The document discusses Pearson's product-moment correlation coefficient (r) and how it is used to examine the relationship between two variables measured at the interval or ratio level. It provides information on how to interpret the strength (weak, moderate, strong) and direction (positive, negative) of relationships based on the r value. It also describes how to calculate the percentage of variance explained from the r value to understand the practical significance of relationships.

1. EXERCISE 23 PEARSON'S PRODUCT-MOMENT
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,

2. 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

3. “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

4. Index 20°/s
Hop index 3
1
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 3
1
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 3
1
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 3
1
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 3
1
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

5. * 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.
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 x- and y-axes. Frequently, letters or numbers are
assigned to each
label, and only the letter or number designator is used to label
one of the axes. To find the r value for a pair of variables, look
both along the
labeled or y-axis in the table below and then along the x-axis,
using the letter designator assigned to the variable you want to
know the
relationship for, and find the cell in the table with the r value.

6. 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,

7. 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.
173
174
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

8. 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

9. 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

10. In Table 2 in p. 175, Hatchett and Park (2004) presented the
correlations among optimism (LOT-R Total and Positive Items);
pessimism
(Negative Items); psychopathology (OQ-45); and coping styles
(Task, Emotion, Avoidance, Avoidance–Distraction, and
Avoidance–Social
Diversion). Table 2 is a mirror-image table with the variables
numbered and labeled on the y-axis and the numbers of the
variables on the x-axis.
The blank spaces in the table are where the variable is
correlated with itself and would be a +1.00 correlation.
TABLE 2 Intercorrelations among Optimism, Psychopathology,
and Coping Styles
Variable 1 2 3 4 5 6 7 8 9
1. OQ-45 (psychopathology) – -0.72** -0.59** 0.74** -0.43**
0.76** -0.22* 0.09 -0.45**
2. LOT-R Total (optimism) – 0.92** -0.94** 0.54** -0.58**
0.11 -0.20* 0.38**
3. Positive Items (from LOT-R) – -0.72** 0.53** -0.48** 0.15

11. -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.