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8
Correlation
Learning Objectives
After reading this chapter, you should be able to:
• Compare the hypothesis of association to the hypothesis of difference.
• Interpret correlation coefficients.
• Explain the coefficient of determination.
• Describe the conditions under which to apply different correlation coefficients.
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CHAPTER 8Introduction
Chapter Outline
8.1 The Hypothesis of Association Picturing Correlation What Correlations Provide Correlation Requirements
8.2 Calculating the Pearson Correlation A Correlation Example Interpreting Results The Relationship Between Degrees of Freedom and Significance The Correlation Hypotheses The Coefficient of Determination
8.3 Correlating Data When One Variable Is Dichotomous Dummy-Coding Correlation in Excel “Degrees of Significance?”
8.4 Spearman’s rho A Correlation Procedure for Ordinal Data
8.5 Correlation Versus Causation
Chapter Summary
Introduction
Correlation is a concept that transcends statistical analysis, and even business. Correla-tion reflects the association between variables. It can be viewed as the extent to which one variable changes as the other changes. Nearly everyone recognizes correlations, or relationships in things. For example, higher rates of smoking are correlated with the more frequent occurrence of lung cancer. This doesn’t necessarily indicate a causal relationship, which, of course, is the position the tobacco companies take. Someone can have lung can- cer without ever smoking, and people smoke without ever getting lung cancer—but the general trends run relative to one another. For example, children’s ages are usually corre- lated with their heights, intelligence is correlated with academic performance, number of years of education is correlated with annual income, and household income is correlated with household spending. None of these relationships is perfect. It just means that the two variables involved generally change together.
Correlations are usually expressed in terms of magnitude and direction. Magnitude rep- resents the strength of the relationship—the extent to which knowing the value of one variable suggests the value of the other. For example, intelligence is likely to be more strongly correlated with academic performance than, say, height or weight would be. Direction represents the nature of the relationship. In positive correlations, such as those
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CHAPTER 8Section 8.1 The Hypothesis of Association
in the above examples, the higher the first variable, the higher the second will likely be. In negative relationships, the higher one variable is, the lower the other is expected to be. For example, education is likely to be negatively correlated with delinquency, job satisfaction is negatively correlated with the employee’s intention to quit, and hours spent at work are negatively correlated with hours spent with family and friends.
Correlation coefficients put these relationships in objective terms. They quantify the magnitude and direction of the relationship between separate vari- ables. There are several kinds of correlation coeffi- cients. Each is specified by conditions relating to the scale and normality of the data, as explained later in this chapter. Before getting to those, however, there are some more general considerations.
8.1 The Hypothesis of Association
Several different statistical tests have been covered to this point in the book. But whether the discussions are about the z-test, one of the t-tests, analysis of variance, or repeated measures tests, they all have this in common: They deal with what is called the hypothesis of difference. They are all analyses of significant differences between groups. But the questions managers want answered are not always about differences. For example, if the service manager at a car dealership is curious about whether the time a qualified mechanic needs to complete a particular kind of service declines as the mechanic gains experience, the question is about the correlation between the time to complete the service and the length of the mechanic’s experience. Questions such as this one are about significant relationships, not significant differences. They fall under a general conceptual umbrella called the hypothesis of association. As a point of departure in the correla- tion discussion, establishing that there is a relationship between two variables is not the
same as determining that one variable is the cause of the other. What correlations establish is that there is co-variation between variables. The correlation coef- ficient is a value that indicates the degree to which they vary together. But because variables vary con- currently does not mean that one variable necessar- ily causes the other.
This distinction between correlation and cause has important implications for the business manager. If
the regional manager of a group of pharmaceutical representatives notices that sales of an over-the-counter medication for congestion rise after a price reduction, would it be safe to conclude that the reduced price is the cause of the sales increase? The difficulty, of course, is that there are likely to be other variables involved. Maybe the time of the years is a fac- tor. If the price reduction happened to correspond with a time when pollen or dust is more prevalent and prompts more congestion, maybe environmental conditions are the cause. Or perhaps the pharmaceutical company conducted an effective advertising campaign at the same time prices were reduced, and sales increased for that reason. Controlling all other potentially relevant variables sometimes makes it very difficult to establish cause in the business world. We will return to this topic toward the end of the chapter.
Key Terms: The correlation coefficient is a value, ranging from 21 to 11, that indicates the strength of the relationship between variables.
Key Terms: Statistical tests under the hypothesis of dif- ference examine whether differ- ences between groups are due to chance. Under the hypothesis of association, the issue is whether relationships are due to chance.
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CHAPTER 8Section 8.1 The Hypothesis of Association
Picturing Correlation
The word correlation makes reference to a co-relation between variables. It indicates that the variables change in concert. As the level of one variable changes, the level of the other variable changes correspondingly. Statistically, this co-relation exists because both mea- sures contain some of the same information. The higher the correlation is, the more infor- mation the measures of the two variables have in common.
