Correlation
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09CH_Sukal_Statistics.pdf
chapter 9
Correlation
Learning Objectives
After reading this chapter, you will be able to. . .
1. explain the hypothesis of association.
2. interpret the correlation coefficient.
3. list the Pearson correlation requirements.
4. describe what the coefficient of determination explains.
5. explain the variables involved in the point-biserial correlation.
6. describe the applications for the Spearman rho correlation.
7. present the data based on correlational output using SPSS.
8. interpret and report results of Pearson correlation and Spearman rho correlation in APA format.
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CHAPTER 9Section 9.1 The Hypothesis of Association
Correlation is a concept that transcends statistical analysis. The study of relationships, which is what correlation involves, has many applications: Cloudy days are related to (correlated with) cooler temperatures. Natural disasters are related to declines in the stock market. An impending test is related to the need to study, and grinding noises in the engine compartment of a car are usually related to repair bills.
Some relationships are stronger than others, so statistical procedures have been developed to quantify, or provide a numerical indicator of, the strength of the relationship between two variables. The numerical indicators are called correlation coefficients, and one of the most common is the Pearson correlation coefficient. The name Pearson refers to Karl Pearson, whose impact not just on studying correlation but also on statistical analysis generally may be unparalleled by that of any other individual.
In the early years of the 20th century, Pearson founded the first department of statisti- cal analysis, at University College London. Under Pearson’s direction, the department attracted, among others, William Sealy Gosset of t-test fame; Ronald A. Fisher, who pro- duced analysis of variance; and Charles Spearman, who developed another correlation approach as well as an elegant statistical procedure called factor analysis. It is difficult to overstate the impact that Pearson had on the evolution of statistical analysis.
A man of fierce independence, Pearson’s education at Cambridge was directed at religion and philosophy rather than mathematics. As a student of religion, he sued the university over the compulsory chapel attendance required of all undergraduates. Winning his suit brought a change to university rules about compulsory attendance, after which Pearson chose to attend chapel! His graduate work (in Germany) emphasized literature, and it is a testimony to his extraordinary breadth of talent that his greatest contributions would be in statistical analysis. Pearson was a contemporary of Einstein. Just as Einstein sought a grand theory that would unite all of physics, Pearson sought a grand theory to unite all of mathematics. Both men were disappointed in these endeavors, but that should not detract from what Pearson did accomplish. Although Pearson’s associations with his col- leagues were not always harmonious, he and the other statisticians who found an aca- demic home in his department virtually defined modern quantitative analysis. No one who “crunches” numbers for any length of time can be unaffected by the procedures and the analytical techniques these men developed.
9.1 The Hypothesis of Association
Previous chapters concentrated on tests of significant difference. The z-test, the inde-pendent samples t-tests, and analysis of variance are tests of the differences between groups. They all fall under a general assumption referred to as the hypothesis of dif- ference. However, some kinds of analyses do not involve questions about significant differences.
If a psychologist asks about the relationship between birth order and achievement moti- vation or about the connection between the amount of time children read and their school grades, the question is about relationships rather than differences. Those ques- tions call for procedures connected to the hypothesis of association, and when results
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CHAPTER 9Section 9.1 The Hypothesis of Association
are statistically significant, it means that the relationship, rather than the difference, prob- ably is not random.
Correlation Versus Causation
Before pursuing correlation, it is important to make a distinction between correlation and causation. Because two characteristics covary, or vary together, does not presume that one necessarily causes the other. Although there may be a causal relationship, it usually cannot be determined just from studying the correlation. One of your author’s statistics profes- sors addressed the risk of confusing correlation with causation this way.
Someone drinks on three successive nights. The first night it is scotch and water, on the second night it is bourbon and water, and on the third he drinks vodka and water. All three mornings after are accompanied by hangovers. Because the water is common to each experience, one might erroneously conclude that water must be the cause.
There is a classic study that demonstrates, among other things, a correlation between the sale of ice cream by vendors on city streets and burglaries in the same city. Someone rush- ing to judgment about cause might wish to curb ice cream sales in order to reduce the number of burglaries, or check the criminal records of ice cream vendors, not recognizing that hotter weather (and the open windows that result) probably drives both. It is not unusual for some third variable to explain an association between a first and a second. Although correlation values provide some evidence for causation, correlation alone is rarely sufficient to demonstrate causality.
Picturing Correlation
If the word correlation is broken down—co-relation—it expresses what is meant: The char- acteristics are related, and the evidence for the relationship is that they vary together, or covary. As the level of one variable changes, the other changes in concert. This happens because both variables contain some of the same information. The higher the correlation, the more information they have in common.
A correlation can be represented visually in what is called a scatterplot. A researcher wishes to study the relationship between verbal ability and intelligence and gathers scores on both of those characteristics for 12 participants. The data is as follows:
Participant 1 2 3 4 5 6 7 8 9 10 11 12
Verbal ability 20 35 42 48 55 60 63 66 72 76 78 85
Intelligence 80 95 90 100 100 100 110 115 120 115 110 125
The first participant has a verbal ability score of 20 and an intelligence score of 80. The data appears to suggest that as the values of one increase, so do the values of the other; there appears to be a positive correlation. But the relationship is easier to see in a scatter- plot, a graph plotting the scores of one variable on the horizontal axis, and the other on the vertical axis with dots indicating the intersection of each pair of scores. There is an Excel scatterplot of the verbal ability/intelligence data in Figure 9.1.
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CHAPTER 9Section 9.1 The Hypothesis of Association
Figure 9.1: The relationship between verbal ability and intelligence
In the scatterplot, intelligence scores are plotted on the vertical/ ordinate or y-axis and the verbal ability scores are plotted on the horizontal/abscissa, or x-axis, with a dot representing where the two scores for each participant intersect; each dot repre- sents an intelligence score and a verbal ability score.
The plot indicates that the hunch about a positive relation- ship was accurate. The general trend is from lower left to upper right. As the value of one variable increases, the value of the other tends to do like- wise. The incline is not dramatic, but there is a general rise in the data points.
Less-than-Perfect Relationships
The relationship certainly is not perfect. The fourth, fifth, and sixth participants all have the same level of intelligence but different levels of verbal ability. The same is true of participants 8 and 10, as well as participants 7 and 11. Still, there is a general lower-left to upper-right relationship, which is what might be expected. Smarter people often have more complex language patterns, perhaps evidenced by higher verbal ability scores.
However, it also is not surprising that the relationship between intelligence and verbal ability (as demonstrated in school performance) is less than perfect. Almost everyone knows someone who is very bright but did not excel in school. Or, at the other extreme, there was that person of fairly average intelligence who performed brilliantly in school. There are names for these exceptions to the general correlation tendency: underachievers and overachievers.
A A single point in a scatterplot represents how many raw scores?
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CHAPTER 9Section 9.1 The Hypothesis of Association
The exceptions point to the fact that people are very complex. Human behavior is rarely explained by one or two variables. Although intelligence is related to verbal aptitude, so are a number of other variables, such as how much the individual reads, how easily the individual is distracted, and how much experience the person has had. One of the reasons correlation values are calculated is to determine the level of agreement when the relation- ships are less than perfect, as they almost always are with people.
The issue for questions under the hypothesis of association is not whether the relationship is perfect, because it would be extremely rare if it was, but rather whether the relationship is statistically significant. Statistically significant correlations tend to reemerge every time new data is gathered and the correlation calculated.
Although perfect correlations are rare when dealing with people, that is not necessarily the case in some disciplines. Mathematicians enjoy the stability of perfect relationships. For example, the formula for the area of a circle, A 5 πr2 (pi times the square of the radius), works for circles of any size because there is a perfect relationship between a circle’s radius and its area.
But even imperfect correlations can be very important. For example, health professionals, knowing that there is a correlation, even a weak one, between the exposure to secondhand smoke and the development of respiratory problems, warn the public against exposure. In this case, as exposure to secondhand smoke increases so too the incidence of respiratory problems increases. The research concerning secondhand smoke and respiratory prob- lems demonstrates a relationship or association between these variables that is not causal but that may, upon further experimental investigation, show a causal element. In a similar example, suppose educators know there is a correlation between the amount of time stu- dents spend on homework and their success on a high school exit exam; as the number of hours spent on homework goes up, high school exit exam scores increase. Based on such a correlation, the educators could encourage students to study more in the expectation that pass rates may rise if they do.
These two examples each illustrate a positive correlation that simply means that as the value of one variable increases (or decreases), the value of the related variable likewise increases (or decreases). In basic terms, the values of the two variables move in the same direction and, when the data is plotted, the slope will be positive; that is, upward to the right (Figure 9.2). The term positive does not imply a value judgment as to whether the outcome is desirable.
A negative correlation is one in which the values of two correlated variables move in opposite directions. As the value of one variable increases, the value of the other decreases and vice versa. When the data is plotted, the slope will be negative; that is, downward to the right (Figure 9.2). Just as a positive correlation does not mean that the relationship between two variables is somehow “good,” neither does a negative correlation suggest that the relationship between two variables is “bad.” Consider the case of a drug reha- bilitation program that demonstrates a negative correlation between the amount of time spent in the program and the rate of recidivism. As the number of days spent in rehabilita- tion increases, the rate of occurrence of relapse decreases.
