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.

Advertisement