Correlations are often illustrated with graphs in which measures of one variable are plot- ted along the horizontal axis and measures of the other variable are plotted along the vertical axis. These graphs are often referred to as scatter plots. Perhaps those at a market- ing agency wish to illustrate the relationship between the amount of exposure a product receives in advertising in a particular week and the corresponding volume of sales for that week. As shown in Figure 8.1 (a), if in one week a series of 20 spots ran, and the volume of sales for the product was 80,000 units, this would be plotted in the graph with a dot at the intersection point of 20 on the horizontal axis and 80,000 on the vertical axis. If in another week there were 30 spots and a sales volume of 100,000 units, this would be plotted with a dot at the (30, 100000) point in the graph. If exposure and sales are plotted for 12 weeks, and the 2 variables are correlated, a pattern will emerge. If the two variables are positively related as the marketing agency expects, the general pattern in the dots will be from lower left to upper right, as shown by the trend line going through the dots in Figure 8.1 (a).
On the other hand, if the ads were so poorly designed that they misrepresented the prod- uct and actually discouraged, rather than encouraged, sales, a negative correlation would emerge. As shown in Figure 8.1 (b), the pattern for a negative correlation runs from the upper left to the lower right. In this situation, the higher the exposure, the lower the sales volume was. Finally, if the campaign were simply ineffective (no correlation), the sales volume trend would be relatively flat in relation to variations in exposure, as shown in Figure 8.1 (c). Alternatively, in a no-correlation situation, the dots could have no particular pattern and just scatter randomly throughout the graph.
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CHAPTER 8Section 8.1 The Hypothesis of Association
Figure 8.1: Advertising exposure and sales volume—a visual depiction of correlation
The relationship between amount of exposure and sales volume will not be perfect. This reflects the fact that there are other factors besides advertising exposure that affect sales volume. Perhaps there is a manufacturing recall in the middle of the ad campaign that receives prominent mention in the news. Perhaps the product is something seasonal and weather-dependent, like swimming apparel, and the weather turns unseasonably cold.
If the relationship between two graphed variables were perfect, the dots in the graph that represent the paired measures would occur in a straight line. The degree to which the dots stray from a straight line reflects the lack of a perfect correlation, as shown in Figure 8.1. Increasing scatter indicates progressive decline in the strength of the relationship.
The question prompted by analyses that relate to the hypothesis of association is not whether a correlation is perfect. Perfect correlations are extremely rare because there are so many factors involved in most relationships that it is very difficult to account for them all. Rather, the issue is whether whatever relationship exists is statistically significant. Imper- fect correlations can be very important. If health professionals know there is a correlation, even a weak one, between the exposure to secondhand smoke and the development of respiratory problems, that correlation will probably affect their advice to their patients. If a pharmaceutical company can demonstrate even a modest relationship between taking a drug to reduce cholesterol and the risk of heart attack or stroke, the correlation will likely be a prominent element in the advertising of the drug.
Number of spots
S a le
s vo
lu m
e (
in t h o u sa
n d s) 200
150
100
50
0 0 10 20 30 40 50 60
(a) A Positive Correlation
Number of spots
(b) A Negative Correlation
S a le
s vo
lu m
e (
in t h o u sa
n d s) 200
150
100
50
0 0 10 20 30 40 50 60
(c) No Correlation
Number of spots
S a le
s vo
lu m
e (
in t h o u sa
n d s) 200
150
100
50
0 0 10 20 30 40 50 60
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CHAPTER 8Section 8.1 The Hypothesis of Association
What Correlations Provide
Calculating a correlation produces a correlation coefficient. The coefficient is a single number that indicates the strength of the relationship between the variables involved. Correlation values range from 21.0 to 11.0. Either of those two extremes indicates a per- fect relationship. Anything between them indicates a less-than-perfect relationship. A cor- relation of zero indicates that the variables are unrelated. As long as the two variables involved can be quantified—reduced to a number—the strength of their correlation can be determined.
Correlation Requirements
There are several different correlation procedures, and which is appropriate in a particular instance depends upon multiple factors, but one of the most basic is the data scale of the measures involved. The Pearson Correlation, which is the most commonly used, requires variables that are measured on either an interval or a ratio scale. When the measures are nominal or ordinal, there are other correlation pro- cedures involved—some of which will be covered later in this chapter. For the Pearson Correlation, besides the scale of the measures, the following requirements apply:
• In their respective populations, the measures of the variables must be nor- mally distributed.
• The populations from which the samples are drawn must be similarly distributed.
• The two samples of measures for which the correlation value is calculated are assumed to be randomly selected from their populations.
• The relationship between the variables must be linear.
When two variables are normally distributed and represented in a scatter plot, 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, the scatter, as we noted earlier, is generally from the lower left to the upper right. If the relationship is nega- tive, the pattern stretches from the upper left to the 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 reflecting the fact that most of the data occur in the middle of the data distributions.
The second requirement in the list above indicates that the populations must be similarly distributed. The characteristic is indicated when the standard deviations account for simi- lar proportions of the entire range of values. It does not mean that the standard deviations of the different variables will necessarily have similar values.
The strength of a correlation can be affected by something called attenuation of range. Attenuation occurs when only an unrepresentative part of the possible range of measures is provided by the sample. The evidence for attenuation might be a standard deviation
Key Terms: The Pearson Cor- relation gauges the strength of the relationship between inter- val or ratio data.
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CHAPTER 8Section 8.1 The Hypothesis of Association
that is substantially smaller than what that value is known to be in the population. For example, if the task were to correlate the tenure of a company’s employees with their productivity, and the tenure data had a standard deviation of 8 points when we know that the population standard deviation for ten- ure in the industry is 15 points, any resulting correla- tion value will be artificially low.
