Fisher's r to z transformation spss

Chapter 7 - BIVARIATE PEARSON CORRELATION

7.1 Research Situations Where Pearson’s r Is Used

Pearson’s r is typically used to describe the strength of the linear relationship between two quantitative variables. Often, these two variables are designated X (predictor) and Y (outcome). Pearson’s r has values that range from −1.00 to +1.00. The sign of r provides information about the direction of the relationship between X and Y. A positive correlation indicates that as scores on X increase, scores on Y also tend to increase; a negative correlation indicates that as scores on X increase, scores on Y tend to decrease; and a correlation near 0 indicates that as scores on X increase, scores on Y neither increase nor decrease in a linear manner. As an example, consider the hypothetical data in Figure 7.1 . Suppose that a time-share sales agency pays each employee a base wage of $10,000 per year and, in addition, a commission of $1,500 for each sale that the employee completes. An employee who makes zero sales would earn $10,000; an employee who sold four time-shares would make $10,000 + $1,500 × 4 = $16,000. In other words, for each one-unit increase in the number of timeshares sold (X), there is a $1,500 increase in wages. Figure 7.1 illustrates a perfect linear relationship between number of units sold (X1) and wages in dollars (Y1).

The absolute magnitude of Pearson’s r provides information about the strength of the linear association between scores on X and Y. For values of r close to 0, there is no linear association between X and Y. When r = +1.00, there is a perfect positive linear association ; when r = −1.00, there is a perfect negative linear association . Intermediate values of r correspond to intermediate strength of the relationship. Figures 7.2 through 7.5 show examples of data for which the correlations are r = +.75, r = +.50, r = +.23, and r = .00.

Pearson’s r is a standardized or unit-free index of the strength of the linear relationship between two variables. No matter what units are used to express the scores on the X and Y variables, the possible values of Pearson’s r range from –1 (a perfect negative linear relationship) to +1 (a perfect positive linear relationship). Consider, for example, a correlation between height and weight. Height could be measured in inches, centimeters, or feet; weight could be measured in ounces, pounds, or kilograms. When we correlate scores on height and weight for a given sample of people, the correlation has the same value no matter which of these units are used to measure height and weight. This happens because the scores on X and Y are converted to z scores (i.e., they are converted to unit-free or standardized distances from their means) during the computation of Pearson’s r.

Figure 7.1 Scatter Plot for a Perfect Linear Relationship, r = +1.00 (Y1 = 10,000 + 1,500 × X1; e.g., for X1 = 4, Y1 = 16,000)

Figure 7.2 Scatter Plot for Correlation r = +.75

Figure 7.3 Scatter Plot for Correlation r = +.50

Figure 7.4 Scatter Plot for Correlation r = +.23

Figure 7.5 Scatter Plot for Unrelated Variables With Correlation r = .00

For X and Y to have a perfect correlation of +1, all the X, Y points must lie on a straight line, as shown in Figure 7.1 . Perfect linear relations are rarely seen in real data. When the relationship is perfectly linear, we can make an exact statement about how values of Y change as values of X increase; for Figure 7.1 , we can say that for a 1-unit increase in X1, there is exactly a 1,500-unit increase in Y1. As the strength of the relationship weakens (e.g., r = +.75), we can make only approximate statements about how Y changes for each 1-unit increase in X. In Figure 7.2 , with r = +.75, we can make the (less precise) statement that the mean value of Y2 tends to increase as the value of X2 increases. For example, for relatively low X2 scores (between 30 and 60), the mean score on Y2 is about 15. For relatively high X2 scores (between 80 and 100), the mean score of the Y2 scores is approximately 45. When the correlation is less than 1 in absolute value, we can no longer predict Y scores perfectly from X, but we can predict that the mean of Y will be different for different values of X. In the scatter plot for r = +.75, the points form an elliptical cluster (rather than a straight line). If you look at an individual value of X, you can see that there is a distribution of several different Y values for each X. When we do a correlation analysis, we assume that the amount of change in the Y mean is consistent as we move from X = 1 to X = 2 to X = 3, and so forth; in other words, we assume that X and Y are linearly related.

As r becomes smaller, it becomes difficult to judge whether there is any linear relationship simply from visual examination of the scatter plot. The data in Figure 7.4 illustrate a weak positive correlation (r = +.23). In this graph, it is difficult to see an increase in mean values of Y4 as X4 increases because the changes in the mean of Y across values of X are so small.

Figure 7.6 Scatter Plot for Negative Correlation, r = −.75

Figure 7.5 shows one type of scatter plot for which the correlation is 0; there is no tendency for Y5 scores to be larger at higher values of X5. In this example, scores on X are not related to scores on Y, linearly or nonlinearly. Whether X5 is between 16 and 18, or 18 and 20, or 22 and 24, the mean value of Y is the same (approximately 10). On the other hand, Figure 7.6 illustrates a strong negative linear relationship (r = −.75); in this example, as scores on X6 increase, scores on Y6 tend to decrease.

Pearson correlation is often applied to data collected in nonexperimental studies; because of this, textbooks often remind students that “correlation does not imply causation.” However, it is possible to apply Pearson’s r to data collected in experimental situations. For example, in a psychophysics experiment, a researcher might manipulate a quantitative independent variable (such as the weight of physical objects) and measure a quantitative dependent variable (such as the perceived heaviness of the objects). After scores on both variables are transformed (using power or log transformations), a correlation is calculated to show how perceived heaviness is related to the actual physical weight of the objects; in this example, where the weights of objects are varied by the researcher under controlled conditions, it is possible to make a causal inference based on a large Pearson’s r. The ability to make a causal inference is determined by the nature of the research design, not by the statistic that happens to be used to describe the strength and nature of the relationship between the variables. When data are collected in the context of a carefully controlled experiment, as in the psychophysical research example, a causal inference may be appropriate. However, in many situations where Pearson’s r is reported, the data come from nonexperimental or correlational research designs, and in those situations, causal inferences from correlation coefficients are not warranted.