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CHAPTER 9Section 9.1 The Hypothesis of Association
Figure 9.2: Degrees of correlational strength
Source: The New Spalding Grammar School, www.sgspsychology.webs.com. Retrieved from http://sgspsychology.webs.com /METHODS/correlation.gif
The Amount of Scatter
The amount of scatter in a scatterplot suggests the strength of the correlation. Scatterplots for strong correlations have very little scatter. The points coalesce around what can appear to be almost a line. More scattered points and a less well-defined line indicate weaker relationships.
Strong Positive
Weak Positive
Strong Negative
Moderate Negative
None Weak Negative
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CHAPTER 9Section 9.1 The Hypothesis of Association
What Correlations Provide
Calculating a correlation produces a single number that indicates the strength of the relationship between the variables involved. Correla- tion values of either 21 or 11 indicate perfect relationships. A correla- tion of 0 indicates no relationship. Values less than the absolute value of 1 indicate an imperfect relationship, with the strength of the rela- tionship declining as the value approaches 0 from either extreme. The sign that precedes the value (1 or 2) indicates whether the correla- tion is positive or negative. In terms of hypothesis testing, the further the value is away from 0, the higher the correlation and the lower the p-value. An indication of p , .05 indicates a 5% probability that there is a significant difference away from 0 (i.e., a significant correlation).
Correlating two variables does not require that they both measure the same characteristic. Often, entirely different kinds of things are correlated. The example of secondhand smoke and respiratory issues involves two completely different variables, but the strength of the relationship between them can be calculated nevertheless. As long as the two vari- ables can be quantified—reduced to a number—the strength of the relationship can be determined. This is also termed a bivariate analysis indicating two variables. As a result, bivariate outliers should be detected when performing these analyses as described in the next section.
Requirements for the Pearson Correlation
There are several different correlation procedures. The appropriate procedure for a par- ticular problem is determined by characteristics such as the scale of the data involved. The Pearson correlation, for example, requires variables of either interval or ratio scale. Nomi- nal or ordinal scale data involves other procedures. In addition to the need for interval or ratio data, the Pearson correlation has the following requirements:
• In their populations, the characteristics are assumed to be normally distributed. Normal distributions can never be reflected in relatively small samples, but there must be reason to believe that the samples come from populations that are normal.
• The distributions from which the samples come must be similarly distributed. • The two samples are assumed to be randomly selected from their populations. • The relationship between the variables must be linear. • Bivariate outliers should be detected and dealt with to account for their influence.
Normality is indicated when the standard deviation is about one-sixth of the range, the measures of central tendency all have about the same value, and other sophisticated methods such as the Shapiro-Wilks or Kolmogorov-Smirnov tests (Chapter 2). The nor- mality of the two variables involved in a correlation is also suggested by the way data is distributed in the scatterplot. When both variables are normal, the points in the plot will be distributed from left to right with the number of points gradually increasing until they reach their greatest frequency in the middle of the graph and then decreasing gradually to the right extreme. If the relationship is positive, that scatter is generally from lower left
B If two variables are normally distributed but uncorrelated, what pattern will their data points make in a scatterplot?
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CHAPTER 9Section 9.1 The Hypothesis of Association
to upper right. If it is negative, the pattern is from upper left to lower right. If there is no correlation between the variables, the points make a circular pattern in the middle of the graph with the greatest density in the middle of the circle. The greater frequency in the middle of the circle reflects central tendency in data distributions.
The similar-distribution requirement does not mean that the standard deviations should be the same. That is not likely to happen unless both variables are measured along the same range. It means that the standard deviations should account for similar proportions of their respective ranges.
The strength of a correlation is affected by range attenuation. When the range of scores in either variable is artificially abbreviated, the correlation value will be artificially low. Range attenuation can be indicated by a standard deviation that is substantially smaller than we know it to be in the population. If we were correlating intelligence scores with reading comprehension, and the intelligence scores have a standard deviation of 8 points when we know that the population standard deviation is 15 points, we can expect any resulting correlation value to be artificially low. One of the advantages of random selec- tion is that random samples of a reasonable size tend to mirror their populations reason- ably well. Range restriction problems are much less likely to occur with randomly selected samples.
Different Types of Correlations
When the relationship between two variables is linear, it means that the degree to which they change in concert with each other is consistent throughout their ranges; if it is low and positive, it is low and positive at low levels of both variables and at higher levels of both variables. Consider the correlation between anxiety and performance on an achieve- ment test. For someone preparing for an important test, a little anxiety is probably a good thing. It prompts the individual to take the test seriously, to prepare study notes, to spend more time reading the textbook, and so on. If there is not any anxiety at all, the test might be disregarded. It seems likely that, at least in the early going, achievement increases with anxiety.
However, there is a point when if the level of anxiety continues to increase, the individual’s performance reaches a plateau and then may even begin to diminish. The test-taker can become so anxious that concentration is difficult and performance no longer improves. Beyond that point, increasing anxiety probably inhibits achievement. These conditions describe a relationship that is curvilinear. It is illustrated in the following data, where anxi- ety is gauged as a function of someone’s increasing pulse rate in beats per minute, and achievement as the number of problems per minute the individual completes successfully.
Anxiety 52 54 58 62 64 67 72 73 75 78 82 86 88
Achievement 3 5 6 6 8 8 9 7 5 5 4 3 1
The scatterplot illustrating the relationship between anxiety and achievement is Fig- ure 9.3.
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CHAPTER 9Section 9.1 The Hypothesis of Association
Figure 9.3: The relationship between achievement and anxiety
Initially, there is a positive relationship between anxiety and perfor- mance. The first few pairs of data have points that rise from left to right. However, a positive relationship becomes negative when as anxiety increases, performance falls off. The correlation is curvilinear. After performance reaches 9 problems completed successfully per minute, more anxiety does not boost this individual’s performance.
The scatterplot also reveals some of the danger associated with range restriction. If someone collected data for only the first half of the sample, so that the first six pairs of scores were the sample, those scores provide very different indicators of the relationship between anxiety and performance than the last six pairs of scores. The first part of the distribution makes the relationship look linear and positive. The latter part of the data makes the relationship look linear but negative. An accurate picture of the relationship requires data throughout the entire ranges of the two variables. Since a curvi- linear relationship is detected in this example, specifically a quadratic curve, a Pearson cor- relation cannot be used as this violates the assumption of linearity. Testing for a quadratic curve can be estimated using the following SPSS steps:
Analyze S Regression S Curve Estimation
Such analyses is beyond the scope of this book, but curve estimation theory, steps, and out- put can be interpreted with the help of most advanced graduate-level statistics textbooks.
Understanding Correlation Values
It is important not to confuse the sign of the correlation with its strength. A correlation of 2.50 contains the same amount of information about the two variables as does a cor- relation of 1.50. The sign makes a great deal of difference to how the relationship is
C What impact does range attenua- tion have on a correlation?
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CHAPTER 9Section 9.2 Calculating the Pearson Correlation
interpreted, but it has nothing to do with the strength of the relationship. With positive correlations, as the value of one variable increases so does the value of the other. When correlations are negative, increasing values of one variable are associated with decreasing values of the other.
Earlier we noted that different scales of data require different types of correlation proce- dures. Different correlation procedures are also needed depending on the number and arrangements of the variables involved:
• Bivariate correlations indicate the relationship between two variables. Bivariate correlations are the focus of this chapter.
• Multiple correlation gauges the relationship between one variable and a combi- nation of others.
• Canonical correlation measures the relationship between two groups of variables.
• Partial correlation measures the relationship between two variables, controlling for the influence of a third variable on both of the first two.
• Semipartial correlation allows one to gauge the relationship between two vari- ables, controlling for the influence of a third on either of the first two.
• Point-biserial correlation allows a correlation with one of the variables having dichotomous data and the other variable with continuous data.
• Tetrachoric correlation allows a correlation when both variables are dichoto- mous data.
All the correlations in this bulleted list, except bivariate correlations, are beyond the scope of this book. Just note that correlation can involve several different configurations and scales of variables.
9.2 Calculating the Pearson Correlation
Also called the Pearson product-moment correlation coefficient, the Pearson correla-tion, or because its symbol is typically a lowercase r, “Pearson’s r” is probably the most often calculated of any correlation value. Thumbing through statistics books and online sources will reveal several formulas. All the formulas provide the same answer, but some are easier to follow than others. Visually, Formula 9.1 is probably simplest:
rxy 5 a 3 1zx2 1zy2 4
n 2 1 Formula 9.1
Note that the r has the subscripts x and y, which refer to the correlation of the x variable with the y variable. Because there is no assertion of cause, it does not matter which vari- able is labeled x and which y. Formula 9.1 indicates that
1. if the x scores are transformed into z scores (Formula 3.1: z 5 x 2 M
s )
2. and the y scores are also transformed into z scores, 3. the value of rxy is the sum of the products of the x and y z scores for each
participant, 4. divided by the number of participants in the data group minus 1.