When the relationship between two variables is linear, it means that the strength of the relationship between them is consistent throughout the ranges of their measures. If the relationship is modest and positive at the lower end of both distributions, it should also be low and positive and the high end of the distribution. When the relationship changes as a different set of paired measures are analyzed, it may indicate that the two variables are not linearly related. As an example, perhaps data are gathered on the relationship between the time since a vehicle is sold and the number of warranty claims made by purchasers. The data for 12 vehicles are as follows:
Owner: A B C D E F G H I J K L
Months since sale: 1 2 3 4 5 6 7 8 9 10 11 12
Number of claims: 0 1 1 2 2 4 3 5 6 4 3 1
Figure 8.2 is a scatter plot of these data.
Figure 8.2: The relationship between the time since sale and the number of warranty claims
Months Since Sale
N o.
W a rr
a n ty
C la
im s
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10 11 12
Key Terms: Attenuation of range refers to circumstances where the true range of values possible for a variable is not reflected in the sample.
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CHAPTER 8Section 8.2 Calculating the Pearson Correlation
Formula 8.1 rxy 5 S1zx 3 zy 2
n 2 1
The scatter plot makes it clear that the relationship between time since the sale of the vehicle and the number of warranty claims is not linear. If it were, the number of claims would continue to rise as the time since sale increases, which is the trend in the first sev- eral months. For some reason, some of those who are at the greatest amount of time since purchasing have made the fewest warranty claims. The trend actually resembles an inverted U-curve, with a positive trend for the first nine months, and a negative trend after- ward. If a Pearson Correlation were calculated for these data, the resulting coefficient would be inaccu- rate because the relationship between the two vari- ables is not linear.
8.2 Calculating the Pearson Correlation
The symbol for the Pearson Correlation is a lowercase r. Often called “Pearson’s r,” this is probably the most often calculated of any correlation value. It has several formulas, all of which can produce a correct coefficient value. The formula adopted here relies on the z scores that you learned to calculate in Chapter 2. There are also “raw-score” formulas that rely on the measures or scores in their original state, such as those upon which the scat- ter plot in Figures 8.1 and 8.2 are based. For the formula used here, however, the original “raw” scores must first be turned into z scores using the Chapter 2 z score transformation.
Recall that the z transformation changes raw scores into scores that fit a distribution where the mean equals 0 and the standard deviation equals 1.0. This is accomplished by calculat- ing the mean and standard deviation of the samples to be correlated and then using the z formula for each group as follows:
z 5 x 2 M
s
Once the original scores have been transformed into z scores, one variable is termed the “x” variable, and the other is called the “y” variable; the assignment of x and y is arbitrary. Verbally, the process for calculating the Pearson Correlation is as follows: “The correlation between x and y (rxy) is equal to the sum of the products of the z score equivalents of each pair of x and y scores, divided by the number of pairs of scores, minus one.”
Symbolically, the formula is:
Key Terms: Relationships that are linear remain consistent throughout the entire range of the variables involved.
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CHAPTER 8Section 8.2 Calculating the Pearson Correlation
The procedure for calculating the correlation coefficient is then as follows:
1. Calculate the mean and standard deviation for the “x” scores. 2. Using the z transformation (Chapter 2), turn each of the raw “x” scores into
z scores. 3. Calculate the mean and standard deviation for the “y” scores. 4. Turn each of the raw “y” scores into z scores. 5. Multiply each pair of zx and zy scores. 6. Sum the z score products. 7. Divide by the sum by the number of pairs, minus one.
A Correlation Example
A quality control specialist is examining the relationship between the amount of feedback assembly line workers are given on their performance in a given day and their mechani- cal efficiency, gauged by the average amount of time in minutes it takes them to success- fully complete a component. Feedback is gauged by the number of times an experienced observer provides direction to the assembler. The data for 10 assemblers are as follows:
Feedback amount (x) 1 2 3 4 5 6 7 8 9 10
Time to complete (y) 5.0 5.5 4.75 4.5 4.25 3.5 2.75 2.0 1.0 .25
We will calculate the Pearson Correlation as follows:
1. Calculate the mean and standard deviation for the “x” scores (feedback amount). Verify that the M 5 5.500 and s 5 3.028.
2. Using the z score formula, turn each of the original “x” scores into z scores. For feedback amounts 1 through 10 the corresponding z values are 21.486, 21.156, 20.826, 20.495, 20.165, 0.165, 0.495, 0.826, 1.156, 1.486. 3. Calculate the mean and standard deviation for the “y” scores. Verify that
M 5 3.350 and s 5 1.788. 4. Using the z score formula for the “y” scores, verify that the z values for time to
complete are 0.923, 1.202, 0.783, 0.643, 0.503, 0.084, 20.336, 20.755, 21.314, 21.733. 5. Multiply each pair of z scores. Determine the products of each pair of z scores 21.486 3 0.923, 21.156 3 1.202
. . . 1.486 3 21.733 and verify that the results for the 10 pairs are 21.371, 21.390, 20.646, 20.319, 20.083, 0.014, 20.166, 20.623, 21.519, 22.577.
6. Sum the z score products and verify that (zx 3 zy) 5 28.682 7. Divide by the number of pairs, minus one (n 2 1), which is 9, and verify that
rxy 5 20.965.
