One type of graphic representation of correlation is referred to by many names, includ-ing a bivariate distribution, a scatter diagram, a scattergram, or—our favorite—a scatterplot. A scatterplot is a simple graphing of the coordinate points for values of the X -variable (placed along the graph’s horizontal axis) and the Y -variable (placed along the graph’s vertical axis). Scatterplots are useful because they provide a quick indication of the direction and magnitude of the relationship, if any, between the two variables.
Figures 4–4 and Figures 4–5 offer a quick course in eyeballing the nature and degree of correlation by means of scatterplots. To distinguish positive from negative correlations, note the direction of the curve. And to estimate the strength of magnitude of the cor-relation, note the degree to which the points form a straight line.
Scatterplots are useful in revealing the presence of curvilinearity in a relationship.
As you may have guessed, curvilinearity in this context refers to an “eyeball gauge” of how curved a graph is. Remember that a Pearson r should be used only if the relation-ship between the variables is linear. If the graph does not appear to take the form of a straight line, the chances are good that the relationship is not linear ( Figure 4–6 ). When the relationship is nonlinear, other statistical tools and techniques may be employed. 8
8. The specifi c statistic to be employed will depend at least in part on the suspected reason for the nonlinearity. For example, if it is believed that the nonlinearity is due to one distribution being highly skewed because of a poor measuring instrument, then the skewed distribution may be statistically normalized and the result may be a correction of the curvilinearity. If—even after graphing the data—a question remains concerning the linearity of the correlation, a statistic called “eta squared” ( 2 ) can be used to calculate the exact degree of curvilinearity.
Figure 4–3
Charles Spearman (1863–1945)
Charles Spearman is best known as the developer of the Spearman rho statistic and the Spearman-Brown prophecy formula, which is used to
“prophesize” the accuracy of tests of different sizes. Spearman is also credited with being the father of a statistical method called factor analysis, discussed later in this text.
Cohen−Swerdlik:
130 Part 2: The Science of Psychological Measurement Figure 4–4
Scatterplots and Correlations for Positive Values of r
Correlation coefficient = 0 Correlation coefficient = .40
Correlation coefficient = .60 Correlation coefficient = .80 0
Correlation coefficient = .90 Correlation coefficient = .95 0
Chapter 4: Of Tests and Testing 131 Figure 4–5
Scatterplots and Correlations for Negative Values of r
Correlation coefficient = – .30 Correlation coefficient = – .50
Correlation coefficient = – .70 Correlation coefficient = – .90 0
(a) (b)
(c) (d)
(e) (f)
0 1 2 3 4 5 6
1 2 3 4 5 6 0
0 1 2 3 4 5 6
1 2 3 4 5 6
0 0 1 2 3 4 5 6
1 2 3 4 5 6 0
0 1 2 3 4 5 6
1 2 3 4 5 6
0 0 1 2 3 4 5 6
1 2 3 4 5 6
Correlation coefficient = – .95 Correlation coefficient = – .99 0
0 1 2 3 4 5 6
1 2 3 4 5 6
Cohen−Swerdlik:
Psychological Testing and Assessment: An Introduction to Tests and Measurement, Seventh Edition
II. The Science of Psychological Measurement
4. Of Tests and Testing
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132 Part 2: The Science of Psychological Measurement Figure 4–7
Scatterplot Showing an Outlier Y
X Outlier
Figure 4–6
Scatterplot Showing a Nonlinear Correlation Y
X
A graph also makes the spotting of outliers relatively easy. An outlier is an extremely atypical point located at a relatively long distance—an outlying distance—from the rest of the coordinate points in a scatterplot ( Figure 4–7 ). Outliers stimulate interpreters of test data to speculate about the reason for the atypical score. For example, consider an outlier on a scatterplot that refl ects a correlation between hours each member of a fi fth-grade class spent studying and their fth-grades on a 20-item spelling test. And let’s say one student studied for 10 hours and received a failing grade. This outlier on the scatter-plot might raise a red fl ag and compel the test user to raise some important questions, such as “How effective are this student’s study skills and habits?” or “What was this student’s state of mind during the test?”
In some cases, outliers are simply the result of administering a test to a very small sample of testtakers. In the example just cited, if the test were given statewide to fi fth-graders and the sample size were much larger, perhaps many more low scorers who put in large amounts of study time would be identifi ed.
As is the case with very low raw scores or raw scores of zero, outliers can some-times help identify a testtaker who did not understand the instructions, was not able to follow the instructions, or was simply oppositional and did not follow the instructions.
In other cases, an outlier can provide a hint of some defi ciency in the testing or scoring procedures.
People who have occasion to use or make interpretations from graphed data need to know if the range of scores has been restricted in any way. To understand why this is so necessary to know, consider Figure 4–8 . Let’s say that graph A describes the rela-tionship between Public University entrance test scores for 600 applicants (all of whom were later admitted) and their grade point averages at the end of the fi rst semester. The s catterplot indicates that the relationship between entrance test scores and grade point average is both linear and positive. But what if the admissions offi cer had accepted only the a pplications of the students who scored within the top half or so on the entrance exam? To a trained eye, this scatterplot (graph B) appears to indicate a weaker c orrelation than that indicated in graph A—an effect attributable exclusively to the restriction of range. Graph B is less a straight line than graph A, and its direction is not as obvious.
Chapter 4: Of Tests and Testing 133
Regression
In everyday language, the word regression is synonymous with “reversion to some previous state.” In the language of statistics, regression also describes a kind of rever-sion—a reversion to the mean over time or generations (or at least that is what it meant originally).
Regression may be defi ned broadly as the analysis of relationships among vari-ables for the purpose of understanding how one variable may predict another. Simple regression involves one independent variable ( X ), typically referred to as the predictor variable, and one dependent variable ( Y ), typically referred to as the outcome variable.
Simple regression analysis results in an equation for a regression line. The regression line is the line of best fi t: the straight line that, in one sense, comes closest to the greatest number of points on the scatterplot of X and Y.
Does the following equation look familiar?
Y⫽ ⫹a bX
In high-school algebra, you were probably taught that this is the equation for a straight line. It’s also the equation for a regression line. In the formula, a and b are regression coeffi cients; b is equal to the slope of the line, and a is the intercept, a constant indicat-ing where the line crosses the Y -axis. The regression line represented by specifi c val-ues of a and b is fi tted precisely to the points on the scatterplot so that the sum of the squared vertical distances from the points to the line will be smaller than for any other line that could be drawn through the same scatterplot. Although fi nding the equation for the regression line might seem diffi cult, the values of a and b can be determined through simple algebraic calculations.
Figure 4–8
Two Scatterplots Illustrating Unrestricted and Restricted Ranges
Grade-point average
0 Entrance test scores 100
Unrestricted range Graph A
Grade-point average
0 Entrance test scores 100
Restricted range Graph B
Cohen−Swerdlik:
Psychological Testing and Assessment: An Introduction to Tests and Measurement, Seventh Edition
II. The Science of Psychological Measurement
4. Of Tests and Testing
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134 Part 2: The Science of Psychological Measurement
The primary use of a regression equation in testing is to predict one score or v ariable from another. For example, suppose a dean at the De Sade School of Dentistry wishes to predict what grade point average (GPA) an applicant might have after the fi rst year at De Sade. The dean would accumulate data about current students’ scores on the dental college entrance examination and end-of-the-fi rst-year GPA. These data would then be used to help predict the GPA ( Y ) from the score on the dental college admissions test ( X ). Individual dental students are represented by points in the scatterplot in Figure 4–9 . The equation for the regression line is computed from these data, which means that the values of a and b are calculated. In this hypothetical case,
GPA⫽0 82 0 03. ⫹ . (entrance exam) This line has been drawn onto the scatterplot in Figure 4–9 .
Using the regression line, the likely value of Y (the GPA) can be predicted based on specifi c values of X (the entrance exam) by plugging the X -value into the equation.
A student with an entrance exam score of 50 would be expected to have a GPA of 2.3.
A student with an entrance exam score of 85 would be expected to earn a GPA of 3.7.
This prediction could also be done graphically by tracing a particular value on the Figure 4–9
Graphic Representation of Regression Line
The correlation between X and Y is 0.76. The equation for this regression line is Y ⫽ 0.82 ⫹ 0.03(X); for each unit increase on X (the dental school entrance examination score), the predicted value of Y (the fi rst-year grade point average) is expected to increase by .03 unit. The standard error of the estimate for this prediction is 0.49.
GPA first year
0.0
5 15 25 35 45 55 65 75 85 95
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Score on dental school entrance exam
Chapter 4: Of Tests and Testing 135
X -axis (the entrance exam score) up to the regression line, and then straight across to the Y -axis, reading off the predicted GPA.
Of course, all students who get an entrance exam score of 50 do not earn the same GPA. This can be seen in Figure 4–8 by tracing from any specifi c entrance exam score on the X -axis up to the cloud of points surrounding the regression line. This is what is meant by error in prediction: Each of these students would be predicted to get the same GPA based on the entrance exam, but in fact they earned different GPAs. This error in the prediction of Y from X is represented by the standard error of the estimate. As you might expect, the higher the correlation between X and Y, the greater the accuracy of the prediction and the smaller the standard error of the estimate.
Multiple regression Suppose that the dean suspects that the prediction of GPA will be enhanced if another test score—say, a score on a test of fi ne motor skills—is also used as a predictor. The use of more than one score to predict Y requires the use of a multiple regression equation.
The multiple regression equation takes into account the intercorrelations among all the variables involved. The correlation between each of the predictor scores and what is being predicted is refl ected in the weight given to each predictor. In this case, what is being predicted is the correlation of the entrance exam and the fi ne motor skills test with the GPA in the fi rst year of dental school. Predictors that correlate highly with the predicted variable are generally given more weight. This means that their regres-sion coeffi cients (referred to as b -values) are larger. No surprise there. We would expect test users to pay the most attention to predictors that predict Y best.
The multiple regression equation also takes into account the correlations among the predictor scores. In this case, it takes into account the correlation between the dental college admissions test scores and scores on the fi ne motor skills test. If many predictors are used and if one is not correlated with any of the other predictors but is correlated with the predicted score, then that predictor may be given relatively more weight because it is providing unique information. In contrast, if two predictor scores are highly correlated with each other then they could be providing redundant information. If both were kept in the regression equation, each might be given less weight so that they would “share” the prediction of Y.
More predictors are not necessarily better. If two predictors are providing the same information, the person using the regression equation may decide to use only one of them for the sake of effi ciency. If the De Sade dean observed that dental school admission test scores and scores on the test of fi ne motor skills were highly correlated with each other and that each of these scores correlated about the same with GPA, the dean might decide to use only one predictor because nothing was gained by the addi-tion of the second predictor.
When it comes to matters of correlation, regression, and related statistical tools, some students are known to “tune out” (for lack of a more sophisticated psycho-logical term). They see such topics as little more than obstacles to be overcome to a career—a statistics-free career—in psychology. In truth, these students might be quite surprised to learn where, later in their careers, the need for knowledge of correlation, regression, and related statistical tools may arise. For example, who would believe that knowledge of regression and how to formulate regression equations might be useful to an NBA team? Certainly, two sports psychologists you are about to meet would believe it ( see Meet an Assessment Professional —make that, Meet a Team of Assess-ment Professionals ).
Cohen−Swerdlik:
Psychological Testing and Assessment: An Introduction to Tests and Measurement, Seventh Edition
II. The Science of Psychological Measurement
4. Of Tests and Testing
148 © The McGraw−Hill
Companies, 2010
136 Part 2: The Science of Psychological Measurement
Meet Dr. Steve Julius and Dr. Howard W. Atlas
he Chicago Bulls of the 1990s is considered one of the great dynasties in sports, as witnessed by their six world championships in that decade . . .
The team benefi ted from great individual contributors, but like all successful organizations, the Bulls were always on the lookout for ways to maintain a competitive edge. The Bulls . . . were one of the fi rst NBA franchises to apply personal-ity testing and behavioral interviewing to aid in the selection of college players during the annual draft, as well as in the evaluation of goodness-of-fi t when considering the addition of free agents. The purpose of this effort was not to rule out psychopathology, but rather to evaluate a range of competencies (e.g., resilience, relation-ship to authority, team orientation) that were deemed necessary for success in the league, in general, and the Chicago Bulls, in particular.
[The team utilized] commonly used and well-validated personality assessment tools and techniques from the world of business (e.g., 16PF-fi fth edition). . . . Eventually, suffi cient data was collected to allow for the validation of a regression formula, useful as a prediction tool in its own right. In addition to selection, the infor-mation collected on the athletes often is used to assist the coaching staff in their efforts to moti-vate and instruct players, as well as to create an atmosphere of collaboration.
Read more of what Dr. Atlas and Dr. Julius had to say—their complete essay—at www.mhhe .com/cohentesting7.
TT
Steve Julius, Ph.D., Sports Psychologist, Chicago Bulls
Howard W. Atlas, Ed.D., Sports Psychologist, Chicago Bulls