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CHAPTER 2. Correlation Intuitively Considered
CHAPTER 3. Standard Scores and Correlation
CHAPTER 4. Characteristics of The Correlation Coefficient
CHAPTER 5. The Vector Approach
CHAPTER 6. The Cartesian Approach
CHAPTER 7. The Problem of External Effects: The Partial Correlation
CHAPTER 8. The Correlation Coefficient
CHAPTER 9. Significance
CHAPTER 10. Different Coefficients of Correlation
CHAPTER 11. Considerations
* I wrote this book for my students at the University of Hawaii who, as I was when I was an undergraduate, were puzzled by this strange animal called the correlation coefficient and the meaning of all those numbers called correlations.
There is little doubt that the correlation coefficient in its many forms has become the workhorse of quantitative research and analysis. And well it should be, for our empirical knowledge is fundamentally of co-varying things. We come to discern relationships among things in terms of whether they change together or separately; we come to impute causes on the basis of phenomena co-occurring; we come to classify as a result of independent variation.
Of course, many of our concepts may be apriori, our frameworks may be projected onto phenomena and create order, and our understanding may be partly intuitive. Our knowledge is a dialectical balance between that sensory reality bearing on us, and our reaching out and imposing on this reality structure and framework.
Whatever the framework within which we order phenomena, however, that reality we perceive is of dependence, concomitance, covariation, coincidence, concurrence; or of independence, disassociation, or disconnectedness. We exist in a field of relatedness: i.e., we come to understand the world around us through the multifold, interlaced and intersecting correlations it manifests. Sometimes we call these relationships cause and effect, sometimes we generalize them into assumed laws, and sometimes we simply call them natural or social uniformities or regularities.
Reflecting this interrelatedness, our theories and ideas about phenomena usually are based on assumed correlations. "Birds of a feather flock together." "Time heals all wounds." "Power kills and absolute power kills absolutely." Or at a different level: "National cohesion increases in wartime." "Prolonged severe inflation disorders society." "Institutional power aggrandizes until checked by opposing institutional power."
No wonder, then, that scientists have tried to make the concept of correlation more precise, to measure and to quantitatively compare correlations. Here I will deal with one such measure, called the product moment correlation coefficient. This is the most widely used technique for assessing correlations and is the basis for techniques determining mathematical functions (such as regression analysis) or patterns of interdependence (such as factor analysis).
Although much used, however, the correlation coefficient
Therefore, the purpose of this book is to convey an understanding of the correlation coefficient to students that will be generally useful. I am especially concerned with providing the student with an intuitive understanding of correlation that will enable him to better comprehend the use of correlation coefficients in the literature, while providing material helpful for applying the correlation coefficient to his own work.
Conveying understanding and facilitating application are my aims. The organization of this book and the topics included mirror these goals, as does my emphasis on figures. A picture is worth a thousand derivations and symbols. Correlation can be beautifully illustrated, but yet many statistical books solely present the mathematical derivations and statistical formula for the correlation coefficient, to the detriment of a student's learning. Here I hope to show the very simple meaning of correlation, both in vector terms and in graphical plots. With this meaning understood, the application of the correlation coefficient is straight forward and intuitively reasonable, while the usual complex statistical formula is then simply a computational tool to that end.
To convey this understanding, the first chapters look at correlation intuitively. Given certain phenomena, what does correlation mean? What about phenomena are we perceiving? How can we approach a measure of correlation? Chapter 5 then looks at correlation in terms of vectors. Correlated phenomena are vector phenomena: they have direction and magnitude. To appreciate the correlation intuitively it is helpful to visualize the geometric or spatial meaning of the coefficient. However, there is an alternative spatial view of correlation as a Cartesian plot. This is the usual picture of correlation presented in the statistical books, and will also be presented in Chapter 6. From this view will be derived the usual formulation of the correlation coefficient.
The remaining chapters move on to more advanced topics concerning the correlation. Chapter 7 describes the partial correlation coefficient. Our ability to portray partial correlation in simple geometric terms exemplifies the usefulness of the geometric approach. Chapter 8 briefly defines the correlation matrix. Chapter 9 considers significance conceptually and untangles two types of significance often confused. Chapter 10 is a discussion of different types of correlation coefficients and should be useful for understanding a particular coefficient in the context of its alternatives. And Chapter 11 recapitulates the major points and adds a few considerations in understanding correlation.
1. Henceforth, unless otherwise specified, correlation coefficient will mean the product moment.
When we perceive two things that covary, what do we see? When we see one thing vary, we perceive it changing in some regard, as the sun setting, the price of goods increasing, or the alternation of green and red lights at an intersection. Therefore, when two things covary there are two possibilities. One is that the change in a thing is concomitant with the change in another, as the change in a child's age covaries with his height. The older, the taller. When higher magnitudes on one thing occur along with higher magnitudes on another and the lower magnitudes on both also co-occur, then the things vary together positively, and we denote this situation as positive covariation or positive correlation.
The second possibility is that two things vary inversely or oppositely. That is, the higher magnitudes of one thing go along with the lower magnitudes of the other and vice versa. Then, we denote this situation as negative covariation or negative correlation. This seems clear enough, but in order to be more systematic about correlation more definition is needed.
Perceived covariation must be covariation across some cases. A case is a component of variation in a thing. For example, the change in the speed of traffic with the presence or absence of a traffic policeman (a negative correlation) is a change across time periods. Different time periods are the cases. Different levels of GNP that go along with different amounts of energy consumption may be perceived across nations. Nations are the cases, and the correlation is positive, meaning that a nation (case) with high GNP has high energy consumption; and one with low GNP has low energy consumption. The degree to which a regime is democratic is inversely correlated with the intensity of its foreign violence. The cases here are different political regimes.
To be more specific, consider the magnitudes shown in Table 2.1. The two things we perceive varying together--the variables--are 1955 GNP per capita and trade. The cases across which these vary are the fourteen nations shown. Although it is not easy to observe because of the many different magnitudes, the correlation is positive, since for more nations than not, high GNP per capita co-occurs with high trade, and low GNP per capita with low trade.
We can summarize this covariation in terms of a four-fold table, as in Table 2.2. Let us define high as above the means (averages) of $481 for GNP per capita in Table 2.1 and $4,975 millions for trade, and low at or below these means. Then we get the positive correlation shown in Table 2.2. The numbers that appear in the cells of the table are the number of nations that have the indicated joint magnitudes. For example, there are nine nations which have both low GNP per capita and low trade.
From the table, we can now clearly see that the correlation between the two variables is positive, since with only one exception (in the upper right cell) high magnitudes are observed together, as are low magnitudes.
If the correlation were negative, then most cases would be counted in the lower left and upper right cells. What if there were about an equal number of cases in all the cells of the fourfold table? Then, there would be little correlation: the two variables would not covary. In other words, sometimes high magnitudes on one variable would occur as often with low as with high magnitudes on the other.
But all this is still imprecise. The four-fold table gives us a way of looking at correlation, but just considering correlation as covarying high or low magnitudes is quite a loss of information, since we are not measuring how high or low the figures are. Moreover, if we are at all going to be precise about a correlation, we should determine some coefficient of correlation--some one number that in itself expresses the correlation between variables. To be a useful coefficient, however, this must be more than a number unique to a pair of variables. It must be a number comparable.between pairs of variables. We must be able to compare correlations, so that we can determine, for example, which variables are more or less correlated, or whether variables change correlation with change in cases. Finally, we want a correlation that indicates whether the correlation is positive or negative. In the next chapter we can intuitively and precisely define such a coefficient.
Although the cases across which two variables covary usually will be the same,
Clearly, we have a classic problem. How can we measure the correlation between different things in different units? We know we perceive covariation between things that are different. But determining common units for different things such that their correlations can be measured and compared to other correlations seems beyond our ability. Yet, we must make units comparable before we can jointly measure variation. But how?
Consider the observations on ten variables in Table 3.1, which include those on GNP per capita and trade in Table 2.1. Note the differences in units between the various variables, both in their nature and average magnitudes. our problem, then, is to determine some way of making the units of such variables comparable, so that we can determine the correlation between any two of them, as well as compare the correlations between various pairs of variables. Is, for example, the covariation of foreign conflict and of defense budget across the fourteen nations greater or less than that of, say, foreign conflict and the freedom of group opposition?
Let us consider this problem for a moment. We want a measure of covariation, of how much two things change together or oppositely. Clearly, then, we have no interest in their different magnitudes. And since magnitudes are irrelevant we can at least make the different magnitudes of the two variables comparable by making their average magnitude the same. This we can do by subtracting the average or mean of each variable from all its magnitudes.
To illustrate this, consider from Table 3.1 the power and defense budget variables, two variables measured in different units and involving quite different magnitudes. As shown in the table, the mean of power is 7.5 and that of defense budget is $5,963.5 (millions). Table 3.2 shows the result of subtracting the mean from each variable. Such data resulting from subtracting the mean is called mean-deviation data, and the mean of mean-deviation data is always zero.
Now that we have made the average magnitude comparable by transforming each variable to a mean of zero, the observations now represent pure variation. It is as though we had taken different statistical profile shots of each variable and overlaid them so that we could better see their covariation. But yet, as we can see from Table 3.2, the mean-deviation data does not give us a very good view of the degree of correlation, although it does enable us to see how the pluses and minuses line up
The problem is that subtracting the means did not change the magnitude of variation, and that of the defense budget is very large compared to power. We still need a transformation to make their variation comparable.
Why not compute an average variation then? This average could be calculated by adding up all the mean-deviations and dividing by the number of cases. However, the summed minus magnitudes equal the summed positive magnitudes (by virtue of the subtraction of the mean), and the average variation always will be zero as shown in Table 3.2, regardless of the absolute magnitude of the variation.
Then why not average the absolute mean-deviation data? That is, eliminate the negative signs. But if we did this, we would run into some mathematical difficulties in eventually determining a correlation coefficient. Absolutes create more problems than they solve.
However, we can invoke a traditional solution. If we square all the mean-deviation data, we eliminate negative signs. Then the average of these squared magnitudes will give us a measure of variation around the mean.
From this point on, I will have to be more precise and use some notation towards that end. I will adopt the following definitions.
- X = anything that varies; a variable;
- Xj = a specific variable j;
- Xk = a specific variable k;
- i = a particular case i
- xij = a magnitude (datum) for case i on variable j;
- n = the number of cases for a variable;
- = the average or mean of variable X;
- X*j = mean-deviation data (the mean has been subtracted from each original magnitude) for variable j;
- xij = the first case (i = 1 for variable j) to the last case (i = n for variable j), that is x1j + x2j + x3j + . . . + xnj = xij.
The last definition
Now, using this notation, we can define the mean as,
- Xj = (xi) / n
And we can also easily define mean-deviation data. Since X*j stands for the mean-deviation data for variable Xj, let x*ij be the corresponding mean-deviation magnitude. Then
- x*ij = xij - j
With these equations I can return to our measure of variation. Remember, we are going to square the mean-deviation data, add, and divide the sum by n to get an average variation. Let 2 (sigma squared) stand for our measure of variation. Then
- 2 = (x*ij2) / n = (xij - j)2/n
- where the last equality follows from Equation 3.2.
We can now introduce two important definitions.
Definition 3.2 :
- 2j = the variance of variable Xj.
- j = the standard deviation of variable Xj, which is the positive square root of the variance.
We now have two measures of variation around the mean and we will find that each of them is a useful tool for understanding correlation. At this point, we can use the standard deviation to help resolve our original problem of making the variation in two variables comparable, since it is expressed in the original units (to get a measure of variation, we squared the mean-deviations; the standard deviation is the square root of the average squared mean-deviations and thus brings us back to the original units).
The standard deviations for the selected sample data are given in Table 3.1. Those for power and defense budgets are the same as for the mean-deviation data in Table 3.2. Subtracting the mean out of a variable does not change its standard deviation.
Now that we have a comparative measure of variation for a variable, we can resolve our original problem of making the variation in two variables comparable: we can norm each by its standard deviation. That is, we can divide each variable's mean-deviation x*ij by its standard deviation. This will transform the data to standard deviation units, and thus make each variable's variation comparable in the same units. Such transformed data is called standardized data, the application of this transformation is called standardization, and the transformed magnitudes are called standard scores. The customary notation for this should be defined.
Definition 3.3 :
- Zj = a standardized variable j,
- zij = a standard score for case i on standardized variable Zj.
And a standard score can now be defined in terms of the other measures we have developed.
- zij = x*ij/j = (xij - j) / j
To return to our two variables of interest, their standard scores are shown in Table 3.3.
Note that the means and standard deviations for the standardized variables are equal. Moreover, the standard deviation of standard scores is equal to unity (1.00), and thus to the variance. For standardized data, our two measures of variation are equal.
At a glance, a standard score tells us how low or high the data for a case on a variable is. Consider the standard scores for the U.S. in Table 3.3. Its standard score of 1.613 on power means that its magnitude is 1.613 times the standard deviation of of the 14 nations on their power; its score of 2.685 on the defense budget means its magnitude on this variable is 2.687 times the standard deviation for the data on the defense budget. Thus a standard score of 1.0 means a nation's variation is one standard deviation from the average, a score of 2.0 means two standard deviations from the average, and so on.
Standard scores enable us to measure the variation of two variables and to compare their variation in common units (of standard scores) and magnitudes, regardless of their original units and magnitudes. We can look at Table 3.3 and see that there is a tendency for the scores to be both high or both low for a nation, and thus for there to be a positive correlation. Now, to measure such a correlation.
Of course, we could use the same averaging approach employed to get a measure of variation. We could add the two scores for each case, sum all these additions and divide this sum by n. However, if we did this we would always end up with zero regardless of the covariation involved.