Despite the inability of nonexperimental studies to provide evidence for making causal inferences, nonexperimental researchers often are interested in the possible existence of causal connections between variables. They often choose particular variables as predictors in correlation analysis because they believe that they might be causal. The presence or absence of a significant statistical association does provide some information: Unless there is a statistical association of some sort between scores on X and Y, it is not plausible to think that these variables are causally related. In other words, the existence of some systematic association between scores on X and Y is a necessary (but not sufficient) condition for making the inference that there is a causal association between X and Y. Significant correlations in nonexperimental research are usually reported merely descriptively, but sometimes the researchers want to show that correlations exist so that they can say that the patterns in their data are consistent with the possible existence of causal connections. It is important, of course, to avoid causal-sounding terminology when the evidence is not strong enough to warrant causal inferences, and so researchers usually limit themselves to saying things such as “X predicted Y” or “X was correlated with Y” when they report data from nonexperimental research.

In some nonexperimental research situations, it makes sense to designate one variable as the predictor and the other variable as the outcome. If scores on X correspond to events earlier in time than scores on Y or if there is reason to think that X might cause Y, then researchers typically use the scores on X as predictors. For example, suppose that X is an assessment of mother/infant attachment made when each participant is a few months old, and Y is an assessment of adult attachment style made when each participant is 18 years old. It would make sense to predict adult attachment at age 18 from infant attachment; it would not make much sense to predict infant attachment from adult attachment. In many nonexperimental studies, the X and Y variables are both assessed at the same point in time, and it is unclear whether X might cause Y, Y might cause X, or whether both X and Y might be causally influenced by other variables. For example, suppose a researcher measures grade point average (GPA) and self-esteem for a group of first-year university students. There is no clear justification for designating one of these variables as a predictor; the choice of which variable to designate as the X or predictor variable in this situation is arbitrary.

7.2 Hypothetical Research Example

As a specific example of a question that can be addressed by looking at a Pearson correlation, consider some survey data collected from 118 university students about their heterosexual dating relationships. The variables in this dataset are described in Table 7.1 ; the scores are in a dataset named love.sav. Only students who were currently involved in a serious dating relationship were included. They provided several kinds of information, including their own gender, partner gender, and a single-item rating of attachment style. They also filled out Sternberg’s Triangular Love Scale (Sternberg, 1997). Based on answers to several questions, total scores were calculated for the degree of intimacy, commitment, and passion felt toward the current relationship partner.

Table 7.1 Description of “Love” Dataset in the File Named love.sav

NOTE: N − 118 college student participants (88 female, 30 male).

Later in the chapter, we will use Pearson’s r to describe the strength of the linear relationship among pairs of these variables and to test whether these correlations are statistically significant. For example, we can ask whether there is a strong positive correlation between scores on intimacy and commitment, as well as between passion and intimacy.

7.3 Assumptions for Pearson’s r

The assumptions that need to be met for Pearson’s r to be an appropriate statistic to describe the relationship between a pair of variables are as follows:

1. Each score on X should be independent of other X scores (and each score on Y should be independent of other Y scores). For further discussion of the assumption of independence among observations and the data collection methods that tend to create problems with this assumption, see Chapter 4 .

2. Scores on both X and Y should be quantitative and normally distributed. Some researchers would state this assumption in an even stronger form: Adherents to strict measurement theory would also require scores on X and Y to be interval/ratio level of measurement. In practice, Pearson’s r is often applied to data that are not interval/ratio level of measurement; for example, the differences between the scores on 5-point rating scales of attitudes probably do not represent exactly equal differences in attitude strength, but it is common practice for researchers to apply Pearson’s r to this type of variable. Harris (2001) summarized arguments about this issue and concluded that it is more important that scores be approximately normally distributed than that the variables satisfy the requirement of true equal interval level of measurement. This does not mean that we should completely ignore issues of level of measurement (see Chapter 1 for further comment on this controversial issue). We can often obtain useful information by applying Pearson’s r even when the data are obtained by using measurement methods that may fall short of the requirements for true equal interval differences between scores; for example, it is common to apply Pearson correlation to scores obtained using 5-point Likert-type rating scales. Pearson’s r can also be applied to data where X or Y (or both) are true dichotomous variables—that is, categorical variables with just two possible values; in this case, it is called a phi coefficient (Φ) . The phi coefficient and other alternate forms of correlation for dichotomous variables are discussed in Chapter 8 .

3. Scores on Y should be linearly related to scores on X. Pearson’s r does not effectively detect curvilinear or nonlinear relationships. An example of a curvilinear relationship between X and Y variables that would not be well described by Pearson’s r appears in Figure 7.7 .

4. X, Y scores should have a bivariate normal distribution. Three-dimensional representations of the bivariate normal distribution were shown in Figures 4.40 and 4.41 , and the appearance of a bivariate normal distribution in a two-dimensional X, Y scatter plot appears in Figure 7.8 . For each value of X, values of Y should be approximately normally distributed. This assumption also implies that there should not be extreme bivariate outliers. Detection of bivariate outliers is discussed in the next section (on preliminary data-screening methods for correlation).

5. Scores on Y should have roughly equal or homogeneous variance across levels of X (and vice versa). Figure 7.9 is an example of data that violate this assumption; the variance of the Y scores tends to be low for small values of X (on the left-hand side of the scatter plot) and high for large values of X (on the right-hand side of the scatter plot).

Figure 7.7 Scatter Plot for Strong Curvilinear Relationship (for These Data, r = .02)

Figure 7.8 Scatter Plot That Shows a Bivariate Normal Distribution for X and Y

SOURCE: www.survo.fi/gallery/010.html

Figure 7.9 Scatter Plot With Heteroscedastic Variance

7.4 Preliminary Data Screening

General guidelines for preliminary data screening were given in Chapter 4 . To assess whether the distributions of scores on X and Y are nearly normal, the researcher can examine a histogram of the scores for each variable. As described in Chapter 4 , most researchers rely on informal visual examination of the distributions to judge normality.

