The correlation coefficient indicates the strength of a linear association. The following sections explain the theory, types and SAS implementation of correlations.

A correlation is a statistic measured between -1 and +1, where:

Figure 8.2 Representation of positive and negative correlations shows a pictorial version of this explanation.

Figure 8.3 Scale of correlation coefficients shows a sample plot of some sales and enquiries data for customers (note this is hypothetical and not related to the core case datasets).

In this scatterplot each customer company is represented by a dot, and the coordinate of the dot is determined by the customer’s enquiries score (the horizontal axis) and sales score (the vertical axis).

The dotted arrow shows what the eye can easily see: the high enquiries scores tend to also be high in sales, and the lows tend to continue to be low. There is a straight-line relationship. This strength is initially measured by the correlation coefficient, which we will learn to measure shortly.

There are different types of correlations. The most common are:

You can generate correlations in SAS using the PROC CORR module. Say that you wish to assess Pearson associations between trust, satisfaction, enquiries and sales. To see this module running, open and run the file “Code08a Correlations”, which is based on the dataset “Data03_Aggregated.” This gives a table like that seen in Figure 8.5 SAS correlation analysis main table.

In Figure 8.5 SAS correlation analysis main table the correlations are given on the top row for each variable. Chapter 12 explains what is meant by the ”p-value” second rows in each line, the third row is the number of observations used in each correlation. (This differs because missing data exists more in some pairs of columns than others.) This table is not the usual way of presenting correlations in a final report – see the next section on how to present correlations properly.

Note that you can easily generate Spearman or other types of correlations too, simply by adding the appropriate keyword (e.g. “SPEARMAN”) before the first semicolon.

As with all statistics, correlations should be arranged in a well-presented table. Figure 8.6 Formatted presentation of a correlation table is an example for our final chapter data.

Notes on this table:

I have written a program that will automatically generate this type of final report. Open and run ”Code08b Gregs correlations.” You merely input your dataset, variables and (optionally) type of correlation to get a formatted table like the above.

How do you know whether a correlation is big or not? In other words how do you know whether or not it indicates a strong association between two variables? Various guidelines for Pearson and Spearman correlations have been suggested, although as with all cut-offs don’t use them too dogmatically. The version in Figure 8.7 Sizes of correlations and their commonly held interpretations is one version:

Look again at the correlations in Figure 8.6 Formatted presentation of a correlation table. Which variable has the strongest association? This would be the biggest correlation (positive or negative), which is .76 between sales and trust. This would be a large positive correlation suggesting strongly that when trust is high or low so are the scores for sales. There are also negative correlations, such as -.59 between small and sales, suggesting that smaller customers have lower sales.

Note also that the .25 correlation between satisfaction and sales is about a third as strong as the .76 between trust and sales. Correlations can be compared as to relative size, including one negative and one positive one.

One point: say you have a correlation that is r = -.11. Some people would say there is a negative relationship between these two variables, just because there is a number and a minus sign. This may be faintly true, but the correlation here is just too small to be sure, and you should not rely on such a small correlation for evidence.

Finally, some readers read correlations as if they are percentages. They are not percentages. The meanings are above and should be adhered to.

The correlation coefficient is not the only assessment of linear association: in fact, it is the simplest but is only the barebones of association. It assesses whether one variable tends to be high when the other is, but it doesn’t really reveal much else. It fails to assess scale (range) within the variables.

Have you ever heard of the ”butterfly effect”? It is a chaos theory picture in which a butterfly flaps its wings in one part of the world, causing a cascade of events that leads to a tidal wave elsewhere in the world.

It is just a metaphor, but consider the following. Say that I tell you that a butterfly flaps its wings. Imagine this as a metaphor for movement in one variable. Now I tell you that every time the butterfly flaps its wings, a body of water moves elsewhere. This is a metaphor for movement in a second variable.

Is there correlation? Yes – if the one variable (the water) tends to change when the other one changes (the butterfly) then you have association between the two and you have correlation.

However, surely you should have further questions? Most importantly, I have told you that when the butterfly flaps its wings the water moves, but I have not told you by how much the water moves. Surely this is the important thing. If the butterfly causes a small ripple in the water every time it flaps, this is very different to whether the change in the water is a tidal wave that kills thousands.

So, correlations deal with associations, but they do not really deal with how big the associations are in the context of the variables.

Covariances are simply correlations that have been adjusted so that they include a sense of the size of the relative association.

Let us use a business example too. Imagine if I told you that I have a variable measured in US dollars, and I tell you that this variable changes by $1 when something else happens, (e.g. when the oil price drops by 10%). What does this mean to you? Surely you would tell me that it depends what a $1 change means in the overall context of my variable. Specifically, it would depend on the spread/range of my variable (i.e. standard deviation). For example the following are two dollar-based variables:

Do you see that a change of $1 means a very different thing depending on what the spread/range of the variable is? In the USD/EUR exchange rate, a change of $1 is huge, whereas in the gold price such a change is small. A correlation coefficient that predicts a linear relationship where $1 is the scale of change is therefore an incomplete statistic until we know more about the standard deviations of the variables.

To adjust for this we sometimes use a measure of association called the “covariance.” The (true) covariance between two variables is a correlation scaled for the standard deviations of the variables. You don’t need to know exactly how this looks, but just remember the basic lesson that covariances reflect how much the one variable moves (varies) given a certain variance in the other variable.

Covariances are better than correlations – they give more information, and pay attention to the relative scales of the variables.

However, unlike the correlation, it is not really possible to look at a set of covariances and see immediately what they mean about the association. Therefore, we use covariances as the real raw material for more complex statistics (like regression as discussed later in the book), but we typically do not analyze them directly.

It’s more important to understand the notion of covariances than to actually calculate them – the essential thing is to know that they underlie many critical statistical procedures. If you do want to generate them, add to keyword COV in the SAS correlation module.