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CHAPTER 9Section 9.2 Calculating the Pearson Correlation
5. The n 2 1 signifies that this is a correlation formula for sample, rather than popu- lation, data. It is the same adjustment for sample data made with the standard deviation calculation in Chapter 1.
• Formula 9.1 can be used to calculate the correlation value of the verbal ability and intelligence scores from the earlier example.
• Transforming the verbal ability and intelligence scores into their equivalent z values with Formula 3.1 produces the following z equivalents for the original raw scores:
Verbal ability (x)
21.991 21.212 2.848 2.537 2.173 0.87 .242 .398 .710 .917 1.021 1.385
Intelli- gence (y)
21.902 20.761 21.141 2.380 2.380 2.380 .380 .761 1.141 .761 .380 1.522
Here, each pair of z scores is multiplied and the products summed:
(21.991 3 21.902) 1 (21.212 3 20.761) 1 … 1 (1.385 3 1.522) 5 10.313
rxy 5 a 3 1zx2 1zy2 4
n 2 1
Because n (the number of pairs of scores) 5 12, n 2 1 5 11:
rxy 5 10.313
11 5 .938
With a maximum possible correlation value of 1.0, rxy 5 .938 indicates a strong relation- ship between verbal ability and intelligence, something that is reflected in the fact that many intelligence measures include measures of verbal ability.
Although Formula 9.1 is visually simple, the need to turn everything into z scores before calculating rxy makes the calculations very time-consuming and tedious. It is the long way around. Formula 9.2 is the formula we will use, and it also turns out to be the for- mula programmed into many hand-calculators. It is visually more complex but much easier to execute.
rxy 5 n a xy 2 ( a x)( a y)
Å e cn a x 2 2 ( a x)
2 d cn a y2 2 ( a y)2 d f Formula 9.2
Where
x 5 one of the scores in each pair
y 5 the other score in each pair (the designation of the x and y pairs must be consistent)
n 5 the number of participants for whom there are scores, or the number of pairs of scores
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CHAPTER 9Section 9.2 Calculating the Pearson Correlation
a xy indicates that the two scores for each participant are multiplied and then summed (This value is referred to as the sum of the cross products.)
a x 2 indicates that each x score is squared, and then the squares summed
( a x) 2 indicates that the x scores are all totaled, and then that total squared
a y 2 indicates that each y score is squared, and then the squares summed
( a y) 2 indicates that the y scores are all totaled, and then that total squared.
The formula is not as daunting as it seems. After calculating a problem or two, the process will become familiar. Probably most of the statistical heavy-lifting will be done with Excel or a hand-calculator that has a built-in correlation function, but it is helpful to prepare for that occasional time when there is no computer and the calculator has no correlation function.
A Correlation Example
A researcher is duplicating a classic experiment by psychologist E. L. Thorndike. The experiment is related to Thorndike’s law of effect, which maintains that behaviors fol- lowed by a satisfying state of affairs are likely to be repeated. In the experiment, the researcher sets up a cage equipped with a door that opens if a cat placed in the cage bats a string suspended inside the cage. According to the law of effect, if batting the string is followed by something satisfying, that behavior should occur more frequently than other behaviors in future trials. A hungry cat is placed in the cage and food is placed outside where it is inaccessible from the inside of the cage. The data represents the several trials and the amount of time in minutes that elapse each time before the cat releases itself. This experiment is repeated 10 times over as many days. The data is as follows:
Trial number 1 2 3 4 5 6 7 8 9 10
Elapsed time 5.0 5.5 4.75 4.5 4.25 3.5 2.75 2.0 1.0 .25
The scatterplot for this data is in Figure 9.4. The data suggests that the relationship is prob- ably negative and quite strong.
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CHAPTER 9Section 9.2 Calculating the Pearson Correlation
Figure 9.4: The relationship between number of trials and elapsed time
The correlation value offers a check of both conclusions. Verify that
n 5 10
a xy 5 137.25
ax 2 5 385
( a x) 2 5 (55)2
ay 2 5 141
( a y) 2 5 (33.5)2
Now, use Formula 9.2:
rxy 5 n a xy 2 ( a x)( a y)
Å e cn a x 2 2 ( a x)
2 d cn a y2 2 ( a y)2 d f
Substituting the relevant values provides
rxy 5 101137.152 2 1552 133.52
"531013852 2 1552 2 4 31011412 2 133.52 2 46
rxy 5 1372.5 2 1842.5
"3 13850 2 30252 4 3 11410 2 1122.252 4
rxy 5 2470
"1825 3 287.752 5 2.965
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CHAPTER 9Section 9.2 Calculating the Pearson Correlation
Interpreting Results
The relationship is indeed negative and because the maximum correlation is 61.0, the relationship is also substantial, but neither of those conclusions indicates whether it is statistically significant. As with z, t, and F, significance is determined by comparing the calculated value to a table value. For the Pearson correlation, it is Table 9.1 (also Table E in the Appendix).
Table 9.1: The critical values of rxy
Number of xy Pairs (n)
Degrees of Freedom (n 2 2)
Alpha (A) Level
0.10 0.05 0.01
3 1 0.988 0.997 1.000
4 2 0.900 0.950 0.990
5 3 0.805 0.878 0.959
6 4 0.729 0.811 0.917
7 5 0.669 0.754 0.875
8 6 0.621 0.707 0.834
9 7 0.582 0.666 0.798
10 8 0.549 0.632 0.765
11 9 0.521 0.602 0.735
12 10 0.497 0.576 0.708
13 11 0.476 0.553 0.684
14 12 0.458 0.532 0.661
15 13 0.441 0.514 0.641
16 14 0.426 0.497 0.623
17 15 0.412 0.482 0.606
18 16 0.400 0.468 0.590
19 17 0.389 0.456 0.575
20 18 0.378 0.444 0.561
21 19 0.369 0.433 0.549
22 20 0.360 0.423 0.537
23 21 0.352 0.413 0.526
24 22 0.344 0.404 0.515
25 23 0.337 0.396 0.505
Source: Brighton Webs Ltd. Statistical and Data Services for Industry. (2006). Critical values of correlation coefficient R. Retrieved from http://www.brighton-webs.co.uk/statistics/critical_values_r.aspx
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CHAPTER 9Section 9.2 Calculating the Pearson Correlation
Like the t and F values, the correct critical value for r is determined by the degrees of free- dom. They are the number of pairs of data minus 2. Be careful not to confuse the number of pairs with the number of scores.
Table 9.1 has columns for a 5 .1, a 5 .05, and a 5 .01. The columns for .05 and .01 are quite typical; the column for a 5 .1 occurs in statistical tables less often. For 8 degrees of freedom (the number of pairs of data minus 2) and with a 5 .05, the critical value is .632, which is written this way:
rxy .05(8) 5 .632
The subscripts help distinguish the table value from the calculated correlation value.
If the calculated value equals or exceeds the table value, the result is statistically sig- nificant. In two-tailed tests, the sign of the correlation value—whether it is positive or negative—is not relevant to whether it is significant. Some tables also include the critical values for one-tailed correlations, which are relevant for determining whether a value with a specified sign is significant. Table 9.1 has only the values for two-tailed tests.
The Relationship Between Degrees of Freedom and Significance
Even with a correlation value as extreme as .956, checking the table for significance is important. In both the t-test and ANOVA, the magnitude of the critical values declines as degrees of freedom (and sample size) increase. It is the same with correlation, but here the decline in critical values is more dramatic. Note from Table 9.1, for example, that if n 5 3 (and, therefore, df 5 1) the correlation would need to be at least rxy 5 .997 (nearly perfect) to be statistically significant. At the other extreme, if n 5 25 (so that df 5 23) a correlation of just rxy 5 .396 is statistically significant!
The Statistical Hypotheses
The null and alternate hypotheses take on different forms with correlation than they had for the difference tests. The null hypothesis is that there is no relationship between the variables. Symbolically, it is written this way:
H0: r 5 0
The r is the Greek letter rho (as in “row your boat”) and is the equivalent of r, the symbol for a Pearson correlation. So the null hypothesis says in effect r, or the correlation, equals 0. More specifically, it means that there is no statistically significant relationship, which is one that is likely to emerge every time data is collected and analyzed. The alternate hypothesis states that the correlation does not equal 0; in this case, there is a statistically significant relationship:
Ha: r ? 0
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CHAPTER 9Section 9.2 Calculating the Pearson Correlation
The Coefficient of Determination
One of the recurring themes in this book is the distinction between statistical significance and practical importance. Measures of practical importance were the reason for calculat- ing Cohen’s d and eta-squared for significant t-test and ANOVA results.
Effect sizes may be even more important for correlation results because relatively small correlations can be statistically significant when the sample sizes are large enough. The effect size for the Pearson correlation is the coefficient of determination (rxy
2). As the notation suggests, the coefficient of determination is simply the square of the correlation coefficient. Squaring the correlation indicates how much of the variance in y is explained by x (or vice versa because there is no judgment about cause).
