Psychology
Correlation Coefficients
Correlation coefficients in psychology measure the strength and direction of the relationship between two variables. They range from -1 to 1, with 0 indicating no correlation and -1 or 1 indicating a perfect negative or positive correlation, respectively. This statistical measure helps psychologists understand how changes in one variable may be associated with changes in another.
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10 Key excerpts on "Correlation Coefficients"
- Harold O. Kiess, Bonnie A. Green(Authors)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
To find if two variables covary, we measure a sample of people and obtain two scores from each person, such as a liking a professor score and an in-class attention score. Then a correlation coefficient is calculated. A correlation coefficient is a statistic that provides a numerical description of the extent of the relatedness of two sets of scores and the direc- tion of the relationship. Values of this coefficient may range from . Statistical hypothesis testing also enters into use with the correlation coefficient. There will always be some chance relationship between scores on two different variables. Thus, the question arises of whether an observed relation, given by the numerical value of the correlation coefficient, is greater than would be expected from chance alone. A statisti- cal test on the correlation coefficient provides an answer for this question. If the two sets of scores are related beyond chance occurrence, then we may be inter- ested in attempting to predict one score from the other. If you knew a subject’s liking of a professor score, could you predict his or her in-class attention score? And, if you could predict the in-class attention score, how accurate would your prediction be? Predicting a score on one variable from a score on a second variable involves using regression analysis. Correlation and regression analysis techniques are widely used in many areas of behav- ioral science to find relationships between variables and to find if one variable predicts another. This chapter introduces correlational studies and the statistics associated with them. Using correlated scores to predict one score from another is introduced in Chapter 14. Let us continue with our example of a possible relationship between liking a professor and in-class attention to introduce concepts of correlation and the correlation coefficient.
eBook - PDF- Donald Ary, Lucy Jacobs, Christine Sorensen Irvine, David Walker(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
As with the other correlations, the phi coefficient indicates both direction and strength of relationships. A variety of Correlation Coefficients are available for use with ordinal and nominal data. These include coefficients for data that are more than just pairs; for example, assessing the agreement of three or more judges ranking the performance of the same subjects. We highly recommend Siegel and Castellan’s Nonparametric Statistics (1988), a remarkably well-organized and easy-to-understand text. 12-3 Considerations for Interpreting a Correlation Coefficient The coefficient of correlation may be simple to calculate, but it can be tricky to inter-pret. It is probably one of the most misinterpreted and/or overinterpreted statistics available to researchers. Various considerations need to be taken into account when evaluating the practical utility of a correlation. The importance of the numerical value of a particular correlation may be evaluated in four ways: (1) considering the nature of its population and the shape of its distribution, (2) its relation to other correlations of the same or similar variables, (3) according to its absolute size and its predictive valid-ity, or (4) in terms of its statistical significance. 12-3a The Nature of the Population and the Shape of Its Distribution The value of an observed correlation is influenced by the characteristics of the popula-tion in which it is observed. For example, a mathematics aptitude test that has a high correlation with subsequent math grades in a regular class where students range widely on both variables would have a low correlation in a gifted class. This is because the math aptitude scores in the gifted class are range restricted (less spread out) compared to those in a regular class. Range restrictions of either the predictor or the criterion scores reduce the strength of the observed correlation.
- Bruce M. King, Patrick J. Rosopa, Edward W. Minium(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
114 CHAPTER 7. CORRELATION Any report of r or r S values, therefore, should include a careful description of the measures used and the circumstances under which the Correlation Coefficients were obtained. Similarly, values reported by others may or may not hold up under the conditions with which you are concerned. Take the others’ results only as working hypotheses, subject to confirmation under your conditions. 7.13 S U M M A R Y In behavioral science, it is often important to determine the degree to which two variables vary together. The two basic tasks are (1) determining the degree of association between two variables and (2) predicting a person’s standing in one variable when we know the standing in an associated variable. In this chapter, we dis- cussed the problem of association; we consider prediction in the next chapter. The Pearson product-moment correlation coefficient, r, is the most commonly used measure of association when two quantita- tive variables are characterized by a linear (straight-line) relation- ship. It reflects agreement between relative standing in one variable and relative standing in the other and so is an index of how well bivariate data points hug the straight line of best fit. The sign of the coefficient specifies whether the two variables are positively or negatively (inversely) related. The magnitude of the coefficient varies from zero when no association is present to ±1.00 when the correlation is perfect. Making a scatter diagram of the data points of a bivariate dis- tribution is an excellent preliminary step to see whether there is any association between two variables. It also provides a rough check on the accuracy of the calculated value of r, and it lets us examine several conditions (such as the linearity of the rela- tionship) that may influence the correlation coefficient and its interpretation. We can use Spearman’s correlation coefficient, r S , in place of Pearson’s r if the paired observations are converted to ranks.
