Four Types of Correlation Coefficients Comparison

Table

Table 1. Four Correlation Coefficients and their Comparisons.

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Statistical Method What the Statistical Method/Test Measures What type of research question would best apply to this method?
Pearson’s r This is the parametric coefficient for determining relationships between variables, including their direction and magnitude, and making inferences about correlations between them (Jackson, 2017; Polit & Beck, 2017). This coefficient is supposed to be used with ratio or interval data with normal distributions; if these assumptions do not apply, the tests below this one should be considered. In general, the method would be used with a question that asks about correlations between variables that are likely to be measured with an interval or ratio scale.
Example: is depression, as measured by the Beck Depression Inventory (ratio), correlated with anxiety, as measured by the Beck Anxiety Inventory (ratio).
Spearman’s rho (ρ) The non-parametric option to the above-described test; it is also associated with testing correlations, but it can only be applied to ordinal data. If nominal data is involved, one should review the tests below. This method is applicable to a question that aims to test for correlations between variables which would use an ordinal scale.
Example: the above-presented question that will employ specific Beck-defined categories (mild, moderate, severe depression and mild, moderate, severe anxiety) instead of ratio data.
Point-Biserial (rpb) A coefficient for correlation-testing that can be used in case one of the variables is nominal and dichotomous. The other one is supposed to be on an interval or ratio scale (Gravetter, Wallnau, & Forzano, 2018; Jackson, 2016). If both variables are nominal, the last option in the table should be used. This coefficient is meant to test the correlations between a nominal and a ratio or interval variable, and its research question needs to reflect this fact.
Example: is depression, as measured using the Beck Depression Inventory (ratio), more prevalent among men or women (nominal, dichotomous)?
phi coefficient (φ) Another correlation coefficient; it is used to test for correlations between “dichotomous and nominal variables” (Jackson, 2016, p. 163). In other words, the phi coefficient is applicable to the variables that use nominal data and have only two values. This coefficient could use a question that focuses on correlations between variables which are nominal and have only two values.
Example: is the adherence to physician recommendations pre-discharge correlated with the adherence to them post-discharge? The variables are nominal and dichotomous (one follows recommendations or does not).

Reflection and Summary

This paper is dedicated to an exercise of comparing correlation coefficients and their specific features. Table 1 summarizes the key information about them and offers examples of the questions that they can respond to; the following summary presents a reflection on the topic. The study of the four coefficients demonstrates that they are very similar, but the specific conditions of their applicability define their use by a researcher.

From the literature on the topic, it is apparent that the four correlation coefficients are meant to work with the same general types of questions; specifically, they are correlation coefficients. As explained by Polit and Beck (2017), this statement means that the coefficients can help to determine if a relationship between variables that is found within a particular sample can be generalized and inferred for the studied population as a whole. Thus, the four coefficients are meant for the inferential testing of correlations between variables, which is why all their research questions will be concerned with particular variables and potential relationships between them.

However, since the coefficients appear to reflect the types of variables that are used, the eventual questions are going to differ. In Table 1, the coefficients are arranged in the following order: the one that is meant for ratio or interval data is the first one, and it is followed by the coefficient that focuses on ordinal data and the two coefficients that can work with nominal data (Jackson, 2016; Polit & Beck, 2017). This arrangement can be useful since, as a parametric coefficient, Pearson’s r requires taking into account a greater number of assumptions and may be inapplicable to many datasets while also being the most powerful option (Polit & Beck, 2017). Upon determining that a dataset cannot use Pearson’s r, a researcher can check the non-parametric alternative (Spearman’s rho) and then the rest of the options.

Since only specific variables are suitable for particular coefficients, the presented example questions attempt to reflect the differences between them. Thus, the first question inquires about correlations between depression and anxiety as measured by well-established inventories that produce ratio data (García-Batista, Guerra-Peña, Cano-Vindel, Herrera-Martínez, & Medrano, 2018; Oh et al., 2018).

Provided that the study is carried out with a large enough sample to ensure normal distribution, this question should be able to use Pearson’s r. On the other hand, a researcher who uses the same question but instead employs ordinal data or has a small, non-normally distributed dataset would have to resort to the Spearman’s rho alternative. With the rest questions, the same conditions should be taken into account. Gender is the most obvious nominal and dichotomous variable (Jackson, 2016), which is why pairing it with another variable that uses a ratio scale would require the application of the point-biserial coefficient. Finally, two nominal variables, like the ones from the last example, require the phi coefficient.

Thus, the presented table can be used to demonstrate the differences between the coefficients in brief, and the example questions appear to illustrate their distinctions. In essence, the coefficients are the same, but they have different requirements for application, which mostly amount to the consideration of the types of data that they can test for correlations. Knowing their specific features is crucial for their correct use.

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References

García-Batista, Z., Guerra-Peña, K., Cano-Vindel, A., Herrera-Martínez, S., & Medrano, L. (2018). Validity and reliability of the Beck Depression Inventory (BDI-II) in general and hospital population of Dominican Republic. PLOS ONE, 13(6), e0199750. Web.

Gravetter, F., Wallnau, L., & Forzano, L. (2018). Essentials of statistics for the behavioral sciences (9th ed.). Boston, MA: Cengage Learning.

Jackson, S. L. (2016). Research methods and statistics: A critical thinking approach (5th ed.). Boston, MA: Cengage Learning.

Jackson, S. L. (2017). Statistics plain and simple (4th ed.). Boston, MA: Cengage Learning.

Oh, H., Park, K., Yoon, S., Kim, Y., Lee, S., Choi, Y., & Choi, K. (2018). Clinical utility of Beck Anxiety Inventory in clinical and nonclinical Korean samples. Frontiers in Psychiatry, 9, 1-10. Web.

Polit, D.F., & Beck, C.T. (2017). Nursing research: Generating and assessing evidence for nursing practice (10th ed.). Philadelphia, PA: Lippincott, Williams & Wilkins.

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