In the problem about number of trials and elapsed time,
rxy 5 2.965, so
rxy 2 5 .931
For that problem, the coefficient of determination is interpreted this way: 93.1% of the variance in time elapsed can be explained by the number of trials, which would be a very important finding with implications for many kinds of performance tasks, except that the numbers were contrived to begin with. Consequently, the coefficient of nondetermina- tion is 1 2 rxy
2, which indicates variance due to unexplained causes or error. In this case it would be
1 2 .931 5 . 069 or 6.9%
The Interpretive Value of rxy 2
The statistic can also indicate how unimportant some low correlations are, even when they are statistically significant. For example, with 23 degrees of freedom, a correlation of rxy 5 .396 is statistically significant. The coefficient of determination for that value is rxy
2 5 .157. One variable in such a relationship explains just 16% of the variance in the other. The other 84% of the variability is related to other factors.
When the variables describe the behavior of people, small coefficients of determination do not surprise us. This is just part of the complexity of human subjects. It is not unusual to expect relatively low values when people are the subjects and only two variables are involved.
However, low correlations and low rxy 2 values are sometimes quite important. If research
revealed that the correlation between the age of first exposure to illegal narcotics and the development of an addiction was rxy 5 2.3 (the younger the subject is at first exposure, the more likely the individual is to develop an addiction), the resulting rxy
2 value would be just .09. But even if just 9% of the variance in addiction was somehow related to age of exposure, because the consequences are so dramatic for the individual and indeed for society, the value would probably be thought of as important. The importance of an rxy
2
value has to be discussed within the context of consequences.
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CHAPTER 9Section 9.2 Calculating the Pearson Correlation
Comparing Correlation Values
In isolation, correlation coefficients can be difficult to interpret because correlation strength does not increase or decrease in consistent increments. The change from rxy 5 .2 to rxy 5 .3 is a less dramatic increase in strength than the increase from rxy 5 .75 to rxy 5 .85, for example. Although the Pearson r requires equal interval data, in the coefficients that are the result, an increase in correlation strength of .1 reflects a very different change when it is from .8 to .9 than it does from .2 to .3. It takes a much stronger increase in the relation- ship to increase by .1 in the upper ranges of correlation values than in the lower ranges, something suggested by the distance between tenths in this number line:
rxy 5 .1 .2 .3 .4 .5 .6 .7 .8 .9
Squaring the correlation coefficient makes the intervals consistent. A change in the coef- ficient of determination from .35 to .5, for example, represents the same increase in pro- portion of variance explained as an increase from .7 to .85, as the following line suggests. Squaring of the r (or r2) indicates the magnitude or effect size. As seen in previous chap- ters, effect size is a statistic that is used to interpret the degree of influence of the IV on the DV. In the case of a correlation, it is the magnitude or strength of the association between two variables. Cohen’s (1988) guidelines for the magnitude of a correlation coefficient are .1 is small, .3 is moderate, and .5 is large.
r2xy 5 .1 .2 .3 .4 .5 .6 .7 .8 .9
Another Correlation Problem
A consultant is asked to solicit political contributions for a politician’s campaign commit- tee. The consultant reviews some of the data available and notes that age appears to be related to the number of dollars donated. To test this observation, a sample is randomly selected from a list of national donors, which yields the following:
Donor 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Age 25 27 32 32 35 38 43 45 45 47 48 52 63 65 66
Amount 20 20 35 25 100 50 75 45 100 150 100 200 50 100 125
The problem is completed in Figure 9.5.
• The correlation of the donor’s age with the amount the donor contributed is rxy 5 .564.
• The critical value at p 5 .05 for 13 degrees of freedom is .514. • Because rxy . r.05(13), the correlation is statistically significant. • The coefficient of determination (rxy
2 5 .318) indicates that about 32% of the variability in how much donors contribute can be explained by their age.
D What is the relationship between degrees of freedom and statistical significance in correlation?
Try It!
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CHAPTER 9Section 9.3 Correlating Data When One Variable Is Dichotomous
Figure 9.5: The Pearson correlation for age and political contribution amount
9.3 Correlating Data When One Variable Is Dichotomous
If the question about donor contributions had been about the relationship between the amount donated and the gender of the donor, the Pearson approach will still provide the answer, but the correlation procedure is called a point-biserial correlation(rpb). The word point refers to the continuous variable, the amount of money donated in this example. The word biserial refers to the other variable, which has only two values or dichotomous data. The change that is required is that the gender variable has to be coded in a way that
Donor’s Age
Contribution Amount
x x2 y xyy2
25
27
32
32
35
38
43
45
45
47
48
52
63
65
66
20
20
35
25
100
50
75
45
100
150
100
200
50
100
125
625
729
1,024
1,024
1,225
1,444
1,849
2,025
2,025
2,209
2,304
2,704
3,969
4,225
4,356
400
400
1,225
625
10,000
2,500
5,625
2,025
10,000
22,500
10,000
40,000
2,500
10,000
15,625
500
540
1,120
800
3,500
1,900
3,225
2,025
4,500
7,050
4,800
10,400
3,150
6,500
8,250
�x = 663 �x2 = 31,737 �y2 = 133,425�y = 1,195 �xy = 58,260
rxy =
=
=
0.564=
rxy 0.05(13) =
15(58,260) � (663)(1,195)
�{[15(31,737) � 6632] [15(133,425) � 1,1952]}
81615
�[(36,486)(573,350)]
n�xy � (�x)(�x)
�{[n�x2 � (�x)2] [n�y2 � (�y)2]}
0.514
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CHAPTER 9Section 9.3 Correlating Data When One Variable Is Dichotomous
reflects its dichotomy. To determine the correlation between the donor’s gender and the amount donated, gender has to be coded either 0 or 1. In terms of the strength of the coef- ficient, it does not matter whether females or males are coded 0 or 1. Since point-biserial is a test of relationships or an association between two variables, it is not the best choice for a causal relationship with a dichotomous independent variable on a continuous dependent variable. For instance, if we wanted to look at the differences between gender (dichoto- mous data) on their level of work-family conflict, an independent-samples t-test or Mann- Whitney U-test will be more robust in testing significant group differences.
The point-biserial correlation has a number of applications. Questions about the relation- ship between gender and income, between public versus private school students and achievement, between Republican versus Democrat and optimism (assuming that opti- mism is measured on at least an interval scale) are all questions that can be analyzed with the point-biserial correlation.
The only adjustment is to code the dichotomous variable as either 0 or 1. When the donors are recoded according to gender, whether the 0 indicates women or men will not affect the strength of the coefficient. It will affect the sign, however, so there needs to be some care in interpreting the result. If donors 3, 5, 6, 7, 9, 10, 11, and 14 are female, and if females are coded 1 and males 0, the following is the result:
Donor (x) 0 0 1 0 1 1 1 0 1 1 1 0 0 1 0
Amount (y) 20 20 35 25 100 50 75 45 100 150 100 200 50 100 125
Calculating the Point-Biserial Correlation
The amounts donated (the y values) remain the same from the age/donor problem and can be retrieved from Figure 9.5, where a y 5 1,195 and a y
2 5 133,425. The other val- ues must be recalculated, although it is much simpler with the gender (x) variable recoded to 1s and 0s. Here is the data in columns:
Gender (x) x2 Amount (y) y2 xy
0 0 20 400 0
0 0 20 400 0
1 1 35 1,225 35
0 0 25 625 0
1 1 100 10,000 100
1 1 50 2,500 50
1 1 75 5,625 75
0 0 45 2,025 0
1 1 100 10,000 100
(continued)
suk85842_09_c09.indd 343 10/23/13 1:43 PM
CHAPTER 9Section 9.3 Correlating Data When One Variable Is Dichotomous
Gender (x) x2 Amount (y) y2 xy
1 1 150 22,500 150
1 1 100 10,000 100
0 0 200 40,000 0
0 0 50 2,500 0
1 1 100 10,000 100
0 0 125 15,625 0
a x 5 8 a x 2 5 8 a y 5 1,195 a y
2 5 133,425 a xy 5 710
rxy 5 n a xy 2 ( a x)( a y)
Å e cn a x 2 2 ( a x)
2 d cn a y2 2 ( a y)2 d f
rxy 5 15(710) 2 (8)(1,195)
"5315(8) 2 (8)2 4 315(133,425) 2 (1,195)2 46
rxy 5 10,650 2 9,560
"3(120 2 64)4 3(2,001,375 2 1,428,025)4
rxy 5 1,090
"32,107,600 5 .19
Still testing at p 5 .05 and with the degrees of freedom still df 5 13,
• the critical value is still rxy .05(13) 5 .514, and • the statistical decision will be to fail to reject H0. The relationship between the
donor’s gender and the amount contributed is not statistically significant. • The rxy 5 .19 is probably a random correlation that is unlikely to be significant if
new data was collected and the analysis repeated.
The interpretation of the point-biserial correlation is the same as it is for a conventional Pearson correlation except that whether the coefficient is positive or negative is a function only of which variable is coded 1. If male donors had been coded with 1s, the correlation would have been negative, rxy 5 2.19.