The researcher also needs to examine a bivariate scatter plot of scores on X and Y to assess whether the scores are linearly related, whether the variance of Y scores is roughly uniform across levels of X, and whether there are bivariate outliers. A bivariate outlier is a score that is an unusual combination of X, Y values; it need not be extreme on either X or Y, but in the scatter plot, it lies outside the region where most of the other X, Y points are located. Pearson’s r can be an inaccurate description of the strength of the relationship between X and Y when there are one or several bivariate outliers. As discussed in Chapter 4 , researchers should take note of outliers and make thoughtful decisions about whether to retain, modify, or remove them from the data. Figure 7.10 shows an example of a set of N = 50 data points; when the extreme bivariate outlier is included (as in the upper panel), the correlation between X and Y is +.64; when the correlation is recalculated with this outlier removed (as shown in the scatter plot in the lower panel), the correlation changes to r = −.10. Figure 7.11 shows data for which a single bivariate outlier deflates the value of Pearson’s r; when the circled data point is included, r = +.53; when it is omitted, r = +.86. It is not desirable to have the outcome of a study depend on the behavior represented by a single data point; the existence of this outlier makes it difficult to evaluate the relationship between the X and Y variables. It would be misleading to report a correlation of r = +.64 for the data that appear in Figure 7.10 without including the information that this large positive correlation would be substantially reduced if one bivariate outlier was omitted. In some cases, it may be more appropriate to report the correlation with the outlier omitted.

It is important to examine a scatter plot of the X, Y scores when interpreting a value of r. A scatter plot makes it possible to assess whether violations of assumptions of r make the Pearson’s r value a poor index of relationship; for instance, the scatter plot can reveal a nonlinear relation between X and Y or extreme outliers that have a disproportionate impact on the obtained value of r. When Pearson correlation is close to zero, it can mean that there is no relationship between X and Y, but correlations close to zero can also occur when there is a nonlinear relationship.

For this example, histograms of scores on the two variables (commitment and intimacy) were obtained by selecting the → procedure; SPSS menu selections for this were outlined in Chapter 4 . An optional box in the Histogram dialog window can be checked to obtain a normal curve superimposed on the histogram; this can be helpful in assessment of the distribution shape.

The histograms for commitment and intimacy (shown in Figures 7.12 and 7.13 ) do not show perfect normal distribution shapes; both distributions were skewed. Possible scores on these variables ranged from 15 to 75; most people rated their relationships near the maximum value of 75 points. Thus, there was a ceiling effect such that scores were compressed at the upper end of the distribution. Only a few people rated their relationships low on commitment and intimacy, and these few low scores were clearly separate from the body of the distributions. As described in Chapter 4 , researchers need to take note of outliers and decide whether they should be removed from the data or recoded. However, these are always judgment calls. Some researchers prefer to screen out and remove unusually high or low scores, as these can have a disproportionate influence on the size of the correlation (particularly in small samples). Some researchers (e.g., Tabachnick & Fidell, 2007) routinely recommend the use of transformations (such as logs) to make nonnormal distribution shapes more nearly normal. (It can be informative to “experiment” with the data and see whether the obtained correlation changes very much when outliers are dropped or transformations are applied.) For the analysis presented here, no transformations were applied to make the distribution shapes more nearly normal; the r value was calculated with the outliers included and also with the outliers excluded.

The bivariate scatter plot for self-reported intimacy and commitment (in Figure 7.14 ) shows a positive, linear, and moderate to strong association between scores; that is, persons who reported higher scores on intimacy also reported higher scores on commitment. Although the pattern of data points in Figure 7.14 does not conform perfectly to the ideal bivariate normal distribution shape, this scatter plot does not show any serious problems. X and Y are approximately linearly related; their bivariate distribution is not extremely different from bivariate normal; there are no extreme bivariate outliers; and while the variance of Y is somewhat larger at low values of X than at high values of X, the differences in variance are not large.

Figure 7.10 A Bivariate Outlier That Inflates the Size of r

NOTE: With the bivariate outlier included, Pearson’s r(48) = +. 64, p < .001; with the bivariate outlier removed, Pearson’s r(47) = −.10, not significant.

Figure 7.11 A Bivariate Outlier That Deflates the Size of r

NOTE: With the bivariate outlier included, Pearson’s r(48) = +.532, p < .001; with the bivariate outlier removed, Pearson’s r(47) = +.86, p < .001.

Figure 7.12 Data Screening: Histogram of Scores for Commitment

NOTE: Descriptive statistics: Mean = 66.63, SD = 8.16, N = 118.

Figure 7.13 Data Screening: Histogram of Scores for Intimacy

NOTE: Descriptive statistics: Mean = 68.04, SD = 7.12, N − 118.

Figure 7.14 Scatter Plot for Prediction of Commitment From Intimacy

7.5 Design Issues in Planning Correlation Research

Several of the problems at the end of this chapter use data with very small numbers of cases so that students can easily calculate Pearson’s r by hand or enter the data into SPSS. However, in general, studies that report Pearson’s r should be based on fairly large samples. Pearson’s r is not robust to the effect of extreme outliers, and the impact of outliers is greater when the N of the sample is small. Values of Pearson’s r show relatively large amounts of sampling error across different batches of data, and correlations obtained from small samples often do not replicate well. In addition, fairly large sample sizes are required so that there is adequate statistical power for the detection of differences between different correlations. Because of sampling error, it is not realistic to expect sample correlations to be a good indication of the strength of the relationship between variables in samples smaller than N = 30. When N is less than 30, the size of the correlation can be greatly influenced by just one or two extreme scores. In addition, researchers often want to choose sample sizes large enough to provide adequate statistical power (see Section 7.1 ). It is advisable to have an N of at least 100 for any study where correlations are reported.

It is extremely important to have a reasonably wide range of scores on both the predictor and the outcome variables. In particular, the scores should cover the range of behaviors to which the researcher wishes to generalize. For example, in a study that predicts verbal Scholastic Aptitude Test (VSAT) scores from GPA, a researcher might want to include a wide range of scores on both variables, with VSAT scores ranging from 250 to 800 and GPAs that range from very poor to excellent marks.

