4.7: Correlations

Up to this point we have focused entirely on how to construct descriptive statistics for a single variable. What we haven’t done is talked about how to describe the relationships between variables in the data. To do that, we want to talk mostly about the correlation between variables. But first, we need some data.

The data

After spending so much time looking at baseball data, you may be getting bored with sports examples. Instead, let’s turn to a topic close to everyone's heart: sleep. The following data set is fictitious, but based on real events. Suppose you are curious to find out how much your infant son’s sleeping habits affect your mood. Let’s say that you can rate your grumpiness very precisely, on a scale from 0 (not at all grumpy) to 100 (grumpy as a very, very grumpy old man). And, let's also assume that you've been measuring your grumpiness, your sleeping patterns, and your son’s sleeping patterns for quite some time now. Let’s say, for 100 days. And, being a nerd, you’ve saved the data as a file called parenthood.sav. If we open the data file in SPSS we'll see that there are four variables:parent.sleep, baby.sleep, parent.grump and day.

First thing, let's calculate some basic descriptive statistics (I've included the command you would type in SPSS if you were using the syntax editor):

FREQUENCIES VARIABLES=parent.sleep baby.sleep grump day
  /NTILES=4
  /STATISTICS=VARIANCE RANGE MINIMUM MAXIMUM MEAN MODE SKEWNESS SESKEW KURTOSIS SEKURT
  /HISTOGRAM
  /FORMAT=DFREQ
  /ORDER=ANALYSIS.

Finally, to give a graphical depiction of what each of the three interesting variables looks like, Figure 5.6 plots histograms.

Figure 5.6: Histograms for the three interesting variables in the parenthood.sav data set

One thing to note: just because SPSS can calculate dozens of different statistics doesn’t mean you should report all of them. If I were writing this up for a report, I’d probably pick out those statistics that are of most interest to me (and to my readership), and then put them into a nice, simple table like the one in Table ??.⁷⁹ Notice that when I put it into a table, I gave everything “human readable” names. This is always good practice.

Table 5.2: Descriptive statistics for the parenthood data.

variable	min	max	mean	median	std. dev	IQR
Grumpiness	41	91	63.71	62	10.05	14
Hours slept	4.84	9	6.97	7.03	1.02	1.45
Son’s hours slept	3.25	12.07	8.05	7.95	2.07	3.21

The Strength and Direction of a Relationship

We can draw scatterplots to give us a general sense of how closely related two variables are. Ideally, though, we might want to say a bit more about it than that. For instance, let’s compare the relationship between parent.sleep and parent.grump (Figure 5.7) with that between baby.sleep and parent.grump (Figure 5.8). When looking at these two plots side by side, it’s clear that the relationship is qualitatively the same in both cases: more sleep equals less grumpiness! However, it’s also pretty obvious that the relationship between parent.sleep and parent.grump is stronger than the relationship between baby.sleep and parent.grump. The plot on the left is “neater” than the one on the right. What it feels like is that if you want to predict what your mood is, it’d help you a little bit to know how many hours your son slept, but it’d be more helpful to know how many hours you slept.

In contrast, let’s consider Figure 5.8 vs. Figure 5.9. If we compare the scatterplot of “baby.sleep v parent.grump” to the scatterplot of “`baby.sleep v parent.sleep”, the overall strength of the relationship is the same, but the direction is different. That is, if your son sleeps more, you get more sleep (positive relationship, but if he sleeps more then you get less grumpy (negative relationship).

The Correlation Coefficient

We can make these ideas a bit more explicit by introducing the idea of a correlation coefficient (or, more specifically, Pearson’s correlation coefficient), which is traditionally denoted by r. The correlation coefficient between two variables X and Y (sometimes denoted rXY), which we’ll define more precisely in the next section, is a measure that varies from −1 to 1. When r=−1 it means that we have a perfect negative relationship, and when r=1 it means we have a perfect positive relationship. When r=0, there’s no relationship at all. If you look at Figure 5.10, you can see several plots showing what different correlations look like.

Figure 5.10: Illustration of the effect of varying the strength and direction of a correlation

The formula for Pearson’s correlation coefficient can be written in several different ways. I think the simplest way to write down the formula is to break it into two steps. Firstly, let’s introduce the idea of covariance. The covariance between two variables X and Y is a generalization of the notion of the variance; it’s a mathematically simple way of describing the relationship between two variables that isn’t terribly informative to humans:

\[
\operatorname{Cov}(X, Y)=\frac{1}{N-1} \sum_{i=1}^{N}\left(X_{i}-\bar{X}\right)\left(Y_{i}-\bar{Y}\right)
\nonumber\]