There are many applications for this procedure. Each of the following questions can be answered with a point-biserial correlation:
• What is the relationship between whether or not a parent earned a college degree and the child’s grades?
• How is whether or not a student is a native speaker of English related to the stu- dent’s test score?
• What is the correlation between blue-collar/white-collar jobs and the amount of leisure time?
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CHAPTER 9Section 9.4 The Pearson Correlation in Excel
If both variables are dichotomous, another bivariate correlation is involved. It is called the phi coefficient and will be covered in Chapter 11.
“Degrees of Significance”?
At rxy 5 .19 and a table value of rxy .05(13) 5 .514, the correlation value is not significant. If it had been rxy 5 .50, and this correlation value was some relationship calculated for your senior thesis, would it be appropriate to refer to it as “almost significant” or “nearly significant”? It is not uncommon to see such qualifiers even in the published literature, but significance decisions should be treated the same way as dichotomous variables are treated. There are just the two possible outcomes: The correlation is significant or it is not significant. To try to make a statement about the nearness to an alternative outcome undermines the principle behind significance testing. There are just two hypotheses, and the outcome is couched in terms of one outcome or the other.
9.4 The Pearson Correlation in Excel
A psychologist is interested in determining the relationship between risk taking and success in solving novel problems. Having devised the Inventory Risk Survey Cat- alog (the I-RiSC), the researcher gauges a group of 16-year-olds’ willingness to do the unconventional and then provides a series of word problems with which the participants are unfamiliar. Scores on the I-RiSC and the problems are as follows for 10 participants:
I-RiSC 2 7 4 5 1 8 7 9 3 6
Problems 14 17 14 16 12 17 16 17 15 15
To complete the problem in Excel, it is easiest to set up the data in two columns. Two rows also will work, but parallel columns are visually simpler.
• Create a label in cell A1 for I-RiSC and in cell B1 ProbSolv so that the I-RiSC data will be in cells A2 to A11 and the ProbSolv data in B2 to B11.
• From the Home tab at the top of the page click Data and then Data Analysis at the far right.
• Select Correlation, which is the second option in the window. • In the Input Range window enter A2:B11, which indicates where the data is
found. Note that the default is that the data is grouped in columns. Had they been entered in rows, this would need to be changed. Had the Labels in First Row box been checked, Excel would have treated the first row in each column (A2 and B2 because that is what is designated) as labels rather than data. That adjustment for the labels was made by indicating that the data begins in A2 rather than A1.
• Enter a cell value below or to the right of the last data entry for the Output Range so that the results do not overwrite the scores—either cell A12 or lower, or to the right of column B.
• Click OK.
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CHAPTER 9Section 9.4 The Pearson Correlation in Excel
The output is deceptively simple. The following box is called a correlation matrix, and it is often used to present correlation results. The intersection of column 1 and column 2 indicates how well the data in column 1—the Excel A column, where I-RiSC data is located—correlates with the data in column 2—the Excel B column, which contains the problem-solving scores.
Column 1 Column 2
Column 1 1
Column 2 0.904203 1
The result of the analysis is a Pearson correlation of rxy 5 .904. The 1s in the diagonal indicate that each variable correlates perfectly with itself (rxy 5 1.0), of course. Note that the output does not indicate whether the calculated value is statistically significant, which makes a check of the critical values table necessary. Table 9.1 indicates that rxy .05(8) 5 .632. The relationship between risk taking and problem solving is statistically significant. Was this not contrived data, it would be quite important to know that about 82% (rxy
2 5 .818) of problem-solving success (.9042) is explained by whatever the I-RiSC measures.
Apply It!
Investigating the Correlation Between Crime and Unemployment
A criminal investigative analyst is interested in studying a possible link between crime and unemployment to help in allocating crime prevention funds. Specif-
ically, she would like to know if murder rates and property crime rates in her state correlate with the state unemployment rate.
The analyst obtains the murder and property rates for her state for the 16 years from 1990 to 2005 from the FBI Uniform Crime Reports. (These rates are given per 100,000 inhabit- ants.) She then consults the Bureau of Labor Statistics for the unemployment rate (in per- centages) in her state during that same time period.
The analyst will compute the Pearson’s r correlation between murder rate and unemploy- ment. She will compute a separate correlation between property crime rate and unemploy- ment. The Pearson’s r requires interval or ratio variables that are normally and similarly distributed. The null hypothesis is that there is no relationship between the variables, writ- ten H0: r 5 0.
(continued)
suk85842_09_c09.indd 346 10/23/13 1:43 PM
CHAPTER 9Section 9.4 The Pearson Correlation in Excel
Apply It! (continued)
Crime rates and unemployment rates for this state are given for the years 1990 through 2005.
Year Murder Rate Property Crime Rate Unemployment
1990 7.1 4462 5.6
1991 6.7 5092 6.8
1992 6.4 4801 7.5
1993 6.4 4662 6.9
1994 6.2 4678 6.1
1995 5.7 4460 5.6
1996 5.8 4438 5.4
1997 5.4 4279 4.9
1998 6.1 4040 4.5
1999 5.5 3852 4.2
2000 5.1 3592 4.0
2001 4.9 3456 4.7
2002 4.3 3412 5.8
2003 4.2 3289 6.0
2004 4.7 3168 5.5
2005 5.0 3081 5.1
Using Excel, the correlation value for murder rate and unemployment is rxy 5 .386.
Using Excel, the correlation value for property crime rate and unemployment is rxy 5 0.551.
Using 14 degrees of freedom (16 data sets 2 2) and with the risk of type I, or alpha, error at p 5 .05, the critical value is rxy .05(14) 5 .497.
If the calculated value equals or exceeds the table value, the result is statistically significant. Therefore, the correlation between murder rate and unemployment is not statistically sig- nificant because
rxy 5 .386 , rxy .05(14) 5 .497.
(continued)
suk85842_09_c09.indd 347 10/23/13 1:43 PM
CHAPTER 9Section 9.5 Nonparametric Tests: Spearman’s Rho
9.5 Nonparametric Tests: Spearman’s Rho
The Pearson correlation requires that both variables must be at least interval scale. The point-biserial correlation requires that one variable must be at least interval scale, and the other must be a variable with only two levels.
Neither of those is helpful when the data is ordinal scale, which describes much of the data that psychologists and other social scientists encounter. Nearly everyone who goes to the mall or answers the telephone has been asked to take a survey, particularly if it happens to be an election year. Sur- vey data is usually ordinal scale. It is common for the ques- tionnaires to have a Likert-type format, where a statement is read and the respondents are asked the degree to which they agree with the statement by selecting from a range of choices such as
Strongly agree
Agree
Neither agree nor disagree
Disagree
Strongly disagree
Although it is common enough to code the responses (Strongly agree 5 1, Agree 5 2, and so on) and then calculate means and standard deviations for all respondents, those sta- tistics assume that the data is at least interval scale. Survey data rarely is. The Likert types of responses are essentially rankings. A response of “Strongly agree” is more positive
The links provided below introduce two sides of a debate: Is Likert scaling ordinal or interval data? Read both sides of the debate to formulate your own conclusion.
Likert scaling is ordinal data, as presented in ASQ’s Quality Progress, Statistics Roundtable:
http://asq.org/quality -progress/2007/07/ statistics/likert-scales -and-data-analyses.html
Likert scaling is interval data, as presented in the Shiken Research Bulletin:
http://jalt.org/test/bro _34.htm
Try It!
Apply It! (continued)
However, the correlation between property crime rate and unemployment is significant. Therefore, the null hypothesis is rejected. There is a relationship between these two variables.
rxy 5 0.551 . rxy .05(14) 5 .497
The coefficient of determination is rxy 2 5 .55122 5 .303.
The coefficient of determination indicates that about 30% of the variance in the property crime rate can be explained by the unemployment rate. The criminal investigative analyst can use this information to divert more funds to preventing property crimes during times of high unemployment.
Apply It! boxes written by Shawn Murphy.
suk85842_09_c09.indd 348 10/23/13 1:43 PM
CHAPTER 9Section 9.5 Nonparametric Tests: Spearman’s Rho
than “Agree” but precisely how much more is not clear. Besides, one respondent’s “Dis- agree” may be another respondent’s “Strongly disagree.” This data is more safely treated as ordinal scale responses.
Back to Correlation
In addition to survey data, other common data also is ordinal scale, such as class rankings and percentile scores. Sometimes one variable is interval/ratio. For example, a researcher might wish to correlate a person’s annual income (ratio data) with subjects’ optimism, which is gauged by a Likert-type survey (ordinal data). Although ratio scale (if your income is 0, it really does mean you have no money), income data is also frequently non- normal. Income data usually has a right skew, which is why reports refer to median rather than mean income. The lack of normality in the ratio variable and a second variable that is ordinal scale both rule out a Pearson’s correlation.
Charles Spearman, Pearson’s colleague at University College London, developed a cor- relation procedure that is tremendously flexible. It will accommodate two variables in a correlation procedure that fit any of the following:
• Both are ordinal scale. • One variable is ordinal scale and one is interval or ratio scale. • Two variables are interval or ratio scale, but one or both fail to meet the Pearson
correlation requirement for normality.