A report of a single correlation is not usually regarded as sufficient to be the basis of a thesis or a publishable paper (American Psychological Association, 2001, p. 5). Studies that use Pearson’s r generally include correlations among many variables and may include other analyses. Sometimes researchers report correlations among all possible pairs of variables; this often results in reporting hundreds of correlations in a single paper. This leads to an inflated risk of Type I error. A more thoughtful and systematic approach involving the examination of selected correlations is usually preferable (as discussed in Chapter 1 ). In exploratory studies, statistically significant correlations that are detected by examining dozens or hundreds of tests need to be replicated through cross-validation or new data collection before they can be treated as “findings.”

7.6 Computation of Pearson’s r

The version of the formula for the computation of Pearson’s r 1 that is easiest to understand conceptually is as follows:

where zX = (X − MX)/sX, zY = (Y − MY)/sY, and N = number of cases (number of X, Y pairs of observations).

Alternative versions of this formula are easier to use and give less rounding error when Pearson’s r is calculated by hand. The version of the formula above is more helpful in understanding how the Pearson’s r value can provide information about the spatial distribution of X, Y data points in a scatter plot. This conceptual formula can be used for by-hand computation; it corresponds to the following operations. First, each X and Y score is converted to a standard score or z score; then, for each participant, zX is multiplied by zY; these products are summed across all participants; and finally, this sum is divided by the number of participants. The resulting value of r falls within the range −1.00 ≤ r ≤ +1.00.

Because we convert X and Y to standardized or z scores, the value of r does not depend on the units that were used to measure these variables. If we take a group of subjects and express their heights in both inches (X1) and centimeters (X2) and their weights in pounds (Y1) and kilograms (Y2), the correlation between X1, Y1 and between X2, Y2 will be identical. In both cases, once we convert height to a z score, we are expressing the individual’s height in terms of a unit-free distance from the mean. A person’s z score for height will be the same whether we work with height in inches, feet, or centimeters.

Another formula for Pearson’s r is based on the covariance between X and Y:

where MX is the mean of the X scores, MY is the mean of the Y scores, and N is the number of X, Y pairs of scores.

Note that the variance of X is equivalent to the covariance of X with itself:

Pearson’s r can be calculated from the covariance of X with Y as follows:

A covariance, 2 like a variance, is an arbitrarily large number; its size depends on the units used to measure the X and Y variables. For example, suppose a researcher wants to assess the relation between height (X) and weight (Y). These can be measured in many different units: Height can be given in inches, feet, meters, or centimeters, and weight can be given in terms of pounds, ounces, kilograms, or grams. If height is stated in inches and weight in ounces, the numerical scores given to most people will be large and the covariance will turn out to be very large. However, if heights are given in feet and weights in pounds, both the scores and the covariances between scores will be smaller values. Covariance, thus, depends on the units of measurement the researcher happened to use. This can make interpretation of covariance difficult, particularly in situations where the units of measurement are arbitrary.

Pearson correlation can be understood as a standardized covariance: The values of r fall within a fixed range from –1 to +1, and the size of r does not depend on the units of measurement the researcher happened to use for the variables. Whether height was measured in inches, feet, or meters, when the height scores are converted to standard or z scores, information about the units of measurement is lost. Because correlation is standardized, it is easier to interpret, and it is possible to set up some verbal guidelines to describe the sizes of correlations.

Table 7.2 Computation of Pearson’s r for a Set of Scores on Heart Rate (HR) and Self-Reported Tension

NOTE: ∑(zX × zY) = 7.41, Pearson’s r = ∑(zX × zY)/(N = 1) = +7.41/9 = .823

Here is a numerical example that shows the computation of Pearson’s r for a small dataset that contains N = 10 pairs of scores on heart rate (HR) and self-reported tension (see Table 7.2 ).

The first two columns of this table contain the original scores for the variables HR and tension. The next two columns contain the z scores for each variable, zHR and ztension (these z scores or standard scores can be saved as output from the SPSS Descriptive Statistics procedure). For this example, HR is the Y variable and tension is the X variable. The final column contains the product of zX and zY for each case, with ∑(zX × zY) at the bottom. Finally, Pearson’s r was obtained by taking ∑(zX × zY)/(N − 1) = + 7.41/9 = .823. (The values of r reported by SPSS use N − 1 in the divisor rather than N as in most textbook formulas for Pearson’s r. When N is large, for example, N greater than 100, the results do not differ much whether N or N − 1 is used as the divisor.) This value of r agrees with the value obtained by running the SPSS bivariate correlation/Pearson’s r procedure on the data that appear in Table 7.2 .

7.7 Statistical Significance Tests for Pearson’s r

7.7.1 Testing the Hypothesis That ρXY = 0

The most common statistical significance test is for the statistical significance of an individual correlation. The population value of the correlation between X and Y is denoted by the Greek letter rho (ρ). Given an obtained sample r between X and Y, we can test the null hypothesis that ρXY in the population equals 0. The formal null hypothesis that corresponds to the lack of a (linear) relationship between X and Y is

When the population correlation ρXY is 0, the sampling distribution of rs is shaped like a normal distribution (for large N) or a t distribution with N – 2 df (for small N), except that the tails are not infinite (the tails end at +1 and −1); see the top panel in Figure 7.15 . That is, when the true population correlation ρ is actually 0, most sample rs tend to be close to 0; the sample rs tend to be normally distributed, but the tails of this distribution are not infinite (as they are for a true normal distribution), because sample correlations cannot be outside the range of −1 to +1. Because the sampling distribution for this situation is roughly that of a normal or t distribution, a t ratio to test this null hypothesis can be set up as follows:

Figure 7.15 Sampling Distributions for r With N = 12

The value of SEr is given by the following equation:

Substituting this value of SEr from Equation 7.7 into Equation 7.6 and rearranging the terms yields the most widely used formula for a t test for the significance of a sample r value; this t test has N − 2 degrees of freedom (df); the hypothesized value of ρ0 is 0.