eBook - PDFQuantitative and Statistical Approaches to Geography
A Practical Manual
- John A. Matthews, W. B. Fisher(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
13 The Strength of Relationships: Correlation Coefficients MANY hypotheses of interest to geographers involve not only questions about overall differences and similarities between data sets but also the degree to which one data set is reflected in, associated with, related to or correlated with variability in another data set. We might show, for example, that within a region of tropical Africa, areas with different quantities of soil nutrients differ significantly in the population that they support. One of the tests previously outlined could be used for this purpose. Such a test does not, however, enable us to say to what degree the variability in SQÜ quality is associated with variability in population. Measures of association or correlation are available for this type of problem, which can be described as a problem involving the strength of relationship between two measurable attributes or variables. The problem becomes one of statistical inference if we wish to be sure that a given strength of relationship differs significantly from 'no relationship' or from a relationship likely to have occurred by chance.
eBook - PDFExperimental Design and Statistics for Psychology
A First Course
- Fabio Sani, John Todman(Authors)
- 2008(Publication Date)
- Wiley-Blackwell(Publisher)
For instance, you might want to explore the pos-sibility that self-esteem and salary are correlated with one another, and that they are both correlated with ‘years of education’. Also, you might want to measure which one among several variables is the best predictor of a given criterion variable. In this case you need to measure all the variables you want to inves-tigate, and apply some advanced procedures (e.g., partial correlation, or multiple regression) that are beyond the scope of this book. (See the books by Howell and Allison in the ‘brief list of recommended books’.) SUMMARY OF CHAPTER • Sometimes researchers hypothesize that two variables are related (i.e., change together), without making claims about which variable influences which. These types of hypotheses are tested by means of ‘correlational studies’. • A relationship between two variables can be explored by means of either correlational analysis or regression analysis. • Correlational analysis is used to (i) describe the relationship between two variables in a visual and numerical fashion, and to (ii) test the statistical significance of the relationship. • To give a visual description of a relationship, researchers use the ‘scattergram’, which helps to form an idea of (i) the direction (whether it is positive or negative), (ii) the strength (magnitude) and (iii) the form (whether it is linear or non-linear), of the relationship under investigation. • The strength of a relationship can be expressed numerically by calculating a descriptive statistic called the ‘coefficient of correlation’ (or r ), whose value ranges from − 1 to + 1. This can be calculated in several ways. Normally, the ‘Pearson’s product–moment correlation coefficient’ (or r ) is used when para-metric assumptions are met (usually when there is an interval scale), while the ‘Spearman rank order correlation’ ( r s or rho ) is used when parametric assumptions are not met (typically, when there is an ordinal scale).
eBook - PDF- Sherri Jackson(Author)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 337 CHAPTER SUMMARY After reading this chapter, you should have an understanding of correlational re-search, which allows researchers to observe relationships between variables; correla-tion coefficients, the statistics that assess that relationship; and regression analysis, a procedure that allows us to predict from one variable to another. Correlations vary in type (positive or negative) and magnitude (weak, moderate, or strong). The pictorial representation of a correlation is a scatterplot. Scatterplots allow us to see the rela-tionship, facilitating its interpretation. When interpreting correlations, several errors are commonly made. These in-clude assuming causality and directionality, the third-variable problem, having a re-strictive range on one or both variables, and the problem of assessing a curvilinear relationship. Knowing that two variables are correlated allows researchers to make predictions from one variable to another. Four different Correlation Coefficients (Pearson’s, Spearman’s, point-biserial, and phi) and when each should be used were discussed. The coefficient of determination was also discussed with respect to more fully understanding Correlation Coefficients. Lastly, regression analysis, which allows us to predict from one variable to another, was described. CHAPTER 9 REVIEW EXERCISES (Answers to exercises appear in Appendix B.) Fill-in Self-Test Answer the following questions. If you have trouble answering any of the questions, restudy the relevant material before going on to the multiple-choice self-test.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
Note that the correlation reflects the noisiness and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear rela-tionships (bottom). N.B.: the figure in the center has a slope of 0 but in that case the correlation coefficient is undefined because the variance of Y is zero. Definition Pearson's correlation coefficient between two variables is defined as the covariance of the two variables divided by the product of their standard deviations: The above formula defines the population correlation coefficient, commonly represented by the Greek letter ρ (rho). Substituting estimates of the covariances and variances based on a sample gives the sample correlation coefficient , commonly denoted r : An equivalent expression gives the correlation coefficient as the mean of the products of the standard scores. Based on a sample of paired data ( X i , Y i ), the sample Pearson correlation coefficient is where , and σ X are the standard score, sample mean, and sample standard deviation respectively. Mathematical properties The absolute value of both the sample and population Pearson Correlation Coefficients are less than or equal to 1. Correlations equal to 1 or -1 correspond to data points lying exactly on a line (in the case of the sample correlation), or to a bivariate distribution entirely supported on a line (in the case of the population correlation). The Pearson correlation coefficient is symmetric: corr ( X , Y ) = corr ( Y , X ). A key mathematical property of the Pearson correlation coefficient is that it is invariant to changes in location and scale. That is, we may transform X to a + bX and transform Y to ________________________ WORLD TECHNOLOGIES ________________________ c + dY , where a , b , c , and d are constants, without changing the correlation coefficient (this fact holds for both the population and sample Pearson Correlation Coefficients).