Intuitively, a reasonable alternative is to multiply the two scores for each case, add all those products, and divide by n to get an average product. Then, joint high plus or joint high negative scores will contribute positively to the sum; if one is high positive and the other high negative, they will contribute negatively to the sum. Thus, the sum will at least discriminate positive and negative covariation. Moreover, multiplication seems to intuitively capture the notion of covariation. For if one case does not vary from the average, its standard score will be zero, and the resulting multiplication of its score by that on another variable will yield zeros, i.e., no covariation. As it should be.
Table 3.4 carries out the multiplication and averaging for our two variables.
We now have something like what we are after. The column of joint products of standard scores measures covariation for each case; the sum measures the overall covariation; and the average of this gives us a measure of average covariation. In truth, this measure of average correlation is the product moment correlation coefficient. We have arrived at that somewhat intimidating coefficient intuitively, simply through searching for some way to systematically compare variation.
- rjk = the product moment correlation between Xj and Xk,
- rjk2 = the coefficient of determination between Xj and Xk.
The squared correlation coefficient is called the coefficient of determination. Since the standard deviation is the square root of the variance (Definition 3.2), and the correlation is computed through data standardized by the standard deviation, we can translate the results back into variance terms by squaring the correlation. The coefficient of determination then defines the proportion of variance in common between two variables.
Let me recap the intuitive process of arriving at the correlation coefficient. First, we recognize the existence of covariation between things and the need to systematically define it. In doing this, the first hurdle is to delimit the cases over which our observations covary. Once this is done, we then have the problem of making our observations comparable. If we are to assess covariation, we must have some way of removing differences between observations due simply to their units. One way of doing this is to make the averages of the variables equal. But, while helpful, this does not remove differences in variation around the mean due to differences in units.
To help with this problem, we then developed a measure of variation, called the standard deviation. Now, if we subtract a variable's mean from each observation and then divide by the standard deviation, we have transformed the variables into scores with equal means and standard deviations. Their variation is now comparable.
To get at covariation, then, it seems most appropriate to find some measure of average covariation of the two sets of standardized scores. The best route to this is to multiply the two scores for each case, add the products, and divide by the number of cases. The result is a coefficient that measures correlation, the characteristics of which will be given in Chapter 4.
Beforehand, a precise definition of the product moment correlation will be helpful. It is
- rjk = (zijzik) / n
Then, replacing zij and zik by Equation 3.4,
- rjk = (((xij - j) / j)((xik - k) / k)) / n = ((xij - j)(xik - k)) / njk
Finally from Equation 3.3 and Definition 3.2, we can rewrite the standard deviation in terms of the raw data, and thus define the correlation entirely as a function of the variables in their original magnitudes. That is
- rjk = ((xij - j)(xik - k)) / njk
- = ((xij - j)(xik - k)) / (((xij - j)2)((xik - k)2))1/2
The terms on the right of the second equality give the formula most often presented in statistical books. It is imposing, but as we have seen, its underlying logic is intuitively reasonable.
1. There are occasions when the cases may differ, such as in lagged correlations where the case for one variable is a time period lagged behind the time period for the other.i=1 In this book the range will always be from i = 1 to n, and so is omitted.
2. I am leaving the range off the , which would be written as
3. Note that the mean of a standardized variable is always zero. Therefore simply adding the standard scores for each case, summing the result across all the cases, and taking the average of that sum will always, regardless of the data, get you:
- ((zij + zik)) / n = (zij) / n + zik) / n
- = (mean of Zj) + (mean of Zk)
- = 0 + 0
On the other hand, if two things vary oppositely and perfectly, then the correlation will equal 1.00.
We therefore have a measure which tells us at a glance whether two things covary perfectly, or near perfectly, and whether positively or negatively. If the coefficient is, say, .80 or .90, we know that the corresponding variables closely vary together in the same direction; if -.80 or -.90, they vary together in opposite directions.
I should point out that there are two ways of viewing independent variation. One is that the more distinct and unrelated the covariation, the greater the independence. Then, a zero correlation represents complete independence and -1.00 or 1.00 indicates complete dependence. Independence viewed in this way is called statistical independence. Two variables are then statistically independent if their correlation is zero.
There is another view of independence, however, called linear independence, which sees independence or dependence as a matter of presence or absence, not more or less. In this perspective, two things varying perfectly together are linear dependent. Thus, variables with correlation of -1.00 or 1.00 are linear dependent. Variables with variation less than perfect are linear independent. Figure 4.1 shows how these two views of dependence overlap.
From my derivation of the correlation coefficient in the last chapter, we know that the squared correlation (Definition 3.3) describes the proportion of variance in common between the two variables. If we multiply this by 100 we then get the percent of variance in common between two variables. That is:
r2jk x 100 = percent of variance in common between Xj and Xk.
For example, we found that the correlation between a nation's power and its defense budget was .66. This correlation squared is .45, which means that across the fourteen nations constituting the sample 45 percent of their variance on the two variables is in common (or 55 percent is not in common). In thus squaring correlations and transforming covariance to percentage terms we have an easy to understand meaning of correlation. And we are then in a position to evaluate a particular correlation.
As a matter of routine it is the squared correlations that should be interpreted. This is because the correlation coefficient is misleading in suggesting the existence of more covariation than exists, and this problem gets worse as the correlation approaches zero. Consider the following correlations and their squares.
Note that as the correlation r decrease by tenths, the r2 decreases by much more. A correlation of .50 only shows that 25 percent variance is in common; a correlation of .20 shows 4 percent in common; and a correlation of .10 shows 1 percent in common (or 99 percent not in common). Thus, squaring should be a healthy corrective to the tendency to consider low correlations, such as .20 and .30, as indicating a meaningful or practical covariation.
1. There are exceptions to this, as when the correlation is computed for dichotomous variables with disparate frequencies. See Section 10.1.
2. I am talking about statistical independence in its descriptive and not inferential sense. In its inferential sense, a nonzero correlation would represent independent variation insofar as its variation from zero could be assumed due to chance, within some acceptable probability of error. If it is improbable that the deviation is due to chance, then the correlation is accepted as measuring statistically dependent variation. See Chapter 9.
The standard scores considered in Chapter 3 provide an intuitive route to developing and understanding correlation. There are two more approaches, each providing a different kind of insight into the nature of the correlation coefficient. One is the vector approach, to be developed here. The other is what I will call the Cartesian approach, to be developed in the next chapter.
Consider again the problem. Two things vary and we wish to determine in some systematic fashion whether they vary together, oppositely, or separately. We have seen that one approach involves a simple averaging and common sense. But let us say that we are the kind of people that must visualize things and draw pictures of relationships before we understand them. How, then, can we visualize covariation? That is, how can we geometrize it?
There are many ways this could be done. For example, we could plot the magnitudes for different cases on each variable to get a profile. And we could then compare the profile curves to see if they moved oppositely or together. While intuitively appealing, however, it does not lead to a precise measure.
But what about this? Variables can be portrayed as vectors in a space. We are used to treating variables this way in visualizing physical forces. A vector is an ordered set of magnitudes; a variable is such an ordered set (each magnitude is for a specific case). A vector can be plotted; therefore a variable can be plotted as a vector. Vectors pointing in the same direction have a positive relationship and those pointing in opposite directions have a negative relationship. Could this geometric fact also be useful in picturing correlation? Let us see.
The first problem is how to plot a variable as a vector. Now consider the variable as located in space spanned by dimensions constituting the cases. For example, consider the stability and freedom of group opposition variables from Table 3.1 and their magnitudes for Israel and Jordan. These two cases then define the dimensions of the space containing these variables--vectors--that are plotted as shown in Figure 5.1. We thus have a picture of variation representing the two pairs of observations for these variables. The variation of the variables across these nations is then indicated by the length and direction of these vectors in this space.
To give a different example, consider again the power and defense budget variables (Table 3.1. There are fourteen cases--nations--for each variable. To consider these variables as vectors, we would therefore treat these nations as constituting the dimensions of a fourteen dimensional space. Now, if we picture both variables as vectors in this space, their relative magnitudes and orientation towards each other would show how they vary together, as in Figure 5.2.
To get a better handle on this, think of the vectors as forces, which is the graphical role they have played for most of us. If the vectors are pointed in a similar direction, they are pushing together. They are working together, which is to say that they are varying together. Vectors oppositely directed are pushing against each other: their efforts are inverse, their covariation is opposite. Then we have vectors that are at right angles, such as one pushing north, another east. These are vectors working independently: their variation is independent.
With this in mind, return again to Figure 5.2. First, the vectors are not pointing oppositely, and therefore their variation is not negative. But they are not pointing exactly in the same direction either, so that they do not completely covary together. Yet, they are pointing enough in the same general direction to suggest some variation in common. But how do we measure this common covariation for vectors?
Again, we have the problem of units to consider. The length and direction of the vectors is a function of the magnitudes and units involved. Thus, the defense budget vector is much longer than the power one.
Since differing lengths are a problem, why not transform the vectors to equal lengths? There is indeed such a transformation called normalization,
- Xj = vector j or variable j
- |Xj| = length of vector j = (x2ij)1/2
That is, the length of a vector is the square root of the squared sum of its magnitudes. Then the normalization of a vector is the division of its separate magnitudes by the vector's length. All vectors so normalized have their lengths transformed to 1.00.
The problem with normalization is that there can be considerable differences in mean magnitudes on the normalized vectors, and these mean differences would then confound the measure of correlation. After all, we want a measure of pure covariation, and differences in average magnitude confound this.
Then, in reconsidering our original vectors, what about transforming the magnitudes on the vectors by subtracting their averages, as we did in Chapter 3 (Equation 3.2)? If we did this we would be translating the origin of the space to the mean for each vector: the zero point on each dimension would be the mean.
By so transforming the vectors to mean-deviation data we eliminate the affect of mean differences, but we still have the differing vector lengths. Need this bother us, now? After all, we are concerned with similarity or dissimilarity in vector direction as an indication of correlation. Let us therefore work for the moment with mean-deviation vectors and see what we get.
Recall that codirectionality for vectors is the essence of covariation. How do we measure codirectionality, then? By the angle between the vectors.
But measuring this angle creates a problem. In degrees it can vary from 0o to 360o and is always positive, where perfect positive covariation would be zero, independent variation would be 90o or 270o and opposite variation 180o. As with the correlation coefficient derived in Chapter 3, it would be desirable to have some measure which would range between something like 1.00 for perfect correlation, -1.00 for perfect negative correlation, and zero for no correlation.
And we do have such a measure given by elementary trigonometry. It is the cosine. The cosine of the angle between vectors will be +1.00 for vectors with an angle of 0o, -1.00 for an angle of 180o (completely opposite directionality), and 0 for an angle of 90o or 270o. What is more, from linear algebra we can compute the cosine directly from the vectors without measuring their angles. For vectors X*j and X*k of mean-deviation data, this is
- Cos jk = (x*ijx*ik) / | X*j||X*k|
Let the magnitudes of power and defense budget be transformed to mean-deviation data. Then Figure 5.3 shows the angle between the vectors given by the cosine (Equation 5.1) and for the mean-deviation data of Table 3.2 we would find the cosine of the angle equals .66.
Thus, the cosine between two variables transformed to mean-deviation data--to data describing the variation around the mean--gives us a measure of correlation. But there is a coincidence here. The .66 cosine for the angle of 49o between the two vectors is the same as the .66 product moment correlation we found for the two variables using standard scores (Table 3.4). Is there in fact a relationship between these alternatives?
To see what this relationship might be, let us expand the formula for the cosine between two mean-deviation vectors so that it is expressed in the original magnitudes.
- Cosine jk = (x*ijx*ik) / | X*j||X*k|
- = ((xij - j)( xik - k) / ( x *ij 2)1/2( x *ik 2)1/2
- = ((xij - j)( xik - k) / (((xij - j)2) ((xik - k)2))1/2
The lengths of the vectors have been expanded using Definition 5.1.
Now compare Equation 5.2 with Equation 3.7 for the correlation. They are the same! Thus,
- cosine jk = rjk
for mean-deviation data. And since standardized data is also mean deviation data, Equation 5.3 holds for standard scores as well.
To sum up, we have found that we can geometrically treat the variation of things as vectors, with the cases across which the variation occurs as dimensions of the space containing the vector. The covariation between two things is then shown by their angle in this space. However, we have to again transform the variation in two things to eliminate the effects of mean-differences. The resulting mean-deviation data reflects pure variation and the covariation between the vectors is then the cosine of their angle. Moreover, this cosine turns out to be precisely the product moment correlation derived from standardized data. Instead of two alternative measures of correlation, we thus have alternative perspectives or routes to understanding correlation: the average cross-products of standardized data, or the cosine of the angle between mean-deviation (or standardized) data vectors.
1. This is not to be confused with transforming distributions to normality, an entirely different procedure.
So far we have approached correlation in intuitive-conceptual and geometric fashion. But there is another approach beloved of introductory statistical tests. This is the Cartesian approach: it relies on a Cartesian coordinate system, where each variable represents a coordinate, and the cases are points plotted in this two-dimensional space.
Let us again begin with trying to determine some measure of the covariation of things. We saw that the variation of a variable could be pictured as a vector and covariation as codirectionality between vectors. We could have, however, pictured variation differently. Rather than fixing the cases as dimensions of the space, we could have treated the variables as the dimensions. Then we could plot each case on these dimensions in terms of their joint magnitudes, as shown in Figure 6.1 for variables Xj and Xk. The variation of the cases in this space then represents the covariation of the variables.
What, then, would perfect covariation look like? Here, we have a fundamental ambiguity. Perfect covariation could look like Figure 6.2a, b, or c. That is, covariation could lie along some curve, like the covariation between time and the height of a ball thrown upward which can be plotted as a parabola. Or, covariation could lie along a straight line. In intuitively assessing covariation, we were concerned with a measure of when magnitudes were both high and low or when one was high, the other low. That is, we wanted to index positive or negative correlation.