The procedure is Spearman’s rho, for which the symbol will be rs. Spearman’s rho is a nonparametric procedure. Nonparametric means that no assumptions need to be made about parameters. For our purposes, it means that rs will accommodate nonnormal data, or data for which normality cannot be judged. The formula is as follows:
rs 5 1 2 6 a d
2
n(n2 2 1) Formula 9.3
Where
d 5 the difference between the rankings for the two variables
n 5 the number of pairs of data
The 1s and the 6 are constant values. They are there every time a Spearman’s correlation is calculated.
The steps to calculating a Spearman’s rho follow:
1. Rank the scores for both variables separately. 2. For each pair of rankings, subtract the second ranking in the pair from the first to
produce a difference score, d. 3. Square each of the d values for d2. 4. Sum the d2 values for a d
2. 5. Solve for rs.
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CHAPTER 9Section 9.5 Nonparametric Tests: Spearman’s Rho
Ranking Tied Scores
There are rules to follow for the ranking procedure. If there are multiples of some of the scores for one of the variables, they must all receive the same ranking. If someone were ranking the following values, for example:
3, 5, 6, 6, 7, 8, 8, 8, 9, 10
ranking the values from smallest to largest would produce
1, 2, 3.5, 3.5, 5, 7, 7, 7, 9, 10
The two 6s and the three 8s are handled as follows:
• Because the two 6s occupy rankings 3 and 4, those two rankings are added together and divided by the number of tied scores, which are two: (3 1 4) 4 2 5 3.5. Both the 6s receive rankings of 3.5, after which the next ranking must then be 5.
• The three 8s occupy rankings 6, 7, and 8. Following the same procedure, (6 1 7 1 8) 4 3 5 7, after which the next ranking must be 9.
An Example
Suppose the preceding data is a measure of economic conservatism among a group of investors, with higher scores indicating the most caution. If the effort is to check for a relationship between investors’ ages and their level of conservatism, the following might be the result:
Conservatism Age
3 26
5 25
6 32
6 35
7 35
8 34
8 37
8 40
9 42
10 39
E Spearman’s rho requires data of what scale?
Try It!
suk85842_09_c09.indd 350 10/23/13 1:43 PM
CHAPTER 9Section 9.5 Nonparametric Tests: Spearman’s Rho
A Spearman’s rho solution is calculated in Figure 9.6. The Spearman’s correlation is rs 5 .852. Table 9.2 is the table of critical values for Spearman’s rho. It is also Table F in the Appendix. There are no degrees of freedom for the Spearman’s value to guide us to the correct critical value, only the number of pairs of data. Note that for p 5 .05 and 10 pairs of data, the critical value is rs .05 (10) 5 .648. The relationship between level of economic conservatism and age in this group of investors is statistically significant; reject H0.
Figure 9.6: The Spearman’s rho correlation: economic conservatism with age
Age
Co ns
er va
tis m
R1 R2 d (R1 � R2) d2
3
5
6
6
7
8
8
8
9
10
26
25
32
35
35
34
37
40
42
39
1
2
3.5
3.5
5
7
7
7
9
10
2
1
3
5.5
5.5
4
7
9
10
8
�1
1
0.5
�2
�0.5
3
0
�2
�1
2
1
1
0.25
4
0.25
9
0
4
1
4
�d2 = 24.50
1. Ranking the scores produces R1 for conservatism and R2 for age.
2. The d score is the difference between the two rankings.
3. The square of the difference score is d2.
6�d2
n(n2 � 1) r5 = 1 �
6(24.5) 10(102 � 1)
= 1 �
= 1 � 0.148
= 0.852
suk85842_09_c09.indd 351 10/23/13 1:43 PM
CHAPTER 9Section 9.5 Nonparametric Tests: Spearman’s Rho
Table 9.2: Critical values for Spearman’s rho
Number of Pairs of Scores p 5 .05 p 5 .01
5 1.0
6 .886 1.0
7 .786 .929
8 .738 .881
9 .683 .883
10 .648 .794
12 .591 .777
14 .544 .715
16 .506 .665
18 .475 .625
20 .450 .591
22 .428 .562
24 .409 .537
26 .392 .515
28 .377 .496
30 .364 .478
Source: Critical values of Spearman's rho (two-tailed). Copyright © Graham Hole, University of Sussex. Retrieved from www.sussex .ac.uk/Users/grahamh/RM1web/Rhotable.htm
Apply It!
Exploring the Correlation Between Job Satisfaction and Commute Times
The human resources manager for a large firm is developing guidelines to allow employees to work at home part time. As part of this project, he would
like to see if there is a correlation between job satisfaction and average commute time.
He asks 10 randomly chosen employees to fill out a questionnaire regarding job satisfaction. Each employee rates his or her overall job satisfaction in one of four categories:
• very satisfied (vs) • somewhat satisfied (ss) • somewhat dissatisfied (sd) • very dissatisfied (vd)
(continued)
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CHAPTER 9Section 9.5 Nonparametric Tests: Spearman’s Rho
Apply It! (continued)
These same employees are also asked to list their average one-way commute time in min- utes on this survey.
Because the job satisfaction data is ordinal scale, the Pearson’s correlation is not appropri- ate, and the HR manager will instead use Spearman’s rho to determine the correlation.
Commute Time
(Minutes)
Commute Rank
Job Satisfaction
Satisfaction Rank
Difference Between
Ranks
Difference2
2 1 vs 2 –1 1
7 2 ss 5 –3 9
11 3 vs 2 1 1
15 4 ss 5 –1 1
17 5 vs 2 3 9
23 6 ss 5 1 1
28 7 sd 7.5 –0.5 0.25
32 8 vd 9.5 –1.5 2.25
36 9 vd 9.5 –0.5 0.25
40 10 sd 7.5 2.5 6.25
ad 2 5 31
N 5 10
The Spearman’s rho formula is
rs 5 1 2 6 ad
2
n(n2 2 1) 5 0.812
For rs 5 .05 and 10 pairs of data, the critical value is rs .05 (10) 5 .648
Because 0.812 . 0.648, the null hypothesis is rejected. The relationship between job satis- faction and average commute time is significant.
This study has shown that the longer the average commute time employees have, the more likely they are to be dissatisfied with their job. The human resources manager can use this information as one reason the company should enact a policy that allows employees to work from home.
Apply It! boxes written by Shawn Murphy.
suk85842_09_c09.indd 353 10/23/13 1:43 PM
CHAPTER 9Section 9.5 Nonparametric Tests: Spearman’s Rho
The Direction of the Ranking
In the example of economic conservatism among investors, the least conservative value received the ranking of 1, and the most, 10. With respect to age, the youngest investor received the ranking of 1. In terms of the value of the statistic, it does not matter whether the rank- ings go from lowest to highest, or from highest to lowest, as long as the same is done for both variables. If we decided that because 10 is the most conservative score in this set that it should be ranked 1, that would work fine, but the same would have to be done for age—the high- est age would need to receive a ranking of 1. If the data in one group ranked from highest down and the other from lowest up, then the cor- relation would appear to be negative!
The Spearman’s Rho in Summary
Spearman’s correlation provides flexibility to the analyst. As long as there is some evi- dence of a relationship, correlations can be calculated for any combination of ordinal, interval, and ratio variables. But of course there has to be some sacrifice for so much lati- tude, and it is statistical power. Note that part of Spearman’s process is the ranking of the data. In the course of ranking values, the amount of difference between any two data points is no longer part of the analysis. When the ages of the investors were ranked,
• the 25-year-old was 1 • the 26-year-old was 2 • and the 32-year-old was 3
As soon as ages are converted to rankings, the amount of age difference between the 26- and 32-year-old was lost. The only information evident from the rankings is that the 26-year-old is 2 and the 32-year-old is 3. That does not happen with a procedure designed specifically for interval or ratio data like Pearson’s r. Consequently, although there some- times is not much difference between their coefficients, a Pearson correlation will some- times be statistically significant when Spearman’s is not. Note the following comparison of critical values at a 5 .05:
No. of Pairs Pearson’s Critical Value* Spearman’s Critical Value
5 .878 1.0
6 .811 .886
10 .632 .648
*For df 5 no. of pairs 2 2.
In each case, the value required for significance with a Spearman correlation is higher than that required for a Pearson correlation.
F If the grade averages of 10 students who were ranked 1 through 10 in their class were correlated with their class rankings and the highest GPA was given a ranking of 1, and so on, what would the resulting coefficient indicate?
Try It!
suk85842_09_c09.indd 354 10/23/13 1:43 PM
CHAPTER 9Section 9.6 Presenting Results
Another limitation is that we cannot square the Spearman value to determine the propor- tion of variance in y explained by x. There is no equivalent of rxy
2 for Spearman’s rho.
When the data does not meet the Pearson requirements, there is not a choice to be made. When it does, a Pearson’s r is going to be more precise than Spearman’s rho. In the next section, we will use real data for a multiple correlation with a Pearson correlation and Spearman rho comparison using SPSS.
