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**STUDY QUESTIONS**

- What is the value of the Pearson
*r*for the relationship between the Hamstring strength index 120Åã/s and the Triple hop index?

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

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

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

- What values for
*r*indicate the strongest possible relationships? What do those values also indicate?

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

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

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

- Which of the following sets of variables has the weakest relationship?
- Quadriceps strength index 60°/s and the Triple hop index
- Hamstring strength index 60°/s and the Carioca test
- Hamstring strength index 120°/s and the Side step test
- Quadriceps strength index 120°/s and the Shuttle run test

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

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

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**ANSWERS TO STUDY QUESTIONS**

*r*= 0.420. The*r-value*is listed in Table 5 for the relationship between the Hamstring strength index 120Åã/s and the Triple hop index.

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*r*= –0.519**. The*r-value*is listed in Table 5 for the relationship between the Quadriceps strength index120Åã/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.

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

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- 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.

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*r*= 0.655**. The*r*value listed for the Quadriceps strength index 60Åã/s and the Hop index indicates a strong, positive relationship, where the Quadriceps strength index 60Åã/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.

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- The relationship between the Quadriceps strength index 60Åã/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.

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- The Triple hop index has the strongest relationship with the Hamstring strength index 60Åã/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.

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- b. Hamstring strength index 60Åã/s and the Carioca test. The weakest relationship is between the

Hamstring strength index of 60Åã/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.

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- 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.

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- 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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**Questions to be graded**

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

- Consider
*r*= 1.00 and*r*= –1.00. Which*r-value*is stronger? Provide a rationale for your answer.

- Describe the direction of the relationship between the Hamstring strength index 60Åã/s and the Shuttle run test.
- Without using numbers, describe the relationship between the Hamstring strength index 120Åã/s and the Triple hop index.

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

- Which of the following sets of variables has the strongest relationship?
- Hamstring strength index 120°/s and the Hop index
- Quadriceps strength index 60°/s and the Carioca test
- Quadriceps strength index 120°/s and the Side step test
- Quadriceps strength index 60°/s and the Triple hop index

- In Table 5, two
*r-values*are reported as*r*= –0.498 and*r*= –0.528. Describes each*r-value*in words, indicating which would be more statistically significant, and provide a rationale for your answer.

- 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.

- The researchers stated that no significant relationship could be described between Hamstring strength indices 60Åã/s and functional stability. Given the data in Table 5, explain why not.

- Consider the relationship reported for the Quadriceps strength index 120Åã/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?

**STUDY QUESTIONS**

**1**. In Table 2, what is the numeric value given for the correlation between LOT-R Total and Negative Items?

- Describe the correlation in Question 1 using words. Is this relationship statistically significant? Provide a rationale for your answer.

- 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.

- Which two variables in Table 2 have the strongest correlation? Provide a rationale for your answer.

**TABLE** Optimism,

- 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.

- As a clinician, does knowledge of the correlation in Question 5 enhance your practice? Provide a rationale for your answer.

- 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.

- Consider two values,
*r*= –0.24 and*r*= 0.78. How would you describe them in relationship to each other?

- Compare the percentages of variance explained for the
*r-values*in Question 8.

- 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.

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**ANSWERS TO STUDY QUESTIONS**

*r*= –0.94**,*p*< 0.01 is the correlation between LOT-R Total and Negative Items.

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*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.

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- 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).

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- 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 signifi cant at *p *< 0.01 as indicated by **.

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*r*= 0.76** indicates the*r*value is statistically signifi cant at*p*< 0.01 as indicated by the key below

Table 2. Percentage of variance = *r**2 *Å~ 100 = (0.76)2 Å~ 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.

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- 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.

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*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 Å~ 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.

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*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.- The percentage of variance explained for
*r*= 0.78 is (0.78)2 Å~ 100 = 60.84%. The percentage of variance explained for*r*= –0.24 is (–0.24)2 Å~ 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.

■ **EXERCISE 24 ****Questions to Be Graded**

- What is the
*r-value*listed for the relationship between variables 4 and 9?

- Describe the correlation
*r*= –0.32** using words. Is this a statistically significant correlation? Provide a rationale for your answer.

- Calculate the percentage of variance explained for
*r*= 0.53. Is this correlation clinically important? Provide a rationale for your answer.

- 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?

- Calculate the percentage of variance explained for
*r*= 0.15. Describe the clinical importance of this relationship.

- 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.

- 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.

- Is the correlation between variables 9 and 4 significant? Is this correlation relevant to practice? Provide a rationale for your answer.

- 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*.

- 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?

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**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.