The Correlation Coefficient (r)

The sample correlation coefficient (r) is a measure of the closeness of association of the points in a scatter plot to a linear regression line based on those points, as in the example above for accumulated saving over time. Possible values of the correlation coefficient range from -1 to +1, with -1 indicating a perfectly linear negative, i.e., inverse, correlation (sloping downward) and +1 indicating a perfectly linear positive correlation (sloping upward).

A correlation coefficient close to 0 suggests little, if any, correlation. The scatter plot suggests that measurement of IQ do not change with increasing age, i.e., there is no evidence that IQ is associated with age.

Calculation of the Correlation Coefficient

The equations below show the calculations sed to compute “r”. However, you do not need to remember these equations. We will use R to do these calculations for us. Nevertheless, the equations give a sense of how “r” is computed.




where Cov(X,Y) is the covariance, i.e., how far each observed (X,Y) pair is from the mean of X and the mean of Y, simultaneously, and and sx
2
and sy
2
are the sample variances for X and Y.

. Cov (X,Y) is computed as:




You don’t have to memorize or use these equations for hand calculations. Instead, we will use R to calculate correlation coefficients. For example, we could use the following command to compute the correlation coefficient for AGE and TOTCHOL in a subset of the Framingham Heart Study as follows:

>
cor(AGE,TOTCHOL)

[1] 0.2917043

Describing Correlation Coefficients

The table below provides some guidelines for how to describe the strength of correlation coefficients, but these are just guidelines for description. Also, keep in mind that even weak correlations can be statistically significant, as you will learn shortly.

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Correlation Coefficient (r) Description



(Rough Guideline )
+1.0 Perfect positive + association
+0.8 to 1.0 Very strong + association
+0.6 to 0.8 Strong + association
+0.4 to 0.6 Moderate + association
+0.2 to 0.4 Weak + association
0.0 to +0.2 Very weak + or no association
0.0 to -0.2 Very weak – or no association
-0.2 to – 0.4 Weak – association
-0.4 to -0.6 Moderate – association
-0.6 to -0.8 Strong – association
-0.8 to -1.0 Very strong – association
-1.0 Perfect negative association

 The four images below give an idea of how some correlation coefficients might look on a scatter plot.

The scatter plot below illustrates the relationship between systolic blood pressure and age in a large number of subjects. It suggests a weak (r=0.36), but statistically significant (p<0.0001) positive association between age and systolic blood pressure. There is quite a bit of scatter, but there are many observations, and there is a clear linear trend.

scatter plot of systolic blood pressure and age. There is a lot of scatter but there appears to be a clear linear trend.

How can a correlation be weak, but still statistically significant? Consider that most outcomes have multiple determinants. For example, body mass index (BMI) is determined by multiple factors (“exposures”), such as age, height, sex, calorie consumption, exercise, genetic factors, etc. So, height is just one determinant and is a contributing factor, but not the only determinant of BMI. As a result, height might be a significant determinant, i.e., it might be significantly associated with BMI but only be a partial factor. If that is the case, even a weak correlation might have be statistically significant if the sample size is sufficiently large. In essence, finding a weak correlation that is statistically significant suggests that that particular exposure has an impact on the outcome variable, but that there are other important determinants as well.

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Beware of Non-Linear Relationships

Many relationships between measurement variables are reasonably linear, but others are not For example, the image below indicates that the risk of death is not linearly correlated with body mass index. Instead, this type of relationship is often described as “U-shaped” or “J-shaped,” because the value of the Y-variable initially decreases with increases in X, but with further increases in X, the Y-variable increases substantially. The relationship between alcohol consumption and mortality is also “J-shaped.”

Source: Calle EE, et al.: N Engl J Med 1999; 341:1097-1105

A simple way to evaluate whether a relationship is reasonably linear is to examine a scatter plot. To illustrate, look at the scatter plot below of height (in inches) and body weight (in pounds) using data from the Weymouth Health Survey in 2004. R was used to create the scatter plot and compute the correlation coefficient.

wey<-na.omit(Weymouth_Adult_Part)



attach(wey)



plot(hgt_inch,weight)



cor(hgt_inch,weight)

[1] 0.5653241

There is quite a lot of scatter, and the large number of data points makes it difficult to fully evaluate the correlation, but the trend is reasonably linear. The correlation coefficient is +0.56.

Beware of Outliers

Note also in the plot above that there are two individuals with apparent heights of 88 and 99 inches. A height of 88 inches (7 feet 3 inches) is plausible, but unlikely, and a height of 99 inches is certainly a coding error. Obvious coding errors should be excluded from the analysis, since they can have an inordinate effect on the results. It’s always a good idea to look at the raw data in order to identify any gross mistakes in coding.

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After excluding the two outliers, the plot looks like this: