Example of how easy it is to mislead yourself: stepwise regression

I am, of course, a statistician. So perhaps it will seem unusual to you when I say I wish there were fewer statistics done. And by that I mean that I’d like to see less statistical modeling done. I am happy to have more data collected, but am far less sanguine about the proliferation of studies based on statistical methods.

There are lots of reasons for this, which I will detail from time to time, but one of the main ones is how easy it is to mislead yourself, particularly if you use statistical procedures in a cookbook fashion. It takes more than a recipe to make an eatable cake.

Among the worst offenders are methods like data mining, sometimes called knowledge discovery, neural networks, and other methods that “automatically” find “significant” relationships between sets of data. In theory, there is nothing wrong with any of these methods. They are not, by themselves, evil. But they become pernicious when used without a true understanding of the data and the possible causal relationships that exist.

However, these methods are in continuous use and are highly touted. An oft-quoted success of data mining was the time a grocery store noticed that unaccompanied men who bought diapers also bought beer. A relationship between data which, we are told, would have gone unnoticed were it not for “powerful computer models.”

I don’t want to appear too negative: these methods can work and they are often used wisely. They can uncover previously unsuspected relationships that can be confirmed or disconfirmed upon collecting new data. Things only go sour when this second step, verifying the relationships with independent data, is ignored. Unfortunately, the temptation to forgo the all-important second step is usually overwhelming. Pressures such as cost of collecting new data, the desire to publish quickly, an inflated sense of certainty, and so on, all contribute to this prematurity.

Stepwise

Stepwise regression is a procedure to find the “best” model to predict y given a set of x’s. The y might be the item most likely bought (like beer) given a set of possible explanatory variables x, like x₁ sex, x₂ total amount spent, x₃ diapers purchased or not, and on and on. The y might instead be total amount spent at a mall, or the probability of defaulting on a loan, or any other response you want to predict. The possibilities for the explanatory variables, the x’s, are limited only to your imagination and ability to collect data.

A regression takes the y and tried to find a multi-dimensional straight line fit between itself and the x’s (e.g., a two-dimensional straight line is a plane). Not all of the x’s will be “statistically significant¹“; those that are not are eliminated from the final equation. We only want to keep those x’s that are helpful in explaining y. In order to do that, we need to have some measure of model “goodness”. The best measure of model goodness is one which measures how well that model does predicting independent data, which is data that in no way was used to fit the model. But obviously, we do not always have such data at hand, so we need another measure. One that is often picked is the Akaike Information Criterion (AIC), which measures how well the model fits the data that was used to fit the model.

Confusing? You don’t actually need to know anything about the AIC other than that lower numbers are better. Besides, the computer does the work for you, so you never have to actually learn about the AIC. What happens is that many combinations of x’s are tried, one by one, an AIC is computed for that combination, and the combination that has the lowest AIC becomes the “best” model. For example, combination 1 might contain (x₂, x₁₇, x₂₂), while combination 2 might contain (x₁, x₃). When the number of x’s is large, the number of possible combinations is huge, so some sort of automatic process is needed to find the best model.

A summary: all your data is fed into a computer, and you want to model a response based on a large number of possible explanatory variables. The computer sorts through all the possible combinations of these explanatory variables, rates them by a model goodness criterion, and picks the one that is best. What could go wrong?

To show you how easy it is to mislead yourself with stepwise procedures, I did the following simulation. I generated 100 observations for y’s and 50 x’s (each of 100 observations of course). All of the observations were just made up numbers, each giving no information about the other. There are no relationships between the x’s and the y². The computer, then, should tell me that the best model is no model at all.

But here is what it found: the stepwise procedure gave me a best combination model with 7 out of the original 50 x’s. But only 4 of those x’s met the usually criterion for being kept in a model (explained below), so my final model is this one:

In classical statistics, an explanatory variable is kept in the model if it has a p-value< 0.05. In Bayesian statistics, an explanatory variable is kept in the model when the probability of that variable (well, of its coefficient being non-zero) is larger than, say, 0.90. Don't worry if you don't understand what any of that means---just know this: this model would pass any test, classical or modern, as being good. The model even had an adjusted R² of 0.26, which is considered excellent in many fields (like marketing or sociology; R² is a number between 0 and 1, higher numbers are better).

Nobody, or very very few, would notice that this model is completely made up. The reason is that, in real life, each of these x’s would have a name attached to it. If, for example, y was the amount spent on travel in a year, then some x’s might be x₇=”married or not”, x₂₁=”number of kids”, and so on. It is just too easy to concoct a reasonable story after the fact to say, “Of course, x₇ should be in the model: after all, married people take vacations differently than do single people.” You might even then go on to publish a paper in the Journal of Hospitality Trends showing “statistically significant” relationships between being married and travel model spent.

And you would be believed.

I wouldn’t believe you, however, until you showed me how your model performed on a set of new data, say from next year’s travel figures. But this is so rarely done that I have yet to run across an example of it. When was the last time anybody read an article in a sociological, psychological, etc., journal in which truly independent data is used to show how a previously built model performed well or failed? If any of my readers have seen this, please drop me a note: you will have made the equivalent of a cryptozoological find.

Incidentally, generating these spurious models is effortless. I didn’t go through 100s of simulations to find one that looked especially misleading. I did just one simulation. Using this stepwise procedure practically guarantees that you will find a “statistically significant” yet spurious model.

¹I will explain this unfortunate term later.

²I first did a “univariate analysis” and only fed into the stepwise routine those x’s which singly had p-values < 0.1. This is done to ease the computational burden of checking all models by first eliminating those x's which are unlikely to be "important." This is also a distressingly common procedure.

Here is the simulation code, to be run in the free and open source R statistical software:

library(MASS); # need be run only once per session
n<-100;
y<-rnorm(n) # "random" response
X<-matrix(rnorm(n*n/2),n,n/2) # "random" x's
f<-0
for (i in 1:(n/2)){
    # univariate analysis; f stores p-values
    f[i]<-anova(lm(y~X[,i]))$Pr[1]
}
i<-(f<.1) # only keep x's with p-values < 0.1
w<-data.frame(y,X[,i]) # w is just those x's with p<0.1 and y
fit<-lm(y~.,data=w) # model object to feed into stepwise
fit.aic <- stepAIC(fit)  # stepwise
summary(fit.aic) # final model summary