EXERCISE 23 PEARSON'S PRODUCTMOMENT CORRELATION COEFFICIENT 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 = 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 ProductMoment 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 Table 5 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 Table 5 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 Table 5. 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 = 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 = 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 = 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. Name:____________________________________________ Class: ____________________ Date: _______________________________________________________________________ __________ 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 Table 5, 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 Table 5, 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 Table 5, 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. EXERCISE 24 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 xand 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 (OQ45); 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 = r 2 100 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 1. Questions to be Graded 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. Answers: Exercise: 23 1. The relationship between Hamstring strength index 60 degree/s and the Shuttle run test is a weak negative relationship due to r = 0.149 listed in Table 5 since r <-0.3 per Pearson's Correlation Coefficient. The r value is not statistically significant since p = 0.424 > 0.01. 2. 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 3. The relationship between Hamstring strength index 60 degree/s and the Shuttle run test is a negative relationship due to r moves away from zero in negative direction and r = 0.149, the strength of this negative relationships would be weak with r <-0.3.means when Hamstring strength index 60 degree/s increases and the Shuttle run test decreases. 4. The relationship between the Hamstring strength index 120/s and the Triple hop index is r = 0.420. This is a moderate positive relationship since the r value falls between 0.3 to 0.5. The r value is also statistically significant since p = 0.019 and this p value is less than = 0.05 5. The weakest relationship between Shuttle run test and Quadriceps strength index 120/s with an r = 0.457** and r value is the closest to 0.00. So the closer the r value is to zero, the smaller the relationship. 6. a. r = 0.431* b. r = 0.474* c. r = 0.519** d. r = 0.619** strongest relationship as its r value is the closest to +1.00. 7. r = 0.498** and p=0.004. The r value listed for the Quadriceps Strength Index 60/s and Shuttle run test and r=-0.498 moderate negative relationship because r falls between- 0.3 to -0. 5. 8. No. Based on the data presented on Table 5, I do not agree with the statement as only two of the functional stability tests,(Hop Index and Triple hop index) have a positive significant relationship with the Quadriceps strength indices as they have positive r values and p values less than the significance level set at = 0.01. The other three functional stability tests (Shuttle run test, Side step test and Carioca test) have a negative significant relationship with the Quadriceps Strength indices as they have negative r values with p values less than the significance level set at = 0.01 9. This is true since the p.values for Hamstring strength indices 60/s and all variables were greater than 0.05. The p-values were 0.08, 0.06, 0.424, 0.506 , and 0.802. These p-values do not represent significance at the 5% level 10.They indicate a strong positive statistical significance, since the r value is clo se to 1.0, and the p value is 0.0 < 0.05. Clinically it shows that as Quadriceps s trength index 120/s increases so does the Hop index. This is important to educate patients on the importance of strength training and exercise Exercise: 24 1. R=-0.32, p<0.01 It is an effective negative size. Percentage of variance =(0.32)sq x100=10.24. This is considered a good relationship with good variance. 2. This represents a moderate negative relationship. It is statistically signifi cant at the 1% level of significance. (** p < 0.01) 3. variance = r2 = (0.53)2 = 0.2809 = 28.1% This is clinically important since any variance over 9% is considered clin ically important 4. This is between variables 3 and 7. It represents correlation between Posit ive Items and Avoidance. There is a weak correlation here because r < 0.3. The effect size is 0.15. The sample size required in this case would be 27 3 (given a power of 0.8). 5. r = 0.15 is listed as the correlation between variables 3 (Positive items) and 7 (Avoidance coping style). There is a weak (r < 0.3) positive relationship between the two variables indicating that the two variables increase and decrease together. r = 0.15 is also the effect size. The effect size is positive and small for the relationship between the Positive items (variable 3) and the Avoidance coping style (variable 7). The effect size is used in the calculation of a power analysis to determine the sample size for future studies. Therefore a large sample size would be needed to detect this relationship in future study because 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. 6. r = 0.15, Percentage of variance = r 2 x 100 Percentage of variance = (0.15) 2 x 100 = 2.25% The relationship represented by r = 0.15 yields a 2.25% of variance explained and is thus not a clinically important relationship as it has a small percentage of variance explained which is lower than the 9% variance explained threshold for clinical importance. 7. The relationship between LOT-R Total scores and Avoidance - Distraction coping style has an r value = 0.2* as listed in Table 2. This represents a weak positive relationship between the two variables indicating that the LOT-R Total scores and Avoidance -Distraction coping styles will either increase or decrease together. The r value has a * next to it, so it is statistically significant at P < 0.05 as indicated by the key below the table. The percentage of variance explained for r = 0.2* is (0.2) 8. r =- 0.32** is statistically significant at the 1% level of significance. It re presents the correlaton between negative items and avoidance-social dive rsion (coping style). This is relevant since it is very possible that this cop ing style may be related to negative items. 