It is also possible to set up an F ratio, with (1, N − 2) df, to test the significance of sample r. This F is equivalent to t2; it has the following form:

Programs such as SPSS provide an exact p value for each sample correlation (a two-tailed p value by default; a one-tailed p value can be requested). Critical values of the t and F distributions are provided in Appendixes B and C . It is also possible to look up whether r is statistically significant as a function of degrees of freedom and the r value itself directly in the table in Appendix E (without having to calculate t or F).

7.7.2 Testing Other Hypotheses About ρXY

It is uncommon to test null hypotheses about other specific hypothesized values of ρXY (such as H0: ρXY = .90). For this type of null hypothesis, the sampling distribution of r is not symmetrical and therefore cannot be approximated by a t distribution. For example, if ρXY = .90, then most sample rs will be close to .90; sample rs will be limited to the range from –1 to +1, so the sampling distribution will be extremely skewed (see the bottom panel in Figure 7.15 ). To correct for this nonnormal distribution shape, a data transformation is applied to r before testing hypotheses about nonzero hypothesized values of ρ. The Fisher r to Z transformation rescales sample rs in a way that yields a more nearly normal distribution shape, which can be used for hypothesis testing. (Note that in this book, lowercase z always refers to a standard score; uppercase Z refers to the Fisher Z transformation based on r. Some books label the Fisher Z using a lowercase z or z′.) The r to Fisher Z transformation is shown in Table 7.3 (for reference, it is also included in Appendix G at the end of this book).

The value of Fisher Z that corresponds to a sample Pearson’s r is usually obtained by table lookup, although Fisher Z can be obtained from this formula:

Table 7.3 Transformation of Pearson’s r to Fisher Z

SOURCE: Adapted from Lane (2001).

A Fisher Z value can also be converted back into an r value by using Table 7.3 .

For the Fisher Z, the standard error (SE) does not depend on ρ but only on N; the sampling distribution of Fisher Z scores has this standard error:

Thus, to test the null hypothesis,

With N = 28 and an observed sample r of .8, the researcher needs to do the following:

1. Convert ρhyp to a corresponding Fisher Z value, Zhyp, by looking up the Z value in Table 7.3 . For ρhyp = .90, Zhyp = 1.472.2.

2. Convert the observed sample r (rsample) to a Fisher Z value (Zsample) by looking up the corresponding Fisher Z value in Table 7.3 . For an observed r of .80, Zsample = 1.099.

3. Calculate SEZ from Equation 7.11 :

4. Compute the z ratio as follows:

z = (Zsample – Zhyp)/SEZ = (1.099 – 1.472)/.20 = –1.865.

5. For α = .05, two-tailed, the reject region for a z test is z > 1.96 and z < −1.96; therefore, do not reject the null hypothesis that ρ = .90.

Following sections discuss the ways in which the Fisher Z transformation is also used when testing the null hypothesis that the value of ρ is equal between two different populations, H0: ρ1 = ρ2. For example, we can test whether the correlation between X and Y is significantly different for women versus men. Fisher Z is also used to set up confidence intervals (CIs) for correlation estimates.

7.7.3 Assessing Differences Between Correlations

It can be quite problematic to compare correlations that are based on different samples or populations, or that involve different variables, because so many factors can artifactually influence the size of Pearson’s r (many of these factors are discussed in Section 7.9 ). For example, suppose a researcher wants to evaluate whether the correlation between emotional intelligence (EI) and drug use is stronger for males than for females. If the scores on drug use have a much more restricted range in the female sample than in the male sample, this restricted range of scores in the female sample might make the correlation between these variables smaller for females. If the measurement of drug use has lower reliability for females than for males, this difference in reliability could also artifactually reduce the magnitude of the correlation between EI and drug use in the female sample. If two correlations differ significantly, this difference might arise due to artifact (such as a narrower range of scores used to compute one r) rather than because of a difference in the true strength of the relationship. Researchers have to be very cautious when comparing correlations, and they should acknowledge possible artifacts that might have led to different r values (sources of artifacts are discussed in Section 7.9 ). For further discussion of problems with comparisons of correlations and other standardized coefficients to make inferences about differences in effect sizes across populations, see Greenland, Maclure, Schlesselman, Poole, and Morgenstern (1991) and Greenland, Schlesselman, and Criqui (1986).

It is useful to have statistical significance tests for comparison of correlations; these at least help to answer whether the difference between a pair of correlations is so small that it could very likely be due to sampling error. Obtaining statistical significance is a necessary, but not a sufficient, condition for concluding that a genuine difference in the strength of relationship is present. Two types of comparisons between correlations are described here.

In the first case, the test compares the strength of the correlation between the same two variables in two different groups or populations. Suppose that the same set of variables (such as X = EI and Y = drug abuse or DA) is correlated in two different groups of participants (Group 1 = males, Group 2 = females). We might ask whether the correlation between EI and DA is significantly different for men versus women. The corresponding null hypothesis is

To test this hypothesis, the Fisher Z transformation has to be applied to both sample r values. Let r1 be the sample correlation between EI and DA for males and r2 the sample correlation between EI and DA for females; N1 and N2 are the numbers of participants in the male and female groups, respectively.

First, using Table 7.3 , look up the Z1 value that corresponds to r1 and the Z2 value that corresponds to r2.

Next, apply the following formula:

The test statistic z is evaluated using the standard normal distribution; if the obtained z ratio is greater than +1.96 or less than –1.96, then the correlations r1 and r2 are judged significantly different using α = .05, two-tailed. This test should be used only when the N in each sample is fairly large, preferably N > 100.

A second situation of interest involves comparison of two different predictor variables. Suppose the researcher wants to know whether the correlation of X with Z is significantly different from the correlation of Y with Z. The corresponding null hypothesis is

This test does not involve the use of Fisher Z transformations. Instead, we need to have all three possible bivariate correlations (rXZ, rYZ, and rXY); N = total number of participants. The test statistic (from Lindeman, Merenda, & Gold, 1980) is a t ratio of this form:

The resulting t value is evaluated using critical values from the t distribution with (N − 3) df. Even if a pair of correlations is judged to be statistically significantly different using these tests, the researcher should be very cautious about interpreting this result. Different size correlations could arise because of differences across populations or across predictors in factors that affect the size of r discussed in Section 7.9 , such as range of scores, reliability of measurement, outliers, and so forth.

