UNDERSTANDING PEARSON’S r, EFFECT SIZE, AND PERCENTAGE OF VARIANCE EXPLAINED
STATISTICAL TECHNIQUE IN REVIEW
Review the statistical information regarding Pearson’s Product-Moment Correlation Coefficient presented in Exercise 23. In this exercise, you will need to apply that information to gain an understanding of interpreting Pearson r results presented in a mirror-image table. A mirror-image table, as the name implies, has the same labels in the same order for both the x- and y-axes. Frequently, letters or numbers are assigned to each label, and only the letter or number designator is used to label one of the axes. To find the r value for a pair of variables, look both along the labeled or y-axis in the table below and then along the x-axis, using the letter designator assigned to the variable you want to know the relationship for, and find the cell in the table with the r value. Below is an example of a mirror-image table that compares hours of class attended, hours studying, and final grade as a percentage. The results in the table are intended as an example of a mirror-image table and are not based on research. If you were asked to identify the r value for the relationship between hours of class attended and the final grade as a percentage, the answer would be r = 0.72, and between hours studying and final grade as a percentage, the answer would be r = 0.78. The dash (–) marks located on the diagonal line of the table represent the variable’s correlation with itself, which is always a perfect positive correlation or r = +1.00.
VARIABLES
A
B
C
A. Hours of class attended
–
0.44
0.72
B. Hours studying
0.44
–
0.78
C. Final grade as a percentage
0.72
0.78
–
Effect Size of an r Value
In determining the strength of a relationship, remember that a weak relationship is r 0.5 or > −0.5. The r value is equal to the effect size or the strength of a relationship. In the table above, the relationship between hours of class attended and hours of studying is r = 0.44 and the effect size = 0.44. The effect size is used in power analysis to determine sample size for future studies. The strength of the effect size is the same as that for the r values, with a weak effect size 0.5 or > −0.5. The smaller the effect size, the greater the sample size needed to detect significant relationships in future studies. Thus the larger the effect size, the smaller the sample size that is needed to determine significant relationships. The determination of study sample sizes with power analysis is presented in Exercise 12.
Percentage of Variance Explained in a Relationship
Percentage of variance explained is a calculation based on a Pearson’s r value. The purpose for calculating the percentage of variance explained is to understand further the relationship or correlation between two variables in terms of clinical importance. To calculate the percentage of variance explained, square the r value then multiply by 100 to determine a percentage.
Formula:
r2 × 100 = % variance explained
Example:
r = 0.78 (correlation between hours studying and final grade as a percentage)
(0.78)2 × 100 = 0.6084 × 100 = 60.84% variance explained
The example above indicates that the hours studying can be used to predict 60.84% of the variance in the final course grade. Calculating the percentage of variance explained helps the researchers and consumers of research better understand the practical implications of reported results. The stronger the r value, the greater the percentage of variance explained. For example if r = 0.5, then 25% of the variance in one variable is explained by an another variable and if r = 0.6, then 36% of the variance is explained. Any Pearson’s r ≥ 0.3, which yields a 9% variance explained, is considered clinically important. Keep in mind that a result may be statistically significant (p9% indicates clinical importance.
10. r = 1.00. The dash recorded on each line forms a diagonal line across Table 2 where each item would be correlated with itself [e.g., 2. LOT-R Total with 2.(LOT-R Total)]. The relationship of an item with itself is always a perfect positive correlation, or r = +1.00.
Name:____________________________________________ Class: ____________________
Date: _________________________________________________________________________________
□ EXERCISE 24 Questions to be Graded
1. What is the r value listed for the relationship between variables 4 and 9?
2. Describe the correlation r = −0.32** using words. Is this a statistically significant correlation? Provide a rationale for your answer.
3. Calculate the percentage of variance explained for r = 0.53. Is this correlation clinically important? Provide a rationale for your answer.
4. According to Table 2, r = 0.15 is listed as the correlation between which two items? Describe this relationship. What is the effect size for this relationship, and what size sample would be needed to detect this relationship in future studies?
5. Calculate the percentage of variance explained for r = 0.15. Describe the clinical importance of this relationship.
6. Which two variables in Table 2, have the weakest correlation, or r value? Which relationship is the closest to this r value? Provide a rationale for your answer.
7. Is the correlation between LOT-R Total scores and Avoidance-Distraction coping style statistically significant? Is this relationship relevant to practice? Provide rationales for your answers.
8. Is the correlation between variables 9 and 4 significant? Is this correlation relevant to practice? Provide a rationale for your answer.
9. Consider two values, r = 0.08 and r = −0.58. Describe them in relationship to each other. Describe the clinical importance of both r values.
10. Examine the Pearson r values for LOT-R Total, which measured Optimism with the Task and Emotion Coping Styles. What do these results indicate? How might you use this information in your practice?
BONUS QUESTION
One of the study goals was to examine the relationship between optimism and psychopathology. Using the data in Table 2, formulate an opinion regarding the overall correlation between optimism and psychopathology. Provide a rationale for your answer.
(Grove 173)
Grove, Susan K. Statistics for Health Care Research: A Practical Workbook. W.B. Saunders Company, 022007. VitalBook file.
The citation provided is a guideline. Please check each citation for accuracy before use.