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CHAPTER 8Section 8.2 Calculating the Pearson Correlation
Interpreting Results
As with the tests of significant differences related to z, t, and F, significance is determined by comparing the calculated coefficient value to a value found in Table 8.1 of critical values.
Table 8.1: The critical values of rxy
Number of xy Pairs (n)
Degrees of Freedom (n 2 2)
Significance
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: Retrieved from http://www.brighton-webs.co.uk/tables/critical_values_r.asp.
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CHAPTER 8Section 8.2 Calculating the Pearson Correlation
As with the t and F tests, the critical value is indexed to degrees of freedom. The degrees of freedom for a Pearson Correlation are the number of pairs of data minus 2. For this prob- lem with 10 pairs of data, degrees of freedom equal 8. When testing at p 5 .05, which is the most common criterion for statistical significance, and 8 degrees of freedom, the critical value is .632. As long as the absolute value of the calculated correlation is equal to or larger than the table value, the correlation is statistically significant. The reason that it is the absolute value that is of interest is because most correlation analyses are two-tailed tests. The question is usually, is there a relationship? Whether the correlation is positive or neg- ative is generally less important than determining whether there is a correlation. The correlation of rxy 5 20.965 just cal- culated is negative. The fact that it is negative indicates that as the amount of feedback increases, the time it takes assemblers to complete a component decreases. If there were a correla- tion of rxy 5 0.965 of some other pair of variables, that coef- ficient would be exactly as strong as what we calculated. The difference is that the two measures would vary in the same direction.
The Relationship Between Degrees of Freedom and Significance
As the degrees of freedom increase, the magnitude of the coefficient value required to be statistically significant drops. While this is also true for t and F tests, it is particularly dra- matic with the Pearson Correlation. Table 8.1 indicates that if we completed a correlation problem with 8 pairs of data (df 5 6), at p 5 .05 the coefficient would have to be at least rxy 5 .707 in order to be statistically significant. At the other end of the table, a problem for which there were 25 pairs of scores (so that degrees of freedom equal 23) would require a correlation coefficient of just rxy 5 .396 in order to be statistically significant when testing at p 5 .05. The higher standard for significance with small samples is designed to protect against confusing a chance relationship with one that is likely to be found every time the two variables are measured.
The Correlation Hypotheses
The null and alternate hypotheses take on different forms with correlation than they had for difference tests. Because these procedures fit under the general hypothesis of asso- ciation, the null hypothesis is that there is no relationship between the variables. The null hypothesis states that “rho” (as in row your boat; r is the Greek equivalent for “r”) equals zero. The alternate hypothesis is that rho doesn’t equal zero, r 0. Failing to reject r 5 0 means that there is apparently no statistically significant relationship between the variables. Rejecting the null hypothesis (r 0) indicates that there is a significant relation- ship. When the calculated value exceeds the table value as it did in the amount of feed- back and assembly time problem just completed, the appropriate statistical decision is to reject the null hypothesis. The evidence in that case indicated that the calculated value is large enough that is likely to emerge every time data for samples of this size are collected and analyzed.
Review Question A: Which is the stronger correla- tion: rxy 5 21.0, or rxy 5 11.0?
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CHAPTER 8Section 8.2 Calculating the Pearson Correlation
The Coefficient of Determination
One of the repetitive themes in this book is about the distinction between statistical sig- nificance and practical importance. Measuring practical importance was the reason behind calculating Cohen’s d and eta-squared for statistically significant results when conducting t-tests and ANOVAs. Sometimes the fact that a result is statistically significant means that it is also important from a practical point of view as well. Any significant relationship between rear-impact car crashes for a certain vehicle and sudden fires that engulf the vehicle and its occupants is also, by definition, important in a practical sense as well. However, it is important not to assume that just because a result is not likely to have occurred by chance, it automatically has important practical significance. One conclusion does not necessarily follow the other.
Calculating the effect size may be more important for correlation results than for signifi- cant t and F results. We noted earlier that relatively weak correlations can become statisti- cally significant if the sample sizes become sufficiently large. So to keep the correlation value in context, a statistically significant rxy value is followed by calculating an effect size, which for the Pearson Correlation is called the coefficient of determination, r2xy. As the symbols suggest, the coefficient of determination is a simple matter of squaring the cor- relation coefficient. The result indicates how much of the variance in one of the variables, either x or y, is explained by changes in the other variable.
In the amount of feedback and assembly time exam- ple, the correlation was 2.965. That would make the coefficient of determination 2.9652, which turns out to be .931. This value indicates that about 93% of the variance in how much time it takes workers to assemble the product can be explained by how much feedback they have received.
The Interpretive Value of rxy 2
In the course of indicating the practical importance of a significant outcome, effect size measures such as the coefficient of determination can also indicate how comparatively unimportant some significant correlations are. For example, we noted above that with 23 degrees of freedom, a correlation of .396 is statistically significant, which means that every time the 2 variables are measured and the coefficient calculated, the relationship is likely to emerge; the relationship is probably not random. Note, however, that the coef- ficient of determination for that value (.3962) is .157. If 2 variables had such a correlation, the coefficient of determination reminds us that either variable explains just about 16% of the variance in the other. The other 84% of the variability is explained by other factors.
