Correlation and regression are powerful tools for describing the relationship between two variables. When you use these tools, you must be aware of their limitations, beginning with the fact that correlation and regression describe only linear relationships. Also remember that the correlation r and the least-squares regression line are not resistant. One influential observation or incorrectly entered data point can greatly change these measures. Always plot your data before interpreting regression or correlation. Here are some other cautions to keep in mind when you apply correlation and regression or read accounts of their use.

Associations for variables can be trusted only for the range of values for which data have been collected. Even a very strong relationship may not hold outside the data’s range.

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Predictions made far beyond the range for which data have been collected can’t be trusted. Few relationships are linear for all values of x. It is risky to stray far from the range of x-values that actually appear in your data.

In general, extrapolation involves using a mathematical relationship beyond the range of the data that were used to estimate the relationship. The scenario described in the previous example is typical: we try to use a least-squares relationship to make predictions for values of the explanatory variable that are much larger than the values in the data that we have. We can encounter the same difficulty when we attempt predictions for values of the explanatory variable that are much smaller than the values in the data that we have.

Careful judgment is needed when making predictions. If the prediction is for values that are within the range of the data that you have, or are not too far above or below, then your prediction can be reasonably accurate. Beyond that, you are in danger of making an inaccurate prediction.

Many regression and correlation studies work with averages or other measures that combine information from many cases. You should note this carefully and resist the temptation to apply the results of such studies to individual cases. Correlations based on averages are usually higher than correlations based on individual cases. This is another reminder that it is important to note exactly what variables are measured in a statistical study.

Correlation and regression describe the relationship between two variables. Often, the relationship between two variables is strongly influenced by other variables. We try to measure potentially influential variables. We can then use more advanced statistical methods to examine all the relationships revealed by our data. Sometimes, however, the relationship between two variables is influenced by other variables that we did not measure or even think about. Variables lurking in the background—measured or not—often help explain statistical associations.

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A lurking variable can falsely suggest a strong relationship between x and y, or it can hide a relationship that is really there. Here is an example of a negative correlation that is due to a lurking variable.

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When we study the relationship between two variables, we often hope to show that changes in the explanatory variable cause changes in the response variable. But a strong association between two variables is not enough to draw conclusions about cause and effect. Sometimes, an observed association really does reflect cause and effect. Natural gas consumption in a household that uses natural gas for heating will be higher in colder months because cold weather requires burning more gas to stay warm. In other cases, an association is explained by lurking variables, and the conclusion that x causes y is either wrong or not proved. Here is an example.

Correlations such as that in Example 2.24 are sometimes called “nonsense correlations.” The correlation is real. What is nonsense is the conclusion that changing one of the variables causes changes in the other. A lurking variable—such as national wealth in Example 2.24—that influences both x and y can create a high correlation, even though there is no direct connection between x and y.

These and other examples lead us to the most important caution about correlation, regression, and statistical association between variables in general.

Chapters 1 and 2 of this book are devoted to the important aspect of statistics called exploratory data analysis (EDA). We use graphs and numerical summaries to examine data, searching for patterns and paying attention to striking deviations from the patterns we find. In discussing regression, we advanced to using the pattern we find (in this case, a linear pattern) for prediction.

Suppose now that we have a truly enormous database, such as all purchases recorded by the cash register scanners of our retail chain during the past week. Surely this mass of data contains patterns that might guide business decisions. If we could clearly see the types of activewear preferred in large California cities and compare the preferences of small Midwest cities—right now, not at the end of the season—we might improve profits in both parts of the country by matching stock with demand. This sounds much like EDA, and indeed it is. Exploring very large databases in the hope of finding useful patterns is called data mining. Here are some distinctive features of data mining:

All of these features point to the need for sophisticated computer science as a basis for data mining. Indeed, data mining is often thought of as a part of computer science. Yet many statistical ideas and tools—mostly tools for dealing with multidimensional data, not the sort of thing that appears in a first statistics course—are very helpful. Like many modern developments, data mining crosses the boundaries of traditional fields of study.

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Do remember that the perils we encounter with blind use of correlation and regression are yet more perilous in data mining, where the fog of an immense database prevents clear vision. Extrapolation, ignoring lurking variables, and confusing association with causation are traps for the unwary data miner.