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< 0.3 or r< −0.3, a moderate relationship is r = 0.3 to 0.5 or −0.3 to −0.5, and a strong 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 < 0.3 or < −0.3, a moderate effect size 0.3 to 0.5 or −0.3 to −0.5, and a strong 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 (p< 0.05), but it may not represent a clinically important finding (Burns & Grove, 2005).
RESEARCH ARTICLE
Source: Hatchett, G. T., & Park, H. L. (2004). Relationships among optimism, coping styles, psychopathology, and counseling outcome. Personality and Individual Differences, 36 (8), 1755–69.
Introduction
Hatchett and Park (2004) conducted a study consisting of 96 college students to determine the relationships between optimism, coping styles, psychopathology, and counseling outcomes. Each participant filled out three questionnaires before beginning counseling: the Outcome Questionnnaire-45 (OQ-45) (measures psychopathology), the Life Orientation Test-Revised (LOT-R) (measures optimism and pessimism), and the Coping Inventory for Stressful Situations (CISS) (measures coping styles). At the termination of treatment, the OQ-45 was re-administered. The researchers reported that optimism “was negatively correlated with psychopathology, emotion-oriented coping, and the avoidance-distraction subscale from the CISS” (Hatchett & Park, 2004, p. 1762). Conversely, they report optimism to be positively correlated with task-oriented coping and the avoidance–social diversion subscales. Pessimism reportedly had the opposite or negative relationships with these same variables. The researchers reported no statistically significant correlation between optimism and counseling outcomes. “Future research might be directed at determining whether the early assessment and subsequent remediation of pessimistic thoughts leads to better outcomes. Furthermore research might ascertain whether optimists and pessimists respond differently to certain types of clinical interventions. [One] might advocate matching clinical interventions to clients’ unique personality characteristics. For example, optimists, who rely more on problem-focused coping strategies, might respond better to more active intervention strategies (e.g., problem-solving skills). On the other hand, pessimists, who report greater use of emotion-oriented coping, might respond better to more expressive and supportive therapeutic techniques” (Hatchett & Park, 2004, pp. 1766–7).
Relevant Study Results
In Table 2 in p. 175, Hatchett and Park (2004) presented the correlations among optimism (LOT-R Total and Positive Items); pessimism (Negative Items); psychopathology (OQ-45); and coping styles (Task, Emotion, Avoidance, Avoidance–Distraction, and Avoidance–Social Diversion). Table 2 is a mirror-image table with the variables numbered and labeled on the y-axis and the numbers of the variables on the x-axis. The blank spaces in the table are where the variable is correlated with itself and would be a +1.00 correlation.
TABLE 2 Intercorrelations among Optimism, Psychopathology, and Coping Styles
Variable
1
2
3
4
5
6
7
8
9
1. OQ-45 (psychopathology)
–
-0.72**
-0.59**
0.74**
-0.43**
0.76**
-0.22*
0.09
-0.45**
2. LOT-R Total (optimism)
–
0.92**
-0.94**
0.54**
-0.58**
0.11
-0.20*
0.38**
3. Positive Items (from LOT-R)
–
-0.72**
0.53**
-0.48**
0.15
-0.16
0.38**
4. Negative Items (from LOT-R)
–
-0.47**
0.58**
-0.06
0.21*
-0.32**
5. Task (coping style)
–
-0.42**
0.08
-0.09
0.22*
6. Emotion (coping style)
–
-0.02
0.21*
-0.24*
7. Avoidance (coping style)
–
0.83**
0.78**
8. Avoidance-Distraction (coping style)
–
0.36**
9. Avoidance-Social Diversion (coping style)
–
* p< 0.05.
** p<0.01.
Hatchett, G. T., & Park, H. L. (2003). Relationships among optimism, coping styles, psychopathology, and counseling outcome. Personality and Individual Differences, 36(8), p. 1762. Copyright © 2003, with permission from Elsevier.
STUDY QUESTIONS
1. In Table 2, what is the numeric value given for the correlation between LOT-R Total and Negative Items?
2. Describe the correlation in Question 1 using words. Is this relationship statistically significant? Provide a rationale for your answer.