Concerning positive correlation, this desire is clearly reflected in Figure 6.2c, for as the magnitude of a case on one variable is high, it is high on the other; as it is low on one, it is low on the other. And uniformly so. We can therefore consider this figure as representing perfect positive correlation. Perfect negative correlation would then be reflected in cases forming a straight line slanting downwards to the right, as shown in Figure 6.3a. If the cases appear randomly distributed as in Figure 6.3b, we have a case of no covariation in common.
But it is our misfortune that most phenomena will have less than perfect covariation and therefore would manifest a plot like that in Figure 6.1. Again, we must develop some measure of all this. And again we can approach this intuitively.
Now, we know that perfect correlation would be represented by the cases forming a straight line and that our real data will vary from this line. Is it not reasonable, then, to think of correlation as the degree to which the cases vary from a straight line?
A problem, however, is determining what this perfect correlation, this straight line amongst the cases plotted on the variables, should be. We know the line should go through the points as in Figure 6.4a, but we want to locate it more precisely than this so that we can develop an exact measure of correlation. One possibility is to locate the line such that the deviation of cases from the line are minimal. Let us consider such deviations as shown in Figure 6.4b, where the deviations are computed as the difference between the case's magnitude xij on one variable (Xj), and the location xij of the case on the line if Xj and Xk were perfectly correlated. I will denote this deviation by di as shown. Deviations d2 and d3 are also shown as examples.
We could then sum these deviations for all cases and locate the line so that the sum is as small as possible. And since deviations above the line would always be positive, those below would be negative, we could always locate the line such that these deviations sum to zero. Unfortunately, however, an infinitude of lines could be fitted to the cases to yield deviations summing to zero, and we cannot define any unique line this way.
We could sum absolute deviations, but then absolutes are again difficult to deal with mathematically. Then why not do what we did in Chapter 3 to get rid of sign differences? Why not square deviations and sum the squares? This in fact provides a solution, for the minimization of di2 leads to a unique line fitting the points. We need not go into the actual minimization. Suffice to say that the equation of the straight line is
Xj = + Xk,
and the line minimizing the squared
Now, let us say that we have such a least squares line in Figure 6.4B, and the squared deviations add to the smallest possible sum. What then? This sum of squared deviations always would be zero in the case of perfect correlation, but otherwise would depend on the original data units. Therefore, different sums of squared deviations for different variables would not be easily comparable or interpretable. When this type of unit-dependent variation occurs, a solution is often to develop a ratio of some sort. For example, national government budgets in national currencies are difficult to compare between nations, but the ratio of national budget to GNP eliminates currency units and provides easily comparable measures.
But to what do we compute a ratio for our sums? Consider that we have so far computed differences from a perfect correlation line fitted to the actual magnitudes of the variables. Could we not also introduce a hypothetical line fitted to the cases as though they had no correlation? We would then have two hypothetical lines, one measuring perfect correlation; the other measuring complete statistical independence, or noncorrelation. Would not some ratio of differences from the two lines give us what we want? Let's see.
To be consistent with the previous approach, squared deviations from the second line would also be calculated. But, where would the line of perfect noncorrelation be placed among the cases in Figure 6.4b?
First, the line would be horizontal, since if there is some variation in common between two variables, the line will angle upward to the right if this correlation is positive, or to the left if negative.
Okay, now where do we place the horizontal line? If there is no correlation between two variables, then from a magnitude on the one variable, we cannot predict to a magnitude in the other. Knowing one variable's variation does not reduce uncertainty about the other's. This is an opposite situation for perfectly correlated variables, where knowing a magnitude on one variable enables a precise prediction as to the magnitude of the other.
When we have complete uncertainty, as in the case of perfect statistical independence, then given the one variable, what is the best estimate of the magnitude on the other?
This is the other variable's mean. In a situation of complete uncertainty about the magnitude of a case on a variable, the best guess as to this magnitude is the variable's mean. Since the line of perfect noncorrelation is a line of complete uncertainty about where a case would lie on the vertical axis, Xj, in Figure 6.4b, given its magnitude on Xkk, the most intuitively reasonable location for the line is at the average of Xj, as in Figure 6.5.
However, not only is this intuitively reasonable, but if in fact the two variables were perfectly uncorrelated, then the line which would minimize d2i would be this horizontal line. In other words, as the correlation approaches zero, the perfect correlation line approaches the horizontal one shown.
Now, given this horizontal line for hypothetically perfect noncorrelation, we can determine the deviation of each case from it. Let me denote such a deviation as gi, which is shown in Figure 6.5, along with g1 as an example. The sum of the squared deviations of gi is g2i , which is a minimum if the variables are perfectly correlated.
Thus, we have two measures of deviation. One from the perfect correlation line fitting the actual data; the other from the line of hypothetical noncorrelation. Surely, there will be a relationship between d2i and g2i.
Now, the less correlated the variables, the more the slanted line approaches the horizontal one. That is, the more d2i approaches g2i. If the variables were perfectly correlated, d2i would be zero, since there would be no deviation from the line of perfect correlation. And therefore the less the observations covary, the closer d2i approaches g2i from zero. If the observations are in fact perfectly uncorrelated, then d2i = g2i.
Thus, it appears that a ratio between d2i and d2i would measure the actual correlation between two variables. If the correlation were perfect, then the ratio would be zero; if there were no correlation, the ratio would be one. But this is the opposite of the way we measured correlation before. Therefore, reverse this measurement by subtracting the ratio from 1.00, and denote the resulting coefficient as h. Then,
- h = 1 - d2i / g2i
To deepen our understanding of h, return to Figure 6.5 and consider again the horizontal line. The deviations gi are from the average j. Therefore,
gi = xij - j,
g2i = (xij - j)2,
g2i / n = (xij - j)2 / n = 2j.
That is, what in fact g2i measures is the total variation of Xj. If we divide this by n, as in Equation 3.3, we would have the variance (2j) of Xj.
The other sum of squares, d2i, measures that part of the variation in Xj that does not covary with Xk--that is independent of covariation with Xk. If Xj and Xk covaried perfectly, there would be no independent variation in Xj and d2i would equal zero.
Therefore, the ratio of d2i to g2i measures the variance in Xj independent of Xk with respect to the total variance of Xj, Thus, when we subtract this ratio from 1.00 we get
h = 1 - d2i / g2i
= (g2i - d2i) / g2i
= (total variance - independent variance) / total variance
= (variance in common) / total variance.
In other words, h measures the proportion of variance between two variables. But this is what the coefficient of determination r2 does (Definition 3.4).
- h = r2,
and our measure of correlation derived through the Cartesian approach is the product moment correlation squared. And the Cartesian approach provides an understanding of why the correlation squared measures the proportion of variance of two variables in common.
1. Through calculus we can determine the values for and that define the minimizing equation. They are- j)(xik - k)) / (xik - k)2= ((xij
= j - k.
A problem with the correlation coefficient arises from the very way the covariation question was phrased in Chapter 2. We wanted some systematic measure of the covariation of two things. And that is precisely what we got with the correlation coefficient: a measure confined wholly to two things and which treats them as an isolated system.
