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Pearson's product moment correlation coefficient, or Pearson's r was developed by Karl Pearson (1948) from a related idea introduced by Sir Francis Galton in the late 1800's. In addition to being the first of the correlational measures to be developed, it is also the most commonly used measure of association. All subsequent correlation measures have been developed from Pearson's equation and are adaptations engineered to control for violations of the assumptions that must be met in order to use Pearson's equation (Burns & Grove, 2005; Polit & Beck, 2006). Pearson's r measures the strength, direction and probability of the linear association between two interval or ratio variables.

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Running head: PEARSON'S PRODUCT MOMENT CORRELATION

Pearson's Product Moment Correlation: Sample Analysis

Jennifer Chee

University of Hawaii at Mānoa School of Nursing

Introduction

Pearson's product moment correlation coefficient, or Pearson's r was developed by Karl

Pearson (1948) from a related idea introduced by Sir Francis Galton in the late 1800's. In

addition to being the first of the correlational measures to be developed, it is also the most

commonly used measure of association. All subsequent correlation measures have been

developed from Pearson's equation and are adaptations engineered to control for violations of

the assumptions that must be met in order to use Pearson's equation (Burns & Grove, 2005; Polit

& Beck, 2006). Pearson's r measures the strength, direction and probability of the linear

association between two interval or ratio variables.

Pearson's Product Moment Correlation Coefficient - Pearson's r

Pearson's r is a measure of the linear relationship between two interval or ratio

variables, and can have a value between -1 and 1. It is the same measure as the point-biserial

correlation; a measure of the relationship between a dichotomous (yes or no, male or female) and

an interval/ratio variable (Cramer, 1998).

The advantage of using Pearson's r is that it is a simple way to assess the association

between two variables; whether they share variance (covary), if the relationship is positive or

negative, and the degree to which they correlate. The disadvantages of using Pearson's r is that it

can not identify relationships that are not linear, and may show a correlation of zero when the

correlation has a relationship other than a linear one. Additionally the types of variable that can

be evaluated are limited.

In addition to Pearson's r, semipartial and partial correlation can be employed in order

to estimate the relationship between an outcome and predictor variable after controlling for the

effects of additional predictors in the equation.

. "Pearson's correlation is the ratio of the variance shared by two variables" (Cramer,

1998, p. 137). A means of illustrating correlation is with a Venn diagram (Figure 1). Each circle

represents the amount of variance of each variable. In Figure 1a the circles do not overlap,

indicating that there is no correlation between the two variables. In Figure 1b the two circles

overlap, the size of the correlation is reflected in the degree of overlapping area. In a perfect

correlation (Figure 1c) the circles would overlap each other completely.

There are multiple formulae that can be used to compute Pearson's r :

For small samples, Pearson's r may be calculated manually using:

!!Σ!" − Σ! Σ!

[!Σ!!! – Σ!! ] [!Σ!!! −! Σ!! ]

Where:

r = Pearson's correlation coefficient

n = number of paired scores

X = score of the first variable

Y = score of the second variable

XY = the product of the two paired scores

Or similarly:

covariance!of ! variable!! !and !!

variance!of! variable!!!×! variance!of ! variable!!

Analogously:

covariance!of! variable!!!and !!

!"#$%#&%!!"#$%&$'( !!" ! !"#$"%&'!! !× !( !"#$%#&%!!"#$%&$'( !!" ! !"#$"%&'!!) !

"The larger the covariance is, the stronger the relationship. The covariance can never be bigger

that the product of the standard deviation of the two variables" (Cramer, 1998, p. 139).

The denominator in the equations is always positive, however the numerator may be

positive, zero, or negative, "…enabling r to be positive, negative, or zero respectively" (Zar,

1999, p. 378).

Assumptions

Of Primary importance are linearity and normality. Pearson's r requires that interval data

be used to determine a linear relationship. Further assumptions must be met in order to establish

statistical significance, "…for the test statistic to be valid the sample distribution has to be

normally distributed" (Filed, 2009, p. 177). Homoscedasticity assumes that the error at each level

of the independent variable is constant. A violation of the assumption of homoscedasticity

increases the chances of obtaining a statistically significant result even though H0 is true.

Significant Testing of Pearson's r

In order to infer that the calculated r is applicable to the population from which the

sample was drawn, statistical analysis must be performed to determine whether the coefficient is

significantly different from zero.