Because we’re multiplying (i.e., taking the “product” of) a quantity that depends on X by a quantity that depends on Y and then averaging⁸⁰, you can think of the formula for the covariance as an “average cross product” between X and Y. The covariance has the nice property that, if X and Y are entirely unrelated, then the covariance is exactly zero. If the relationship between them is positive (in the sense shown in Figure@reffig:corr) then the covariance is also positive; and if the relationship is negative then the covariance is also negative. In other words, the covariance captures the basic qualitative idea of correlation. Unfortunately, the raw magnitude of the covariance isn’t easy to interpret: it depends on the units in which X and Y are expressed, and worse yet, the actual units that the covariance itself is expressed in are really weird. For instance, if X refers to the parent.sleep variable (units: hours) and Y refers to the parent.grump variable (units: grumpiness), then the units for their covariance are “hours × grumps”. And I have no freaking idea what that would even mean.

The Pearson correlation coefficient r fixes this interpretation problem by standardizing the covariance, in pretty much the exact same way that the z-score standardizes a raw score: by dividing by the standard deviation. However, because we have two variables that contribute to the covariance, the standardization only works if we divide by both standard deviations.⁸¹ In other words, the correlation between X and Y can be written as follows:

\[
r_{X Y}=\frac{\operatorname{Cov}(X, Y)}{\hat{\sigma}_{X} \hat{\sigma}_{Y}}
\nonumber\]

By doing this standardization, not only do we keep all of the nice properties of the covariance discussed earlier, but the actual values of r are on a meaningful scale: r=1 implies a perfect positive relationship, and r=−1 implies a perfect negative relationship. I’ll expand a little more on this point later, in Section@refsec:interpretingcorrelations. But before I do, let’s look at how to calculate correlations in SPSS.

Calculating Correlations in SPSS

Calculating correlations in SPSS is alarmingly simple. Seriously. You're going to think I'm joking.

Let's start by making sure we have the parenthood.sav data file opened in SPSS. Then we'll head on over to the Analyze menu and select Correlate... and then Bivariate...:

This will open the following dialog window:

In the left column are all of the available variables for correlation, and as usual, on the right, you will place the variables you would like to correlate. Let's start with Parent Sleep Hours and Grumpiness:

Note, too, that the Show only the lower triangle has been checked. This will make the output easier to read. This dialog also shows that SPSS will compute the Pearson correlation and do a two-tailed test of significance. More on that later. At this point, click OK. Here's the output you should see:

So, what does this tell us? The output indicates that the correlation between Parent Sleep Hours and Grumpiness is -.903, which SPSS indicates is significant. So, we know the relationship is negative, meaning that the more sleep the parent gets, the less grumpy they are. Makes sense.

Interpreting a Correlation

Naturally, in real life, you don’t see many correlations of 1. So how should you interpret a correlation of, say r=.4? The honest answer is that it really depends on what you want to use the data for, and on how strong the correlations in your field tend to be. A friend of mine in engineering once argued that any correlation less than .95 is completely useless (I think he was exaggerating, even for engineering). On the other hand, there are real cases – even in psychology – where you should really expect correlations that strong. For instance, one of the benchmark data sets used to test theories of how people judge similarities is so clean that any theory that can’t achieve a correlation of at least .9 really isn’t deemed to be successful. However, when looking for (say) elementary correlates of intelligence (e.g., inspection time, response time), if you get a correlation above .3 you’re doing very very well. In short, the interpretation of a correlation depends a lot on the context. That said, the rough guide in Table ?? is pretty typical.

Correlation	Strength	Direction
-1.0 to -0.9	Very strong	Negative
-0.9 to -0.7	Strong	Negative
-0.7 to -0.4	Moderate	Negative
-0.4 to -0.2	Weak	Negative
-0.2 to 0	Negligible	Negative
0 to 0.2	Negligible	Positive
0.2 to 0.4	Weak	Positive
0.4 to 0.7	Moderate	Positive
0.7 to 0.9	Strong	Positive
0.9 to 1.0	Very strong	Positive

However, something that can never be stressed enough is that you should always look at the scatterplot before attaching any interpretation to the data. A correlation might not mean what you think it means. The classic illustration of this is “Anscombe’s Quartet” (Anscombe, 1973), which is a collection of four data sets. Each data set has two variables, an X and a Y. For all four data sets the mean value for X is 9 and the mean for Y is 7.5. The standard deviations for all X variables are almost identical, as are those for the Y variables. And in each case, the correlation between X and Y is r=0.816.

Given that the four data sets have the same correlation and almost identical standard deviations, you’d think that they would look pretty similar to one another. They do not. If we draw scatterplots of X against Y for all four variables, as shown in Figure 5.11 we see that the four are spectacularly different.

