Explain why the researcher's conclusion is incorrect. Diagnose the statistical error made by the researcher, explain what shape the data points would form on a scatterplot, and explain how the researcher should interpret both the Pearson's value and the actual pattern of the data.
Case context: A psychologist collects data on college students' nightly sleep hours and self-reported depression levels. Upon analysis, the researcher finds that students who sleep about eight hours have the lowest depression scores, while those sleeping less than five hours or more than ten hours have significantly higher depression scores. The researcher calculates Pearson's and finds a value of . Based on this correlation coefficient, the researcher concludes in their paper that sleep duration has no relationship to depression levels.
Question: Explain why the researcher's conclusion is incorrect. Diagnose the statistical error made by the researcher, explain what shape the data points would form on a scatterplot, and explain how the researcher should interpret both the Pearson's value and the actual pattern of the data.
Sample answer: The researcher's conclusion is incorrect because they failed to realize that Pearson's only detects linear relationships. The relationship described is nonlinear, forming an upside-down 'U' shape on a scatterplot where the lowest depression occurs around eight hours of sleep, and higher depression occurs at both extremes of sleep. Because the pattern is curved, Pearson's is close to zero (0.03), which misleadingly suggests no relationship. The researcher should interpret the Pearson's as indicating the absence of a linear relationship, but recognize that a strong nonlinear relationship is present.
Key points:
- Pearson's only evaluates the strength and direction of linear relationships.
- The data points form a curved, upside-down 'U' shape on a scatterplot.
- A correlation coefficient near zero misleadingly suggests no relationship due to the curved pattern.
- The researcher should conclude that a strong nonlinear relationship exists, rather than no relationship.
Rubric: Students must demonstrate comprehension by explaining that: 1) Pearson's only measures linear relationships; 2) the sleep-depression relationship is nonlinear and forms an upside-down 'U' shape; and 3) a low Pearson's (near zero) does not mean there is no relationship, but rather that a linear model is incorrect for this curved pattern.
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Research Methods in Psychology - 4th American Edition @ KPU
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After plotting nightly sleep hours against depression scores and observing an inverted-U pattern in the scatterplot, a researcher wants to evaluate whether Pearson's is the right statistic to report. Arrange the following reasoning steps in the order that best represents a sound scientific evaluation.
Describe the hypothetical nonlinear relationship between nightly hours of sleep and depression levels. In your description, recall the specific shape the data points form on a scatterplot, the sleep duration associated with the lowest depression levels, how other sleep durations relate to depression, and why Pearson's is an inappropriate statistic for summarizing this relationship.
Explain why the researcher's conclusion is incorrect. Diagnose the statistical error made by the researcher, explain what shape the data points would form on a scatterplot, and explain how the researcher should interpret both the Pearson's value and the actual pattern of the data.
A researcher is studying nightly sleep hours and depression levels. If they calculate a Pearson's close to zero, apply your understanding of this relationship to identify what visual step they must take with their data and what specific curved pattern they should look for on the resulting plot.