**Lessons in biostatistics:**

**Martina Udovičić**What we need to know when calculating the coefficient of correlation? Biochemia Medica 2007;17(1):10-5. http://dx.doi.org/10.11613/BM.2007.002

^{1}, Ksenija Baždarić^{1}, Lidija Bilić-Zulle^{1,2}, Mladen Petrovečki^{1,3}.

^{1}Department of Medical Informatics, School of Medicine, University of Rijeka, Rijeka, Croatia

^{2}Institute of Laboratory Diagnosis, Rijeka Clinical Hospital Center, Rijeka, Croatia

^{3}Clinical Institute of Laboratory Diagnosis, Dubrava Clinical Hospital, Zagreb, Croatia

Corresponding author: umartina [at] medri [dot] hr

**Abstract**

Correlation is a statistical procedure applied to calculate association between two variables. The value of correlation is numerically shown by a coefficient of correlation, most often by Pearson’s or Spearman’s coefficient, while the significance of the coefficient is expressed by

*P*value. The coefficient of correlation shows the extent to which changes in the value of one variable are correlated to changes in the value of the other. A sign preceding the coefficient of correlation (+ or -) indicates the direction of correlation. The most frequent errors in calculating correlation are related to conditions for calculation, interpretation of the coefficient and correlation significance, high correlation coefficients, assumption of causal relationship, the strength of correlation (coefficient of determination), and comparison of two correlation coefficients.**Key words**: correlation, Pearson’s correlation coefficient, Spearman’s correlation coefficient, coefficient of determination, error, statistics

**Introduction**

The statistical procedure of calculating correlation is one of the most frequently used procedures in biomedicine. Correlation is agreement of values from two data sets, and it expresses the degree of association between investigated phenomena. Biomedical studies often examine the correlation between two data sets as, e.g. the correlation between concentrations of blood glucose and glycated hemoglobin, or between biological age and cholesterol level. The use of the coefficient of correlation depends on the type of data, i.e. on the scale where the data are available. The Pearson’s and Spearman’s coefficients of correlation are used most frequently (1).

The Pearson’s coefficient of correlation is employed for variables on an interval or ratio scale (numerical data) that are in linear relation. The linear relation of variables may be read from a scatterplot and it implies that the points follow and scatter around the straight line. The data may sometimes be interconnected without being in linear relation and then the Pearson’s correlation coefficient cannot be calculated (1). For instance, if we observe - in accordance with the Michaelis Menten model of enzyme kinetics, - the correlation between enzyme reaction velocity and substrate concentration in a solution, we can conclude that this correlation is very high but not linear, and the relation between the two variables is described by a curve.

Pearson’s coefficient of correlation is denoted by a small letter

*r*or*r*, and its values may range from -1 to +1. The value of the correlation coefficient from 0 to 1 is positive correlation and it designates proportional growth of values in both data sets. An example of positive correlation is the duration of diabetes mellitus and the degree of damage of eye capillaries. The longer the duration of the disease, the higher the damage to eye capillaries. The correlation coefficient value from 0 to -1 indicates negative correlation, i.e. a rise in the value of one variable that is proportional to a decline in the value of the other; e.g. oxygen concentration in the air drops with the rise in altitude above sea level. Perfect correlations, i.e. the values of the coefficient of correlation r = ± 1 are not characteristical for biological systems and most frequently refer to theoretical models. The zero value of the coefficient of correlation indicates absence of linear correlation, i.e. by knowing the values of one variable, we can conclude nothing on the values of the other. Thus, for instance, if we observe the correlation between the size of the pupil of the eye and calcium ion concentration in the blood, we can conclude that there is no correlation, i.e. each size of the pupil could be associated to any calcium ion concentration (understandably, within physiological limits) (2)._{p}Spearman's coefficient of correlation (rho, r

_{s}) or rank correlation is calculated when one of the data sets is on ordinal scale, or when data distribution significantly deviates from normal distribution and data are available that considerably diverge from most of those measured (outliers) (3). Linear correlation, implied by the Pearson’s coefficient of correlation, is not required for the Spearman’s correlation coefficient which can also be calculated for small samples (N<35). In case of r_{s}= 0, it may be concluded that there is no actual correlation between variables (1).The procedure of calculating correlation is frequently applied incorrectly. Prior to calculation, it is therefore necessary to understand the concept and types of correlation, conditions for calculating correlation and interpreting association in order to avoid wrong conclusions.

