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Interval-Level Variables

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The Measurement of Association

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Abstract

This chapter describes permutation statistical methods for measures of association designed for two or more interval-level variables. Included in this chapter are simple and multiple ordinary least squares (OLS) regression, simple and multiple least absolute deviation (LAD) regression, point-biserial correlation, and biserial correlation. Fisher’s Z transform for non-normal distributions is examined and evaluated. This chapter concludes with a discussion of the intraclass correlation coefficient.

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Notes

  1. 1.

    In this chapter, a caret (∧) over a symbol such as \(\hat {\alpha }\) or \(\hat {\beta }\) indicates an OLS regression model predicted value of a corresponding population parameter, while a tilde (∼) over a symbol such as \(\tilde {\alpha }\) or \(\tilde {\beta }\) indicates a LAD regression model predicted value of a corresponding population parameter.

  2. 2.

    For comparison, the top of Mount Everest is approximately 8.85 km with a pressure of about 300 millibars.

  3. 3.

    For real-life applications and comparisons of OLS and LAD regression applied to meteorological forecasting, see two articles in Weather and Forecasting by Mielke, Berry, Landsea, and Gray [39, 40].

  4. 4.

    It is probably safe to assume that in any actual research situation, the population correlation coefficient is always not equal to zero.

  5. 5.

    The problem is not confined to r pb. In general, the problem is called the base-rate problem or the marginal-dependent problem. See two excellent discussions of the problem by Goodman [19] and McGrath and Meyer [31].

  6. 6.

    For many years height has been considered as normally distributed, but recent research indicates that this is not necessarily the case [30, pp. 205–207].

  7. 7.

    Note that the sum of squared deviation is divided by N, not N − 1 and the symbol for the standard deviation is S y with an uppercase letter S to distinguish it from the usual sample standard deviation denoted by s y.

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Authors and Affiliations

  1. Department of Sociology, Colorado State University, Fort Collins, Colorado, USA

    Kenneth J. Berry

  2. Alexandria, Virginia, USA

    Janis E. Johnston

  3. Department of Statistics, Colorado State University, Fort Collins, Colorado, USA

    Paul W. Mielke Jr.

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  1. Kenneth J. Berry
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  2. Janis E. Johnston
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  3. Paul W. Mielke Jr.
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Berry, K.J., Johnston, J.E., Mielke, P.W. (2018). Interval-Level Variables. In: The Measurement of Association. Springer, Cham. https://doi.org/10.1007/978-3-319-98926-6_7

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