Nonparametric procedures, are not only available for hypothesis tests, they are also nonparametric ways, to measure, for instance, the association between variables. The most important of these is the Spearman rank correlation coefficient. It's often treated, as a nonparametric counterpart of the Pearson correlation coefficient. In this video, I will explain when to apply, how to calculate, and how to interpret the Spearman correlation coefficient. A correlation coefficient, is a standardized measure to express the degree by which two variables are associated. The standardization implies that it has a fixed range over which it varies, and it doesn't change when you change, for example, the units of the variables by adding a constant, or multiplication. Most correlation coefficients vary between -1 and +1. The Pearson correlation coefficient, also called the product moment correlation coefficient, is the coefficient that is used most frequently. It measures a linear association between two numerical variables. If you'd like to test the Pearson correlation coefficient, for example, to evaluate whether it's different from zero, you have to additionally assume that the two variables are bivariate normally distributed. Which implies that the scatterplot of the data, shows an approximate ellipsoidal shape. And that each of the variables separately follows a normal distribution. In general, the Pearson correlation coefficient is sensitive to outliers, and skewedness of the distribution in one or both variables. The Spearman correlation coefficient is a good replacement of the Pearson correlation, if one of the following conditions applies to your variable. They're not numerical, but one or both of the variables are ordinal. They are not linearly related, they contain one or more outliers, they don't follow a bivariate normal distribution, or you cannot check this distribution, due to lack of data. To understand the Spearman correlation, it's necessary to know what a monotonic function is. A monotonic function is one that either never increases, or never decreases, as it's independent variable increases. The following graphs illustrate monotonic functions. At the left, you see a monotonically decreasing function. In the middle a monotonically increasing function. And at the right, a function that is not monotonic. The Spearman correlation, measures the strength of a monotonic relationship between paired data. This implies, that the Spearman correlation coefficient for this data, would be +1, but also for this data, or even this data. And the same applies of course to negatively related data. You can apply the Spearman correlation coefficient, to both ordinal and numerical variables. And to interpret it, you assume that these variables are monotonically related. But to test it, for example, to see whether its value is significantly different from zero, there is no requirement on the distribution of the data, like bivariate normality, in the case of Pearson's correlation. And therefore, it's a nonparametric statistic. The Spearman's correlation is calculated, by first ranking the variables, whereby average ranks are assigned in the case of ties. Subsequently, by calculating the Pearson correlation on the ranked values of this data. Because it works on ranked data, the experiment correlation coefficient is also called the rank correlation coefficient. Let's look at an example. Here the number of ingredients, in the price at which a cake is being sold is shown. This is the Pearson correlation coefficient, expressing the strength of the linear relation between the variables. To calculate the Spearman correlation coefficient, we first determine the ranks for each variable. Using an average of the ranks for tied values. And next, we apply the same correlation equation to the ranked data, to obtain the Spearman correlation. This expresses the strength of the monotonic relation between the two variables. The Spearman correlation coefficient is higher. Let's have a look at the shape of the relationship. As you see the relation is non-linear. Up to ten ingredients, and the price increases strongly. But beyond this point, the number of ingredients does not influence price much anymore. In this case, it would be better to use the Spearman correlation, to express the strength of the relationship. There are other cases, where the Spearman correlation coefficient, would be preferred over the Pearson correlation coefficient as well. For example, in this case, where you would have ordinal data. Or the situation, with outliers in your numerical data. The interpretation of the Spearman correlation coefficient, is furthermore very similar to the Pearson correlation coefficient. Whereby, visualizing the relationship between the variables is crucial A correlation coefficient of zero does, for instance, not imply that there's no relation. Consider this case, where there's a clear parabolic relation between two variables, both the Spearman and the Pearson coefficients will be zero. Furthermore, a correlation coefficient is influenced by both the effect size, and the spread around relation that is being evaluated. So even though, these correlation coefficients are the same, the different data set tell a different story. In the graph, at the left, there is more scatter. But at the same time, a larger effect size. That is, a larger change in y with a change in x. Like any sampling statistic, correlation coefficient can be tested. And also confidence intervals can be calculated. The most frequently applied test, is to evaluate if a value for correlation coefficient is significantly different from zero. The null hypothesis for this test states, that the correlation coefficient has a population parameter of zero and the alternative hypothesis can be one-sided, that it is higher or lower than zero, or two sided that is different from zero. There are several equations to describe the sampling distribution of the Spearman correlation, as well as statistical tables, which gives the cumulative probabilities associated with the value of correlation coefficient, and the number of data paths used for its calculation. We will not go further into the details of these calculations here. They are equivalent to those of a one sample proportion. I hope you understood the following from this video: The experiment coalition coefficient measures the degree of monotuous association between two variables with ordinal or quantitive measurement levels. It's value ranges from -1 for a negative association, to +1 for a positive association. It's calculated by first ranking each of the variables, and then determining the Pearson correlation coefficient for this rank. In comparison to the Pearson correlation coefficient, the Spearman correlation is less to outliers. But at the downside, also about 10% less powerful when measuring a linear trend. Even though, the interpretation of the two correlation coefficient, appears to be similar when considered superficially, they both measure a degree of association. They're quite different when considering the details. The Pearson correlation measures the degree of linear association. Whereas the Spearman correlation coefficients, measured the degree of monotonous association which can be both linear and non-linear.