RES 710 Week 6 Correlation Example

How Closely Do Waits and Satisfaction Move Together? A Correlation Analysis

[Student Name]

University of Phoenix

RES/710: Statistical Research Methods and Design I

Week 6 Assignment

[Instructor Name]

[Date]

The learner, the credit union, its members and all data are composites written for a model paper.

Week 5 compared branch members who waited more or less than 10 minutes and found a sizable gap in satisfaction. Splitting wait time at one point, though, discards most of its information. This paper treats wait as continuous and asks how strongly it is associated with satisfaction across the full range, for the 992 members of the composite Grand Rapids credit union who had a recorded wait.

Start with the Plot

Anscombe (1973) constructed four small data sets with nearly identical means, variances and correlation coefficients that looked completely different when plotted: one linear, one curved, one driven by a single outlier and one with no relationship except for one extreme point. The point was that a coefficient alone can mislead and that graphs are essential.

The scatterplot of satisfaction against recorded wait shows a downward pattern that is steep across the first 10 minutes and flattens afterward. The cloud widens at short waits, since members with short waits vary widely in satisfaction for other reasons. A few members waited over 40 minutes during the outages identified in Week 2.

Choosing a Coefficient

Pearson's r measures the strength of a linear relationship between two interval or ratio variables. Spearman's rho applies Pearson's formula to ranks and so measures any consistently increasing or decreasing relationship. Schober et al. (2018) explain that coefficients should be chosen by the type of data and the shape of the relationship and that common verbal labels for strength are arbitrary; interpretation should consider the field and the stakes.

Bishara and Hittner (2012) compared approaches for testing correlations with nonnormal data through simulation and found that Pearson's r on raw skewed data could produce inflated false positive rates and that Spearman's rho or a transformation toward normality generally performed better. Because wait time is strongly skewed, the learner compares three options.

Results

Pearson's r, raw waits: r = -.34, p < .001.

Pearson's r, log of wait time: r = -.41, p < .001, 95% CI [-.46, -.36].

Spearman's rho: rho = -.40, p < .001.

The log transformation straightens the curved relationship, which is why its coefficient is stronger than the raw value and close to Spearman's. The learner reports the log-based Pearson coefficient as primary, with rho as confirmation. The confidence interval uses Fisher's z transformation, which converts r to an approximately normal scale, computes the interval there and converts back.

Strength and Shared Variance

A correlation of -.41 means that longer waits are moderately associated with lower satisfaction. Squaring it gives .17: about 17 percent of the variation in satisfaction is shared with log wait time in a linear relationship. That leaves 83 percent associated with other factors, such as the transaction's outcome, staff courtesy or members' general attitudes. Wait time matters, but it is far from the whole story.

Waiting explains about a sixth of the variation in satisfaction: enough to act on, not enough to blame for everything.

Perceived Versus Recorded Wait

Week 5 found that members perceive waits as longer than recorded. Perceived wait is even more strongly related to satisfaction: rho = -.52, compared with -.40 for recorded wait. Squared, the shared variance rises from about 16 percent to 27 percent. This pattern is consistent with the idea that how a wait feels matters more than its length. It could also reflect the direction running the other way: members unhappy with a visit for other reasons may remember the wait as longer.

Are the Two Correlations Really Different?

Because recorded and perceived wait come from the same members, the gap between -.40 and -.52 cannot be judged by looking at the two intervals separately. A test for dependent correlations, which accounts for the correlation of .58 between the two wait measures, showed that perceived wait's association with satisfaction is significantly stronger, p < .001. The learner reports this test rather than simply noting that one number is larger.

Reporting the Results

In APA style, the primary result reads: log wait time was negatively correlated with satisfaction, r(990) = -.41, p < .001, 95% CI [-.46, -.36]. Degrees of freedom for a correlation are the sample size minus two. A correlation matrix in the appendix lists all pairs among wait, perceived wait, satisfaction, net promoter likelihood, age and tenure, with Spearman coefficients for the skewed and ordinal variables.

Controlling for a Third Variable

Older members wait longer, since they use branches at busier times, and they also rate satisfaction differently. A partial correlation measures the relationship between two variables after removing the linear influence of a third. The partial correlation of log wait with satisfaction, controlling for age, was -.39, barely changed from -.41. Age is therefore not producing the relationship, though other variables, such as branch staffing, might.

Restricted Range and Outliers

Correlation depends on the range of values in the sample. Within the 645 branch members, the correlation is -.43; within phone members, whose waits are shorter and less variable, it is -.29. That difference could partly reflect the narrower range of phone waits rather than a weaker underlying relationship. Removing the 12 outage waits changed the log-based coefficient from -.41 to -.40, so those cases do not drive the result.

Why This Is Not Causation

The correlation does not show that shortening waits would raise satisfaction. Three alternatives remain. Branches with long waits may also be understaffed in ways that affect courtesy and accuracy, so staffing could drive both. Members with complicated transactions wait longer and are more often frustrated by the transaction itself. And members who are dissatisfied may overstate how long they waited. An experiment, such as adding staff at randomly chosen branches during peak hours, could test cause directly; the learner notes this as a possible pilot for the credit union.

A First Look at Prediction

A simple regression line fitted to raw waits estimates that satisfaction declines by about 0.06 points for each additional minute, or 0.6 points for 10 extra minutes. Because the relationship is curved, the decline is steeper over the first few minutes than this average slope suggests. RES 720 will build multiple regression models that include channel, branch, age and transaction type together and test whether wait time's association holds once these are considered.

What the Results Mean for the Research Question

For the credit union, the results support the branch managers' view that waits are linked to satisfaction, with the steepest losses early in a wait. They also suggest that improving how waits feel, through information about expected wait and comfortable seating, may help as much as reducing minutes. These are hypotheses for the dissertation's later analyses and possibly a field test.

Conclusion

Plotting first revealed a curved, fan-shaped relationship; a log transformation and Spearman's rho gave consistent moderate negative correlations around -.40; perceived wait was more strongly related than recorded wait; and controlling for age left the relationship intact. Correlation established association, not cause. Week 7 will turn to appraising published quantitative studies.

References

Anscombe, F. J. (1973). Graphs in statistical analysis. The American Statistician, 27(1), 17-21. https://doi.org/10.1080/00031305.1973.10478966

Bishara, A. J., & Hittner, J. B. (2012). Testing the significance of a correlation with nonnormal data: Comparison of Pearson, Spearman, transformation, and resampling approaches. Psychological Methods, 17(3), 399-417. https://doi.org/10.1037/a0028087

Schober, P., Boer, C., & Schwarte, L. A. (2018). Correlation coefficients: Appropriate use and interpretation. Anesthesia & Analgesia, 126(5), 1763-1768. https://doi.org/10.1213/ANE.0000000000002864