics has started to sink its teeth more and more into the world of advancement, it seems there is a group of folks out there who wrestle with the same issue. And the issue seems to be this:

“If I have a binary dependent variable (e.g., major giver/ non major giver, volunteer/non-volunteer, reunion attender/non-reunion attender, etc.), which technique should I use? Logistic regression or multiple regression?”

We considered a number of ways to try to answer this question:

• We could simply assert an opinion based on our bank of experience with both techniques.
• We could show you the results of a number of data sets using both techniques and then offer our opinion.
• We could show you a way to compare both techniques using some of your own data.

We chose the third option because we think there is no better way to learn about a statistical technique than by using the technique on real data. Whenever we’ve done this sort of exploring ourselves, we’ve been humbled by how much we’ve learned.

Before we show you a way to compare the two techniques, we’ll offer some thoughts on why this question (“Should I use logistic regression or multiple regression?”) is so tough to find an answer to. If you’re anxious to move on to our comparison process, you can skip this section. But we hope you don’t.

Why This Is Not an Easy Question to Find an Answer To

We see at least two reasons why this is so:

• Multiple regression has lived in the neighborhood a long time; logistic regression is a new kid on the block.
• The articles and books we’ve read on comparisons of the two techniques are hard to understand.

Multiple regression is a longtime resident; logistic regression is a new kid on the block.

When World War II came along, there was a pressing need for rapid ways to assess the potential of young men (and some women) for the critical jobs that the military services were trying to fill. It was in this flurry of preparation that multiple regression began to see a great deal of practical application by behavioral scientists who had left their academic jobs and joined up for the duration. The theory behind multiple regression had been worked out much earlier in the century by geniuses like Ronald Fisher, Karl Pearson, and Edward Hotelling. But the method did not get much use until the war effort necessitated that use. The computational effort involved was just too forbidding.

Logistic regression is a different story. From the reading we’ve done, logistic regression got its early practical use in the world of medicine where biostatisticians were trying to predict binary outcomes like survived/did not survive, contracted disease/did not contract disease, had a coronary event/did not have a coronary event, and the like. It’s only been within the last fifteen or twenty years that logistic regression has found its way into the parlance of statisticians in the behavioral sciences.

These two paragraphs are a long way around of saying that logistic regression is (in our opinion) nowhere near as well vetted as is multiple regression by people like us in advancement who are interested in predicting behavior, especially giving behavior.

The articles and books we’ve read on comparisons of the two techniques are hard to understand.

Since I (Peter) was pushing to do this piece, John and I decided it would be my responsibility to do some searching of the more recent literature on logistic regression as it relates to the substance of this project.

To start off, I reread portions of texts I have accumulated over the years that focus on multiple regression as a general data analytic technique. Each text has a section on logistic regression. As I waded back into these sections, I asked myself: “Is what I’m reading here going to enlighten more than confuse the folks we have in mind for this piece?”  Without exception, my answer was, “Nope, just the reverse.” There was altogether too much focus on complicated equations and theory and nowhere near enough emphasis on the practical use of logistic regression. (This, in spite of the fact that each text had an introduction ensuring us the book would go light on math and heavy on application.)

Then, using my trusty iPad, I set about seeing what I could find on the web. Not surprisingly, I found a ton of articles (and even some full length books) that had found their way into the public domain. I downloaded a bunch of them to read whenever I could find enough time to dig into them. I’m sorry to report that each time I’d give one of these things a try, I would hear my father’s voice (dad graduated third in his class in engineering school) as he paged through my own science and math texts when I was in college: “They oughta teach the clowns who wrote these things to write in plain English.” (I always tried to use such comments as excuses for bad grades. Never worked.)

Levity aside, it is hard to find clearly written articles or books on the use of logistic versus multiple regression in the behavioral sciences. I think it’s a bad situation that needs fixing, but that fixing won’t occur anytime soon. On the other hand, I think dad was right not to let me off easy for giving up on badly written material. And you shouldn’t let my pessimism dissuade you from trying out some of these same articles and books. (If enough of you are interested, perhaps Kevin and John and I can put together a list of suggested readings.)

A Way to Compare Logistic Regression with Multiple Regression

As promised we’ll take you through a set of steps you can use with some of your own data:

1. Pick a binary dependent variable and a set of predictors.
2. Compute a predicted probability value for every record in your sample using both multiple regression and logistic regression.
3. Draw three random subsamples of 20 records each from the total sample so that each subsample includes the predicted multiple regression probability value and the predicted logistic regression probability value for every record.
4. Display each subsample of these records in a table and a graph.
5. Do an eyeball comparison of the probability values in both the tables and the graphs.