Comparing Correlation Values
The Pearson r requires data that are at least interval scale. Remember that the name “interval data” means that there are equal distances between consecutive numbers any- where along the range for the measure. For a characteristic measured on an interval (or ratio) scale, the difference in the quantity of the thing measured from 3.0 to 4.0 is the
Key Terms: The coefficient of determination is the square of the Pearson Correlation. It indi- cates how much of the variance in either variable is explained by the other variable.
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CHAPTER 8Section 8.3 Correlating Data When One Variable Is Dichotomous
same amount as from 22.50 to 23.50, for example, or any other difference of 1.0 anywhere along the scale. The same does not hold for Pearson Correlation coefficients, however. An increase in a correlation coefficient from .1 to .2 represents a modest increase in the strength of the relationship. An increase from .8 to .9, on the other hand, represents a substantial increase in the correlation between the variables. As a result, it would not be strictly accurate to say that a correlation of .90 represents a relationship twice as strong as a correlation of .45.
One of the contributions the coefficient of determination makes to interpretation is that the intervals between the related values become consistent. A change in the r2 value from .35 to .5, for example, represents the same difference in the proportion of variance explained as a change from .72 to .87. An r2 5 .6 explains twice as much of the variance as an r2 5 .3.
8.3 Correlating Data When One Variable Is Dichotomous When the scale of the variables in a correlation problem changes, so does the correlation procedure. Suppose that a beef producer wants to correlate the gender of the purchaser with the amount of money spent on beef purchases in a major grocery chain. The amount of money spent is a ratio variable and so fits the requirement for a Pearson Correlation. The problem is the other variable. Gender is neither interval nor ratio scale. Indeed, it is a nominal variable with only two categories. The characteristics of these two variables require a procedure called a point-biserial correlation. The “point” in the title refers to the continuous variable, the amount of money that the consumer spends in beef pur- chases. The “biserial” refers to the other variable in the relationship, which can have only two manifestations. In the example they are female and male.
The point-biserial correlation has a number of applications. Questions about the relationship between marital status and income among employ- ees, about the relationship between shift (day ver- sus night) and productivity, about the correlation between type of purchase (cash versus credit card), and the amount of the purchase all lend themselves to point-biserial analyses.
Dummy-Coding
The point-biserial calculations are actually the same as those for a conventional Pearson Correlation. The change is that the dichotomous variable must be dummy-coded. Sim- ply put, when a dichotomous variable is dummy-coded, it is represented by the values of zero and one. The coder picks one of the manifestations of the dichotomous variable, and whenever that manifestation is present, it is coded as a one. When it is not present,
the coding is zero. Back to the beef purchases and the gender of the purchaser example, if the decision is to code female purchasers as ones, every time the purchaser is male, the coding will be zero. In terms of the absolute value of the coefficient, it makes no difference which is coded zero, and which one. If the
Key Terms: The point-biserial correlation is a procedure for relating an interval/ratio vari- able with a nominal variable that has two categories.
Key Terms: Dummy-coding uses zeros and ones to code nominal scale variables.
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CHAPTER 8Section 8.3 Correlating Data When One Variable Is Dichotomous
coding is reversed, positive correlations become negative correlations of the same mag- nitude, and likewise, any negative correlations become positive correlations of the same absolute value.
We can illustrate the point-biserial correlation and the Excel approach to calculating cor- relation with a gender and beef purchases example. Perhaps the beef producer hires an aspiring business student to analyze the relationship between the gender of the purchaser and the amount spent on beef before the Memorial Day weekend barbecue events. For 12 shoppers, 6 women and 6 men, the beef purchases to the nearest dollar are as follows:
Women: 12, 0, 14, 22, 21, 18 Men: 15, 32, 45, 38, 25, 20
Correlation in Excel
To complete the point-biserial correlation in Excel, the data can be arranged in columns or rows. For this example, we will use Column A for the gender of the purchaser, and code it using a 1 for females and a 0 for males. In Column B, labeled Cost, enter the amount spent on beef purchases. Once the data have been entered there will be entries in cells A1 through A13 and B1 through B13. At that point, the process is:
• Click the Data tab at the top of the page. • Click the Data Analysis option at the extreme right. • Select Correlation from the list of options. • Click OK. • Indicate that the input range is A1:B13. • Indicate that there are Labels in the first row. • Select an output range that won’t overwrite the data set, perhaps D1. • Click OK.
The result is the Excel screenshot that is Figure 8.3.
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CHAPTER 8Section 8.3 Correlating Data When One Variable Is Dichotomous
Figure 8.3: A point-biserial correlation in Excel
The point-biserial correlation of gender with price expenditure is rxy 5 2.633. Although Excel produces the correlation value, the output does not indicate whether the value is statistically significant. To determine significance, consult Table 8.1. Note that for df 5 10 and p 5 .05, any value of rxy 5 .576 or greater should prompt a decision to reject the null hypothesis; there is a statistically significant relationship.
Recall that a negative correlation indicates that the higher values of one variable are con- nected with the lower values of the other. In the case of the point-biserial correlation, the fact that the coefficient is negative simply indicates that the lower values of the continu- ous variable are associated with whichever manifestation of the dichotomous variable was coded “1,” which in the above example were the female buyers. In other words, women (coded 1) bought significantly less beef than men (hence the negative correlation). If women had been coded “0” and men “1,” the correlation would have been identical in absolute value but in the positive direction, indicating that men (now coded 1) significantly bought more beef than women (hence the positive correlation).