9.6 Presenting Results
Using a public data set from Pew Research (2010), YouthEconomy_dataset.sav file, a multiple correlation will be performed between four variables, Specifically age, q1 (happiness), q5 (optimism), and q11 (job satisfaction). Both a Pearson correlation and a Spearman rho will be performed simultaneously using SPSS.
To perform the analysis, go to AnalyzeSCorrelateSBivariate. Input age, q1 (happiness), q5 (optimism), and q11 (job satisfaction) into the Variables box (your screen should look like that in Figure 9.7). Check both Pearson and Spearman boxes below. Then click OK. The resulting SPSS output tables are provided in Figure 9.8.
Figure 9.7: SPSS steps in a multiple correlation
Source: Pew Research Social & Demographic Trends. (2011). Youth and Economy. Retrieved from http://www.pewsocialtrends.org /category/datasets/
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CHAPTER 9Section 9.6 Presenting Results
Figure 9.8: SPSS output in a multiple correlation
Source: Pew Research Social & Demographic Trends. (2011). Youth and Economy. Retrieved from http://www.pewsocialtrends.org/ category/datasets/
Correlations
Sig. (2-tailed)
Pearson Correlation
N
Sig. (2-tailed)
Pearson Correlation
N
Q1.
AGE.
Q.5
Q.11
**. Correlation is significant at the 0.01 level (2-tailed). *. Correlation is significant at the 0.05 level (2-tailed).
Sig. (2-tailed)
Pearson Correlation
N
Sig. (2-tailed)
Pearson Correlation
N
AGE.
9197
0.091**
1
0.000
9197
0.000
9197
�0.083**
0.334**
0.000
5366
Q1.
0.000
9197
1
0.901**
9197
0.000
9197
0.127**
0.180**
0.000
5366
Q.5
0.000
9197
0.180**
0.334**
0.000
9197
1
9197
�0.034*
0.013
5366
Q.11
0.000
5366
5366
0.127**
�0.083**
0.000
0.013
5366
1
�0.034*
5366
Correlations
AGE. What is your age?
Spearman’s rho
N
Sig. (2-tailed)
Correlation Coefficient
N
Q1.
Q.5
Q.11
**. Correlation is significant at the 0.01 level (2-tailed).
Sig. (2-tailed)
Correlation Coefficient
N
Sig. (2-tailed)
Correlation Coefficient
N
Sig. (2-tailed)
Correlation Coefficient
AGE.
9226
0.056**
0.000
9226
0.000
9226
�0.136**
0.410**
0.000
5385
1.000
Q1.
9226
1.000
9226
0.000
9226
0.208**
0.119**
0.000
5385
0.000
0.506**
Q.5
9226
0.119**
0.000
9226
1.000
9226
�0.019
0.168
5385
0.000
0.410**
Q.11
5385
5385
0.208**
0.000
0.168
5385
1.000
�0.019*
5385
0.000
�0.136**
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CHAPTER 9Summary
9.7 Interpreting Results
Refer to the most recent edition of the APA manual for specific detail on formatting statistics; Table 9.3 may be used as a quick guide in presenting the statistics covered in this chapter.
Table 9.3: Guide to APA formatting of statistics results
Abbreviation or Term Description
r Pearson’s correlation coefficient
rs Spearman’s correlation coefficient
r Rho
Source: Publication Manual of the American Psychological Association, 6th edition. © 2009 American Psychological Association, pp. 119–122.
Using the results from the preceding SPSS example, we present the results (Figure 9.8), in the following way:
• There is a significant Pearson correlation between age with q1-happiness [r(9197) 5 .091, p , .05] and q5-optimism [r(9197) 5 .334, p , .05]. The same is true for Spearman rho with age and q1-happiness [rs(9226) 5 .056, p , .05] and q5-optimism [rs(9226) 5 .410, p , .05]. In other words, as age increases, so do happiness and optimism.
• There was a significant negative correlation between age and q11-job satisfaction for both Pearson and Spearman [r(5366) 5 2.083, p , .05] [rs(5385) 5 2.136, p , .05]. As age increases, job satisfaction decreases.
• There is a significant negative Pearson correlation between q5-optimism and q11- job satisfaction [rs(5366) 5 2.034, p , .05] but not statistically significant Spearman rho for q1-happiness and q11-job satisfaction [r(5385) 5 2.19, p 5 .168]. According to the significant correlation, as optimism increases, job satisfaction decreases.
Summary Many of the questions researchers and scholars ask are about the relationships between variables. To accommodate them, the discussion in this chapter shifted to statistical pro- cedures that fall under the hypothesis of association umbrella (Objective 1). Three of the many correlation procedures that respond to the hypothesis of difference are the Pearson correlation, the point-biserial correlation, and Spearman’s rho. In each case, their pos- sible values range from 21.0 to 11.0, and all their coefficients are interpreted the same way. Positive correlations indicate that as the values in one variable increase, the values in the other also increase. Negative correlations indicate that as one increases, the other decreases. The sign of the coefficient, however, is unrelated to its strength (Objective 2).
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CHAPTER 9Key Terms
The differences among the correlation procedures in this chapter are in the kinds of vari- ables they accommodate. The Pearson correlation requires interval or ratio variables that are normally and similarly distributed (Objective 3). A special application of Pearson, the point-biserial correlation, requires an interval/ratio variable and a second variable that has only two manifestations, or a dichotomously scored variable (Objective 5). Spearman’s rho will accommodate any combination of ordinal, interval, or ratio variables (Objec- tive 6). Because the data that is used in a Pearson correlation contains more information than the rankings that make up the data for Spearman’s approach, the Pearson value provides more information about the nature of the relationship between the variables. This is evident because the Pearson correlation value can be squared to produce the coef- ficient of determination. The rxy
2 value indicates the proportion of one variable that can be explained by changes in the other (Objective 4). Results for both the Pearson correlation and Spearman rho were presented with the use of SPSS (Objective 7). The results from the SPSS output were then interpreted and reported in proper APA format (Objective 8).
Two variables are correlated when each contains information common also to the other. The amount of one explained by the other is what that rxy
2 value, the coefficient of deter- mination, indicates. This concept provides a foundation for regression, which is the focus of Chapter 10. Regression allows what is known of y from analyzing x to predict the value of y from x. It involves calculations and thinking with which you are already familiar, so work the end-of-chapter problems, reread any of the sections in Chapter 9, and prepare for Chapter 10!
Key Terms
bivariate correlations All procedures that test for significant relationships between two variables.
canonical correlation Measure of the rela- tionship between two groups of variables.
coefficient of determination The propor- tion of one variable in a Pearson correlation that can be explained by the other.
correlation matrix A box in which the vari- ables involved are listed in rows as well as in columns and each variable is correlated with all variables, including itself.
dichotomous data A binary variable with two choices such as gender (i.e., male/ female) or answer (i.e., yes/no).
hypothesis of association The umbrella term for significance tests that deal with the correlation between variables.
hypothesis of difference The umbrella term for significance tests dealing with dif- ferences between groups.
linear When the relationship between two variables is linear, the strength of the relationship is consistent throughout their ranges. With curvilinear relationships, the strength and sometimes even the nature of the relationship (positive or negative) changes depending upon where in the vari- ables’ ranges they are measured.
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CHAPTER 9Chapter Exercises
multiple correlation Gauge of the strength of the relationship between one variable and two or more other variables.
nonparametric Statistical procedures that make no assumptions about data normality or data distribution.
partial correlation Measure of the relation- ship between two variables, controlling for the influence of a third in both of the first two.
Pearson correlation coefficient Indication of the strength of the relationship between interval- or ratio-scale variables.
point-biserial correlation A special appli- cation of the Pearson correlation for those instances where one of the variables, such as gender or marital status, has just two manifestations.
range attenuation Occurs when a vari- able is not measured throughout its entire range. Attenuated range artificially reduces the strength of any resulting correlation value.
scatterplot A graph representing two vari- ables, one on the horizontal axis and the other on the vertical axis. Each point in the graph indicates the measure of both vari- ables for one individual.
semipartial correlation Gauge of the rela- tionship between two variables, controlling for a third in just one of the first two.
Spearman’s rho A correlation procedure for two ordinal variables, one ordinal and one interval or ratio variable, or two inter- val or ratio variables that fail to meet Pear- son correlation requirements for normality.
tetrachoric correlation Allows a correla- tion when both variables are dichotomous data.
Chapter Exercises
Answers to Try It! Questions The answers to all Try It! questions introduced in this chapter are provided below.
A. A single point in a scatterplot represents two raw scores, one for x and one for y. B. If the two variables are normally distributed but uncorrelated, their combined
scatterplot will be circular with greatest density in the middle of the plot because of the tendency for most of the data to be in the middle of either distribution.
C. Range attenuation diminishes the strength of the correlation value in linear rela- tionships. It produces an artificially low correlation coefficient.
D. As degrees of freedom increase, the correlation value required to reach signifi- cance diminishes.
E. Spearman’s rho will accommodate variables that have any combination of ordi- nal, interval, or ratio scale.
F. The coefficient would indicate that the higher the ranking, the lower the GPA. If a ranking of 1, is “best,” the best (highest) GPA must also receive a class ranking of 1. Otherwise, the relationship looks negative when it is not.
