Correlation analysis

What is a correlation analysis?

Correlation analysis is a statistical method used to evaluate the relationship between two variables, such as the association between body size and shoe size.

The strength of this relationship is measured by the correlation coefficient, which ranges from -1 to +1. A coefficient close to +1 indicates a strong positive correlation, while a value near -1 signifies a strong negative correlation. Values around zero suggest little to no relationship. Correlation analyses can thus be used to make a statement about the strength and direction of the correlation.

Correlation and causality

If correlation analysis reveals a relationship between two variables, it is possible to further investigate whether one variable can be used to predict the other. For instance, if a correlation is found, one could examine whether the age at which a child first speaks sentences can be used to predict their future academic success through linear regression analysis.

However, caution is necessary! Correlations do not imply causation. Any identified correlations should be examined in greater detail and not immediately interpreted as causal relationships, even if a connection seems obvious.

However, in some cases, the nature of the variables allows for a causal relationship to be assumed from the outset. For example, if a correlation is found between age and salary, it is evident that age influences salary rather than the reverse—otherwise, it would imply that reducing one's salary could somehow make a person younger, which is clearly nonsensical.

Interpret correlation

With the help of correlation analysis two statements can be made:

• one about the direction
• and one about the strength

of the linear relationship between two metric or ordinally scaled variables. The direction indicates whether the correlation is positive or negative, while the strength indicates whether the correlation between the variables is strong or weak.

Scatter plot and correlation

Just as important as the consideration of the correlation coefficient is the graphical consideration of the correlation of two variables in a scatter diagram.

The scatter plot gives you a rough estimate of whether there is a correlation, whether it is linear or nonlinear, and whether there are outliers.

Test correlation for significance

If there is a correlation in the sample, it is still necessary to test whether there is enough evidence that the correlation also exists in the population. Thus, the question arises when a correlation coefficient can be considered statistically significant.

The significance of correlation coefficients can be tested using a t-test. As a rule, it is tested whether the correlation coefficient is significantly different from zero, i.e. linear independence is tested. In this case, the null hypothesis is that there is no correlation between the variables under consideration. In contrast, the alternative hypothesis assumes that there is a correlation.

As with any other hypothesis test, the significance level is first set, usually at 5%. If the calculated p-value is below 5 %, the null hypothesis is rejected and the alternative hypothesis applies. Thus, if the p-value is below 5%, it is assumed that there is a relationship between the variables in the population.

The t-value for testing the hypothesis is given by

where n is the sample size and r is the determined correlation in the sample. The corresponding p-value can be easily calculated in the correlation calculator on DATAtab.

Pearson correlation analysis

With the Pearson correlation analysis you get a statement about the linear correlation between metric scaled variables. The respective covariance is used for the calculation. The covariance gives a positive value if there is a positive correlation between the variables and a negative value if there is a negative correlation. The covariance is calculated as:

However, the covariance is not standardized and can assume values between plus and minus infinity. This makes it difficult to compare the strength of relationships between different variables. For this reason, the correlation coefficient, also called product-moment correlation coefficient, is calculated. The correlation coefficient is obtained by normalizing the covariance. For this normalization, the variances of the two variables involved are used and the correlation coefficient is calculated as

The Pearson correlation coefficient can now take values between -1 and +1 and can be interpreted as follows

• The value +1 means that there is an entirely positive linear relationship (the more, the more).
• The value -1 indicates that an entirely negative linear relationship exists (the more, the less).
• With a value of 0 there is no linear relationship, i.e. the variables do not correlate with each other.

Now finally the strength of the relationship can be interpreted. This can be illustrated by the following table:

To check in advance whether a linear relationship exists, scatter plots should be considered. This way, the respective relationship between the variables can also be checked visually. The Pearson correlation is only useful and purposeful if linear relationships are present.

Pearson Correlation assumptions

For Pearson correlation to be used, the variables must be normally distributed and there must be a linear relationship between the variables. The normal distribution can be tested either analytically or graphically with the Q-Q plot. Whether the variables have a linear correlation is best checked with a scatter plot. If these conditions are not met, then the Spearman correlation is used.

Spearman rank correlation

Spearman correlation analysis is used to calculate the relationship between two variables that have ordinal level of measurement. Spearman rank correlation is the non-parametric equivalent of Pearson correlation analysis. This procedure is therefore used when the prerequisites for a correlation analysis (=parametric procedure) are not met, i.e. when there is no metric data and no normal distribution. In this context it is often referred to as "Spearman correlation" or "Spearman's Rho" if Spearman rank correlation is meant.

The questions that can be treated by Spearman rank correlation are similar to those of the Pearson correlation coefficient, i.e. "Is there a correlation between two variables or characteristics". For example: "Is there a correlation between age and religiousness in the France population?

The calculation of the rank correlation is based on the ranking system of the data series. This means that the measured values are not used for the calculation, but are transformed into ranks. The test is then performed using these ranks.

For the rank correlation coefficient ρ, values between -1 and 1 are possible. If there is a value less than zero (ρ < 0), there is a negative linear correlation. If a value is greater than zero (ρ > 0), there is a positive linear relationship. If the value is zero (ρ = 0), there is no relationship between the variables. As with the Spearman correlation coefficient, the strength of the correlation can be classified as follows:

Point biserial correlation

The point biserial correlation is used when one of the variables is dichotomous, e.g. studied and not studied, and the other has metric scale level, e.g. salary.

The calculation of a point biserial correlation is the same as the calculation of the Pearson correlation. To calculate it, one of the two categories of the dichotomous variable is coded as 0 and the other as 1.

Calculate correlation analysis with DATAtab

A student wants to know if there is a correlation between the height and weight of the participants in the statistics course. For this purpose, the student drew a sample, which is described in the table below.

To analyze the linear relationships by means of a correlation analysis, you can calculate a correlation with DATAtab. First copy the table above into the statistics calculator. Then click on "Correlation" and select the two variables from the example. Finally you will get the following results.

First, you will get the null and the alternative hypothesis. The null hypothesis is: "There is no correlation between height and weight". Then you get the correlation coefficient and the p value. If you click on Summary in words, you will get the following interpretation:

Directional (one-sided) correlation hypothesis

Of course, in DATatab you can also choose to calculate a directional hypothesis.

In this case, you must first check whether the correlation is at all in the direction of the alternative hypothesis, i.e. that height and weight are positively correlated. If this is the case, the calculated p-value must be divided by two, since only one side of the distribution is considered. However, DATAtab takes care of these two steps for you. The summary in words then looks like this: