For everyone who slept through Stats 101, Charles Wheelan’s Naked Statistics is a lifesaver. From batting averages and political polls to Schlitz ads and medical research, Wheelan “illustrates exactly why even the most reluctant mathophobe is well advised to achieve a personal understanding of the statistical underpinnings of life” (New York Times). What follows is adapted from the book, out now in paperback.
Behind every important study there are good data that made the analysis possible. And behind every bad study . . . well, read on. People often speak about “lying with statistics.” I would argue that some of the most egregious statistical mistakes involve lying with data; the statistical analysis is fine, but the data on which the calculations are performed are bogus or inappropriate. Here are some common examples of “garbage in, garbage out.”
Pauline Kael, the longtime film critic for The New Yorker, is alleged to have said after Richard Nixon’s election as president, “Nixon couldn’t have won. I don’t know anyone who voted for him.” The quotation is most likely apocryphal, but it’s a lovely example of how a lousy sample (one’s group of liberal friends) can offer a misleading snapshot of a larger population (voters from across America). And it introduces the question one should always ask: How have we chosen the sample or samples that we are evaluating? If each member of the relevant population does not have an equal chance of ending up in the sample, we are going to have a problem with whatever results emerge from that sample. One ritual of presidential politics is the Iowa straw poll, in which Republican candidates descend on Ames, Iowa, in August of the year before a presidential election to woo participants, each of whom pays $30 to cast a vote in the poll. The Iowa straw poll does not tell us that much about the future of Republican candidates. (The poll has predicted only three of the last five Republican nominees.) Why? Because Iowans who pay $30 to vote in the straw poll are different from other Iowa Republicans; and Iowa Republicans are different from Republican voters in the rest of the country.
“Nixon couldn’t have won. I don’t know anyone who voted for him.”
Selection bias can be introduced in many other ways. A survey of consumers in an airport is going to be biased by the fact that people who fly are likely to be wealthier than the general public; a survey at a rest stop on Interstate 90 may have the opposite problem. Both surveys are likely to be biased by the fact that people who are willing to answer a survey in a public place are different from people who would prefer not to be bothered. If you ask 100 people in a public place to complete a short survey, and 60 are willing to answer your questions, those 60 are likely to be different in significant ways from the 40 who walked by without making eye contact.
Positive findings are more likely to be published than negative findings, which can skew the results that we see. Suppose you have just conducted a rigorous, longitudinal study in which you find conclusively that playing video games does not prevent colon cancer. You’ve followed a representative sample of 100,000 Americans for twenty years; those participants who spend hours playing video games have roughly the same incidence of colon cancer as the participants who do not play video games at all. We’ll assume your methodology is impeccable. Which prestigious medical journal is going to publish your results?
Most things don’t prevent cancer.
None, for two reasons. First, there is no strong scientific reason to believe that playing video games has any impact on colon cancer, so it is not obvious why you were doing this study. Second, and more relevant here, the fact that something does not prevent cancer is not a particularly interesting finding. After all, most things don’t prevent cancer. Negative findings are not especially sexy, in medicine or elsewhere.
The net effect is to distort the research that we see, or do not see. Suppose that one of your graduate school classmates has conducted a different longitudinal study. She finds that people who spend a lot of time playing video games do have a lower incidence of colon cancer. Now that is interesting! That is exactly the kind of finding that would catch the attention of a medical journal, the popular press, bloggers, and video game makers (who would slap labels on their products extolling the health benefits of their products). It wouldn’t be long before Tiger Moms all over the country were “protecting” their children from cancer by snatching books out of their hands and forcing them to play video games instead.
Of course, one important recurring idea in statistics is that unusual things happen every once in a while, just as a matter of chance. If you conduct 100 studies, one of them is likely to turn up results that are pure nonsense—like a statistical association between playing video games and a lower incidence of colon cancer. Here is the problem: The 99 studies that find no link between video games and colon cancer will not get published, because they are not very interesting. The one study that does find a statistical link will make it into print and get loads of follow-on attention. The source of the bias stems not from the studies themselves but from the skewed information that actually reaches the public. Someone reading the scientific literature on video games and cancer would find only a single study, and that single study will suggest that playing video games can prevent cancer. In fact, 99 studies out of 100 would have found no such link.
Memory is a fascinating thing—though not always a great source of good data. We have a natural human impulse to understand the present as a logical consequence of things that happened in the past—cause and effect. The problem is that our memories turn out to be “systematically fragile” when we are trying to explain some particularly good or bad outcome in the present. Consider a study looking at the relationship between diet and cancer. In 1993, a Harvard researcher compiled a data set comprising a group of women with breast cancer and an age-matched group of women who had not been diagnosed with cancer. Women in both groups were asked about their dietary habits earlier in life. The study produced clear results: The women with breast cancer were significantly more likely to have had diets that were high in fat when they were younger.