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CHAPTER 9Chapter Exercises
Review Questions The answers to the odd-numbered items are in the answers appendix.
1. What values indicate the strongest and weakest values for a Pearson’s r?
2. What is the equivalent in a Pearson correlation for h2 or Cohen’s d?
3. What are the requirements for calculating Pearson’s r?
4. What is range attenuation, and how does it affect correlation values for linear relationships?
5. A university counselor gathers data on students’ grades and whether or not they are employed. What statistical procedure will gauge that relationship?
6. What procedure will indicate whether there is a significant relationship between sales representatives’ sales rank and their attitudes about the product they sell?
7. What procedure will gauge the relationship between university students’ grade averages and their scores on, for example, a statistics test?
8. Refer to Question 7: What statistic will indicate the proportion of the students’ test scores that is a function of their GPA?
9. A forensic psychologist gathers data on the average time juveniles go to bed and whether or not they have an arrest record.
a. What type of correlation (Spearman, Pearson, or point-biserial) is appropriate in this instance?
b. What is the correlation? c. How much of variability in arrest records can be explained by what time the
juvenile goes to bed?
Retire Arrest
1 9.0 No
2 9.5 No
3 11.0 Yes
4 11.5 Yes
5 10.0 Yes
6 9.75 No
7 10.0 No
8 10.25 Yes
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CHAPTER 9Chapter Exercises
10. A group of consumers has just taken two surveys on (a) their attitude about the economy and (b) their attitude about those in government. In both, higher scores mean more optimism. The data is ordinal scale. Are the two attitudes related?
Economy Government
1 15 10
2 5 4
3 16 11
4 10 8
5 11 13
6 3 4
7 12 10
8 11 8
9 10 7
10 14 9
11. A group of students has been told that reading will help them in a test of verbal ability required by the university they wish to attend. The x variable indicates the minutes per day spent reading. The y variable is their scores on the test.
Minutes Score
1 15 57
2 80 84
3 0 60
4 75 92
5 30 65
6 10 60
7 22 75
8 15 68
a. Is the relationship statistically significant? b. How much of the variance in test scores can be explained by differences in the
amount of time spent reading?
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CHAPTER 9Chapter Exercises
12. An educational psychologist is working with developmentally disabled students in a special education setting and is curious about the relationship between stu- dents’ persistence on puzzle tasks (measured in the number of minutes they remain on task) and their number of absences from class.
Persist Absent
1 12 3
2 4 3
3 15 5
4 18 7
5 12 1
6 5 4
7 8 3
8 9 4
Is the relationship between persistence and attendance statistically significant at p 5 .05?
13. An employer wishes to analyze the relationship between stress and job perfor- mance. Stress is reflected in systolic blood pressure. Job performance is the number of sales per day.
a. What is the appropriate correlation procedure? b. Is the relationship statistically significant?
Sales Blood Pressure
1 1 150
2 4 140
3 3 140
4 6 110
5 2 140
6 4 130
7 0 160
8 3 110
9 5 120
10 7 160
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CHAPTER 9Chapter Exercises
14. An industrial/organizational psychologist is determining the relationship between workers’ willingness to embrace new manufacturing procedures, gauged with a dogmatism scale (higher scores indicate greater dogmatism) and their level of job satisfaction (higher scores indicate greater satisfaction). The satisfaction data is at least ordinal scale.
a. What is the relationship? b. What is the null hypothesis? c. Do you reject or fail to reject? d. What is the relationship between dogmatism and job satisfaction? e. Is the correlation statistically significant?
Dogmatism Satisfaction
1 8 4
2 4 12
3 3 14
4 5 15
5 7 5
6 2 14
7 3 15
8 1 15
Analyzing the Research Review the article abstracts provided below. You can then access the full articles via your university’s online library portal to answer the critical thinking questions. Answers can be found in the answers appendix.
Using Pearson Correlation in a Schema Study
Bamber, M., & McMahon, R. (2008). Danger—Early maladaptive schemas at work!: The role of early maladaptive schemas in career choice and the development of occu- pational stress in health workers. Clinical Psychology & Psychotherapy, 15(2), 96–112.
Article Abstract
The schema-focused model of occupational stress and work dysfunctions (Bamber & Price, 2006; Bamber, 2006) hypothesizes that individuals with EMS (unconsciously) gravi- tate toward occupations with similar dynamics and structures to the toxic early environ- ments and relationships that created them. They subsequently re-enact these EMS and their associated maladaptive coping styles in the workplace. For most individuals, this results in “schema healing”, but for some individuals with more rigid and severe EMS, schema healing is not achieved and the structures and relationships of the workplace, together with the utilization of maladaptive coping styles, serve to perpetuate their EMS. The model hypothesizes that it is these individuals who are most vulnerable to develop- ing occupational stress syndromes. To date, this model has been subjected to very lit- tle empirical investigation, so the main aim of this study was to address this gap in the
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CHAPTER 9Chapter Exercises
literature by testing out some of its main assumptions and to provide empirical data, which would either support or reject the model using a population of health workers. Spe- cifically, it was hypothesized that “occupation-specific” EMS would be found in health workers from a range of different healthcare professions. It was also hypothesized that the presence of higher levels of EMS would be predictive of raised levels of occupational stress, psychiatric caseness, and increased sickness absence in those individuals. A cross- sectional study design was employed and a total of 249 staff working within a NHS Trust, belonging to one of five occupational groups (medical doctors, nurses, clinical psycholo- gists, IT staff and managers), participated in the study. All participants completed the Young Schema Questionnaire-Short Form (Young, 1998); the Maslach Burnout Inventory- Human Services Form (Maslach & Jackson, 1981), and the General Health Questionnaire- 28-item version (Goldberg, 1978). A demographic questionnaire and sickness absence data was also collected. The results of a between groups analysis of variance and further post hoc statistical analyses identified a number of occupation specific EMS. Also, the results of a series of multiple linear regression analyses indicated the presence of some EMS to be predictive of higher levels of burnout, psychiatric caseness, and sickness absence in health workers. In conclusion, the findings of this study provide empirical support for the schema-focused model of occupational stress and work dysfunctions (Bamber & Price, 2006; Bamber, 2006), and it appears that the existence of underlying EMS may constitute a predisposing vulnerability factor to developing occupational stress.
Critical Thinking Questions
1. In the study it was stated that it was estimated that a sample size of 85 would give an 80% power to detect a correlation of r 5 .3 (p , 0.05, two-tailed). Define the concept of power.
2. If emotional deprivation is r 5 .15 (p , 0.05, two-tailed), what is the direction and significance?
3. There is a significant correlation between GHQ total, MBI, and sickness absence. What variable causes the other?
Using a Non-parametric Correlation in a Quality of Life Study
Arpawong, T. E., Richeimer, S. H., Weinstein, F., Elghamrawy, A., & Milam, J. E. (2013). Posttraumatic growth, quality of life, and treatment symptoms among cancer chemotherapy outpatients. Health Psychology, 32(4), 397–408.
Article Abstract
Objective: The experiences of positive adjustment and growth, termed Posttraumatic Growth (PTG), are commonly reported among cancer survivors in the years after treat- ment. However, few studies have examined PTG among patients in active treatment for cancer. This study examined both positive and negative valenced change in PTG and rela- tionships with treatment-related symptoms and mental and physical quality of life (QOL) among adults in active cancer treatment. Methods: In this cross-sectional study, adult outpatients (n 5 114) completed a self-administered questionnaire. Hierarchical linear regression modeling (HLM) was performed to examine unique associations between posi- tive and negative valenced change in PTG and QOL subscales and symptom reporting,
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CHAPTER 9Chapter Exercises
controlling for theoretically relevant sociodemographic variables. Results: The majority of participants (87%) reported at least one positive life change, whereas half (50%) reported at least one negative life change across PTG items. In HLM analysis of QOL subscales, nega- tive valence PTG scores were positively associated with Physical Functioning and Bodily Pain and inversely associated with General Health, Role Physical, and Mental Health (F(12, 71) 5 5.13; p , .0001). In HLM analysis of treatment symptom burden, positive valence PTG scores were inversely associated with age at diagnosis and reports of nausea (F(8, 83) 5 2.93; p 5 .007). Conclusions: Reports of positive and negative life changes since diagnosis are common among adults actively receiving treatment for cancer. Assessments of both valenced PTG scores can provide a broader profile of biopsychosocial adjustment and symptom reporting during cancer treatment.
Critical Thinking Questions
1. Why did this study use a Spearman’s test versus a Kendall’s tau?
2. The Spearman’s correlation coefficient between anxiety and BMI is (2.04). If the significance value of this coefficient is (.001), what can be concluded?
3. The Spearman’s correlation coefficient between dyspnea on exertion and exercise capacity peak is .27 with p , .01. What does this mean?
4. Why is it a good thing to check that the N value corresponds to the number of observations that were made in the Spearman’s correlation output?
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