7.7.4 Reporting Many Correlations: Need to Control Inflated Risk of Type I Error

Journal articles rarely report just a single Pearson’s r; in fact, the Publication Manual of the American Psychological Association (American Psychological Association, 2001) states that this is not sufficient for a reportable study. Unfortunately, however, many studies report such a large number of correlations that evaluation of statistical significance becomes problematic. Suppose that k = 20 variables are measured in a nonexperimental study. If the researcher does all possible bivariate correlations, there will be k × (k − 1)/2 different correlations, in this case (20 × 19)/2 = 190 correlations. When we set our risk of Type I error at α = .05 for the statistical significance test for each individual correlation, this implies that out of 100 statistical significance tests that are done (on data from populations that really have no relationship between the X and Y variables), about 5 tests will be instances of Type I error (rejection of H0 when H0 is true). When a journal article reports 200 correlations, for example, one would expect that about 5% of these (10 correlations) should be statistically significant using the α = .05 significance level, even if the data were completely random. Thus, of the 200 correlations, it is very likely that at least some of the significance tests (on the order of 9 or 10) will be instances of Type I error. If the researcher runs 200 correlations and finds that the majority of them (say, 150 out of 200) are significant, then it seems likely that at least some of these correlations are not merely artifacts of chance. However, if a researcher reports 200 correlations and only 10 are significant, then it’s quite possible that the researcher has found nothing beyond the expected number of Type I errors. It is even more problematic for the reader when it’s not clear how many correlations were run; if a researcher runs 200 correlations, hand selects 10 statistically significant rs after the fact, and then reports only the 10 rs that were judged to be significant, it is extremely misleading to the reader, who is no longer able to evaluate the true magnitude of the risk of Type I error.

In general, it is common in exploratory nonexperimental research to run large numbers of significance tests; this inevitably leads to an inflated risk of Type I error. That is, the probability that the entire research report contains at least one instance of Type I error is much higher than the nominal risk of α = .05 that is used for any single significance test. There are several possible ways to deal with this problem of inflated risk of Type I error.

7.7.4.1 Limiting the Number of Correlations

One approach is to limit the number of correlations that will be examined at the outset, before looking at the data, based on theoretical assumptions about which predictive relations are of interest. The possible drawback of this approach is that it may preclude serendipitous discoveries. Sometimes, unexpected observed correlations do point to relationships among variables that were not anticipated from theory but that can be confirmed in subsequent replications.

7.7.4.2 Cross-Validation of Correlations

A second approach is cross-validation. In a cross-validation study, the researcher randomly divides the data into two batches; thus, if the entire study had data for N = 500 participants, each batch would contain 250 cases. The researcher then does extensive exploratory analysis on the first batch of data and decides on a limited number of correlations or predictive equations that seem to be interesting and useful. Then, the researcher reruns this small set of correlations on the second half of the data. If the relations between variables remain significant in this fresh batch of data, it is less likely that these relationships were just instances of Type I error. The main problem with this approach is that researchers often don’t have large enough numbers of cases to make this possible.

7.7.4.3 Bonferroni Procedure: A More Conservative Alpha Level for Tests of Individual Correlations

A third approach is the Bonferroni procedure. Suppose that the researcher plans to do k = 10 correlations and wants to have an experiment-wise alpha (EWα) of .05. To keep the risk of obtaining at least one Type I error as low as 5% for a set of k = 10 significance tests, it is necessary to set the per-comparison alpha (PCα) level lower for each individual test. Using the Bonferroni procedure, the PCα used to test the significance of each individual r value is set at EWα/k; for example, if EWα = .05 and k = 10, each individual correlation has to have an observed p value less than .05/10, or .005, to be judged statistically significant. The main drawback of this approach is that it is quite conservative. Sometimes the number of correlations that are tested in exploratory studies is quite large (100 or 200 correlations have been reported in some recent papers). If the error rate were adjusted by dividing .05 by 100, the resulting PCα would be so low that it would almost never be possible to judge individual correlations significant. Sometimes the experiment-wise α for the Bonferroni test is set higher than .05; for example, EWα = .10 or .20.

If the researcher does not try to limit the risk of Type I error in any of these three ways (by limiting the number of significance tests, doing a cross-validation, or using a Bonferroni-type correction for alpha levels), then at the very least, the researcher should explain in the write-up that the p values that are reported probably underestimate the true overall risk of Type I error. In these situations, the Discussion section of the paper should reiterate that the study is exploratory, that relationships detected by running large numbers of significance tests are likely to include large numbers of Type I errors, and that replications of the correlations with new samples are needed before researchers can be confident that the relationships are not simply due to chance or sampling error.

7.8 Setting Up CIs for Correlations

If the researcher wants to set up a CI using a sample correlation, he or she must use the Fisher Z transformation (in cases where r is not equal to 0). The upper and lower bounds of the 95% CI can be calculated by applying the usual formula for a CI to the Fisher Z values that correspond to the sample r.

The general formula for a CI is as follows:

To set up a CI around a sample r value (let r = .50, with N = 43, for example), first, look up the Fisher Z value that corresponds to r = .50; from Table 7.3 , this is Fisher Z = .549. For N = 43 and df − N − 3 = 40,

For N = 43 and a 95% CI, tcrit is approximately equal to +2.02 for the top 2.5% of the distribution. The Fisher Z values and the critical values of t are substituted into the equations for the lower and upper bounds:

Lower bound of 95% CI = Fisher Z − 2.02 × SEZ = .549 − 2.02 × .157 = .232, Upper bound of 95% CI = Fisher Z + 2.02 × SEZ = .549 + 2.02 × .157 = .866.

Table 7.3 is used to transform these boundaries given in terms of Fisher Z back into estimated correlation values:

Fisher Z = .232 is equivalent to r = .23, Fisher Z = .866 is equivalent to r = .70.