1. Pick a binary dependent variable and a set of predictors.

For this example, we used a private four year institution with about 13,000 solicitable alums. Here are the variables we chose:

Dependent variable. Each alum who had given $31 or more lifetime was defined as 1, all others who had given less than that amount were defined as 0. There were 6,293 0’s and 6,204 1’s. Just about an even fifty/fifty split.

Predictor variables:

• CLASS YEAR
• SQUARE OF CLASS YEAR
• EMAIL ADDRESS LISTED (YES/NO, 1=YES, 0=NO)
• MARITAL STATUS (SINGLE =1, ALL OTHERS=0)
• HOME PHONE LISTED (YES/NO, 1=YES, 0=NO)
• UNIQUE ID NUMBER

Why did we use ID number as one of the predictors? Over the years we’ve found that many schools use all-numeric ID numbers. When these numbers are entered into a regression analysis, they often work as predictors. More importantly, they help to create very granular predicted scores that can easily be binned into equal size groups.

2. Compute a predicted probability value for every record in your sample using both multiple regression and logistic regression.

This is where things start to get a bit technical and where a little background reading on both multiple regression and logistic regression wouldn’t hurt. Again, most of the material you’ll find will be tough to decipher. Here we’ll keep it as simple as we can.

For both techniques the predicted value you want to generate is a probability, a number that varies between 0 and 1.  In this example, that value will represent the probability that a record has given $31 or more lifetime to the college.

Now here’s the rub, the logistic regression model will always generate a probability value that varies between 0 and 1. However, the multiple regression model will almost always generate a value that varies between something less than 0 (a negative number) and a number greater than 1. In fact, in this example the range of probability values for the logistic regression model extends from .037 to .948. The range of probability values for the multiple regression model extends from -.122 to 1.003.

(By the way, this is why so many statisticians advise the use of logistic regression over multiple regression when the dependent variable is binary. In essence they are saying, “A probability value can’t exceed 1 nor can it be less than 0. Since multiple regression often yields values less than 0 and greater than 1, use logistic regression.” To be fair, we’re exaggerating a bit, but not very much.)

3. Draw three random subsamples of 20 records each from the total sample so that each subsample includes the predicted multiple regression probability value and the predicted logistic regression probability value for all 20 records.

The size and number of these subsamples is, of course, arbitrary. We decided that three subsamples were better than two and that four or more would be overkill. Twenty records, as you’ll see a bit further on, is a number that allows you to see patterns in a table or graph without overcrowding the picture.

4. Display each subsample of these records in a table and a graph.

Tables 1-3 and Figures 1-3 below show how we took this step for our example. To make sure we’re being clear, let’s go through some of the details in Table 1 and Figure 1 (which we constructed for the first subsample of twenty randomly drawn records).

In Table 1 the probability values for multiple regression for each record are displayed in the left-hand column. The corresponding probability values for the same records for logistic regression are displayed in the right-hand column. For example, the multiple regression probability for the first record is .078827109. The record’s logistic regression probability is .098107437. In plain English, that means the multiple regression model for this example is saying that this particular alum has about eight chances in a hundred of giving $31 or more lifetime. The logistic regression model is saying that the same alum has about ten chances in a hundred of giving $31 or more lifetime.

Table 1: Predicted Probability Values Generated from Using Multiple Regression and Logistic Regression for the First of Three Randomly Drawn Subsamples of 20 Records

Figure 1 shows the pairs of values you see in Table 1 displayed graphically in a scatterplot. You’ll notice that the points in the scatterplot appear to fall along what roughly looks like a straight line. This means that the multiple regression model and the logistic regression model are assigning very similar probabilities to each of the 20 records in the subsample. If you study Table 1, you can see this trend, but the trend is much easier to discern in the scatter plot.

Table 2: Predicted Probability Values Generated from Using Multiple Regression and Logistic Regression for the Second of Three Randomly Drawn Subsamples of 20 Records

Table 3: Predicted Probability Values Generated from Using Multiple Regression and Logistic Regression for the Third of Three Randomly Drawn Subsamples of 20 Records

 

5. Do an eyeball comparison of the probability values in both the tables and the graphs.

We’ve already done such a comparison in Table 1 and Figure 1. If we do the same comparison for Tables 2 and 3 and for Figures 2 and 3, it’s pretty clear that we’ll come to the same conclusion: Multiple regression and logistic regression (for this example) are giving us very similar answers.

