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Quantifying R2 bias in the presence of measurement error

Author

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  • Karl Majeske
  • Terri Lynch-Caris
  • Janet Brelin-Fornari

Abstract

Measurement error (ME) is the difference between the true unknown value of a variable and the data assigned to that variable during the measuring process. The multiple correlation coefficient quantifies the strength of the relationship between the dependent and independent variable(s) in regression modeling. In this paper, we show that ME in the dependent variable results in a negative bias in the multiple correlation coefficient, making the relationship appear weaker than it should. The adjusted R2 provides regression modelers an unbiased estimate of the multiple correlation coefficient. However, due to the ME induced bias in the multiple correlation coefficient, the otherwise unbiased adjusted R2 under-estimates the variance explained by a regression model. This paper proposes two statistics for estimating the multiple correlation coefficient, both of which take into account the ME in the dependent variable. The first statistic uses all unbiased estimators, but may produce values outside the [0,1] interval. The second statistic requires modeling a single data set, created by including descriptive variables on the subjects used in a gage study. Based on sums of squares, the statistic has the properties of an R2: it measures the proportion of variance explained; has values restricted to the [0,1] interval; and the endpoints indicate no variance explained and all variance explained respectively. We demonstrate the methodology using data from a study of cervical spine range of motion in children.

Suggested Citation

  • Karl Majeske & Terri Lynch-Caris & Janet Brelin-Fornari, 2010. "Quantifying R2 bias in the presence of measurement error," Journal of Applied Statistics, Taylor & Francis Journals, vol. 37(4), pages 667-677.
  • Handle: RePEc:taf:japsta:v:37:y:2010:i:4:p:667-677
    DOI: 10.1080/02664760902814542
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    References listed on IDEAS

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    1. Kumar, Mahesh & Patel, Nitin R., 2007. "Clustering data with measurement errors," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6084-6101, August.
    2. Schechtman, E. & Spiegelman, C., 2007. "Mitigating the effect of measurement errors in quantile estimation," Statistics & Probability Letters, Elsevier, vol. 77(5), pages 514-524, March.
    3. Hernandez, Monica & Pudney, Stephen, 2007. "Measurement error in models of welfare participation," Journal of Public Economics, Elsevier, vol. 91(1-2), pages 327-341, February.
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    Cited by:

    1. Wen He & Hwee Cheng Tan & Leon Wong, 2020. "Return windows and the value relevance of earnings," Accounting and Finance, Accounting and Finance Association of Australia and New Zealand, vol. 60(3), pages 2549-2583, September.

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    More about this item

    Keywords

    measurement error; regression analysis; R2; bias correction; gage R&R;
    All these keywords.

    JEL classification:

    • R2 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - Household Analysis

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