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The authors have declared that no competing interests exist.

Wrote the paper: BD JM. Software development: BD.

A valid comparison of the magnitude of two correlations requires researchers to directly contrast the correlations using an appropriate statistical test. In many popular statistics packages, however, tests for the significance of the difference between correlations are missing. To close this gap, we introduce

Determining the relationship between two variables is at the heart of many research endeavours. In the social sciences, the most popular statistical method to quantify the magnitude of an association between two numeric variables is the Pearson product-moment correlation. It indicates the strength of a linear relationship between two variables, which may be either positive, negative, or zero. In many research contexts, it is necessary to compare the magnitude of two such correlations, for example, if a researcher wants to know whether an association changed after a treatment, or whether it differs between two groups of interest. When comparing correlations, a test of significance is necessary to control for the possibility of an observed difference occurring simply by chance. However, many introductory statistics textbooks [

Even when recognizing the importance of a formal statistical test of the difference between correlations, the researcher has many different significance tests to choose from, and the choice of the correct method is vital. Before picking a test, researchers have to distinguish between the following three cases: (1) The correlations were measured in two independent groups A and B. This case applies, for example, if a researcher wants to compare the correlations between anxiety and extraversion in two different groups A and B (_{A} = _{B}). If the two groups are dependent, the relationship between them needs further differentiation: (2) The two correlations can be overlapping (_{A12} = _{A23}), i.e., the correlations have one variable in common. _{A12} and _{A23} refer to the population correlations in group A between variables 1 and 2 and variables 2 and 3, respectively. For instance, a researcher may be interested in determining whether the correlation between anxiety and extraversion is smaller than between anxiety and diligence within the same group A. (3) In the case of two dependent correlations, the two correlations can also be nonoverlapping (_{A12} = _{A34}), i.e., they have no variable in common. This case applies, for example, if a researcher wants to determine whether the correlation between anxiety and extraversion is higher than the correlation between intelligence and creativity within the same group. A researcher also faces nonoverlapping dependent correlations when investigating whether the correlation between two variables is higher before rather than after a treatment provided to the same group.

For each of these three cases, various tests have been proposed. An overview of the tests for comparing independent correlations is provided in

Fisher’s [ |
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Zou’s [ |
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Pearson and Filon’s [ |
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Hotelling’s [ |
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Williams’ [ |
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Olkin’s [ |
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Dunn and Clark’s [ |
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Hendrickson et al.’s [ |
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Steiger’s [ |
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Meng, Rosenthal, and Rubin’s [ |
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Hittner et al.’s [ |
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Zou’s [ |
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Pearson and Filon’s [ |
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Dunn and Clark’s [ |
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Steiger’s [ |
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Raghunathan, Rosenthal, and Rubin’s [ |
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Silver, Hittner, and May’s [ |
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Zou’s [ |
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Many popular statistics programs do not provide any, or only a subset of the significance tests described above. Moreover, existing programs that allow for statistical comparisons between correlations are isolated stand-alone applications and do not come with a graphical user interface (GUI). For example, DEPCOR [

With

For each case, an example of the formula passed as an argument to the

A comparison of

Some limitations of

There are two convenient ways to use

Thus,

In the following, using fictional data, examples are given for all three cases that may occur when comparing correlations.

The first example presents code for the comparison of the correlations between a score achieved on a logic task (

^{∼}logic + intelligence.a | logic + intelligence.a, + aptitude)

In this example, the test result indicates that the difference between the two correlations

The second example code determines whether the correlation between a score achieved on general knowledge questions (

^{∼}knowledge + intelligence.a | logic + + intelligence.a, aptitude [[“sample1”]])

The results of all tests lead to the convergent conclusion that the difference between the two correlations

The third example code tests whether the correlation between a score achieved on general knowledge questions (

^{∼}knowledge + intelligence.a | logic + + intelligence.b, aptitude [[“sample1”]])

Also in this example, the test results converge in showing that the difference between the two correlations

In this article, we introduced

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We would like to thank Meik Michalke for his valuable and constructive suggestions during the development of the