Th e hypothesis that the population correlation coefficient is 0, and may be computed by

calculating t :

!!−2

1−!!

Where:

r = Pearson's product-moment correlation coefficient

n = sample size of paired scores

df = n - 2

If the sample size is small, a high correlation coefficient (close to -1 or 1), may be non-

significant. Contrariwise, a large sample may have a statistically significant r but have no clinical

significance. Subsequently it is important to consider both the magnitude of the correlation

coefficient, the significance of the t-test, and the context of the research question.

Coefficient of Determination

By squaring the correlation coefficient r, the total variability in Y can be accounted for

after regressing Y on X; r 2 can be considered to be a measure of the strength of the linear

relationship. The resulting value when multiplied by 100 results in a percent variance, e.g., if the

correlation coefficient for X and Y is r = .50, then r2 = (.50)(.50) = .25 = .25(100) = 25%. X

explains 25% of the variability in Y (Zar, 1999).

The Matrix

The output from a statistical program such as SPSS contains a single table, the

correlation matrix. The correlation analysis computes the correlation coefficient of a pair of

variables - or if the analysis contains multiple variables - the matrix contains the results for all

variables under consideration, and is obtained by performing bivariate correlational analysis on

every possible pairwise combination of variables in the dataset. "On the matrix, the r value and

the p value are given for each pair of variables. At a diagonal through the matrix are variables

correlated with themselves" (Burns & Grove, 2005, p. 488), the value of r for these correlations

is equal to 1 and the p value is .000 because a variable is perfectly related to itself.

Pearson's Partial Correlation

A statistically significant association between two measures does not mean that the two

variables have a causal relationship, e.g. if students who spend more time in the simulation

laboratory were found to have higher scores during skills evaluations, this association does not

necessarily imply a causal link between hours in the 'sim lab' and higher marks on evaluations,

it is conceivable that the relationship is specious (Zar, 1999). Such relationships are referred to as

spurious. Such spurious associations are the result of one or more other factors that are

"genuinely related to these two variables" (Cramer, 1998, p. 156). To continue the example, a

student's curricular progression may be related to both hours in the 'sim lab' and higher marks

on evaluations. When curricular progression is taken into consideration there may be no

association between the other two variables. A more likely example of a spurious relationship is

one in which part but not all of the association between two measures may be due to one or more

other factors, e.g., students who spend more time in the simulation laboratory may have had

more exposure to nursing skills secondary to their progression through the curriculum. If the

influence of the student's position within the curriculum on the association between hours in

'sim lab' and higher marks on evaluations is partialled out, it can be presumed that the remaining

association between these two variables is not a direct result of position within the curriculum.

Similarly, removing the influence of hours in 'sim lab' from the relationship between position

within the curriculum and higher marks on evaluations to determine how much of the

relationship between the two variables was not due to hours in 'sim lab'.

It is important to note that a significant reduction in the size of an association between

two variables by partialling out the influence of a third variable does not necessarily

mean that that reduction indicates the degree of spuriousness in the original relationship

between the two variables. (Cramer, 1998, p. 157)

To perform a partial correlation to control for one variable – first -order partial

correlation ( r12.3 ) the following formula may be used:

!".!=

!" −(!

!"!×!!!

!")

(1−2!"

!)!× !(1−!2 !"!

To perform a partial correlation to control for two variables – second-order partial

correlation ( r12.34 ) the following formula, which is based on the formula for first order

correlations, may be used:

!".!=

!".!−(!

!". !!×!!!

!". !)!

1−2 !".!

!)!×!(1−!2 !".!!

And third-order partial correlations are similarly "based on second-order partial

correlations and so on" (Cramer, 1998, p. 160; Pedhazur, 1982).

Pearson's Semipartial Correlation

Whereas partial correlation allows for partialling out a variable from two other variables,

semipartial correlation (or part correlation) allows for the partialling out of a variable from only

one the variables that are being correlated. Put another way, semipartial correlation is the

correlation between the outcome variable and another predictor variable that has been controlled

for by another predictor variable. For example assume an admissions committee at a school of

nursing is considering three variables, high school grade point average (HS_GPA), score on the

Test of Essential Academic Skills (TEAS), and intelligence quotient (IQ). It would not be

unreasonable to assume IQ and TEAS are positively correlated. In this situation the admissions

committee is interested in the relationship between TEAS and HS_GPA, while controlling for

IQ. The formula for r12.3 (correlation of HS_GPA and TEAS while controlling for IQ) could

provide some information. Also r13.2 will indicate the correlation for IQ and HS_GPA while

controlling for TEAS.

