CORRELATION COEFFICIENT – PEARSON’S PRODUCT-MOMENT

CORRELATION COEFFICIENT – PEARSON’S PRODUCT-MOMENT

STATISTICAL TECHNIQUE IN REVIEW

Many studies are conducted to identify relationships between or among variables. The correlational coefficient is the mathematical expression of the relationship studied. Three common analysis techniques are used to examine relationships: Spearman Rank-Order Correlation or rho, Kendall’s Tau or tau, and the Pearson’s Product-Moment Correlation Coefficient or r. Spearman and Kendall’s Tau are used to examine relationships with ordinal level data. Pearson’s Correlation Coefficient is the most common correlational analysis technique used to examine the relationship between two variables measured at the interval or ratio level.

Relationships are discussed in terms of direction and strength. The direction of the relationship is expressed as either positive or negative. A positive or direct relationship exists when one variable increases as does the other variable increases, or when one variable decreases as the other decreases. Conversely, a negative or inverse relationship exists when one variable increases and the other variable decreases. The strength of a relationship is described as weak, moderate, or strong. Pearson’s r is never greater than −1.00 or +1.00, so an r value of −1.00 or +1.00 indicates the strongest possible relationship, either negative or positive, respectively. An r value of 0.00 indicates no relationship. To describe a relationship, the labels weak (r< 0.3), moderate (r = 0.3 to 0.5), and strong (r> 0.5) are used in conjunction with both positive and negative values of r. Thus, the strength of the negative relationships would be weak with r< −0.3, moderate with r = −0.3 to −0.5, and strong with r> −0.5 (Burns & Grove, 2007).

RESEARCH ARTICLE

Source:Keays, S. L., Bullock-Saxton, J. E., Newcombe, P., &Keays, A. C. (2003). The relationship between knee strength and functional stability before and after anterior cruciate ligament reconstruction. Journal of Orthopedic Research, 21 (2), 231–7.

Introduction

Keays et al. (2003) conducted a correlational study to determine “the relationship between muscle strength and functional stability in 31 patients pre- and postoperatively, following a unilateral anterior cruciate ligament rupture” (Keays et al., 2003, p. 231). The results of the study showed a significant positive correlation between quadriceps strength indices and functional stability, both before and after surgery. No significant relationship was demonstrated between hamstring strength indices 60°/s and functional stability, as presented in table 5.

Relevant Study Results

“Patients with an unstable knee as a result of an anterior cruciate ligament (ACL) rupture rely heavily on muscle function around the joint to maintain dynamic stability during functional activity. It is uncertain which muscles play the decisive role in functional stability or exactly which aspect of muscle function is most critical” (Keays et al., 2003, p. 231). “The aim of this study was to assess the relationship between muscle strength and functional stability of 31 patients pre- and postoperatively, following unilateral ACL ligament rupture” (Keays et al., 2003, p. 231). “To assess the relationship between maximum isokinetic strength and functional performance Pearson’s correlations (r) were computed. … Due to the number of correlations computed, and therefore the increased likelihood that chance results may be evident, a more conservative significance level of α = 0.01 was adopted to control for increased Type 1 error” (see table 5; Keays et al., 2003, pp. 232–3).

TABLE 5 Pearson’s Product-Moment Correlation between Strength Indices and Function after Surgery

n

Quadriceps Strength Index 60°/s

Hamstring Strength Index 60°/s

Quadriceps Strength Index 120°/s

Hamstring Strength Index 20°/s

Hop index

31

r = 0.655**

r = 0.247

r = 0.744**

r = 0.431*

Sig. (two tailed)

p = 0.000

p = 0.080

p = 0.000

p = 0.016

Triple hop index

31

r = 0.619**

r = 0.342

r = 0.742**

r = 0.420*

Sig. (two tailed)

p = 0.000

p = 0.060

p = 0.000

p = 0.019

Shuttle run test

31

r = −0.498**

r = −0.149

r = −0.457**

r = −0.178

Sig. (two tailed)

p = 0.004

p = 0.424

p = 0.010

p = 0.338

Side step test

31

r = −0.528**

r = −0.124

r = −0.519**

r = 0.238*

Sig. (two tailed)

p = 0.002

p = 0.506

p = 0.003

p = 0.198

Carioca test

31

r = −0.474*

r = −0.047

r = −0.510**

r = 0.267

Sig. (two tailed)

p = 0.000

p = 0.802

p = 0.003

p = 0.146

* Correlation is significant at the 0.05 level (two tailed).

** Correlation is significant at the 0.01 level (two tailed).

Keays, S. L., Bullock-Saxton, J. E., Newcombe, P., &Keays, A. C. (2003). The relationship between knee strength and functional stability before and after anterior cruciate ligament reconstruction. Journal of Orthopaedic Research, 21(2), 235. Copyright © 2003, with permission from The Orthopaedic Research Society.

STUDY QUESTIONS

1. What is the value of the Pearson r for the relationship between the Hamstring strength index 120°/s and the Triple hop index?

2. What is the value of the Pearson r for the relationship between the Quadriceps strength index 120°/s and the Side step test? Is this r value significant?

3. The closer the value of r to 0.00 the stronger the relationship in a study. Is this statement true or false? Provide a rationale for your answer.

4. What values for r indicate the strongest possible relationships? What do those values also indicate?

5. Without using numbers, describe the relationship between the Quadriceps strength index 60°/s and the Hop index.

6. Describe the direction and strength of the relationship between the Quadriceps strength index 60°/s and the Triple hop index.