But few pairs of things are isolated. Most things are part of larger wholes or complexes of interrelated and covarying parts, and these wholes in turn are still parts of other wholes, as a cell is a part of the leaf, which is part of the tree that is part of a forest. Things usually covary in large clusters or patterns of interdependence and causal influences (which is the job of factor analysis to uncover). To then isolate two things, compute the correlation between them, and to interpret that correlation as measuring the relationship between the two variables alone can be misleading.
Consider two examples. First, let us assess the correlation between illiteracy and infant mortality. We may feel that a lack of education means poor infant care, resulting in higher mortality. Of course, we want to be as comparative as possible so we compute the correlation for all nations. The actual correlation for 1955 is .61, which is to say that illiteracy and infant mortality have 37 percent of their variance in common. Are we then to conclude that education helps prevent infant mortality? If we were unwary, we might. But these two variables are not isolated. They are aspects of larger social systems and both subject to extraneous influences; and in this case, these influences flow from economic development. Many in the least developed countries have insufficient diets and inadequate health services, and live in squalid conditions. High infant mortality follows. Also by virtue of nonexistent or poor educational systems, many people are illiterate. This is quite the opposite of the developed nations, which usually have good medical services, a varied and high protein diet, and high literacy. Thus the correlation between illiteracy and infant mortality. It has less to do with any intrinsic relationship between them then to the joint influence on illiteracy and infant mortality of national development.
A second example has to do with the covariation between economic growth rate and social conflict. Let us hypothesize that they have a negative correlation due to economic growth increasing opportunities, multiple group membership, and cross-pressures, thus draining off conflict. Let us find, however, that the actual correlation is near zero. Before rushing out into the streets to proclaim that economic growth is independent of conflict, however, we might consider whether exogenous influences are dampening the real correlation. In this case, we could argue that the educational growth rate is the depressant. Increasing education creates new interests, broadens expectations, and generates a consciousness of deprivations. Thus, if education increases faster than opportunities, social conflict would increase. To assess the correlation between economic growth rate and conflict, therefore, we should hold constant the educational growth rate.
How do we handle these kinds of situations? There is an approach called partial correlation, which involves calculating the correlation between two variables holding constant the external influences of a third.
Before looking at the formula for computing the partial correlation, three intuitive approaches to understanding what is involved may be helpful. First, consider the first example of the correlation between illiteracy and infant mortality. How can we hold constant economic development? of course, before we can do anything, we must have some measure of economic development, and I will use GNP (gross national product) per capita for this purpose. Given this measure, then, a reasonable approach would be to divide the sample of nations into three groups: those with high, with moderate, and with low GNP per capita. Then the correlation between illiteracy and infant mortality rate can be calculated separately within each group. We would then have three different correlations between illiteracy and infant mortality, at different economic levels. We could then calculate the average of these three correlations (weighting by the number of nations in each sample) to determine an overall correlation, holding constant economic development. The result would be the partial correlation coefficient.
As a second intuitive approach, consider the deviations di between the actual magnitudes and the perfect correlation line for two variables, as shown in Figure 6.4b. This deviation measures the amount of variation in Xj unrelated to Xk. It is called the residual.
Now, consider two separate plots, one for the correlation between GNP per capita and illiteracy; the other for the correlation between GNP per capita and infant mortality. In each case, GNP per capita would be Xk, the horizontal axis. For each plot there will be a perfect correlation line, from which can be calculated the residuals for GNP per capita and illiteracy, and for GNP per capita and infant mortality. These two sets of residuals, di, measure the independence of illiteracy and infant mortality from GNP per capita.
With this in mind, consider. What we are after is the correlation between the two variables with the effect of economic development, as measured by GNP per capita, removed. But are not these residuals exactly the two variables with these effects taken out? Of course. Therefore, it seems reasonable to define the partial correlation as the product moment correlation between these residuals--between the di for illiteracy and the di for infant mortality--which would be .13. And in fact, this is the partial correlation.
The third intuitive approach is geometrical. Before, I showed that the cosine of the angle between two variables, each interpreted as a vector, was identical to the correlation, when the means were subtracted from the magnitudes. Figure 7.1 represents the actual correlations between the variables (vectors) GNP per capita, illiteracy, and infant mortality for mean-deviation data. It can be seen that GNP per capita has negative correlations (angles over 90o) with the other two variables, and is in fact -.67 for GNP per capita with infant mortality and -.83 with illiteracy.
Recall that what two vectors are at right angles, they have zero correlation--they have no variation in common. If GNP per capita were at right angles to the other two variables, we would know that their correlation between these two variables would be independent of GNP per capita. But, unfortunately, we are not blessed with such a simple reality, as we can see in Figure 7.1. However, is there not a way to create this independence? Can we not determine what this correlation would be if there were this independence? Yes, by using a geometric trick.
First, create a plane at a right angle to the GNP per capita vector as shown in Figure 7.2. Since the plane is at a right angle to GNP per capita, anything lying on the plane will also be at a right angle to GNP per capita. Second, therefore, project the illiteracy and infant mortality vectors onto the plane such that the lines of projection are also at right angles to the plane (and thus parallel to the GNP per capita vector). These projections are also shown in the Figure. The projected vectors will now be at right angles to the GNP per capita vector and thus independent of it. It follows, therefore, that the angle between these projected vectors will also be independent of GNP per capita, and indeed, this is the case.
Now, since we are after the correlation between illiteracy and infant mortality, holding constant (independent of) GNP per capita, it appears intuitively reasonable to consider the cosine of between the projected vectors as this correlation. Cosine is, as we know, the product moment correlation for mean-deviation data (which we are assuming), and is the angle independent of GNP per capita. In fact, cosine ( = 82.53o) is the partial correlation between illiteracy and infant mortality and equals .13.
To conclude, the three approaches--grouping, residuals, and vectors--to understanding partial correlation provide insight into what it means to assess the variation between two variables independent of a third. What remains is to present the formula for doing this, which is
- rjk.m = (rjk - rjmrkm) / ((1 - r2jm)(1 - r2km))1/2,
- rjk.m = partial correlation between Xj and Xk, holding Xm constant;
- rjk, rjm, rkm = product moment correlations between Xj, Xk, and Xm.
For illiteracy and infant mortality, partialling out the influence of GNP per capita, the formula equations of GNP per capita, the formula equals
rjk.m = (.61 - (-.67)(-.83)) / ((1 - (-.67)2)(1 - (-.83)2))1/2 = .13.
The partial correlation of .13 is quite a drop from the original correlation of .61 between illiteracy and infant mortality, a sharp decrease from 37 percent to 2 percent covariation. Thus, the hypothesis of an underlying economic development influencing the correlation between these two variables surely has substance.
The discussion of partial correlation has been only in terms of one external influence--a third variable. Partial correlation, however, can be generalized to any number of other variables, and formulas for their calculation are readily available in standard statistical texts. My concern here was not to present these extensions, but to provide a description of the underlying logic. Once this logic is clear for a third variable, then understanding what is involved when holding constant more than one variable is straight forward.