So Where Does This All Take Us?

We’d like to cover several topics in this closing section:

• A frequent objection to using multiple regression versus logistic regression when the dependent variable is binary
• Trying our approach on your own
• The conclusion we think you’ll eventually arrive at
• How we’ve just scratched the surface here

A frequent objection to using multiple regression versus logistic regression when the dependent variable is binary

Earlier we said that many statisticians seem to advise the use of logistic regression over multiple regression by invoking this logic: “A probability value can’t exceed 1 nor can it be less than 0. Since multiple regression often yields values less than 0 and greater than 1, use logistic regression.” We also said we were exaggerating the stance of these statisticians a bit (but not very much).

While we can understand this argument, our feeling is that, in the applied fields we toil in, that argument is not a very practical one. In fact a seasoned statistics professor we know says (in effect): “What’s the big deal? If multiple regression yields any predicted values less than 0, consider them 0. If multiple regression yields any values greater than 1, consider them 1. End of story.” We agree.

Trying our approach on your own

In this piece we’ve shown the results of one comparison between multiple and logistic regression on one set of data. It’s clear that the results we got for the two techniques were very similar. But does that mean we’d get such similar results with other examples? Not necessarily.

So here’s what we’d recommend. Try doing your own comparisons of the two techniques with:

• Different data sets. If you’re a higher education institution, you might pick a couple of data sets, one for alums who’ve been out for more than 25 years and one for folks who’ve been out less than 10 years. If you’re a non-profit, you can use a set of members from the west coast and one from the east coast.
• Different variables. Try different binary dependent variables like those we mentioned earlier: major giver/non major giver, volunteer/non-volunteer, reunion attender/non-reunion attender, etc. And try different predictors. Try to mix categorical variables like marital status with quantitative variables like age. If you’re comfortable with more sophisticated stats, try throwing in cross products and exponential terms.
• Different splits in the dependent variable. In our example piece the dependent variable was almost an exact 50/50 split. Since the underlying variable we used was quantitative (lifetime giving), we could have adjusted those splits in a number of ways: 60/40, 75/25, 80/20, 95/5, and on and on the list could go. Had we tried these different kinds of splits, would we have the same kinds of results for the two techniques? Since we actually did look at different splits like these, we can report that the results for both techniques were pretty much the same. But that’s for this example. That could change with a different data set and different variables.

The conclusion we think you’ll eventually arrive at

We’re very serious about having you compare multiple regression and logistic regression on a variety of data sets with a variety of variables and with different splits in the dependent variable. If you do, you’ll learn a ton. Guaranteed.

On the other hand, if we put ourselves in your shoes, it’s easy to imagine your saying, “Come on guys. I’m not gonna do that. Just tell me what you think about which technique is better when the dependent variable is binary. Pick a winner.”

Given our experience, we can’t pick a winner. In fact, if pushed, we’re inclined to opt in favor of multiple regression for a couple of reasons. It not only seems to perform about as well as logistic regression, but more importantly (with the stats software we use) multiple regression is simply faster and easier to use than logistic regression. But we still use logistic regression for models with dependent variables. And we continue to compare its efficacy against multiple regression when we can. And we rarely see a meaningful difference between the results.

Why do we still use both modeling techniques? Because we think taking a hard and fast stance when you’re doing applied science is not a good idea. Too easy to end up with egg on your face. Our best advice is to use whichever method is most familiar and readily available to you.

As always, we welcome your comments and reactions. Maybe even more so with this one.

Stepwise, model-foolish?

My approach to building predictive models using multiple linear regression might seem plodding to some. I add predictor variables to the regression one by one, instead of using stepwise methods. Even though the number of predictor variables I use has greatly increased, and the time needed to build a model has lengthened, I am even less likely to use stepwise regression today than I was a few years ago.

Stepwise regression, available in most stats software, tosses all the predictor variables into the analysis at once and picks the best for you. It’s a semi-automated process that can work forwards or backwards, adding or deleting variables until it’s satisfied a statistical rule of thumb. The software should give you some control over the process, but mostly your computer is making all the big decisions.