7. Which variable has the strongest relationship with the Hamstring strength index 60°/s? Explain the basis for your answer. Is this r value significant?

8. Which of the following sets of variables has the weakest relationship?

a. Quadriceps strength index 60°/s and the Triple hop index

b. Hamstring strength index 60°/s and the Carioca test

c. Hamstring strength index 120°/s and the Side step test

d. Quadriceps strength index 120°/s and the Shuttle run test

9. Can the Pearson r prove causality between variables? Provide a rationale for your answer.

10. Consider r = −0.72 and r = –.72. Describe any differences or similarities between these r values.

ANSWERS TO STUDY QUESTIONS

1. r = 0.420. The r value is listed in for the relationship between the Hamstring strength index 120°/s and the Triple hop index.

2. r = −0.519**. The r value is listed in for the relationship between the Quadriceps strength index 120°/s and the Side step test. The ** indicate that the r value is statistically significant since its probability or p = 0.003, which is smaller than the significance level set at 0.01. The ** indicate the level of significance that is identified in the key below

3. False. An r value of 0.00 indicates no relationship exists, so the closer the r value is to zero, the smaller the relationship.

4. The r values of +1.00 and −1.00 both indicate the strongest possible relationships among variables. Positive (+) 1.00 is the strongest or perfect positive relationship and indicates that variables change together, either increasing or decreasing simultaneously. Negative (–) 1.00 is the strongest or perfect negative relationship and indicates that variables change in opposite directions: as one variable increases another variable decreases. These extreme values are not found in studies since no variables have perfect positive or negative relationships.

5. r = 0.655**. The r value listed for the Quadriceps strength index 60°/s and the Hop index indicates a strong, positive relationship, where the Quadriceps strength index 60°/s increases as the Hop index increases. Also, the relationship is significant at p< 0.000, and this p value is less than α = 0.01, so the r value is statistically significant. 6. The relationship between the Quadriceps strength index 60°/s and the Triple hop index is r = 0.619**. A positive or direct relationship exists between these two variables, indicating that the Quadriceps strength and Triple hop indices either increase or decrease together. This is a strong relationship since the r> 0.5. The r value is also statistically significant since p = 0.000 and this p value is less than α = 0.01.

7. The Triple hop index has the strongest relationship with the Hamstring strength index 60°/s with an r = 0.342. Recall that the closer the value of r to 1.00 or −1.00, the stronger the relationship being described. This relationship is not significant since it has probability or p = 0.060 and this value is greater than α = 0.01.

8. b. Hamstring strength index 60°/s and the Carioca test. The weakest relationship is between the Hamstring strength index of 60°/s and the Carioca test with an r = −0.047. The Answers a, c, and d had r values of 0.619, 0.238, and −0.457, respectively. Answer b is correct as its r value is the closest to 0.00.

9. The Pearson r does not prove causality between variables; it merely explains the strength and direction of the relationship between two variables. Relationships indicate that two variables are linked to each other but not that one variable brings about or causes the other. Causality indicates a strong relationship between two variables, but one of the variables must always precede the other in time and be present when the effect occurs. With causality, you manipulate the independent variable to create an effect on the dependent variable.

10. Both r values (r = −0.72 and r = –.72) have the same mathematical meaning, signifying a strong, negative relationship between two variables. Researchers are trending toward dropping the leading zeros before decimal points. Clinically, it has become important to use a leading zero prior to decimal points. In fact, the Joint Commission on Healthcare Organizations has mandated that the leading zero be present before decimals to alert the health care professional that the number is a decimal. Following this in clinical practice decreases the number of medication errors made.

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□ EXERCISE 23 Questions to be Graded

1. What is the r value for the relationship between Hamstring strength index 60°/s and the Shuttle run test? Is this r value significant? Provide a rationale for your answer.

2. Consider r = 1.00 and r = −1.00. Which r value is stronger? Provide a rationale for your answer.

3. Describe the direction of the relationship between the Hamstring strength index 60°/s and the Shuttle run test.

4. Without using numbers, describe the relationship between the Hamstring strength index 120°/s and the Triple hop index.

5. Which variable has the weakest relationship with the Quadriceps strength index 120°/s? Provide a rationale for your answer.

6. Which of the following sets of variables has the strongest relationship?

a. Hamstring strength index 120°/s and the Hop index

b. Quadriceps strength index 60°/s and the Carioca test

c. Quadriceps strength index 120°/s and the Side step test

d. Quadriceps strength index 60°/s and the Triple hop index

7. In, two r values are reported as r = −0.498 and r = −0.528. Describe each r value in words, indicating which would be more statistically significant, and provide a rationale for your answer.

8. The researchers stated that the study showed a positive, significant correlation between Quadriceps strength indices and pre- and postoperative functional stability. Considering the data presented in the do you agree with their statement? Provide a rationale for your answer.

9. The researchers stated that no significant relationship could be described between Hamstring strength indices 60°/s and functional stability. Given the data in explain why not.

10. Consider the relationship reported for the Quadriceps strength index 120°/s and the Hop index (r = 0.744**, p = 0.000). What do these r and p values indicate related to statistical significance and clinical importance?

(Grove 167)

Grove, Susan K. Statistics for Health Care Research: A Practical Workbook. W.B. Saunders Company, 022007. VitalBook file.

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