3. Calculate the percentage of variance explained by the relationship of OQ-45 or psychopathology and Task coping style. Is this correlation clinically important? Is the correlation statistically significant? Provide a rationale for your answers.
4. Which two variables in Table 2 have the strongest correlation? Provide a rationale for your answer.
5. Is the correlation between Emotion coping style and OQ-45 or psychopathology scores statistically significant? Is it clinically important? Provide a rationale for your answers.
6. As a clinician, does knowledge of the correlation in Question 5 enhance your practice? Provide a rationale for your answer.
7. What is the effect size of the relationship between variables 3 and 8? Describe the strength of this effect size. What is the value of knowing the effect size? Discuss the percentage of variance explained by this relationship.
8. Consider two values, r = −0.24 and r = 0.78. How would you describe them in relationship to each other?
9. Compare the percentages of variance explained for the r values in Question 8.
10. What r value would you expect to have been recorded in place of each dash (–) had the researchers chosen to record a number? Provide a rationale for your answer.
ANSWERS TO STUDY QUESTIONS
1. r = −0.94**, p< 0.01 is the correlation between LOT-R Total and Negative Items.
2. r = −0.94** represents a strong, negative relationship between LOT-R (optimism) and Negative Items; therefore, as LOT-R values or optimism increase, the values of the Negative Items decrease. This r value has ** next to it, so it is statistically significant at p< 0.01, as indicated by the key below the table.
3. The correlation between OQ-45 or psychopathology and Task coping style is r = −0.43**.
Percentage of variance =r2 × 100 Percentage of variance = (-0.43)2 × 100 = 18.49 %
The relationship represented by r = −0.43 is clinically important. Scores on the OQ-45 questionnaire measuring psychopathology can be used to explain 18.49% of the variance in the Task coping style scores. The r = −0.43** is also statistically significant at p< 0.01 (see the key at the bottom of Table 2).
4. LOT-R Total (optimism) and Negative Items have the strongest relationship with r = −0.94**. This r value is the closest to −1 and the farthest value from 0.00, which indicates it is the strongest relationship in the table. The relationship is significant at p< 0.01 as indicated by **.
5. r = 0.76** indicates the r value is statistically significant at p< 0.01 as indicated by the key below Table 2. Percentage of variance = r2 × 100 = (0.76)2 × 100 = 57.76%. This correlation is clinically important with a percentage of variance greater than 9% and is actually 57.76%, indicating that the OQ-45 scores can be used to predict 57.76% of the variance in the Emotion coping style scores.
6. Knowing that scores on the psychopathology scale, OQ-45, allows the prediction of 57.76% of the variance in the emotion-based coping style scores. Thus, knowing the scores on one scale can allow prediction of scores on another scale, and that would be helpful to practicing professionals who might have time to administer one scale but not both. So the scores on the psychopathology scale provide understanding and prediction of the scores on the emotion-based coping style scale.
7. r = −0.16 is also the effect size. The effect size is negative and small for the relationship between positive items and avoidance-distraction coping style. The effect size is used in the calculation of a power analysis to determine sample size for future studies. Percentage of variance = (–0.16)2 × 100 = 2.56%. The positive items scores can only predict 2.56% of the variance in avoidance-distraction coping style scores, so this is clinically not a very important relationship due to its weak effect size and small percentage of variance explained.
8. r = 0.78 is a strong positive relationship and is the stronger relationship of the two, as r = −0.24 indicates a weak negative relationship. The r value closest to 0.00 is considered the weakest relationship. Also, r = 0.78** is more significant at p< 0.01, where r = −0.24* is significant at p< 0.05. The smaller the p value, the more significant the result.
9. The percentage of variance explained for r = 0.78 is (0.78)2 × 100 = 60.84%. The percentage of variance explained for r = −0.24 is (–0.24)2 × 100 = 5.76%. Thus, the first relationship is much more useful in clinical practice in understanding the relationship between two variables, since 60.84% of the variance is explained with this relationship versus 5.76% by the second relationship. Recall that percentage of variance >9% 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.

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