I understand the allure. We’re all looking for ways to save time, and generally anything that automates a repetitive process is a good thing. Given a hundred variables to choose from, I wouldn’t be surprised if my software was able to get a better-fitting model than I could produce on my own.

But in this case, it’s not for me.

Building a decent model isn’t just about getting a good fit in terms of high R square. That statistic tells you how well the model fits the data that the model was built on — not data the model hasn’t yet seen, which is where the model does its work (or doesn’t). The true worth of the model is revealed only over time, but you’re more likely to succeed if you’ve applied your knowledge and judgement to variable selection. I tend to add variables one by one in order of their Pearson correlation with the target variable, but I am also aware of groups of variables that are highly correlated with each other and likely to cause issues. The process is not so repetitive that it can always be automated. Stepwise regression is more apt to select a lot of trivial variables with overlapping effects and ignore a significant predictor that I know will do the job better.

Or so I suspect. My avoidance of stepwise regression has always been due to a vague antipathy rather than anything based on sound technical concerns. This collection of thoughts I came across recently lent some justification of this undefined feeling: Problems with stepwise regression. Some of the authors’ concerns are indeed technical, but the ones that resonated the most for me boiled down to this: Automated variable selection divorces the modeler from the process so that he or she is less likely to learn things about their data. It’s just not as much fun when you’re not making the selections yourself, and you’re not getting a feel for the relationships in your data.

Stepwise regression may hold appeal for beginning modellers, especially those looking for push-button results. I can’t deny that software for predictive analysis is getting better and better at automating some of the most tedious aspects of model-building, particularly in preparing and cleaning the data. But for any modeller, especially one working with unfamiliar data, nothing beats adding and removing variables one at a time, by hand.

Does your astrological sign predict whether you’ll give?

Last weekend, with so many other pressing things I could have been doing, I got it in my head to analyze people’s astrological signs for potential association with propensity to give. I don’t know what came over me; perhaps it was the Supermoon. But when you’ve got a data set in front of you that contains giving history and good birth dates for nearly 85,000 alumni, why not?

Let me say first that I put no stock in astrology, but I know a few people who think being a Libra or a Gemini makes some sort of difference. I imagine there are many more who are into Chinese astrology, who think the same about being a Rat or a Monkey. And even I have to admit that an irrational aspect of me embraces my Taurus/Rooster nature.

If one’s sign implies anything about personality or fortune, I should think it would be reflected in one’s generosity. Ever in pursuit of the truth, I spent a rather tedious hour parsing 85,000 birth dates into the signs of the zodiac and the animal signs of Chinese astrology. As you will see, there are in fact some interesting patterns associated with birth date, on the surface at least.

Because human beings mate at any time of year, the alumni in the sample are roughly equally distributed among the 12 signs of the zodiac. There seem to be slightly more births in the warmer months than in the period of December to February: Cancer (June 21 to July 22) represents 8.9% of the sample while at the lower end, Capricorn (Dec 22 to Jan 20) represents 7.6% of the sample — a spread of less than two percentage points.

What we want to know is if any one sign is particularly likely to give to alma mater. I coded anyone who had any giving in their lifetime as ‘1’ and all never-donors as ‘0’. At the high end, Taurus natives have a donor rate of 30.7% and at the low end, Aries natives have a donor rate of 29.0%. All the other signs fall between those two rates, a range of a little more than one and a half percentage points.

That’s a very narrow range of variance. If I were seriously evaluating the variable ‘Astrological sign’ as a predictor, I would probably stop right there, seeing nothing exciting enough to make me continue.

But have a look at this bar chart. I’ve arranged the signs in their calendar order, which immediately suggests that there’s a pattern in the data: A peak at Taurus, gradually falling to Scorpio, peaking again at Sagittarius, then falling again until Taurus comes around once more.

The problem with the bar chart is that the differences in giving rates are exaggerated visually, because the range of variance is so limited. What appears to be a pattern may be nothing of the sort.

In fact, the next chart tells a conflicting tale. The Tauruses may have the highest participation rate, but among donors they and three other signs have the lowest median level of lifetime giving ($150), and Aries have the highest median ($172.50). The calendar-order effect we saw above has vanished.