Ah, but this wasn’t actually a study of how diet affects the likelihood of getting cancer. This was a study of how getting cancer affects a woman’s memory of her diet earlier in life. All of the women in the study had completed a dietary survey years earlier, before any of them had been diagnosed with cancer. The striking finding was that women with breast cancer recalled a diet that was much higher in fat than what they actually consumed; the women with no cancer did not.
Women with breast cancer recalled a diet that was much higher in fat than what they actually consumed; the women with no cancer did not.
The New York Times Magazine described the insidious nature of this recall bias:
The diagnosis of breast cancer had not just changed a woman’s present and the future; it had altered her past. Women with breast cancer had (unconsciously) decided that a higher-fat diet was a likely predisposition for their disease and (unconsciously) recalled a high-fat diet. It was a pattern poignantly familiar to anyone who knows the history of this stigmatized illness: these women, like thousands of women before them, had searched their own memories for a cause and then summoned that cause into memory.
Recall bias is one reason that longitudinal studies are often preferred to cross-sectional studies. In a longitudinal study the data are collected contemporaneously. At age five, a participant can be asked about his attitudes toward school. Then, thirteen years later, we can revisit that same participant and determine whether he has dropped out of high school. In a cross-sectional study, in which all the data are collected at one point in time, we must ask an eighteen-year-old high school dropout how he or she felt about school at age five, which is inherently less reliable.
Suppose a high school principal reports that test scores for a particular cohort of students has risen steadily for four years. The sophomore scores for this class were better than their freshman scores. The scores from junior year were better still, and the senior year scores were best of all. We’ll stipulate that there is no cheating going on, and not even any creative use of descriptive statistics. Every year this cohort of students has done better than it did the preceding year, by every possible measure: mean, median, percentage of students at grade level, and so on. Would you (a) nominate this school leader for “principal of the year” or (b) demand more data?
If you have a room of people with varying heights, forcing the short people to leave will raise the average height in the room, but it doesn’t make anyone taller.
I say “b.” I smell survivorship bias, which occurs when some or many of the observations are falling out of the sample, changing the composition of the observations that are left and therefore affecting the results of any analysis. Let’s suppose that our principal is truly awful. The students in his school are learning nothing; each year half of them drop out. Well, that could do very nice things for the school’s test scores—without any individual student testing better. If we make the reasonable assumption that the worst students (with the lowest test scores) are the most likely to drop out, then the average test scores of those students left behind will go up steadily as more and more students drop out. (If you have a room of people with varying heights, forcing the short people to leave will raise the average height in the room, but it doesn’t make anyone taller.)
Healthy User Bias
People who take vitamins regularly are likely to be healthy—because they are the kind of people who take vitamins regularly! Whether the vitamins have any impact is a separate issue. Consider the following thought experiment. Suppose public health officials promulgate a theory that all new parents should put their children to bed only in purple pajamas, because that helps stimulate brain development. Twenty years later, longitudinal research confirms that having worn purple pajamas as a child does have an overwhelmingly large positive association with success in life. We find, for example, that 98 percent of entering Harvard freshmen wore purple pajamas as children (and many still do) compared with only 3 percent of inmates in the Massachusetts state prison system.
The purple pajamas do not matter.
Of course, the purple pajamas do not matter; but having the kind of parents who put their children in purple pajamas does matter. Even when we try to control for factors like parental education, we are still going to be left with unobservable differences between those parents who obsess about putting their children in purple pajamas and those who don’t. As New York Times health writer Gary Taubes explains, “At its simplest, the problem is that people who faithfully engage in activities that are good for them—taking a drug as prescribed, for instance, or eating what they believe is a healthy diet—are fundamentally different from those who don’t.” This effect can potentially confound any study trying to evaluate the real effect of activities perceived to be healthful, such as exercising regularly or eating kale. We think we are comparing the health effects of two diets: kale versus no kale. In fact, if the treatment and control groups are not randomly assigned, we are comparing two diets that are being eaten by two different kinds of people. We have a treatment group that is different from the control group in two respects, rather than just one.
If statistics is detective work, then the data are the clues. My wife spent a year teaching high school students in rural New Hampshire. One of her students was arrested for breaking into a hardware store and stealing some tools. The police were able to crack the case because (1) it had just snowed and there were tracks in the snow leading from the hardware store to the student’s home; and (2) the stolen tools were found inside. Good clues help.
Like good data. But first you have to get good data, and that is a lot harder than it seems.
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Charles Wheelan is the author of the internationally best-selling Naked Economics and Naked Statistics and a former correspondent for The Economist, and founder of The Centrist Party. He teaches public policy and economics at Dartmouth College and lives in Hanover, New Hampshire, with his family. For more, visit nakedeconomics.com.