Thus, if a researcher obtains a sample r of .50 with N = 43, the 95% CI is from .23 to .70. If this CI does not include 0, then the sample r would be judged statistically significant using α = .05, two-tailed. The SPSS Bivariate Correlation procedure does not provide CIs for r values, but these can easily be calculated by hand.

7.9 Factors That Influence the Magnitude and Sign of Pearson’s r

7.9.1 Pattern of Data Points in the X, Y Scatter Plot

To understand how the formula for correlation can provide information about the location of points in a scatter plot and how it detects a tendency for high scores on Y to co-occur with high or low scores on X, it is helpful to look at the arrangement of points in an X, Y scatter plot (see Figure 7.16 ). Consider what happens when the scatter plot is divided into four quadrants or regions: scores that are above and below the mean on X and scores that are above and below the mean on Y.

The data points in Regions II and III are cases that are “concordant”; these are cases for which high X scores were associated with high Y scores or low X scores were paired with low Y scores. In Region II, both zX and zY are positive, and their product is also positive; in Region III, both zX and zY are negative, so their product is also positive. If most of the data points fall in Regions II and/or III, it follows that most of the contributions to the ∑ (zX × zY) sum of products will be positive, and the correlation will tend to be large and positive.

Figure 7.16 X, Y Scatter Plot Divided Into Quadrants (Above and Below the Means on X and Y)

The data points in Regions I and IV are “discordant” because these are cases where high X went with low Y and/or low X went with high Y. In Region I, zX is negative and zY is positive; in Region IV, zX is positive and zY is negative. This means that the product of zX and zY for each point that falls in Region I or IV is negative. If there are a large number of data points in Region I and/or IV, then most of the contributions to ∑(zX × zY) will be negative, and r will tend to be negative.

If the data points are about evenly distributed among the four regions, then positive and negative values of zX × zY will be about equally common, they will tend to cancel each other out when summed, and the overall correlation will be close to zero. This can happen because X and Y are unrelated (as in Figure 7.5 ) or in situations where there is a strongly curvilinear relationship (as in Figure 7.7 ). In either of these situations, high X scores are associated with high Y scores about as often as high X scores are associated with low Y scores.

Note that any time a statistical formula includes a product between variables of the form ∑(X × Y) or ∑(zX × zY), the computation provides information about correlation or covariance. These products summarize information about the spatial arrangement of X, Y data points in the scatter plot; these summed products tend to be large and positive when most of the data points are in the upper right and lower left (concordant) areas of the scatter plot. In general, formulas that include sums such as ∑X or ∑Y provide information about the means of variables (just divide by N to get the mean). Terms that involve ∑X2 or ∑Y2 provide information about variability. Awareness about the information that these terms provide makes it possible to decode the kinds of information that are included in more complex computational formulas. Any time a ∑(X × Y) term appears, one of the elements of information included in the computation is covariance or correlation between X and Y.

Correlations provide imperfect information about the “true” strength of predictive relationships between variables. Many characteristics of the data, such as restricted ranges of scores, nonnormal distribution shape, outliers, and low reliability, can lead to over- or underestimation of the correlations between variables. Correlations and covariances provide the basic information for many other multivariate analyses (such as multiple regression and multivariate analysis of variance). It follows that artifacts that influence the values of sample correlations and covariances will also affect the results of other multivariate analyses. It is therefore extremely important for researchers to understand how characteristics of the data, such as restricted range, outliers, or measurement unreliability, influence the size of Pearson’s r, for these aspects of the data also influence the sizes of regression coefficients, factor loadings , and other coefficients used in multivariate models.

7.9.2 Biased Sample Selection: Restricted Range or Extreme Groups

The ranges of scores on the X and Y variables can influence the size of the sample correlation. If the research goal is to estimate the true strength of the correlation between X and Y variables for some population of interest, then the ideal sample should be randomly selected from the population of interest and should have distributions of scores on both X and Y that are representative of, or similar to, the population of interest. That is, the mean, variance, and distribution shape of scores in the sample should be similar to the population mean, variance, and distribution shape.

Suppose that the researcher wants to assess the correlation between GPA and VSAT scores. If data are obtained for a random sample of many students from a large high school with a wide range of student abilities, scores on GPA and VSAT are likely to have wide ranges (GPA from about 0 to 4.0, VSAT from about 250 to 800). See Figure 7.17 for hypothetical data that show a wide range of scores on both variables. In this example, when a wide range of scores are included, the sample correlation between VSAT and GPA is fairly high (r = +.61).

However, samples are sometimes not representative of the population of interest; because of accidentally biased or intentionally selective recruitment of participants, the distribution of scores in a sample may differ from the distribution of scores in the population of interest. Some sampling methods result in a restricted range of scores (on X or Y or both variables). Suppose that the researcher obtains a convenience sample by using scores for a class of honors students. Within this subgroup, the range of scores on GPA may be quite restricted (3.3 – 4.0), and the range of scores on VSAT may also be rather restricted (640 – 800). Within this subgroup, the correlation between GPA and VSAT scores will tend to be smaller than the correlation in the entire high school, as an artifact of restricted range. Figure 7.18 shows the subset of scores from Figure 7.17 that includes only cases with GPAs greater than 3.3 and VSAT scores greater than 640. For this group, which has a restricted range of scores on both variables, the correlation between GPA and VSAT scores drops to +.34. It is more difficult to predict a 40- to 60-point difference in VSAT scores from a .2- or .3-point difference in GPA for the relatively homogeneous group of honors students, whose data are shown in Figure 7.18 , than to predict the 300- to 400-point differences in VSAT scores from 2- or 3-point differences in GPA in the more diverse sample shown in Figure 7.17 .

In general, when planning studies, researchers should try to include a reasonably wide range of scores on both predictor and outcome variables. They should also try to include the entire range of scores about which they want to be able to make inferences because it is risky to extrapolate correlations beyond the range of scores for which you have data. For example, if you show that there is only a small correlation between age and blood pressure for a sample of participants with ages up to 40 years, you cannot safely assume that the association between age and blood pressure remains weak for ages of 50, 60, and 80 (for which you have no data). Even if you find a strong linear relation between two variables in your sample, you cannot assume that this relation can be extrapolated beyond the range of X and Y scores for which you have data (or, for that matter, to different types of research participants).