- David Kleinbaum, Lawrence Kupper, Azhar Nizam, Eli Rosenberg(Authors)
- 2013(Publication Date)
- Cengage Learning EMEA(Publisher)
108 6 The Correlation Coefficient and Straight-line Regression Analysis 6.1 Definition of r The correlation coefficient is an often-used statistic that provides a measure of how two ran-dom variables are linearly associated in a sample and has properties closely related to those of straight-line regression. We define the sample correlation coefficient r for two variables X and Y by the formula r 5 a n i 5 1 1 X i 2 X 21 Y i 2 Y 2 a a n i 5 1 1 X i 2 X 2 2 a n i 5 1 1 Y i 2 Y 2 2 b 1/2 5 SSXY 1 SSX ? SSY (6.1) where SSXY 5 a n i 5 1 1 X i 2 X 21 Y i 2 Y 2 , SSX 5 a n i 5 1 1 X i 2 X 2 2 , and SSY 5 a n i 5 1 1 Y i 2 Y 2 2 . An equivalent formula for r that illustrates its mathematical relationship to the least-squares estimate of the slope of a fitted regression line is 1 r 5 S X S Y b ˆ 1 (6.2) 1 S 2 X 5 1 n 2 1 SSX and S 2 Y 5 1 n 2 1 SSY are the estimated sample variances of the X and Y variables, respectively. Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 6.2 r as a Measure of Association 109 ■ Example 6.1 For the age–systolic blood pressure data in Table 5.1, r is 0.66. This value can be obtained from the SAS output on page 59 by taking the square root of the R -square value of 0.4324. Alternatively, using (6.2), we have r 5 15.29 22.58 1 0.97 2 5 0.66 Three important mathematical properties are associated with r : 1. The possible values of r range from 2 1 to 1. 2. r is a dimensionless quantity; that is, r is independent of the units of measure-ment of X and Y . 3. r is positive, negative, or zero as b ˆ 1 is positive, negative, or zero; and vice versa.
eBook - PDFStatistics for the Social Sciences
A General Linear Model Approach
- Russell T. Warne(Author)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
Check Yourself! • Why does a correlation between two variables not imply that the independent variable causes the dependent variable? • Explain the third variable problem. Summary Pearson’ s correlation is a statistic that quantifies the relationship between two interval- or ratio-level variables. It is calculated with Formula 12.1: r ¼ Σ x i X ð Þ y i Y ð Þ n 1 ^ σ x ^ σ y 356 Correlation There are two components of the Pearson’ s r value to interpret. The first is the sign of the correlation coefficient. Positive values indicate that individuals who have high X scores tend to have high Y scores (and that individuals with low X scores tend to have low Y scores). A negative correlation indicates that individuals with high X scores tend to have low Y scores (and that individuals with low X scores tend to have high Y scores). The second component of a correlation coefficient that is interpreted is the correlation’ s number, which indicates the strength of the relationship and the consistency with which the relationship is observed. The closer an r value is to +1 or –1, the stronger the relationship between the variables. The closer the number is to zero, the weaker the relationship. For many purposes, calculating the r value is enough to answer the research questions in a study. But it is also possible to test a correlation coefficient for statistical significance, where the null hypothesis is r = 0. Such a null hypothesis statistical significance test (NHST) follows the same steps as all NHSTs. The effect size for Pearson’ s r is calculated by squaring the r value (i.e., obtaining r 2 ). The data used to calculate Pearson’ s r can be visualized with a scatterplot, which was introduced in Chapter 4. In a scatterplot each sample member is represented as a dot plotted in a position corresponding to the individual’ s X and Y scores. Scatterplots for strong correlations tend to have a group of dots that are closely grouped together.
eBook - PDFStatistics for the Social Sciences
A General Linear Model Approach
- Russell T. Warne(Author)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
This is probably surprising because the correlation with just 10 individuals was much smaller: +0.221. This discrepancy occurred because Pearson’ s r values are unstable when sample sizes are small – a characteristic that r shares with every other sample statistic. Therefore, to obtain stable correlations you must have a large sample size. Stronger correlations stabilize with smaller samples than do weaker correlations. Larger sample sizes are also required for 348 Correlation Warning: Correlation Is Not Causation A correlation quantifies the strength of a relationship between an independent and a dependent variable. It is because of this function of a correlation that we can interpret the Pearson’ s r values in Table 12.8 and understand the strength and direction of a correlation between two variables. There is a temptation, though, to interpret a relationship between two variables as an indication that changes in the independent variable cause changes in the dependent variable. For example, Table 12.8 indicates that the correlation between depression and job satisfaction is r = –0.43 (Faragher et al., 2005), meaning that people with higher levels of depression tend to be less satisfied with their jobs. Some readers may believe that this correlation indicates that higher levels of depression cause individuals to be less satisfied with their jobs. Although this interpretation is seductive, it is not the only possible interpretation of a correlation. There are three different interpretations (i.e., models) that a correlation can support. These possible interpretations are shown in Figure 12.6. The first possibility is that the changes in the independent variable (X) cause changes in the dependent variable (Y). The second possibility is that changes in the dependent variable (Y) cause changes in the independent variable (X). The final possibility is that a third, unknown variable (Z) can cause changes in X and Y.
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