Figure 5.11: Anscombe’s quartet. All four of these data sets have a Pearson correlation of r=.816, but they are qualitatively different from one another.

The lesson here, which so very many people seem to forget in real life is “always graph your raw data”. This will be the focus of the next chapter.

Spearman’s Rank Correlations

Figure 5.12: The relationship between hours worked and grade received, for a toy data set consisting of only 10 students (each circle corresponds to one student). The dashed line through the middle shows the linear relationship between the two variables. This produces a strong Pearson correlation of r=.91. However, the interesting thing to note here is that there’s actually a perfect monotonic relationship between the two variables: in this toy example at least, increasing the hours worked always increases the grade received, as illustrated by the solid line. This is reflected in a Spearman correlation of rho=1. With such a small data set, however, it’s an open question as to which version better describes the actual relationship involved.

The Pearson correlation coefficient is useful for many things, but it does have shortcomings. One issue, in particular, stands out: what it actually measures is the strength of the linear relationship between two variables. In other words, what it gives you is a measure of the extent to which the data all tend to fall on a single, perfectly straight line. Often, this is a pretty good approximation to what we mean when we say “relationship”, and so the Pearson correlation is a good thing to calculate. Sometimes, it isn’t.

One very common situation where the Pearson correlation isn’t quite the right thing to use arises when an increase in one variable X really is reflected in an increase in another variable Y, but the nature of the relationship isn’t necessarily linear. An example of this might be the relationship between effort and reward when studying for an exam. If you put in zero effort (X) into learning a subject, then you should expect a grade of 0% (Y). However, a little bit of effort will cause a massive improvement: just turning up to lectures means that you learn a fair bit, and if you just turn up to classes, and scribble a few things down so your grade might rise to 35%, all without a lot of effort. However, you just don’t get the same effect at the other end of the scale. As everyone knows, it takes a lot more effort to get a grade of 90% than it takes to get a grade of 55%. What this means is that, if I’ve got data looking at study effort and grades, there’s a pretty good chance that Pearson correlations will be misleading.

To illustrate, consider the data plotted in Figure 5.12, showing the relationship between hours spent studying and grade received for 10 students taking some class. The curious thing about this – highly fictitious – data set is that increasing your effort always increases your grade. It might be by a lot or it might be by a little, but increasing effort will never decrease your grade. The data are stored in effort.sav.

The raw data look like this:

If we run a standard Pearson correlation, it shows a strong relationship between hours worked and grade received:

but this doesn’t actually capture the observation that increasing hours worked always increases the grade. There’s a sense here in which we want to be able to say that the correlation is perfect but for a somewhat different notion of what a “relationship” is. What we’re looking for is something that captures the fact that there is a perfect ordinal relationship here. That is, if student 1 works more hours than student 2, then we can guarantee that student 1 will get a better grade. That’s not what a correlation of r=.91 says at all.

How should we address this? Actually, it’s really easy: if we’re looking for ordinal relationships, all we have to do is treat the data as if it were ordinal scale! So, instead of measuring effort in terms of “hours worked”, let's rank all 10 of our students in order of hours worked. That is, student 1 did the least work out of anyone (2 hours) so they get the lowest rank (rank = 1). Student 4 was the next laziest, putting in only 6 hours of work in over the whole semester, so they get the next lowest rank (rank = 2). Notice that I’m using “rank =1” to mean “low rank”. Sometimes in everyday language, we talk about “rank = 1” to mean “top rank” rather than “bottom rank”. So be careful: you can rank “from smallest value to largest value” (i.e., small equals rank 1) or you can rank “from largest value to smallest value” (i.e., large equals rank 1). In this case, I’m ranking from smallest to largest, but in real life, it’s really easy to forget which way you set things up, so you have to put a bit of effort into remembering!

Okay, so let’s have a look at our students when we rank them from worst to best in terms of effort and reward:

	rank (hours worked)	rank (grade received)
student	1	1
student	2	10
student	3	6
student	4	2
student	5	3
student	6	5
student	7	4
student	8	8
student	9	7
student	10	9

Hm. These are identical. The student who put in the most effort got the best grade, the student with the least effort got the worst grade, etc.

Now, if you want, you can mess around and create a new data set in SPSS lie the one above and then run a Pearson correlation, OR you could simply ask SPSS to run a Spearman’s rank order correlation. Which of those two processes do you suppose is easier?

To do this in SPSS, make like you're doing a Pearson correlation, except you will uncheck the Pearson correlation and check the Spearman correlation. That's it. Seriously.

When you click on OK, we get a result that looks like this:

SPSS has told us that the Spearman rank order correlation between hours spent studying and grade earned is 1.000, which we would expect since people who study more always do better. That's a quick look at correlation. We'll revisit this later as well.