, measures that part of the variation in Xj that does not covary with Xk.
1. Recall from Chapter 6 that the residual, di
2. For simplicity in Figure 7.2, I have rounded 82.53o to 83o.
When the correlation between many variables are computed, they are often organized in matrix form as in Table 8.1 for the selected sample data. Since the correlation rjk between Xj and Xk is the same as rkj between Xk and Xj, only the bottom triangular portion of the matrix is given.
The matrix provides a way of easily comparing correlations (this shows the virtue of having a correlation which is comparative, i.e., which is not dependent on the units of the original data, and which has the same upper and lower bounds of +1.00 and same interpretation regardless of the variables) and determining clusters of variables that covary. Such is often aided visually by noting the high correlations, as done by parentheses in the matrix. And more systematic methods are available for defining the interrelationships among the variables as displayed in the table, such as factor analysis.
The correlation matrix is basic to many kinds of analysis. It is a bridge over which scientists can move from their data to sophisticated statistical analyses of patterns, dimensions, factors, causes, dependencies, discriminations, taxonomies, or hierarchies. In its own right, however, the correlation matrix contains much useful knowledge.
The correlation matrix can be directly computed from the original data matrix. The fundamentals matrix algebra for doing this is outside the scope of this book,
1. For a description of factor analysis, see my Applied Factor Analysis (1970).
2. For a helpful book on matrices, see Horst (1963).
- (1/n)Z'mxnZnxm = Rmxm
- R = the 14 x 14 variable correlation matrix of Table 8.1 (were the upper triangle of correlations filled in);
- R = a symmetric, Gramian matrix.
For empirical data, R is usually (but not always) nonsingular.
The correlation coefficient is descriptive. It measures the covariation in the magnitudes of two things. Often, however, this covariation is only a spring board to saying something about the population (or universe) from which the cases were taken, such as of all nations from the correlation for the variables in Table 3.1.
But chance affects our observations in many ways, and we should have some systematic method of assessing the likelihood of our results being accidental before rushing to categorical generalizations.
Then, we are concerned with one of two questions:
The first is a question of likelihood, given a particular set of observations; the second is a question of sample significance, given a particular sample of observations from a larger universe.
I will consider both questions in turn, and my approach will be intuitive and conceptual rather than statistical.
The same approach is applicable to assessing the role of chance on the correlation calculated. Consider the defense budget and trade magnitudes for the fourteen nations in Table 3.1 Given these magnitudes, what is the likelihood of getting by chance the joint combination of trade and defense budget magnitudes that would be correlated .71 (as shown in Table 8.1) or greater?
Mathematical statisticians have developed complex formulas for determining such probabilities, from which standard tables have been computed for statistical reference books. A simplified version of such a table is Table 9.1, which divides the correlation coefficients into the five probability levels shown for differing numbers of cases N.
Using this table, we can see that a correlation of .71 or greater for 14 nations (N) has a probability of less than .005 of occurring by chance. That is, less than one out of 200 random combinations of our observations should yield a .71 correlation or greater. It follows that the .71 correlation between trade and defense budget is meaningful, in the sense of being unlikely a chance result, given all the possible paired combinations of the data for the 14 cases on the two variables.
But, of course, any result can still occur by chance. One could on a first try blindly select the one of two white balls out of 50. One could win a lottery with odds of a million to one, also. So the .71 correlation could still be a chance happening. But if on different observations for different years and nations, we continue to get such a correlation, then our confidence in discarding chance as a possibility increases--our conviction grows that there is some underlying relationship or cause, as we would suspect something other than chance if a person won the Irish sweepstakes three years in a row.
However, assume we had hypothesized that a non-zero correlation exists between trade and defense budget, that we selected nations and observation in a way not to favor our hypothesis, and then we computed the correlation of .71. The probability of getting by chance such a correlation, or higher, is less than one out of two-hundred times, if in fact the correlation should be zero. This suggests that our hypothesis is correct. Correlations among data collected to test previously stated hypothesis always have more power than correlations which are simply assessed (exploratory). Babe Ruth's famous home run slammed over the centerfield he had just pointed to, gave him stature unattainable by any unpredicted home run.
Now, assume the fourteen nations we have used to assess the correlation between trade and defense budget is a good sample, i.e., well reflects all nations. Then what inference about all nations can be made from a correlation of .71 between the two variables?
A useful way of answering this is in terms of an alternative hypothesis.
The answer to this is given by the probability levels in Table 9.1, where we find the probability to be less than .005, or less than one out of two hundred. This can be interpreted as follows: The probability is less than .005 that we would be wrong in rejecting the hypothesis that the population correlation is zero or negative. With such a low probability of error, we might confidently reject this hypothesis, and accept that there is a positive correlation between trade and defense budgets for all nations. In other words, we can infer that our sample results reflect the nature of the population. They are statistically significant.
What if the alternative hypothesis were that a zero correlation exists between the two variables? Then, our concern would be with the probability of getting a plus or minus sample correlation of .71, or absolutely greater, were the alternative hypothesis true. This is a "two-tailed" probability in the sense that we are after the chance of a plus or minus correlation. Reference to Table 9.1 would inform us that the two-tailed probability is double that for the one-tailed probability, or 2 x .005 = .01. The probability of wrongfully rejecting the hypothesis of a population zero correlation is thus less than one out of a hundred. Therefore, most would feel confident in inferring that a non-zero correlation exists between trade and defense expenditures.
Therefore, very small correlations can be significant at very low probabilities of their being chance results, even though the variance in common is nil. Table 9.2 compares significance and variance in common for correlations at a probability level of .05 for selected sample sizes.
Clearly, one can have significant results statistically, when there is very little variation in common. A high significance does not mean a strong relationship. Even though for 1,000 cases, 99.75 percent of the variation between two variables is not in common, the small covariation that does exist can significantly differ from zero.
Which should one consider, then? Significance or variance in common? This depends on what one is testing or concerned about. If one wants results from which to make forecasts or predictions, correlations of even .7 or .8 may not be sufficient, no matter how significant, since there is still much unrelated variance. If one's results are to be a base for policy decisions, only a high percent of variance in common may be acceptable. But if one is interested in uncovering relationships, no matter how small, then significance is of concern.
Yes, when we keep in mind the two types of significance. Clearly, this is not a sample correlation and sample significance is meaningless. But, we can assess the likelihood of this being a chance correlation between the two sets of magnitudes for all nations, as described in Section 9.1.
Fortunately, both types of significance can be assessed using the same probability table, such as Table 9.1.
Both types of significance described here assume a normal distribution for both variables, i.e., that the magnitudes approximate a bell-shaped distribution.
When sampling significance is of concern, the observations are assumed drawn from a bivariate normal population. That is, were the frequencies of observation plotted for both variables for the population, then they would be distributed in the shape of a bell placed on the middle of a plane, with the lower flanges widening out and merging into the flat surface.
