14.2: Significant Correlations

Last updated
Save as PDF

Page ID: 22042

Bradley H. Dowden
California State University Sacramento

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Given an observed correlation, how can you tell whether it is significant rather than accidental? Well, the problem of telling when an association is significant is akin to the problem of telling when any statistic is significant. The point is that the correlation is significant if you can trust it in making your future predictions. Conversely, an observed correlation is not significant if there is a good chance that it is appearing by accident and thus wouldn't be a reliable sign of future events. However, you usually aren't in a position to perform this kind of calculation of the significance of the correlation. If you are faced with a set of data that show a correlation, you might never be able to figure out whether the correlation is significant unless you collect more data. If the correlation is just an accident, it will disappear in the bigger pool of data when more data are collected.

If you cannot collect more data, and cannot ask a statistician to calculate the correlation coefficient for you, then the only other way to tell whether the correlation is significant is to see whether accepted scientific theories about how the world works enable you to predict that the correlation will hold. For example, suppose you have all the data available about Napoleon Bonaparte's childhood and you discover a positive correlation between his height as a child and his age as a child. Is the correlation significant? You cannot collect any more data about Napoleon's childhood; you have all there is to have. You cannot start his life over again and collect data about his childhood during his second lifetime. Nevertheless, you know that the correlation between his height and age is no accident, because you hold a well-justified theory about human biology implying that height in childhood increases with age. If this connection holds for everybody, then it holds for French emperors, too. So the correlation in the Napoleon data is significant. In summary, to decide whether a correlation is accidental, you might try to collect more data, or you might look to our theoretical knowledge for an indication of whether there ought to be a correlation.

Exercise \(\PageIndex{1}\)

What is the story behind the correlation in the passage below?

Amid Wall Street's impenetrable jargon and stone-faced forecasts for the stock market in 1984 comes this nugget of foolishness: If the Washington Redskins win the Super Bowl this Sunday, the market is going up. Or, conversely, if you think the market is going up this year, you ought to put your money on the Redskins.

If this method of market forecasting sounds like nonsense, you're in good company. But consider this: The results of the Super Bowl have become a... signal of future market activity.

In the 17 Super Bowl games that have been played since 1967, every year [in which] the National Football Conference team won, the New York Stock Exchange composite index ended the year with a gain. And in every year in which the American Football Conference team won, the market sank.¹

This correlation is caused by heavy betting on the Super Bowl results.
There is a correlation because game results caused stock market investments by football players.
The correlation is a coincidence (that is, not a significant association).
The correlation can be used as a reliable indicator that if the NFC won the Super Bowl last time, then the AFC will win next time.

Answer: Answer (c). The pattern will not continue in the future. It is an accident. Answer (a) is unlikely to be correct because bets are unlikely to affect the Stock Exchange so radically

The stronger a correlation, the more likely that some causal connection or some causal story exists behind it. Investigators are usually interested in learning the details of this causal story; that is the central topic of the next section, but here’s a word of caution: The rooster crows, then the sun comes up. He didn’t cause it, did he?

¹ R. Foster Winans, "Wall Street Ponders If Bull Market in '84 Hinges on Redskins," The Wall Street Journal, January 18, 1984, p. 49.