These two charts fail to tell the same tale, which indicates to me that although we may observe some variance in giving between astrological signs, the variance might well be due to mere chance. Is there a way to demonstrate this statistically? I was discussing this recently with Peter Wylie, who helped me sort this out. Peter told me that the supposed pattern in the first chart reminded him of the opening of Malcolm Gladwell’s book, Outliers, in which the author examines why a hugely disproportionate number of professional hockey and soccer players are born in January, February and March. (I won’t go farther than that — read the book for that discussion.)

In the case of professional hockey players, birth date and a player’s development (and career progress) are definitely associated. It’s not due to a random effect. In the case of birth date and giving, however, there is room for doubt. Peter took me through the use of chi-square, a statistic I hadn’t encountered since high school. I’m not going into detail about chi-square — there is plenty out there online to read — but briefly, chi-square is used to determine if a distribution of observed frequencies of a value for a categorical or ordinal variable differs from the theoretical expected frequencies for that variable, and from there, if the discrepancy is statistically significant.

Figuring out the statistical significance part used to involve looking up the calculated value for chi-square in a table based on something called degrees of freedom, but nowadays your stats software will automatically provide you with a statistic telling you whether the result is significant or not: the p statistic, which will be familiar to you if you’ve used linear regression. The rule of thumb for significance is a p-value of 0.05 or less.

As it turns out, the observed differences in the frequency of donors for each astrological sign has a significance value of p = 0.3715. This is way above the 0.05 confidence level, and therefore we cannot rule out the possibility that these variations are due to mere chance. So astrology is a bust for fundraisers.

Now for something completely different. We haven’t looked at the Chinese animal signs yet. Here is a table showing a breakdown by Chinese astrological sign by the percentage of alumni with at least some giving, and median lifetime giving. The table is sorted by donor participation rate, lowest to highest.

Hmm, it would seem that being a Horse is associated with a higher level of generosity than the norm. And here’s the biggest surprise: A Chi-square test reveals the differences in donor frequencies between animals to be significant! (p-value < 0.0001).

What’s going on here? Shall we conclude that the Chinese astrologers have it all figured out?

Let’s go back to the data. First of all, how were alumni assigned an animal sign in the first place? You may be familiar with the paper placemats in Chinese restaurants that list birth years and their corresponding animal signs. Anyone born in the years 1900, 1912, 1924, 1936, 1948, 1960, 1972, 1984, 1996 or 2008 is a Rat. Anyone born in 1901, 1913, 1925, etc. etc. is an Ox, and so on, until all the years are accounted for. Because the alumni in each animal category are drawn from birth years with an equal span of years between them, we might assume that each sign has roughly the same average age. This is key, because if the signs differ on average age, then age might be an underlying cause of variations in giving.

My data set does not include anyone born before 1930, and goes up to 1993 (a single precocious alum who graduated at a very young age). Tigers, with the lowest participation rate, are drawn from the birth years 1938, 1950, 1962, 1974 and 1986. Horses, with the highest participation rate, are drawn from the birth years 1930, 1942, 1954, 1966 and 1978, plus only a handful of young alumni from 1990. For Tigers, 77% were born in 1974 or earlier, but for Horses, 99% of alumni were born in 1978 or earlier.

The bottom line is that the Horses in my data set are older than the Tigers, as a group. The Horses have a median age of 45, while the Tigers have a median age of 37. And we all know by now that older alumni are more likely to be donors.

Again, my conversation with Peter Wylie helped me figure this out statistically. The short answer is: After you’ve accounted for the age of alumni, variations in giving by animal sign are no longer significant.

(The longer answer is: If you perform a linear regression of Age on Lifetime Giving (log-transformed) and compute residuals, then run an Analysis of Variance (ANOVA) for the residuals and Animal Sign, the variance is NOT significant, p = 0.1118. The residuals can be thought of as Lifetime Giving with the explanatory effect of Age “washed out,” leaving only the unexplained variance. Animal Sign fails to account for any significant amount of the remaining variance in LT Giving, which is an indication that Animal Sign is just a proxy for Age.)

Does any of this matter? Mostly no. First of all, a little common sense can keep you out of trouble. Sure, some significant predictors will be non-intuitive, but it doesn’t hurt to be skeptical. Second, if you do happen to prepare some predictors based on astrological sign, their non-significance will be evident as soon as you add them to your regression analysis, particularly if you’ve already added Age or Class Year as a predictor in the case of the Chinese signs. Altogether, then, the risk that your models will be harmed by such meaningless variables is very low.