It is possible however, that of greater interest to the [admissions committee] is the

predictive power of the [TEAS] after that of [IQ] has already been taken into

account…the interest is in the increment in the proportion of the variance in [HS_GPA]

accounted for by the [TEAS] that is over and above the proportion of variance accounted

for by [IQ]. In such a situation [IQ] should be partialled out from [TEAS], but not from

[HS_GPA] where it belongs [by performing a semipartial correlation.] (Pedhazur, 1982,

p. 115)

Sample Problem

The flexibility permitted by simulation has made it attractive to institutions as a

supplement to real-world clinical experience (Murray, Grant, Howarth, & Leigh, 2008; Rosseter,

2011). Resource constraints by academic institutions, fiscal reforms and societal pressures have

promoted a safety-conscious-litigation-avoiding culture where simulation provides a means of

risk free applied practice (Health care at the crossroads: Strategies for improving the medical

liability system and preventing patient injury, 2005).

Effective use of simulation is dependent on a complete understanding of the role of

simulation's central conceptual elements. Deliberate practice, a constituent of Ericsson's theory

of expertise, has been identified as a central concept in effective simulation learning (Issenberg,

McGaghie, Petrusa, Gordon, & Scalese, 2005).

Deliberate practice is compatible with simulation frameworks already being suggested for

use in nursing education (Jeffries, 2006). In a review of the websites of leading nursing schools

in the United States ("Top Nursing Schools and Best Ranked Nursing Colleges," 2011), each

mentioned access to simulation centers or labs. In an informal non-scientific survey of these

same schools, respondents reported use of low, moderate, and high fidelity simulation in addition

to task trainers and moulage/tabletop models. The respondents reported that, depending on the

students' class standing, they are required to complete anywhere from 100 – 280 real-world

clinical hours, and a minimum of 4 simulated clinical hours (.04 and .014% respectively) in the

simulation lab during adult health medical/surgical semesters. The same institutions describe

simulation as a compliment to rather than a replacement for clinical experience (Chee, 2011).

A small undergraduate-nursing program was considering a substantial increase in the

amount of curricular time the nursing students spent in the simulation laboratory ('sim lab').

Before making the changes the school wished to evaluate the impact of simulation on the

biannual skills evaluations. The 'sim lab' was made available to the students on a voluntary

basis. The 'sim lab' was equipped with evidence-based multimedia learning modules for the

purpose of independent deliberate practice of clinical nursing skills.

Data was collected regarding the number of hours each of the undergraduate-nursing

students spent in the simulation laboratory participating in deliberate practice, pre- and posttest

evaluation scores, and some demographic information were collected. The course instructors

hypothesized that there was no difference in the evaluation scores depending on the amount of

time students spent participating in deliberate practice for the purpose of learning nursing skills.

SPSS 21 was used to perform all assumption testing and analysis.

Null and Alternative Hypothesis

The null and alternative hypothesis for the correlation is:

H0 : ρ = 0, the population correlation coefficient is equal to zero

And the alternative hypothesis is:

HA : ρ ≠ 0, the population correlation coefficient is not equal to zero.

Where ρ is the population correlation coefficient

Results

Testing of Assumptions

A Pearson correlation is appropriate only when a linear relationship exists between the

two variables. To determine if a linear relationship existed between the primary variables of

interest a scatter plot (Figure 2) was generated between the predictor (sim_hours) and the

outcome variable (post_test). The relationship approximated a straight line; the scatterplot

suggests that the relationship of the two variables is positive and linear; the assumption of

linearity has not been violated.

Next the assumption of bivariate normality was evaluated. Both variables were tested for

normality using the Shapiro-Wilk test (Table 1), which was appropriate for the sample size. Both

variables are normally distributed, as assessed by Shapiro-Wilk's test (p > .05).

Pearson's Product Moment Correlation Coefficient

The main analysis was performed with sim_hours as the predictor variable and post_test

as the outcome variable (Table 2). There was a positive correlation between hours spent

practicing in the 'sim lab' and post-test scores, r(98) = .751, p < .0005. The relationship was

statistically significant; therefore we reject the null hypothesis and accept the alternative

hypothesis; there is a correlation between time spent in the 'sim lab' and the successful execution

of psychomotor skills in a posttest evaluation.

