Application
of Correlation
Abstract
Correlation is a
statistical method used to assess a possible linear association between two
continuous variables. It is simple both to calculate and to interpret. However,
misuse of correlation is so common among researchers that some statisticians
have wished that the method had never been devised at all. The aim of this
article is to provide a guide to appropriate use of correlation in medical
research and to highlight some misuse. Examples of the applications of the correlation
coefficient have been provided using data from statistical simulations as well
as real data. Rule of thumb for interpreting size of a correlation coefficient
has been provided.
Definitions
of Correlation and Clarifications
The term correlation is
sometimes used loosely in verbal communication. Among scientific colleagues,
the term correlation is used to refer to an association, connection, or any
form of relationship, link or correspondence. This broad colloquial definition
sometimes leads to misuse of the statistical term “correlation” among
scientists in research. Misuse of correlation is so common that some
statisticians have wished that the method had never been devised.1
Webster's Online
Dictionary defines correlation as a reciprocal relation between two or more
things; a statistic representing how closely two variables co-vary; it can vary
from −1 (perfect negative correlation) through 0 (no correlation) to +1
(perfect positive correlation).2
In statistical terms,
correlation is a method of assessing a possible two-way linear association
between two continuous variables.1 Correlation is measured by a statistic
called the correlation coefficient, which represents the strength of the
putative linear association between the variables in question. It is a dimensionless
quantity that takes a value in the range −1 to +13. A correlation coefficient
of zero indicates that no linear relationship exists between two continuous
variables, and a correlation coefficient of −1 or +1 indicates a perfect linear
relationship. The strength of relationship can be anywhere between −1 and +1.
The stronger the correlation, the closer the correlation coefficient comes to
±1. If the coefficient is a positive number, the variables are directly related
(i.e., as the value of one variable goes up, the value of the other also tends
to do so). If, on the other hand, the coefficient is a negative number, the
variables are inversely related (i.e., as the value of one variable goes up,
the value of the other tends to go down).3 Any other form of relationship
between two continuous variables that is not linear is not correlation in
statistical terms. To emphasise
this point, a mathematical relationship does not necessarily mean that there is
correlation. For example, consider the equation y=2×2. In statistical terms, it
is inappropriate to say that there is correlation between x and y. This is so
because, although there is a relationship, the relationship is not linear over
this range of the specified values of x. It is possible to predict y exactly
for each value of x in the given range, but correlation is neither −1 nor +1.
Hence, it would be inconsistent with the definition of correlation and it
cannot therefore be said that x is Types of correlation coefficients(‘Applications
of correlation techniques | SpringerLink’, n.d.).
Pearson's
Product Moment Correlation Coefficient
Pearson's product
moment correlation coefficient is denoted as ϱ for a population parameter and
as r for a sample statistic. It is used when both variables being studied are
normally distributed. This coefficient is affected by extreme values, which may
exaggerate or dampen the strength of relationship, and is therefore
inappropriate when either or both variables are not normally distributed. For a
correlation between variables x and y, the formula for calculating the sample
Pearson's correlation coefficient is given by3a high positive correlation
(Table 1). The Pearson's correlation coefficient for these variables is 0.80.
In this case the two correlation coefficients are similar and lead to the same
conclusion, however in some cases the two may be very different leading to
different statistical conclusions. For example, in the same group of women the
spearman's correlation between haemoglobin level and parity is 0.3 while the
Pearson's correlation is 0.2. In this case the two coefficients may lead to
different statistical inference. For example, a correlation coefficient of 0.2
is considered to be negligible correlation while a correlation coefficient of
0.3 is considered as low positive correlation (Table 1), so it would be
important to use the most appropriate one. The most appropriate coefficient in
this case is the Spearman's because parity is skewed.
Table
1
Rule
of Thumb for Interpreting the Size of a Correlation Coefficient4
Size of correlation
|
Interpersonal
|
|
90 to 1.00 (−.90 to −1.00)
|
Very high positive (negative)
correlation
|
|
50 to .70 (−.50 to −.70)
|
Moderate positive (negative)
correlation
|
|
.70
to .90 (−.70 to −.90)
|
High positive (negative) correlation
|
In another dataset of
251 adult women, age and weight were log-transformed. The reason for
transforming was to make the variables normally distributed so that we can use
Pearson's correlation coefficient. Then we analysed the data for a linear
association between log of age (agelog) and log of weight (wlog). Both
variables are approximately normally distributed on the log scale. In this case
Pearson's correlation coefficient is more appropriate. The coefficient is
0.184. This shows that there is negligible correlation between the age and
weight on the log scale (Table 1).
In Fig. 5 the pattern
changes at the higher values of parity. Table 2 shows how Spearman's and
Pearson's correlation coefficients change when seven patients having higher
values of parity have been excluded. When the seven higher parity values are
excluded, Pearson's correlation coefficient changes substantially compared to
Spearman's correlation coefficient. Although the difference in the Pearson
Correlation coefficient before and after excluding outliers is not
statistically significant, the interpretation may be different. The correlation
coefficient of 0.2 before excluding outliers is considered as negligible
correlation while 0.3 after excluding outliers may be interpreted as weak
positive correlation (Table 1). The interpretation for the Spearman's
correlation remains the same before and after excluding outliers with a
correlation coefficient of 0.3. The difference in the change between Spearman's
and Pearson's coefficients when outliers are excluded raises an important point
in choosing the appropriate statistic. Non-normally distributed data may
include outlier values that necessitate usage of Spearman's correlation
coefficient.
Simple application of
the correlation coefficient can be exemplified using data from a sample of 780
women attending their first antenatal clinic (ANC) visits. We can expect a
positive linear relationship between maternal age in years and parity because
parity cannot decrease with age, but we cannot predict the strength of this
relationship. The task is one of quantifying the strength of the association.
That is, we are interested in the strength of relationship between the two
variables rather than direction since direction is obvious in this case.
Maternal age is continuous and usually skewed while parity is ordinal and
skewed. With these scales of measurement for the data, the appropriate
correlation coefficient to use is Spearman's. The Spearman's coefficient is
0.84 for this data. In this case, maternal age is strongly correlated with
parity, i.e. has a high positive correlation (Table 1). The Pearson's
correlation coefficient for these variables is 0.80. In this case the two
correlation coefficients are similar and lead to the same conclusion, however
in some cases the two may be very different leading to different statistical
conclusions. For example, in the same group of women the spearman's correlation
between haemoglobin level and parity is 0.3 while the Pearson's correlation is
0.2. In this case the two coefficients may lead to different statistical
inference. For example, a correlation coefficient of 0.2 is considered to be
negligible correlation while a correlation coefficient of 0.3 is considered as
low positive correlation (Table 1), so it would be important to use the most
appropriate one. The most appropriate coefficient in this case is the
Spearman's because parity is skewed(‘A
guide to appropriate use of Correlation coefficient in medical research’, n.d.).
In another dataset of
251 adult women, age and weight were log-transformed. The reason for
transforming was to make the variables normally distributed so that we can use
Pearson's correlation coefficient. Then we analysed the data for a linear
association between log of age (agelog) and log of weight (wlog). Both
variables are approximately normally distributed on the log scale. In this case
Pearson's correlation coefficient is more appropriate. The coefficient is
0.184. This shows that there is negligible correlation between the age and
weight on the log scale (Table 1).
In Fig. 5 the pattern
changes at the higher values of parity. Table 2 shows how Spearman's and
Pearson's correlation coefficients change when seven patients having higher
values of parity have been excluded. When the seven higher parity values are
excluded, Pearson's correlation coefficient changes substantially compared to
Spearman's correlation coefficient. Although the difference in the Pearson
Correlation coefficient before and after excluding outliers is not
statistically significant, the interpretation may be different. The correlation
coefficient of 0.2 before excluding outliers is considered as negligible
correlation while 0.3 after excluding outliers may be interpreted as weak
positive correlation (Table 1). The interpretation for the Spearman's correlation
remains the same before and after excluding outliers with a correlation
coefficient of 0.3. The difference in the change between Spearman's and
Pearson's coefficients when outliers are excluded raises an important point in
choosing the appropriate statistic. Non-normally distributed data may include
outlier values that necessitate usage of Spearman's correlation coefficient.
In summary, correlation
coefficients are used to assess the strength and direction of the linear
relationships between pairs of variables. When both variables are normally
distributed use Pearson's correlation coefficient, otherwise use Spearman's
correlation coefficient. Spearman's correlation coefficient is more robust to
outliers than is Pearson's correlation coefficient. Correlation coefficients do
not communicate information about whether one variable moves in response to
another. There is no attempt to establish one variable as dependent and the
other as independent. Thus, relationships identified using correlation coefficients
should be interpreted for what they are: associations, not causal
relationships.5 Correlation must not be used to assess agreement between
methods. Agreement between methods should be assessed using Bland-Altman
plots6.
References
1. Altman DG. Practical Statistics for
Medical Research. Chapman & Hall/CRC; [Google Scholar]
2. Webster's Online Dictionary. [Google
Scholar]
3. Swinscow TDV. In: Statistics at
square one. Nineth Edition. Campbell M J, editor. University of Southampton;
Copyright BMJ Publishing Group 1997. [Google Scholar]
4. Hinkle DE, Wiersma W, Jurs SG.
Applied Statistics for the Behavioral Sciences. 5th ed. Boston: Houghton
Mifflin; 2003. [Google Scholar]
5. Clarke GM, Cooke D. A basic course in
Statistics. 3rd ed [Google Scholar]
6. Altman DG, Bland JM. Measurement in
Medicine: The Analysis of Method Comparison Studies. The Statistician.
1983;32:307–317. [Google Scholar]
Articles from Malawi Medical Journal are
provided here courtesy of Medical Association of Malawi
Reference
- A guide to appropriate use of Correlation coefficient in medical research. (n.d.). Retrieved 20 August 2019, from https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3576830/
- Applications of correlation techniques | SpringerLink. (n.d.). Retrieved 20 August 2019, from https://link.springer.com/article/10.1007/BF01984652
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