What follows are some of the most frequent mistakes made while calculating correlations, and their explanations.

**Conditions for calculating correlation**

**Question:**Is it correct to calculate the Pearson’s correlation coefficient for the degree of burns on the body and the duration of hospitalization expressed by the number of days?

**Answer**: It is not correct.

**Explanation:**Initial step in calculating correlation is to check if the measured data meet the conditions for calculating the Pearson’s correlation. The degree of burns on the body can be ranked on a scale from 1 to 4; such data are categorical (classifying subjects in predefined “classes”) and they follow an ordinal scale. The duration of hospital therapy expressed in the number of days is on a ratio scale and is suitable for calculating the Pearson’s correlation coefficient if the other variable is on an interval or ratio scale. The Pearson’s coefficient of correlation can be calculated only if the following conditions are met: the data for both examined variables are on an interval or ratio scale, the data for at least one variable have normal, i.e symmetrical distribution, the examined sample is large (N > 35), and the condition of linear correlation is met, which may be read from a scatterplot (1).

Unless the conditions for calculation of the Pearson’s coefficient of correlation are met, the Spearman’s correlation coefficient can be applied. In the example described above, the degree of burns is measured on an ordinal scale, and therefore the condition for Pearson’s correlation is not fulfilled but rather the Spearman’s rank coefficient of correlation should be calculated.

**Interpretation and significance of the coefficient of correlation**

**Question:**In a study of correlation between the mood and the amount of liquid consumed by daily drinking, the correlation r = 0.12;

*P*= 0.003 was obtained. Is it correct to conclude that there is a significant correlation between the mood and the amount of the consumed liquid?

**Answer:**It is not correct.

**Explanation:**After calculating the coefficient of correlation, it is important to know how to interpret the result, that is, the real meaning of the correlation coefficient. In presenting the results of correlation, the coefficient of correlation “r” should be expressed by a number with two decimal places, and the significance of the coefficient of correlation “P” in a number with three decimal places (4). If the coefficient of correlation is significant in regard to the set limit of significance (commonly

*P*< 0.05), we may conclude that the coefficient of correlation is significant and may be interpreted. If the value is

*P*> 0.05, we can conclude that the coefficient of correlation is not significant and in this case it may not be interpreted regardless of its value. When interpreting the value of the corrrelation coefficient, the same rules are valid for both Pearson’s and Spearman’s coefficient, and r values from 0 to 0.25 or from 0 to -0.25 are commonly regarded to indicate the absence of correlation, whereas r values from 0.25 to 0.50 or from -0.25 to -0.50 point to poor correlation between variables. r values ranging from 0.50 to 0.75 or -0.50 to -0.75 indicate moderate to good correlation, and r values from 0.75 to 1 or from -0.75 to -1 point to very good to excellent correlation between the variables (1).

Accordingly, it is wrong to conclude that there is a significant correlation between the mood and the amount of liquid taken during a day. Correct conclusion is as follows: there is no correlation between the examined variables (r = 0.12), which may be claimed because the correlation coefficient is significant (P = 0.003) (5,6).

**High value of the correlation coefficient**

**Question:**The correlation value obtained in a study of correlation between body height and biological age was r = 0.97. May we conclude that height and age are definitely excellently correlated?

**Answer**: No, at least not beyond doubt.

**Interpretation**: If the correlation coefficient calculated for biological variables is r > 0.95, an error in measurement and sampling or possible alteration of measured results should be suspected. Due to natural variety of biological systems, it is virtually impossible to obtain such a high correlation coefficient if measurements have been done correctly (representative sample, sufficiently sensitive instrument, etc.) (1). The type of data collected by measurements and processed statistically should always be taken into account. For example, if comparison is made of the values of glucose measured in a series of blood samples by two different instruments, i.e. biochemical analyzers, the coefficient of correlation may be expected to be very high (even up to r = 0.99), which in this case indicates good agreement between the two instruments.

**Correlation and causal relationship**

**Question:**r = 0.78 and

*P*= 0.002 were determined in a study of correlation between blood alcohol level and traffic accidents. Are we allowed to conclude that alcohol consumption is the cause of traffic accidents, i.e. that the observed traffic accidents are the consequence of alcohol consumption?

**Answer:**No, we are not.

**Explanation:**Correlation provides information on association rather than a cause- and-effect relationship between variables. Thus, if there is a high correlation between alcohol consumption and traffic accidents, we may not conclude that one variable affects the other, i.e. that alcohol consumption causes traffic accidents. It is possible that increased amount of alcohol causes the increased number of accidents, yet there is a possibility of a considerable effect of other uninvestigated factors or rare events (7,8). In the example described above, these factors or events could be road condition, proper operation of a vehicle, potential illness of a driver unrelated to alcohol, action of other pharmacologically active substances, and the like.

In research, correlation should be primarily employed to build hypotheses rather than to test them, the latter being a frequent and entirely wrong application (9). If, e.g., correlation is established between variables, causal relationship should be demonstrated by scientific experiment (10). The only experiment to demonstrate such relationship in biomedicine is a randomized controlled clinical trial (10).

**The strength of correlation**

**Question:**By comparing catalytic concentration of two enzymes in the blood, the correlation r = 0.52;

*P*= 0.002 was obtained. Can we conclude that enzyme values share 52% of catalytic concentration values?

**Answer:**No, we cannot.

**Explanation:**The coefficient of correlation is not a measure of the strength of correlation. The correlation coefficient value r = 0.52 cannot be interpreted as 52% correlation, i.e. 52% of the joint values for the two catalytic enzyme concentrations. The proportion of shared values, i.e. the strength of linear correlation is expressed by the coefficient of determination. The coefficient of determination is calculated simply by squaring the correlation coefficient, and is denoted by r

^{2}. It can be calculated only for the Pearson’s correlation (3). Therefore the strength of correlation (coefficient of determination) in this example is r

^{2 }= 0.52 x 0.52 = 0.27, ie. the catalytic concentrations of two enzymes share 27% of common values. Twice as high correlation does not imply the twofold strength of correlation; e.g., if the correlation was r

_{1}= 0.26, the strength of correlation would be r

_{1}

^{2}= 0.07 (7%); also, it would be r

_{2}

^{2}= 0.27 (27%) for the twofold higher correlation, r

_{2}= 0.52.

**Comparison of two correlation coefficients with the same properties on two subject samples**

**Question:**Correlation between the time spent at computer work and the speed of typing a text into computer has been examined for women (N

_{1}= 60) and men (N

_{2}= 40). The coefficient of correlation, for women is r

_{1}= 0.70 and for men r

_{2}= 0.50: both are statistically significant. Can we conclude that r

_{1}> r

_{2}, i.e. that the correlation between the time spent at computer work and computer typing speed is higher in women?

**Answer:**No, we cannot.

**Explanation:**The two coefficients of correlation should by no means be directly compared but the significance of difference between the correlations for two data sets should be examined. The procedure of establishing the significance of the difference between two coefficients of correlation takes into account the value of correlation coefficients and the size of both samples (8).

By comparison of the two correlation coefficients in the example above, the correlation between the time spent in computer work and the typing speed of women has not been found to differ significantly from the correlation of the same variables in men (P = 0.132) (11).

**Conclusion**

Determination of association, i.e. correlation between phenomena (variables) is an important tool in scientific study. The associations observed between two phenomena only allow us to pose a hypothesis in a scientific experiment that is, actually, itself an attempt to establish causal relationship (which is never demonstrated by correlation). Aside from biological systems (particularly laboratory medicine), the coefficient of correlation is important in the study and comparison of two analytical systems (methods, instruments, etc.) when we can, on the basis of its high value, replace a more complex method by, e.g. an easier or cheaper one. Being frequently used in data processing in scientific studies, correlation is also often misused mostly due to ignorance or negligence of the rules for using a correlation test. The consequence is faulty conclusions on scientific hypotheses that lead to fallacies rather than new insights.

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