Strength of Association

The absolute value!of the Pearson coefficient determines the strength of the correlation.

Using Cohen's (1988) guidelines for interpreting strength of association (.1 < !! < .3 a small

correlation, .3 < !! < .5, a moderate correlation, !>. 5 a strong correlation), the results of this

analysis show a strong positive correlation between the predictor and outcome variables.

Coefficient of Determination

Hours spent in the 'sim lab' explained 60% of the variability in post-test scores: r 2 =

(.751)(.751) = .564 = .564(100) = 56%.

Partial Correlation

The instructors suspected that the correlation between time spent in the 'sim lab' and

posttest evaluation scores was only a spurious correlation caused by a difference in the baseline

characteristics of the students. Subsequently they investigated the relationship of the variables

sim_hours and post_test further by evaluating the influence of other data that had been collected.

A second correlational analysis was performed (Table 3) which included a second

predictor variable, pre_test, suspected of having a possible influence on post_test.

From this matrix (Table 3) it was known that post-test scores are strongly positively

correlated with hours spent in the sim_lab, and it was also known that post_test scores were

highly positively correlated with pre_test . But what was unknown was what the correlation

would be when pre_test was controlled for.

The evaluation was accomplished with a partial correlation. The output (Table 4)

displayed the zero order correlation coefficients, which were the three Pearson's correlation

coefficients without any control variable taken into account. The zero-order correlations

supported the hypothesis that more hours spent in the 'sim lab' increased the post-test score. A

strong association of r = .751, which was highly significant. The variable pre_test was also

significantly correlated with both sim_hours and post_test score (r = .665 and r = .927). The

second part of the output showed the correlation coefficient between sim_hours and post_test

when pre_test was controlled for. The correlation coefficient was now, r12.3 = .479, and was

statistically significant, p < .005.

Thus when the influence of the variable pre_test was controlled for the sim_lab

accounted for only 23% of the variability in post_test.

Discussion

The second (partial) correlation was useful in understanding the impact of factors other

than sim_hours on students' nursing skill evaluations. The instructors should continue to look at

elements of student experience and curriculum that contribute to pre- and posttest scores in an

effort to make intelligent decisions about how best to focus resources.

Conclusion

Pearson's product moment correlation coefficient (Pearson's r) is not conceptually

demanding; which underlies it's utility. It is one of the foundational elements of modern

statistical analysis, and is a simple means of evaluating the linear relationships between

variables. The direction (positive or negative) and strength of the relationship (coefficient of

determination) may also be evaluated. While it does not show causal relationships it is an

pragmatic way to understand the relationship of variables of interest.

References

Burns, N., & Grove, S. (2005). The Practice of Nursing Research: Conduct, Critique, and

Utilization (5 ed.). St. Louis: Elsevier Saunders.

Chee, J. (2011). [Simulation Use in Undergraduate Nursing Education].

Cohen, J. (1988). Statistical power analysis for the behavioral sciences. New York: Academic.

Cramer, D. (1998). Fundamental Statistics for Social Research. London: Routledge.

Crown, W. (1998). Statistical Models for the Social and Behavioral Sciences: Multiple

Regression and Limited-Dependent Variable Models. London: Praeger.

Issenberg, S., McGaghie, W., Petrusa, E., Gordon, D., & Scalese, R. (2005). Features and uses of

high-fidelty medical simulations that lead to effective learning: A BEME systematic

review. . Medical Teacher, 27(1), 10-28. doi: 10.1080/01421590500046924

Jeffries, P. (2006). Developing evidenced-based teaching strategies and practices when using

simulations. Clinical Simulation in Nursing, 2(1), e1-e2. doi: 10.1016/j.ecns.2009.05.014

Murray, C., Grant, M., Howarth, M., & Leigh, J. (2008). The use of simulatoin as a teaching and

learning approach to support practice learning. Nurse Education in Practice, 8, 5-8.

Pearson, K. (1948). Early Statistical Papers. Cambridge, England: University Press.

Pedhazur, E. (1982). Multiple Regression in Behavioral Research: Explanation and Prediction.

New York: Holt, Rinehart and Winston.

Polit, D., & Beck, C. (2006). Essentials of Nursing Research: Methods, Appraisal, and

Utilization (6 ed.). Philadelphia: Lippincott Williams & Wilkins.