9. r = 0.08 represents a weak positive relationship, while r = -0.58 represent s a strong negative relationship. The absolute value of the r value closer to 1.0 represents the strongest relationship, meaning that r =- 0.58 shows stronger/correlation. In this study r = -0.58 has statistical significance at t he 1% level while r = 0.08 doesn't have any statistical significance. 10.The results indicate a strong positive relationship between positivity and t ask (coping style) where r = 0.54**, and is also statistically significant at the 1% level. It also indicates a strong negative relationship. .....It was nice working for you. Please rate my answer..... EXERCISE 23 PEARSON'S PRODUCTMOMENT CORRELATION COEFFICIENT 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 = 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 ProductMoment 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 Table 5 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 Table 5 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 Table 5. 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 = 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 = 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 = 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. Name:____________________________________________ Class: ____________________ Date: _______________________________________________________________________ __________ 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 Table 5, 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 Table 5, 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 Table 5, 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. EXERCISE 24 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 xand 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 (OQ45); 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 = r 2 100 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 1. Questions to be Graded 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. Answer to bonus question: From Table 2, the relationship between psychopathology (OQ-45) and optimism (LOT-R Total and positive items) are listed with an r value = -0.72** (LOT-R Total) and r = -0.59** (Positive Items). This shows that there is a strong negative relationship between psychopathology and optimism (both LOT-R Total and Positive Items); therefore the variables change in opposite directions; as one variable increases, the other variable decreases. The LOT-R Total which measured optimism, r = -0.72**. The r value has ** next to it, so it is statistically significant at p < 0.01, as indicated by the key below the table. The percentage of variance explained for r = -0.72** is (-0.72)2x 100 = 51.84%. This means that he LOT-R Total which measured optimism can be used to predict 51.84% of the variance in the psychopathology scores. This relationship is clinically important as it is greater than the 9%variance explained threshold for clinical importance. Therefore, this relationship is very relevant to practice. For the positive items optimism relationship with psychopathology, r = -0.59**. This value has** next to it, so it is statistically significant at p < 0.01, as indicated by the key below the table. The percentage of variance explained for r = -0.59** is (-0.59)2x 100 = 34.81%. This means that the LOT-R Total which measured optimism can be used to predict 34.81% of the variance in the psychopathology scores. This relationship is clinically important as it is greater than the 9%variance explained threshold for clinical importance. Therefore, this relationship is very relevant to practice. Answers: Exercise: 23 1. The relationship between Hamstring strength index 60 degree/s and the Shuttle run test is a weak negative relationship due to r = 0.149 listed in Table 5 since r <-0.3 per Pearson's Correlation Coefficient. The r value is not statistically significant since p = 0.424 > 0.01. 2. 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 3. The relationship between Hamstring strength index 60 degree/s and the Shuttle run test is a negative relationship due to r moves away from zero in negative direction and r = 0.149, the strength of this negative relationships would be weak with r <-0.3.means when Hamstring strength index 60 degree/s increases and the Shuttle run test decreases. 4. The relationship between the Hamstring strength index 120/s and the Triple hop index is r = 0.420. This is a moderate positive relationship since the r value falls between 0.3 to 0.5. The r value is also statistically significant since p = 0.019 and this p value is less than = 0.05 5. The weakest relationship between Shuttle run test and Quadriceps strength index 120/s with an r = 0.457** and r value is the closest to 0.00. So the closer the r value is to zero, the smaller the relationship. 6. a. r = 0.431* b. r = 0.474* c. r = 0.519** d. r = 0.619** strongest relationship as its r value is the closest to +1.00. 7. r = 0.498** and p=0.004. The r value listed for the Quadriceps Strength Index 60/s and Shuttle run test and r=-0.498 moderate negative relationship because r falls between- 0.3 to -0. 5. 8. No. Based on the data presented on Table 5, I do not agree with the statement as only two of the functional stability tests,(Hop Index and Triple hop index) have a positive significant relationship with the Quadriceps strength indices as they have positive r values and p values less than the significance level set at = 0.01. The other three functional stability tests (Shuttle run test, Side step test and Carioca test) have a negative significant relationship with the Quadriceps Strength indices as they have negative r values with p values less than the significance level set at = 0.01 9. This is true since the p.values for Hamstring strength indices 60/s and all variables were greater than 0.05. The p-values were 0.08, 0.06, 0.424, 0.506 , and 0.802. These p-values do not represent significance at the 5% level 10.They indicate a str