On the whole, the four correlation coefficients in question are used to assess the relationship between two variables. All these analyses produce a number ranging from -1 to 1, where -1 denotes a perfect negative correlation (a linear or monotonic function), 1 indicates a perfect positive correlation (a linear or monotonic function), and 0 indicates no correlation; the closer the coefficient is to -1 or 1 and the farther it is from 0, the more the relationship approaches the function (Warner, 2013).
Which correlation coefficient to use depends on the properties of the variables to be analyzed. Often, a Pearson correlation coefficient will be used. However, it requires several assumptions to be met, namely: (1) interval/ratio levels of measurement of both variables; (2) a linear relationship between the variables (and not, e.g., curvilinear); (3) the absence of significant outliers; and (4) approximately normal distribution of each of the variables (Field, 2013; Warner, 2013).
If only the first assumption is not met, namely, in the case when one of the variables is dichotomous, then the point-biserial correlation should be used. It should be noted that in this case, the direction of the relationship will depend on the coding of the dichotomous variable (Field, 2013).
If the first assumption is not met, and at least one of the variables is at least ordinal; or if there are many significant outliers in the data and it is not desired to remove them; or if the distribution is not normal – then, Spearman correlation will often be used. It is a non-parametric version of the Pearson correlation; it ranks the data and then calculates the Pearson correlation coefficient for that ranked data (Field, 2013). However, due to the ranking, part of the information is lost, and the Spearman correlation coefficient only shows how close the relationship is to some monotonic function (instead of just a linear function).
Finally, Kendall rank correlation should be used in circumstances similar to those when a Spearman correlation ought to be utilized, but when, in addition to that, a ranking of the data produces many tied ranks (i.e., many values in the data have the same rank). The interpretation of this coefficient is similar to that of the Spearman correlation coefficient (Field, 2013).
References
Field, A. (2013). Discovering statistics using IBM SPSS Statistics (4th ed.). Thousand Oaks, CA: SAGE Publications. Web.
Warner, R. M. (2013). Applied statistics: From bivariate through multivariate techniques (2nd ed.). Thousand Oaks, CA: SAGE Publications. Web.