Rosseter, R. (2011). Nursing shortage fact Sheet: American Associaltion of Colleges of Nursing.

Zar, J. H. (1999). Biostatistical Analysis. Upper Saddle River, NJ: Prentice Hall.

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Yasmin Ahmad

This study was conducted with the aim to identify the relationship availability facilities at home with compulsive Internet use among primary school student in Taiping Perak. This study use quantitative method. The questionnaire were distributed to 2 primary school and involve 100 primary school student in Taiping Perak as a sample in this study. The findings of this study indicates that there is a positive relationship between the availability of facilities at home and compulsive Internet use among primary school students.

1969 to 2003, 34 years. Simulations are now in widespread use in medical education and medical personnel evaluation. Outcomes research on the use and effectiveness of simulation technology in medical education is scattered, inconsistent and varies widely in methodological rigor and substantive focus. Review and synthesize existing evidence in educational science that addresses the question, 'What are the features and uses of high-fidelity medical simulations that lead to most effective learning?'. The search covered five literature databases (ERIC, MEDLINE, PsycINFO, Web of Science and Timelit) and employed 91 single search terms and concepts and their Boolean combinations. Hand searching, Internet searches and attention to the 'grey literature' were also used. The aim was to perform the most thorough literature search possible of peer-reviewed publications and reports in the unpublished literature that have been judged for academic quality. Four screening criteria were used to reduce the initial pool of 670 journal articles to a focused set of 109 studies: (a) elimination of review articles in favor of empirical studies; (b) use of a simulator as an educational assessment or intervention with learner outcomes measured quantitatively; (c) comparative research, either experimental or quasi-experimental; and (d) research that involves simulation as an educational intervention. Data were extracted systematically from the 109 eligible journal articles by independent coders. Each coder used a standardized data extraction protocol. Qualitative data synthesis and tabular presentation of research methods and outcomes were used. Heterogeneity of research designs, educational interventions, outcome measures and timeframe precluded data synthesis using meta-analysis. HEADLINE RESULTS: Coding accuracy for features of the journal articles is high. The extant quality of the published research is generally weak. The weight of the best available evidence suggests that high-fidelity medical simulations facilitate learning under the right conditions. These include the following: providing feedback--51 (47%) journal articles reported that educational feedback is the most important feature of simulation-based medical education; repetitive practice--43 (39%) journal articles identified repetitive practice as a key feature involving the use of high-fidelity simulations in medical education; curriculum integration--27 (25%) journal articles cited integration of simulation-based exercises into the standard medical school or postgraduate educational curriculum as an essential feature of their effective use; range of difficulty level--15 (14%) journal articles address the importance of the range of task difficulty level as an important variable in simulation-based medical education; multiple learning strategies--11 (10%) journal articles identified the adaptability of high-fidelity simulations to multiple learning strategies as an important factor in their educational effectiveness; capture clinical variation--11 (10%) journal articles cited simulators that capture a wide variety of clinical conditions as more useful than those with a narrow range; controlled environment--10 (9%) journal articles emphasized the importance of using high-fidelity simulations in a controlled environment where learners can make, detect and correct errors without adverse consequences; individualized learning--10 (9%) journal articles highlighted the importance of having reproducible, standardized educational experiences where learners are active participants, not passive bystanders; defined outcomes--seven (6%) journal articles cited the importance of having clearly stated goals with tangible outcome measures that will more likely lead to learners mastering skills; simulator validity--four (3%) journal articles provided evidence for the direct correlation of simulation validity with effective learning. While research in this field needs improvement in terms of rigor and quality, high-fidelity medical simulations are educationally effective and simulation-based education complements medical education in patient care settings.

Simulation is an approach to teaching and learning which is gaining a greater emphasis within nurse education. This has been fueled by the Nursing and Midwifery Council's (NMC) decision to identify a baseline standard for using simulation safely and its inclusion as a contributory part to practice learning [Nursing and Midwifery Council (2006a). This paper presents some of the advantages and issues for consideration in relation to its effectiveness as a teaching and learning method. Of particular concern is the limited empirical evidence to support its effect on clinical practice. Debate and further research is needed to help consolidate our knowledge and develop an evidence base for its continued use.

Nursing shortage fact Sheet: American Associaltion of Colleges of Nursing