Correlation and you

If you read books and blogs on statistics, eventually your understanding of even the most basic concepts will start to smear. Things we think ought to be well-established by now are matters of controversy. Ranking high on the list of most slippery concepts is correlation. In today’s post, I’m going to make the concept very complicated, and then I’m going to dismiss the complexity and make it all simple again. I’m telling you this in advance so I don’t lose you partway through.

Correlation is the foundation of predictive modeling. The degree to which one variable x, changes its value in relation to a second variable y, either positively or negatively, is the very definition of x‘s usefulness in predicting the value of y. The tool I use to quantify the strength of that relationship is Pearson Product-Moment Correlation, or Pearson’s r, which some statistics texts simply call “correlation”. (It would help if you read the blog post on Pearson before reading this one.)

Before we go any farther, I need to explain why I want to quantify the strength of relationships. At the start of any modeling project, I have a hundred or so potential predictor variables in my data file. Some are going to be excellent predictors, most will be only so-so, and others will have little or no association with the outcome variable. I want to introduce variables into the regression analysis in an order that makes sense, so that the best predictors are added first. Due to the complexity of interactions among variables, there is no telling in advance what will actually happen as variables are added, so any list of variables ordered by strength of correlation is merely a rough guide.

What I DON’T use Pearson’s r for much is exploring variables. At the exploration stage, before modeling begins, I will look at a variable a number of ways in order to get a sense of how valuable the variable will be, and how I might want to transform it or re-express it to make it better. I will look at how the variable is distributed, and compare average and median giving between groups (for example, Home Phone Present, Y/N). These and other techniques, as described in Peter Wylie’s book “Data Mining for Fundraisers,” are simpler, more direct, and often more helpful than abstract measures of correlation.

It’s only after I’ve done the exploration work and tweaked the variables for maximum effect that I’m ready to rank them in order by their correlation values. So, with that out of the way, let’s look at Pearson’s r in more detail.

The textbooks make it abundantly clear: Pearson’s r quantifies the relationship between two continuous variables that are linearly related. Right away, we’ve got a problem: Very few of the variables I work with are continuous, and most of the relationships I see do not meet the definition of “linear”. Yet, I use Pearson’s r exclusively. Does this mean I’ve been misusing and abusing the method?

I don’t see it that way, but it is interesting to read how these things are discussed in the literature.

You can assess whether a relationship between two variables is roughly linear by looking at a scatterplot of the variables. Below is a scatterplot of ‘Giving’ (log-transformed) and ‘Age’, created in Data Desk. It’s a big, messy cloud of points (some 80,000 of them!). A lot of the relationship is hidden by overplotting (the overlapping of points) along the bottom line — that row of points at the zero giving mark represents non-donors, and there are many more of them near the young end of that line than there are near the older end. Still, at least you can see a vague linear relationship in the upward fanning of the data: As age increases, so does lifetime giving. A best-fit line through the data would slope upward from left to right, and therefore the Pearson correlation value is high.

We’ve got two continuous variables, and a linear relationship, so we get the Statistician’s Seal of Approval: It’s okay to use Pearson’s r to measure strength of correlation between these two variables.

Unfortunately, as I said, most of the variables we use in predictive modeling are not continuous, and they don’t look like much of anything in a scatterplot. Here’s a scatterplot of a Likert-scale survey response. The survey question asked alumni how likely they are to donate to alma mater, and the scale runs from 1 to 5, with 5 being “very likely.” This is not a continuous variable, because there are no possible intermediate values among the five levels. It’s ordinal. The plot with Lifetime Giving is difficult to interpret, but it sure isn’t linear:

Yes, the line of points for the highest response, 5, does extend higher than any other line in terms of lifetime giving. But due to overplotting, there is no way to tell how many nondonors lurk in the single dots that appear at the foot of every line and which indicate zero dollars given. This in no way resembles a cloud of points through which one can imagine a best-fit line being drawn. I can TELL you that a positive response to this question is strongly associated with high levels of lifetime giving, and it is, but you could be forgiven for remaining unconvinced by this “evidence”.

Even worse are the most common predictor variables: indicator variables, in binary form (0/1). For example, let’s say I express the condition of being ‘Married’ as a binary variable. A scatterplot of ‘Giving’ and ‘Married’ is even less useful than the one for the survey question:

Yuck! We’ve got 80,000 data points all jammed up in two solid lines at zero (not married) and 1 (married). We can’t tell from this plot, but it just so happens that the not-married line contains a lot of points sitting at lifetime giving of zero, far more than the married line. Being married is associated with giving, but who could tell from this? There’s no way this relationship is linear.

One of the tests for the appropriateness of using Pearson’s r is whether a scatterplot of the variable looks like a “straight enough” line. Another test is that both variables are quantitative and continuous — not categorical (or ordinal) and discrete. The fact that I can take a categorical variable (Married) and re-express it as a number (0/1) makes no difference. Turning it into a number makes it possible to calculate Pearson’s r, but that doesn’t make it okay to do so.

So the textbooks tell us. What else do they have to say? Read on.

There are methods other than Pearson’s r which we can use to measure the degree of association between two variables, which do not require the presence of a linear relationship. One of these is Spearman’s Rho, which is the correlation between the ranks of two variables. Rho replaces the data values themselves with their ranks within each variable — so the lowest value in each variable becomes ‘1’, the next lowest becomes ‘2’, and so on — and then calculates to what degree those values are related between the two variables, either negatively or positively.

Spearman’s Rho is sometimes called Spearman Rank Correlation, which muddies the waters a bit as it implies that it’s a measure of correlation. It is, but the correlation is between the ranks, not the data values themselves. The bottom line is that a statistician would tell us that Spearman is the appropriate calculation for putting a value to the strength of association between two variables that are not linearly related but which may show a consistently increasing or decreasing trend. Unlike Pearson’s r, it is a nonparametric method — it is free of any requirement that the distribution of the variables look a certain way.

Spearman’s Rho has some special properties which I won’t get into, but overall it looks a lot like Pearson’s r. It can take on values between -1 and 1 (a perfect negative relationship to a perfect positive relationship, both called monotone relationships); values near zero indicate the absence of a relationship — just like Pearson’s r. And your stats software makes it equally easy to compute.

So, great. We’ve got Spearman’s Rho, which we are told is just the thing for analyzing the Likert scale variable I showed you earlier. What about the indicator variable (for ‘Married’)? Well, no, we’re told: You can’t use Pearson or Spearman’s to calculate correlation for categorical variables. For that, you need the point-biserial correlation coefficient.

Huh?

That’s right, another measure of correlation. In fact, there are many types of measures out there. I’ve got a sheet in front of me right now that lists more than eight different measures, the choice of which depends on the combination of variables you’re analyzing (two continuous variables, one continuous and one ordinal variable, two binary variables, one binary and one ordinal, etc. etc.). And that list is not exhaustive.

Another wrinkle is that some texts don’t call these relationships “correlation” at all. By a strict definition, if a relationship between variables it isn’t appropriate for Pearson’s r, then it ain’t correlation. We are supposed to call it by the more vague term “association.”

Hey, I’m cool with that. I’ll call it whatever you want. But what are the practical implications of all this?

As near as I can tell, ZERO.

Uh-huh, I’ve just made you read more than a thousand words on how complicated correlation is. Now I’m going to dismiss all of that with an imperial wave of my hand. I need to you ignore everything I’ve just said, for two reasons which I will elaborate on in a moment:

1. As I stated at the outset, the only reason I calculate correlations is to explore which variables are most likely to figure prominently in a regression analysis. For a rough ranking of variables, Pearson’s r is a “good enough” tool.
2. The correlation r is the basis of linear regression, which is our end-goal. Not Spearman, not point-biserial, nor any other measure.

Regarding point number one: We are not concerned about the precise value of the calculated association, only the approximate ranking of our variables. All we want to know is: which variables are probably most valuable for our model and should be added to the regression first? As it turns out, rankings using other measures of correlation (sorry — association) hardly vary from a Pearson’s r ranking. It’s extra bother for nothing.

And to point number two: There’s a real disconnect between what the textbooks say about correlation analysis and what they say about regression. It seems to me that if Pearson’s r is inappropriate for all but continuous, linearly-related variables, then we would also be told that only continuous, linearly-related variables can be used in regression. That’s not the case: Social-science researchers and modelers toss ordinal and binary variables into regressions with wild abandon. If we didn’t we’d have almost nothing left to work with.

The disconnect is bridged with an explanation I found in one university stats textbook. It’s touched on only briefly, and towards the end of the book. The gist is this: For 0/1 indicator variables added to a linear regression, the coefficient of correlation is not the slope of a line, as we are always told to understand it. The indicator acts to vertically shift the line, so that instead of one regression slope, we have two: The unshifted line if the indicator variable is equal to zero, and another line shifted vertically up or down (depending on the sign of the coefficient), if the value is 1. This seems like essential information, but hardly rates any discussion at all. That’s stats for you.