By virtue of these distributive requirements, the assessment of significance also demands that the data be interval. or ratio measurements, i.e., data like that for trade, GNP per capita, defense budget, GNP for defense, or U.S. agreement shown in Table 3.1 Dichotomous data, as that for stability and foreign conflict, or rank order data, as that for power, cannot have the significance of their product moment correlations assessed. For this, one must use a different type of correlation coefficient, of which the next chapter will give examples.
1. Statisticians have formulated a systematic design, called "tests of hypotheses," for making a decision to reject or accept statistical hypotheses. Most elementary statistical text books have a chapter or so dealing with this topic.
I have focused upon the product moment correlation coefficient throughout. It is the most widely used coefficient and for many scientists the only one. Indeed, most computer programs computing correlations employ the product moment without so informing their users in the program write up.
But useful alternatives do exist. And a good understanding of correlations requires an appreciation of these alternatives and their rationale. Here I will describe the most popular alternatives conceptually, leaving the statistical and computational details to the literature.
Both utilize the same amount of information in the observations, although not as much as the product moment.
Other alternative correlation coefficients are applicable to dichotomous data--observations in two variables that comprise only two magnitudes, as for stability and foreign conflict in Table 3.1 There are the phi, phi-ove-phi-max, and tetrachoric coefficients.
The phi is the product moment applied to dichotomous data and is also a function of the chi-square of a fourfold table, such as Table 2.2, thus enabling the statistical significance of the phi to be assessed. The range of phi is between -1.00 at +1.00 if the margins of the fourfold table are equal.
The maximum possible value of phi for given marginals can be computed and used to form the phi-over-phi-mix correlation coefficient, which is the ratio of phi to the maximum possible phi given the marginals. Regardless of marginal values, then, phi-over-phi-max will be plus or minus 1.00 in the case of perfect correlation, and these coefficients will be comparable for different variables.
However, the phi-over-phi-max makes a strong assumption that the underlying bivariate distribution in the data is rectangular. Moreover, this coefficient has an increasingly steep approach to 1.00 as the number of cases with the two magnitudes become increasingly disproportionate.
In addition to the phi-over-phi-max coefficient for dichotomous data, the tetrachoric coefficient could be computed. This estimates the value of the product moment correlation, if the dichotomous data are drawn from a normal distribution. The basic assumption of the tetrachoric, therefore, is that the underlying distribution of the data is bivariate normal. However, in contrast to the phi coefficient, the tetrachoric does not have its range affected by unequal data marginals and its significance can be assessed using appropriate tables.
Figure 10.1 may help to make the distinction between pattern and magnitude clear.
A number of pattern-magnitude correlation coefficients have been developed. One that is particularly useful is the intraclass correlation coefficient, which can be applied to any number of variables. Just restricting it to two variables, however, the intraclass divides their variance into two parts. One is the variance of each case on the two variables and is called the within-class variance. If the magnitudes for each case on the two variables are the same, this variance is zero. The second part is the variance of the variable across the cases, which is the between-class variance. The intraclass correlation is then simply the between-class minus the within-class variance as a ratio to the sum of the two kinds of variance. And the significance of this ratio can be assessed.
For two variables, the intraclass will range between -1.00 and +1.00. When it is 1.00, each case has identical magnitudes for each variable--all the variation is across cases. When it is -1.00, all the variation is due to each case having different magnitudes on each variable.
1. I provide more detail with references to specific sources in my Applied Factor Analysis (1970, Section 12.3).
2. Assuming a bivariate normally distributed population, the efficiency of the Spearman and Kendall rank correlation coefficient will be 91 percent that of the product moment.
3. The margins of the fourfold table are the number of cases for each of the two values of a variable. For example, in Table 2.2, the marginals for trade are 10 cases with a low value, 4 cases with a high; for GNP per capita there are 9 cases with low values, 5 with high. Clearly, these marginals are unequal.
In this final chapter I will pull together several aspects of correlation and some pertinent considerations, as well as add a few final comments.
Two things are correlated if they covary positively or negatively. And we have a widely used, mathematically based, coefficient called the product moment for determining this correlation.
This coefficient will describe the correlation between any two variables, regardless of their type of measurement. If, however, the statistical significance of the correlation is of concern, then to assess this significance interval or ratio measurement and normal distributions are assumed.
Alternative correlation coefficients are available for specific purposes and types of data. Moreover, if the data do not meet the assumptions required for assessing the significance of the product moment, then these alternative coefficients may enable significance to be determined.
The product moment, and indeed, most alternative coefficients, measure pattern correlation. Alternatives, especially the intraclass, are available to also measure both pattern at magnitude correlation, however.
Whether describing the data or regarding significance, the correlation coefficient measures linear correlations, i.e., that along a straight line as in Figure 6.2c or Figure 6.3a. Even were observations to fall exactly on a curve as in Figure 6.2a or Figure 6.2b, the product moment or alternative coefficients would show a zero or low linear correlation.
The product moment correlation is sensitive to extreme magnitudes. Extreme cases can have many times the effect of other cases on the correlation coefficient. The correlation coefficient may thus "hang" on a few cases with unusually large or small magnitudes, and data transformation or alternative correlations might be used to avoid this problem.
Errors in the data on two variables can make the correlation coefficient higher or lower than it should be. However, error that is random--uncorrelated among themselves or with the true magnitudes--only depresses the absolute correlation. Thus the correlation coefficient for data with random error understates the real correlation and is thus a conservative measure.
The correlation between two variables may be influenced by other variables. Formulas are available, fortunately, to determine these influences and remove them from the correlation. Therefore, if outside influences are suspected, a high, low, or zero correlation between two variables should not be accepted at face value and partial correlations, holding constant these extraneous influences, can be calculated.
There is a geometry of correlation in terms of vectors that provides an intuitively and heuristically powerful picture of correlation. This is simply the cosine of the angle between two variables of mean-deviation data plotted as vectors.
Correlation between magnitudes on variables for a specific time period (such as power and defense budget for 1955) do not indicate the correlations between these variables over time (such as for power and defense budget by year in the U.S., 1946-75). Similarly, over time correlations do not indicate what the correlations would be for cases at a point in time. That is, for the same variables cross-sectional and time series correlations are independent.
Finally, the correlation coefficient is a useful and potentially powerful tool. It can aid understanding reality, but it is no substitute for insight, reason, and imagination. The correlation coefficient is a flashlight of the mind. It must be turned on and directed by our interests and knowledge; and it can help gratify and illuminate both. But like a flashlight, it can be uselessly turned on in the daytime, used unnecessarily beneath a lamp, employed to search for something in the wrong room, or become a play thing.
Understanding correlations is understanding both the nature of the coefficient and its dependence on human intelligence, intuition, and competence.
1. The logic and details underlying this paragraph are elaborated in Rummel (1972, Chapter 6).
2. The picture of this independence is shown in Rummel (1970, Figure 8.1, p. 201).
Horst, P., Matrix Algebra for Social Scientists. New York: Holt, Rinehart, and Winston, 1963.
Rummel, R. J., Applied Factor Analysis. Evanston, Ill.: Northwestern University Press, 1970.
____________,Dimensions of Nations. Beverly Hills: Sage, 1972.