Fig. 11.1. Different kinds of correlations. The horizontal axis represents one variable (X) and the vertical axis represents a different variable (Y), with values of X and Y increasing according to the distance from the origin. Models A, B & C show correlations for continuous variables which can take on a range of values (e.g., height, weight), whereas Model D reveals a correlation for discrete variables (variable X might be gender, variable Y presence or absence of the Y chromosome). Model A reveals a strong positive correlation, Model B a strong negative correlation, and Model C a weak positive correlation. The correlation in Model D would be regarded as positive if values could be assigned to X and Y, but if values cannot be assigned (e.g., gender and presence of Y chromosome), we would not refer to the correlation as being positive or negative.

Correlations are common in Business . Businesses often obtain large quantities of correlational data as they go about their activities (Table 11.1). An insurance company in the course of doing business obtains data about which types of customers are more often involved in accidents. These data are purely observational - the company can't force a 68 year old grandmother to drive a pickup if she doesn't want to. The data consist of driver age, sex, make and model of car, zip code, street address and so forth. In addition, the company knows how many and the type of accidents for each customer. These correlations are clearly quite useful in predicting what customers will have more accidents.

Correlated variables having substantial impact on profits and losses or on the efficiency of government operations

Correlations are used to manipulate us. Most advertisements, sales pitches, and political speeches invoke correlations to influence our behavior. A company tends to display its product in favorable settings to build an imaginary correlation between its product and the desirable surroundings (e.g., beer commercials using attractive members of the opposite sex, 4WD autos being pictured with a backdrop of remote, montane scenery). Negative campaigning usually involves describing some unfavorable outcome that occurred during an opponent's tenure in office to develop a correlation in the viewer's mind between the candidate and bad consequences of their election to office.

The reason that correlations are used so often in commercials is that they work-- people make the causal extrapolation from correlations. We tend to blame our current president for many social problems, even though the president has little control over many of them. In a well-known but unfortunate psychological experiment of some decades ago, a child was encouraged to develop a close attachment to a white rate, whereupon the experimenters intentionally frightened the child with the rat. Thereafter, the child avoided white objects -- a rather surprising correlate of the rat. Other studies have shown that people respond differently to an item of clothing according to what they are told about an imaginary person who wore it: the response is more favorable if the supposed previous wearer is famous than if the person is infamous. The information thus established a correlation between the clothing and a desirable or undesirable person, and the subjects mentally extrapolated that correlation to some kind of causation of good or bad from wearing the object. And some of our responses to correlations are very powerful. The experience of getting overly drunk on one kind of alcoholic beverage is often enough to cause a person to avoid that beverage years into the future but not to avoid other kinds of alcoholic beverages.

Negative uses. A more negative context for the application of correlation to influence behavior is the practice known as character assassination. A person can be denigrated in one aspect of their life by identifying an unfavorable characteristic in some other (and perhaps trivial) aspect of their life We automatically extrapolate the negative correlation to them as a whole.

The problem with correlations: Hidden variables

The problem that underlies evaluation of correlations is extremely common in science. We observe an association, or correlation, between two or more variables. In the nuclear power plant example, there is a correlation between residential proximity to a nuclear plant and cancer, because people near power plants are more likely to get cancer than those who live away from power plants. And we try to infer the causation from that correlation (does the plant actually cause cancer?). Time and again, science has learned the hard way that we cannot infer causation from correlation: correlation does not imply causation.

What does this mean? Say that you observe a correlation between smoking and lung cancer. To infer that smoking CAUSES lung cancer, you would argue that people should stop smoking to lower their lung cancer rates. If smoking does not cause lung cancer, however, then stopping smoking would actually have no effect on lung cancer rates (we are very confident, however, that smoking causes lung cancer). How can a correlation not reflect causation? Consider a plot of the number of churches in a town (city) and the number of bars in a town:

 

This is drawn so that there tend to be more churches than bars in a town, but as the number of churches increases, so does the number of bars on average. Although these data were made up for illustration, the correlation is almost certainly true. To argue causation from these data, we would either have to say that churches cause people to drink more (whether intentionally or unintentionally), or argue that lots of drinkers in a town causes more churches to be built (e.g., churches move in where there are sinners). Furthermore, causation would suggest either that banning bars would reduce the number of churches in the town, or that the way to cut down on the number of bars was to close down churches (depending on which way the causation went). In reality, the correlation is due to a hidden variable -- population size. That is, larger towns have more demand for churches and for bars, as well as other social institutions.

To reiterate the theme of this chapter, the major difficulty with all correlations is that there are many models consistent with any correlation: the correlation between two variables may be caused by a third, fourth, or dozens of variables other than the two being compared. Thus we are left with countless alternative models in addition to the obvious ones. For example, we initially think that the correlation between cancer and residence near a power plant shows that nuclear power plants cause cancer. Then we learn that another factor, site of the power plant, may be important. It appears that the important factor is not the power plant itself, but rather some characteristic of sites chosen for power plants (one obvious possibility is that nuclear power plants are situated in low income areas that have higher cancer rates than suffered by the general population). That is, there are correlations between all sorts of other variables besides just residence and cancer.

There are many issues in society that hinge on correlations (see table). In some cases, a correlation may identify a causal relationship, such as health defects being caused by environmental toxins. Yet because the correlational data don't reject countless alternatives models, no action is taken to correct the problem. In other cases, a correlation may be assumed to reflect the cause when it does not.

 

Public policy issues that involve understanding the cause of a correlation

Issue	Possible causation
High cancer incidence near industrial sites, toxic waste dumps, nuclear power plants.	If the increased cancer rate is actually caused by the hazard, there would be compelling motivation for taking action. But it is often difficult to rule out the alternative explanation that those living near the hazard have different diets or for other reasons are more susceptible to cancer than the general population.
Racial differences in standardized test scores.	There are two opposing positions in this acrimonious debate: i) a person's race, per se, causes them to have low test scores, or ii) minorities often have low incomes, and it is income rather than race that determines test score. The first explanation states that a person is born with a certain intellectual ability, the second states that they acquire it.

 

Correlations complicate studying diet and heart disease

The medical news over the last decade or so has been obsessed with the relationship between diet and heart disease. (Heart disease is chiefly the build-up of deposits inside blood vessels, hardening the arteries and enabling the vessels to rupture and clog.) A report that dietary fiber lowered heart attack risk led to an avalanche of pills and breakfast cereals high in fiber. More recently, a trendy topic has been iron levels in the blood. It is not clear what to make of these reports, but we can be confident that associations between diet and heart disease will continue to be the subject of studies for decades to come. However, let's consider the problems such studies pose.

Your diet consists of literally hundreds of correlated components. For example, people who eat a lot of meat also tend to eat a lot of fat, and people that eat lots of vitamin C tend to also eat much fiber. These, and numerous similar correlations, create huge problems in determining what diet you should eat to avoid heart disease. A study that found an correlation between heart disease and fat, for example, would be hard to interpret because we would not know if it was the fat, per se, or the meat that was the problem. The problem in this example is not as great as it is in other cases, because we can actually conduct experiments with human diets to explore causal relationships. But even in these experiments, it is difficult to control and randomize all relevant factors.

Why do we bother with correlations at all?

Given the problems with interpreting correlational data, one might reasonably ask: why do we bother with them at all if it is a causal relationship that we seek? Why not just gather data that could provide a more definite answer, or otherwise just ignore correlations? The reason is pragmatism. Correlational data are usually relatively easy and inexpensive to obtain, at least in comparison to experimental data. Also, many cause-effect relationships are so subtle that we often first learn of them through correlations detected in observational data. That is, they are often useful.