Note that the instructions above can also be used to calculate Pearson’s Correlation in Excel. The only difference is that the first column will include the values of a continuous variable x, rather than a dummy-coded nominal variable. For example, in the quality con- trol problem above, feedback amounts would have been in the first column, and average component assembly time would have been in the second column. All other steps would have been identical.
Review Question B: How many values does dummy-coding involve?
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CHAPTER 8Section 8.4 Spearman's rho
“Degrees of Significance?”
In the above example, the correlation was 2.633 and the critical value was .576. Given those circumstances, is it appropriate to note that the correlation was “barely significant”? If the coefficient value falls short of the critical value from the table, should it be referred to as “almost significant” or “nearly significant”? It isn’t 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. 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 or the other.
8.4 Spearman’s rho The Pearson Correlation requires both variables to be at least interval scale. The point- biserial requires one variable to be at least interval scale, and the other to be dichotomous. Neither of those is helpful when the data are ordinal scale, and ordinal data are common in business settings. Nearly everyone who goes to the mall or answers the telephone has been asked at some point to take a survey (particularly if it happens to be an election year). Survey data are usually ordinal. It is common to be asked to rank a series of options, soft drinks, for example, in terms of the individual’s preference for them. When responses are rankings, they are ordinal scale data. It is also quite common for questionnaires to have what is called Likert-type format, where the respondent is asked the degree to which she/he agrees with a statement by selecting choices such as:
• Strongly Agree • Agree • Neither Agree nor Disagree • Disagree • Strongly Disagree
Although it is common to assign numeric values to survey responses (Strongly Agree 5 1, Agree 5 2, and so on) and then calculate means and standard deviations for all respon- dents, those statistics assume that the data are at least interval scale. Survey data rarely are. The responses above are essentially rankings. A response of “Strongly Agree” is more positive than just “Agree,” but it is not clear precisely how much more. Besides, one respondent’s “Disagree” may be another respondent’s “Strongly Disagree.” Data that are based on surveys and questionnaires are more characteristic of ordinal than of interval data.
If one desires to explore the relationship between ordinal variables, what should be assessed is the extent to which the variable rankings co-vary, rather than their values, as was the case in Pearson’s r. Spearman’s rho is essentially a correlation between variable rankings. It assesses the extent to which rankings on one variable correspond to rankings on another.
Key Terms: Spearman’s rho is a correlation procedure for ordinal scale data.
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CHAPTER 8Section 8.4 Spearman's rho
Where
rs 5 the symbol for Spearman’s rho 6 is a constant value
d 5 the difference between the rankings of the pairs of values n 5 the number of pairs of values
Constant values such as the “6” in the Spearman’s formula are quite common in nonpara- metric procedures. They are always included in the formula regardless of sample size or any other factor. Calculating the Spearman’s rho coefficient is:
• 1 minus the following: · 6 times the sum of the squares of the ranking difference scores, with that
value · divided by the number of pairs of scores times the square of the number of
pairs of scores minus 1.
The calculations follow these steps:
1. Place scores in parallel columns and rank the scores within each column. 2. In each pair, subtract the second score’s rank from the first score’s rank to get
the d value. 3. Square each of the d values, d2.
Formula 8.2 rs 5 1 2 6Sd2
n1n2 2 12
A Correlation Procedure for Ordinal Data
Besides survey data, class rankings, percentile scores, and a good deal of other common data are measured using ordinal scales. Spearman’s rho (the same rho that is used in the correlation hypotheses) will accommodate two variables in a correlation procedure that fit any of the following criteria:
• 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 normality
and homoscedasticity requirements.
With respect to the last criterion, Spearman’s rho is more flexible than the Pearson’s r because it is a nonparametric correlation procedure. This means that it is not calculated using parameters such as m and s, so no assumptions need to be made about issues such as data normality. The formula is as follows:
Key Terms: Nonparametric means without assumptions about parameters such as normality.
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CHAPTER 8Section 8.4 Spearman's rho
4. Sum the d2 values. 5. Divide 6 3 d2 by the product of number of pairs of scores (n) 3 (n2 2 1) 6. Take 1 minus the result of step 5 to determine the correlation coefficient.
Ranking Tied Scores
If there are multiples of a score for one of the variables, they must all receive the same rank- ing. Assume the task is to rank the following values:
3, 5, 6, 6, 7, 8, 8, 8, 9, 10
The rankings would be as follows:
Score Rank
3 10
5 9
6 7.5
6 7.5
7 6
8 4
8 4
8 4
9 2
10 1
• The highest score, 10, receives a rank of 1. • The 9 is 2. • The 8s involve ranks 3, 4, and 5, since there are 3 of them. If we add 3 1 4 1 5
and divide by the number of tied scores (3), we have 12 4 3 5 4. All 8s receive ranks of “4.”
• Since the 8s take up place rankings 3, 4, and 5, the next score, which happens to be 7, will be ranked “6.”
• The 6 scores take up the next 2 places, which will be ranks 7 and 8. • These are figured as 7 1 8 5 15, and 15 ÷ 2 5 7.5. The 6s will both be ranked 7.5. • The process is completed by giving the 5 a ranking of 9, and a 10 ranking for
the 3.
An Example
An organization offers a retirement plan as a benefit for its employees. Ten portfolios are available for employees to invest in. The portfolios vary in their levels of risk, with port- folio 1 being the riskiest and portfolio 10 being the most conservative. The organization
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CHAPTER 8Section 8.4 Spearman's rho
wishes to know if its employees become more conservative in their investments as they age and approach retirement. Below are the ages and portfolio choices of 10 employees.
