When interest resides in the bivariate relationship between just two variables or among a small number of variables, researchers typically present their rs within the text of their article. (This reporting strategy is shown in Excerpts 3.2 through 3.5.) When interest centers on the bivariate relationships among many variables, how-ever, the resulting rs are often summarized within a special table called a correlation matrix.
It should be noted that several bivariate correlations can be computed among a set of variables, even for relatively small sets of variables. With eight variables,
Scores on MIDAS School Math subscale were moderately correlated (r .58) with scores on Ohio State Math Achievement Test. . . .
Source: Shearer, C. B., & Darrell A. L. (2009). Exploring the application of multiple intelli-gences theory to career counseling. The Career Development Quarterly, 58(1), 3–13.
The strong correlation (.716) between age at graduation and pre-MBA work experience is reasonable: Older graduates are likely to have more pre-MBA work experience.
Source: Yeaple, R. N., Johnston, M. W., & Whittingham, K. L. (2009). Measuring the economic value of pre-MBA work experience. Journal of Education for Business, 85(1), 13–20.
1In many research studies, the focus is on the difference between means. Beginning in Chapter 10, our dis-cussion of t-test and F-tests show how researchers compare means.
EXCERPTS 3.3–3.5
• (continued)
for example, 28 separate bivariate rs can be computed. With 10 variables, there are 45 rs. In general, the number of bivariate correlations is equal to , where k indicates the number of variables.
In Excerpt 3.6, we see a correlation matrix that summarizes the measured bivariate relationships among six variables. In the study associated with this excerpt, 90 college students took a test of creativity (the Idea Generation Test) and filled out a personality survey that measured each student on the “Big 5” dimension of Open-ness, ConscientiousOpen-ness, Extraversion, AgreeableOpen-ness, and Neuroticism. As you can see, this correlation matrix contains rs arranged in a triangle. Each r indicates the correlation between the two variables that label that r’s row and column. For example, the value of .38 is the correlation between Openness and Agreeableness.
k(k - 1)>2
EXCERPT 3.6
• A Standard Correlation Matrix
TABLE 3 Correlations Between the Idea Generation Test and Personality Dimensions
Test 1 2 3 4 5 6
1. Idea Generation 1
2. Openness .10 1
3. Conscientiousness .04 .34 1
4. Extraversion .22 .02 .08 1
5. Agreeableness .00 .38 .28 .41 1
6. Neuroticism .01 .01 .01 .39 .10 1
Source: Ellwood, S., Pallier, G., Snyder, A., & Gallate, J. (2009). The incubation effect: Hatching a solution? Creativity Research Journal, 21(1), 6–14.
Two things are noteworthy about the correlation matrix shown in Excerpt 3.6.
First, when a row and a column refer to the same variable (as is the case with the top row and the left column, the second row and the second column, etc.), there is a “1” positioned at the intersection of that row and column. Clearly, the correlation of any variable with itself is perfect. Thus, the correlation coefficients (each equal to 1) in the diagonal are not informative; the “meat” of the correlation matrix lies elsewhere.
Second, there are no correlation coefficients above the diagonal. If correla-tions appear there, they would be a mirror image of the rs positioned below the diagonal. The value .10 would appear on the top row in the second column,-.04 would appear on the top row in the third column, and so on. Such rs, if they were put into the correlation matrix, are fully redundant with the rs that already are pre-sent; accordingly, they add nothing.
In Excerpt 3.6, the correlation matrix is set up with the 15 bivariate correla-tion coefficients posicorrela-tioned below the diagonal of 1s. At times, you will come across a correlation matrix in which (1) the values of the correlation coefficients are posi-tioned above rather than below the diagonal, or (2) each diagonal element has either a dash or nothing at all. Such alternative presentations should not cause you any dif-ficulty, because they still contain all possible bivariate correlations that are inter-preted in the same way that we interinter-preted the rs in Excerpt 3.6.
Excerpt 3.7 illustrates how two correlation matrices can be combined into one table. In the study associated with this table, 572 Dutch adults filled out three personality inventories (dealing with self-efficacy, intention, and action planning) and also answered questions about their current and past consumption of fruit. A similar group of 585 individuals did the same thing, except their consumption questions dealt with snacks, not fruit. After collecting the data, the researchers computed, separately for each group, bivariate correlation coefficients among the five variables. Using the note beneath the correlation matrix as a guide, we can look to see how the correlation between any two variables compares across the two groups. The two correlations between current and past consumption were quite similar (0.76 and 0.60). However, the correlations between self-efficacy and consumption were quite different (0.57 for the fruit group; 0.36 for the snack group).
EXCERPT 3.7
• Two Correlation Matrices Shown in a Single Table
TABLE 1 Pearson Correlations between Cognitions, Past Behavior and Current Outcome Behaviorsa,b
1 2 3 4 5
1. Self-efficacy — 0.58 0.40 0.63 0.57
2. Intention 0.38 — 0.48 0.42 0.36
3. Action planning 0.17 0.57 — 0.31 0.33
4. Past fruit/snack consumption 0.42 0.31 0.18 — 0.76
5. Fruit/snack consumption 0.36 0.29 0.22 0.60 —
aAll correlations between variables in the fruit consumption study are depicted above the diagonal; correlations between variables in the snack consumption study are depicted below the diagonal.
bAll correlations are significant at the 0.001 level (two-tailed).
Source: van Osch, L., Beenackers, M., Reubsaet, A., Lechner, L., Candel, M., & de Vries, H.
(2009). Action planning as predictor of health protective and risk behavior: An investigation of fruit and snack consumption. International Journal of Behavior Nutrition and Physical Activity, 69(6), 1–10.
I should point out that researchers sometimes put things in the diagonal other than 1s (as in Excerpt 3.6), dashes (as in Excerpt 3.7), or empty spaces. Occasion-ally, you may see numbers in the diagonal that are not correlation coefficients but rather reliability coefficients. (We consider the topic of reliability in Chapter 4.) Also, you are likely to come across a correlation matrix in which the row or col-umn that contains no correlation coefficients has been deleted. For example, in Excerpt 3.6, the top row (for Test 1) could be eliminated without altering the amount of statistical information being communicated, as could the sixth column. When this is done, correlation coefficients appear in the table’s diagonal. (This practice of eliminating a row or a column, however, never takes place in conjunction with a correlation matrix such as that shown in Excerpt 3.7, wherein two correlation ma-trices are combined into a single table.)