To summarize:

1. Don’t be sidetracked by warnings regarding measures of association/correlation that have specific uses but do not relate to your end goal: Building a regression model for the pragmatic purpose of making predictions.
2. Most of the time, scatterplots can’t tell you what you need to know, because most of our data is categorical.
3. Indicator variables (and ordinal variables) are materially different from the pretty, linearly-related variables, and they are absolutely OK for use in regression.
4. SAY “association”, but DO “correlation”.
5. Don’t feel bad if you’re having difficulty learning predictive modeling from a stats textbook. I can’t see how anyone could.

What the heck IS “Y”, anyway?

At the APRA Conference in Anaheim a month ago, a session attendee was troubled by something he saw during a presentation given by David Robertson of Syracuse. The attendee was focused on the “constant” value in David’s example of a multiple linear regression model for propensity to give. This constant, which I will talk about shortly, was some significant figure, say “50”. Because the Y value (i.e., the dependent variable, or outcome variable, or predicted value) was expressed in dollars (of giving), then this seemed to indicate that the “floor” for giving, the minimum value someone could be predicted to give, was $50.

How do you figure that?, this attendee wanted to know. It’s a reasonable question, for which I will try to provide my own answer. (More knowledgeable stats people may wish to weigh in; it would be appreciated.) There are implications here for how we interpret the predicted value of “Y”.

When you do a regression analysis, your software will automatically calculate this “constant,” which is simply the first term (“a”) in the regression equation:

In other words, if all your predictor variables (X’s) calculate out to zero, then Y will equal ‘a’. The part of this that the attendee found hard to swallow was that the minimum possible amount an alum could donate, as predicted by the model, was something greater than zero dollars. It seemed nonsensical.

Well, yes and no. First of all, the constant is no such thing. If you were to plot a regression line, that straight line has to cross the Y axis somewhere. The value of Y when the sum of X’s is zero is that crossing point (a.k.a. the Y-intercept). But that doesn’t mean it’s the minimum. Y does equal zero at a point: When the sum of predictors is negative — that is, when the regression line passes to the left of the Y axis and down, and crosses the X axis (the X-intercept).

So, really, you’re not learning much by looking at the constant. It’s a mathematical necessity — it describes an important aspect of what any line looks like when plotted — but that’s all. While the constant is always present in our regression analysis for predictive modeling, we tend to ignore it.

But all this is leading me to an even more fundamental question, the one posed in the headline: What is Y?

In David’s example, the one that so perplexed my fellow conference attendee, Y was expressed in real dollars. This is valid modeling practice. However, I have never looked at Y in real units (i.e., dollars), due to difficulty in interpreting the result. For example, the output of multiple linear regression can be negative: Does that mean the prospect is going to take money from us? As well, when we work with a transformed version of the DV (such as the natural log, which is very common), the output will need to be transformed back in order to make sense.

I sidestep issues of interpretation by simply assuming that the predicted value is meaningless in itself. What I am primarily interested in is relative probability, and where a value ranks in comparison with the values predicted for other individuals in the sample. In other words, is a prospect in the top 10% of alumni? Or the top 0.5%? Or the bottom 20%? The closer an individual is to the top of the heap, the more likely he or she is to give, and at higher levels.

I rank everyone in the sample by their predicted values, and then chop the sample up into deciles and percentiles. Percentiles, I am careful to explain, are not the same thing as probabilities: Someone in the 99th percentile is not 99% likely to make a gift. They might be 60% likely — it depends. The important thing is that someone in the 98th percentile will be slightly less likely to give, and someone in the 50th percentile will be MUCH less likely to give.

This highlights an important difference between multiple linear regression, which I’m talking about here, and binary logistic regression. The output of the latter form of regression is “probability”; very useful, and not so difficult to interpret. Not so with multiple linear regression — the output in this case is something different, which we may interpret in various ways but which will not directly give us values for probability.

Fortunately, fundraisers are already very familiar with the idea of ranking prospects in descending order by likelihood (or capacity, or inclination, or preferably some intelligent combination of these). Most people can readily understand what a percentile score means. For us data modelers, though, getting from “raw Y” to a neat score takes a little extra work.