Figure 7.17 Correlation Between Grade Point Average (GPA) and Verbal Scholastic Aptitude Test (VSAT) Scores in Data With Unrestricted Range (r = +.61)

Figure 7.18 Correlation Between Grade Point Average (GPA) and Verbal Scholastic Aptitude Test (VSAT) Scores in a Subset of Data With Restricted Range (Pearson’s r = +.34)

NOTE: This is the subset of the data in Figure 7.17 for which GPA > 3.3 and VSAT > 640.

A different type of bias in correlation estimates occurs when a researcher purposefully selects groups that are extreme on both X and Y variables. This is sometimes done in early stages of research in an attempt to ensure that a relationship can be detected. Figure 7.19 illustrates the data for GPA and VSAT for two extreme groups selected from the larger batch of data in Figure 7.17 (honors students vs. failing students). The correlation between GPA and VSAT scores for this sample that comprises two extreme groups was r = +.93. Pearson’s r obtained for samples that are formed by looking only at extreme groups tends to be much higher than the correlation for the entire range of scores. When extreme groups are used, the researcher should note that the correlation for this type of data typically overestimates the correlation that would be found in a sample that included the entire range of possible scores. Examination of extreme groups can be legitimate in early stages of research, as long as researchers understand that the correlations obtained from such samples do not describe the strength of relationship for the entire range of scores.

Figure 7.19 Correlation Between Grade Point Average (GPA) and Verbal Scholastic Aptitude Test (VSAT) Scores Based on Extreme Groups (Pearson’s r = +.93)

NOTE: Two subsets of the data in Figure 7.17 (low group, GPA < 1.8 and VSAT < 400; high group, GPA > 3.3 and VSAT > 640).

7.9.3 Correlations for Samples That Combine Groups

It is important to realize that a correlation between two variables (for instance, X = EI and Y = drug use) may be different for different types of people. For example, Brackett et al. (2004) found that EI was significantly predictive of illicit drug use behavior for males but not for females (men with higher EI engaged in less drug use). The scatter plot (of hypothetical data) in Figure 7.20 illustrates a similar but stronger interaction effect —“different slopes for different folks.” In Figure 7.20 , there is a fairly strong negative correlation between EI and drug use for males; scores for males appear as triangular markers in Figure 7.20 . In other words, there was a tendency for males with higher EI to use drugs less. For women (data points shown as circular markers in Figure 7.20 ), drug use and EI were not significantly correlated. The gender differences shown in this graph are somewhat exaggerated (compared with the actual gender differences Brackett et al. [2004] found in their data), to make it clear that there were differences in the correlation between EI and drugs for these two groups (women vs. men).

A spurious correlation can also arise as an artifact of between-group differences. The hypothetical data shown in Figure 7.21 show a positive correlation between height and violent behavior for a sample that includes both male and female participants (r = +.687). Note that the overall negative correlation between height and violence occurred because women were low on both height and violence compared with men; the apparent correlation between height and violence is an artifact that arises because of gender differences on both the variables. Within the male and female groups, there was no significant correlation between height and violence (r = −.045 for males, r = −.066 for females, both not significant). A spurious correlation between height and violence arose when these two groups were lumped together into one analysis that did not take gender into account.

In either of these two research situations, it can be quite misleading to look at a correlation for a batch of data that mixes two or several different kinds of participants’ data together. It may be necessary to compute correlations separately within each group (separately for males and for females, in this example) to assess whether the variables are really related and, if so, whether the nature of the relationship differs within various subgroups in your data.

7.9.4 Control of Extraneous Variables

Chapter 10 describes ways of statistically controlling for other variables that may influence the correlation between an X and Y pair of variables. For example, one simple way to “control for” gender is to calculate X, Y correlations separately for the male and female groups of participants. When one or more additional variables are statistically controlled, the size of the X, Y correlation can change in any of the following ways: It may become larger or smaller, change sign, drop to zero, or remain the same. It is rarely sufficient in research to look at a single bivariate correlation in isolation; it is often necessary to take other variables into account to see how these affect the nature of the X, Y relationship. Thus, another factor that influences the sizes of correlations between X and Y is the set of other variables that are controlled, either through statistical control (in the data analysis) or through experimental control (by holding some variables constant, for example).

Figure 7.20 Scatter Plot for Interaction Between Gender and Emotional Intelligence (EI) as Predictors of Drug Use: “Different Slopes for Different Folks”

NOTE: Correlation between EI and drug use for entire sample is r(248) = −.60, p < .001; correlation within female subgroup (circular markers) is r(112) = −.11, not significant; correlation within male subgroup (triangular markers) is r(134) = −.73, p < .001.

7.9.5 Disproportionate Influence by Bivariate Outliers
Like the sample mean (M), Pearson’s r is not robust against the influence of outliers. A single bivariate outlier can lead to either gross overestimation or gross underestimation of the value of Pearson’s r (refer to Figures 7.10 and 7.11 for visual examples). Sometimes bivariate outliers arise due to errors in data entry, but they can also be valid scores that are unusual combinations of values (it would be unusual to find a person with a height of 72 in. and a weight of 100 lb, for example). Particularly in relatively small samples, a single unusual data value can have a disproportionate impact on the estimate of the correlation. For example, in Figure 7.10, if the outlier in the upper right-hand corner of the scatter plot is included, r = +.64; if that outlier is deleted, r drops to −.10. It is not desirable to have the outcome of a study hinge on the scores of just one or a few unusual participants. An outlier can either inflate the size of the sample correlation (as in Figure 7.10) or deflate it (as in Figure 7.11). For Figure 7.11, the r is +.532 if the outlier in the lower right-hand corner is included in the computation of r; the r value increases to +.86 if this bivariate outlier is omitted.

Figure 7.21 Scatter Plot of a Spurious Correlation Between Height and Violence (Due to Gender Differences)

NOTE: For entire sample, r(248) = +.687, p < .001; male subgroup only, r(134) = −.045, not significant; female subgroup only, r(112) = −.066, not significant.