Portfolio Age
3 26
5 25
6 32
6 35
7 35
8 34
8 37
8 40
9 42
10 39
The portfolios have already been ranked in the previous section, when we illustrated ranking tied scores. For the age data, we have the following rankings if we rank from old- est to youngest: 9, 10, 8, 5.5, 5.5, 7, 4, 2, 1, 3.
If the two sets of rankings are placed into parallel columns, the difference between rank- ings is determined, the difference is squared, and the squared differences are totaled, the result is:
Portfolio Rank Age Rank d d2
10 9 1 1
9 10 21 1
7.5 8 2.5 .25
7.5 5.5 2 4
6 5.5 .5 .25
4 7 23 9
4 4 0 0
4 2 2 4
2 1 1 1
1 3 22 4
24.5
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CHAPTER 8Section 8.4 Spearman's rho
Solving for rs provides this:
rs 5 1 2 6Sd2
n1n2 2 12
rs 5 1 2 6124.52
101102 2 12 5 .852
The Spearman’s Correlation is equal to .852. Like Pearson results, the value is interpreted by comparing it to a critical value from a table. The critical value in Table 8.2 (Table F in the Appendix) is a function of the number of pairs of data—not the number of pairs minus 2, as was the case with Pearson. When testing at p 5 .05 with 10 pairs of data, the critical value for Spearman’s rho is .648; rs.05(10) 5 .648. The relationship between level of portfolio conservatism and age in this group of investors is statistically significant. The correct deci- sion is to reject the null (no relationship) hypothesis.
Table 8.2: 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 .929
9 .683 .833
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: Retrieved from www.sussex.ac.uk/Users/grahamh/RM1web/Rhotable.htm
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CHAPTER 8Section 8.4 Spearman's rho
In the example, the conservatism scores and the ages were ranked from highest to lowest, with the most conservative portfolios and the oldest employees receiving the lower rank- ings. In terms of the coefficient value it makes no difference whether the rankings go from lowest to highest or from highest to lowest, as long as the same is done for both variables. If we decided to follow the rankings provided by the portfolio numbers so that the more con- servative portfolios receive a higher ranking, that would work fine, but the same would have to occur with age—the highest age would need to receive the higher ranking. Were the data in one group ranked from highest down, and the other from lowest up, the cor- relation would appear to be negative, which is fine as long as you keep track of the direc- tion of your rankings.
Sometimes common sense has to dictate the ranking approach. Perhaps there is a question about the relationship between the salaries of top executives and the profitability of their companies. Further, perhaps salaries are reported as percentile values. If several execu- tives’ incomes were reported as 5th percentile, 12th percentile, 15th percentile, and so on, accurate rankings would need to reflect that for this group, the person at the 5th percentile has the top ranking because the numerically lowest percentile score is also the person with the highest salary.
Spearman’s correlation provides a great deal of flexibility to the analyst. As long as there is some evidence of a relationship, correlations can be calculated for any combination of ordinal, interval, and ratio variables. There are no statistical “free lunches,” however, and the sacrifice for so much analytical flexibility 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 taken into consideration. If the salaries of top executives are ranked and the top earner receives $5 million a year and the next highest earner receives $3.5 million, those become rankings 1 and 2. The fact that $1.5 million separates them is lost to further analysis. Contrarily, in the Pearson Correlation, the size of the interval between individuals remains part of the analysis, which increases the probability for a correlation to be statistically significant when the nonparametric procedure shows that it is not.
The lack of data sophistication also emerges in the interpre- tation of the coefficient. The Pearson value can be squared to provide the coefficient of determination, which indicates how much of the variance in one variable can be explained by manipulating the other. There is no equivalent of r2 for Spear- man’s rho.
Review Question C: What correlation procedure would be used for correlating sales generated for the month with the salesperson’s ranking in sales for the prior month?
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CHAPTER 8Section 8.5 Correlation Versus Causation
8.5 Correlation Versus Causation
At the conclusion of the correlation discussion, it is important to recognize that cor-relation does not necessarily imply causation. What correlation coefficients measure is co-variation between variables—the degree to which they vary together. However, just because variables vary concurrently, that is usually not sufficient evidence for concluding that one variable necessarily causes the other.
As an example, a manufacturing company implements an extensive training program for its workers, and company profits rise the same year the program is implemented. It would be a mistake, however, to conclude from only this much information that the training caused the improvement in profits. In business, it can be very difficult to control all of the relevant variables. Perhaps that same year, the company also began to automate some of the more repetitive assembly procedures, which reduced labor costs and increased prof- itability. Or maybe the company received a particularly lucrative contract that involved products with unusually low overhead costs. Maybe the inventory of components was high and very little needed to be purchased to meet manufacturing demands. In fact, the causal direction could have been the opposite. In anticipation of higher profits due to the above reasons or any number of others, the company may have decided to implement the training program that year. In that case, it is profitability that caused training rather than the other way around! A thorough investigation may indeed demonstrate that the reason for better profits is the training program, but it would be a mistake to make that determination based on the strength of the correlation alone.
