SCH7864_4CORRELATION APPLICATION AND INTERPRETATION

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Week 4: Correlation Application

Proper Reporting of Correlations

Reporting a correlation requires an understanding of the following elements: the statistical notation for a Pearson's correlation (r), the degrees of freedom (df), the correlation coefficient, and the probability value. For example, you might report:

“There was a statistically significant positive correlation between test anxiety and test scores, r(98) = .32, p < .01 (two-tailed).” 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. The degrees of freedom for Pearson's r is N − 2. For example, if there were 100 participants in the sample, then the df would be 98 (100 − 2 = 98). 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).

So, you might have a correlation of r(98) = +.20, p < .05.

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 nondirectional 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.

APA Focus of the Week: Reporting Standards in APA Format

Here are the most common statistical notations we will be using in this class, based off of Table 6.5 in the APA 7 manual (APA, 2020, pp. 183-186).

APA abbreviation or symbol	Definition
M	Mean
SD	Standard Deviation
df	Degrees of Freedom
p	Probability or Significance value
skewness	Skewness
kurtosis	Kurtosis
r	Pearson’s correlation coefficient
t	t-Test value
Levene’s F	Levene’s test of homogeneity of variance
W	Shapiro-Wilk
F	ANOVA statistic
ANOVA	Analysis of Variance

Reporting examples:

Mean and standard deviation	Group A (M = 25.63, SD= 1.77) had similar…
Pearson’s correlation	r(103) = 76.3, p <.001. There is a significant correlation between…
Levene’s assumption of homogeneity of variance	Levene’s F = .84, p = .67. The assumption of homogeneity of variance is met.
t-Test	t(98) = 22, p = .048. There is a significant difference between…
Shapiro-Wilk test of normality	W = .95, p = .03. The assumption of normality is violated.
ANOVA	F(3, 96) = 25.64, p = .032. There is a significant difference in at least one…