Application of Correlation

Proper Reporting of Correlations Reporting a correlation in proper APA style requires an understanding of the following elements: the statistical notation for a Pearson’s correlation ( r), the degrees of freedom, the correlation coefficient, the probability value, and the effect size. Consider the following example from Warner (2013, pp. 309–310): Only the correlation between commitment and length of relationship was statistically significant, r(116) = +.20, p < .05 (two-tailed). The r2 was .04; thus, only about 4% of the variance in commitment scores could be predicted from the length of the relationship; this is a weak positive relationship. r, Degrees of Freedom, and Correlation Coefficient The statistical notation for Pearson’s correlation is r, and following it is the degrees of freedom for this statistical test (116). The degrees of freedom for Pearson’s r is N − 2. There were 118 participants in the sample cited above (118 − 2 = 116). Note that SPSS output for Pearson’s r provides N, so you must subtract 2 from N to correctly report degrees of freedom. Next is the actual correlation coefficient including the sign. After the correlation coefficient is the probability value ( p). Probability Values Prior to the widespread use of SPSS and other statistical software programs, p values were often calculated by hand. The convention in reporting p values was to simply state, p < .05 to reject the null hypothesis and p > .05 to not reject the null hypothesis. However, SPSS provides an exact probability value that should be reported instead. Hypothetical examples would be p = .02 to reject the null hypothesis and p = .54 to not reject the null hypothesis (round exact p values to two decimal places). One confusing point of SPSS output is that highly significant p values are reported as .000, because SPSS only reports probability values out to three decimal places. Remember that there is a “1” out there somewhere, such as p = .000001, as there is always some small chance that the null hypothesis is true. When SPSS reports a p value of .000, report p < .001 and reject the null hypothesis. The “(two-tailed)” notation after the p value indicates that the researcher was testing a non-directional alternative hypothesis ( H1: rXY ≠ 0). He or she did not have any a priori justification to test a directional hypothesis of the relationship between commitment and length of the relationship. In terms of alpha level, the region of rejection was therefore 2.5% on the left side of the distribution and 2.5% on the right side of the distribution (2.5% + 2.5% = 5%, or alpha level of .05). A “(one-tailed)” notation indicates a directional alternative hypothesis. In this case, all 5% of the region of rejection is established on either the left (negative) side ( H1: rXY < 0) or the right (positive) side ( H1: rXY > 0) of the distribution. A directional hypothesis must be justified prior to examining the results. In this course, we will always specify a two-tailed (non-directional) test, which is more conservative relative to a one-tailed test. The advantage is that a non-directional test detects relationships or differences on either side of the distribution, which is recommended in exploratory research. Effect Size Effect sizes provide additional context for the strength of the relationship in correlation. Effect sizes are important because any non-zero correlation will be statistically significant if the sample size is large enough. After the probability value is stated, provide the r2 effect size and interpret it as small, medium, or large. It is good form to report the effect size for both significant and non-significant statistics for meta-analyses (that is, statistical studies that combine the results across multiple independent research studies), but in journal articles where space is limited, authors will often report effect sizes only for statistics that reject the null hypothesis. The Warner text provides a “Results” example at the end of each chapter for all statistics studied in this course. You are encouraged to review these examples and follow their structure when writing up Section 4, “Interpretation,” of the DAA Template. Reference Warner, R. M. (2013). Applied statistics: From bivariate through multivariate techniques (2nd ed.). Thousand Oaks, CA: Sage. Use your IBM SPSS Statistics Step by Step text to complete the following: Read Chapter 10, “Bivariate Correlation,” pages 139–148. This reading addresses the following topics: Interpreting correlations. Linear versus curvilinear relationships. Causality. SPSS instructions for calculating correlations. Generating SPSS output. Learning Components This activity will help you achieve the following learning components: Analyze the assumptions of correlation. Interpret the correlation output.