A good deal of evidence shows people making exactly that error. For example:
• During Sir Isaac Newton’s time, the bubonic plague struck Britain again. Probably because of the increase in rats, the cat population flourished. Peo- ple in London mistakenly assumed that the increase in cats was somehow responsible for the plague and set about destroying the agent that might have diminished the problem.
• American school children always suffer by comparison to Japanese school children in international comparisons. Japanese school children have a longer school day. There have been many calls for a longer school day and a longer school year to help American students be more competitive.
• The confusion of correlation and cause also occurs in business. It seems logical that higher prices correlate with greater profits. But does increasing prices necessarily improve the bottom line? Sam Walton, who built a retail- ing juggernaut in Walmart did so because he couldn’t convince his bosses at a company he worked for previously that lowering prices increases profits. It seemed counterintuitive to them, but Walton recognized that that the volume of sales can influence profits quite as readily as the price of the product.
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CHAPTER 8Chapter Formulas
Chapter Summary
Many of the questions researchers and scholars ask are about the relationships between variables, which requires a hypothesis of association (Objective 1). Three correlation procedures that respond to the hypothesis of association are the Pearson Correlation, the point-biserial correlation, and Spearman’s rho. In each case their possible values range from 21.0 to 11.0, and all their coefficients are interpreted the same way. The larger the absolute value of the correlation, the stronger the association is between the two variables. The sign of the correlation coefficient indicates the direction of the relationship. Positive correlations indicate that as the values in one variable increase, the values in the other do likewise. Negative correlations indicate that as one increases, the other decreases (Objec- tive 2).
The differences among the correlation procedures in this chapter are in the kinds of vari- ables they accommodate. Pearson requires interval or ratio variables that are normally and similarly distributed. A special application of Pearson, the point-biserial correlation, requires an interval/ratio variable and a second variable that has only two categories, a dichotomously scored variable. Spearman will accommodate any combination of ordinal, interval, or ratio variables but will only assess the correlation between the rankings on the variables, rather than their actual values (Objective 4). Because the data that are used in a Pearson Correlation contain 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. Evidence of this is that the Pearson Correlation value can be squared to produce the coefficient of determination. That value indicates the proportion of one variable that can be explained by the other (Objective 3). There is no equivalent of this statistic for Spearman values.
Answers to Review Questions
A. The sign of the correlation has nothing to do with its strength. It is only relevant to its interpretation. Correlations of 21 and 11 are both perfect correlations.
B. Dummy-coding involves just two values, zeros and ones. C. Correlating a ratio scale variable (sales) with an ordinal scale variable (sales
ranking for the prior month) calls for Spearman’s rho.
Chapter Formulas
Formula 8.1 rxy 5 S1zx 3 zy 2
n 2 1 This is the z score formula for calculating a Pearson
Correlation.
Formula 8.2 rs 5 1 2 6Sd2
n1n2 2 12 This is the formula for calculating a Spearman’s rho correlation.
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CHAPTER 8Management Application Exercises
Management Application Exercises
Unless otherwise stated, use p 5 .05 in all your answers.
1. An employment agency gathers the following data on its clients: • Age • Gender • Educational level (no high school, high school, associate’s, bachelor’s,
master’s) • Years of past experience • Whether or not they have been successfully placed in employment by the
agency
For those who have been successfully placed, the following data is gathered:
• Starting salary • Current salary • Tenure in months
a. Which type of correlation procedure would be most appropriate to gauge that relationship between each pair of variables?
b. Do you expect each pair of variables to be significantly correlated or not? Why?
c. For the variables you expect to be significantly correlated, what do you expect the direction of the relationship to be? Why?
2. Data are gathered regarding the length of tenure top executives have at a major corporation and whether those executives have been divorced. The data for eight executives are as follows:
Tenure Divorce
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
a. What’s the most appropriate procedure for evaluating the relationship be- tween tenure and divorce?
b. What is the correlation and how can it be interpreted in terms of magni- tude, direction, and practical importance?
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CHAPTER 8Key Terms
c. How much of whether executives have been divorced can be accounted for by their length of tenure with the organization? How much of tenure can be explained by whether there has been a divorce?
d. Make a logical argument for why tenure may be causing divorce, and then make another logical argument for why divorce may be causing tenure.
3. Ten employees have just taken two surveys on a) their job satisfaction and b) their life satisfaction. For both variables, higher scores indicate more satisfaction. The data are ordinal. Is the relationship random?
Job Life
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
Key Terms
• The correlation coefficient is a value, usually ranging from 21 to 11, which indi- cates the strength and direction of the relationship between variables.
• Statistical tests under the hypothesis of difference examine whether differences between groups are due to chance. Under the hypothesis of association the issue is whether relationships are due to chance.
• Attenuation of range refers to circumstances where the true range of values pos- sible for a variable is not reflected in the sample.
• Relationships which are linear remain consistent throughout the entire range of the variables involved.
• The Pearson Correlation gauges the strength and direction of the relationship between interval or ratio data.
• The coefficient of determination, r2, indicates the proportion of variability in one of the variables in a Pearson Correlation that can be explained by changes in the other.
• The point-biserial correlation is a procedure for relating an interval/ratio variable with a nominal variable that has two categories.
• Dummy-coding uses zeros and ones to code nominal scale variables. • Spearman’s rho is a correlation procedure for ordinal scale data. • Nonparametric means without assumptions about parameters such as data nor-
mality and homoscedasticity.
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