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Correcting the t statistic for measurement error

Marketing Letters volume 23, pages 671–682 (2012)Cite this article

Abstract

Studies in marketing often involve application of multi-item scales to measure latent constructs. Once the psychometric properties of a scale have been assessed, responses to individual scale items are often summed to form a composite score, which then is compared across groups by performing statistical tests such as a t test. In this note, we draw researchers’ attention to an often overlooked fact that the t test is attenuated by imperfect measures. As a solution, we propose the disattenuated t statistic and discuss how it would increase accuracy of estimates and affect decisions in the marketing discipline.

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Fig. 1

Notes

  1. The F statistic, which is more appropriate when comparing mean differences among three or more groups, is also likely to be attenuated by measurement error. However, the focus of this research is on the two-group case and how to disattenuate the t statistic.

  2. For the sake of clarity, the group superscript “g” and respondent subscript “i” are not shown in the equation.

  3. Parenthetically, reliability is itself affected by the number of scale items. From Eq. 3, when the error variance is kept constant across scale items, increasing the number of scale items (p) will result in a decrease in the total variance. For smaller values of total variance, the t statistic increases. So, larger scales tend to drive the t statistic higher whereas smaller scales lead to lower t statistics. This is because all things being equal, longer scales have higher reliability. However, it is not the objective of this paper to discuss the merits of employing smaller vs. larger scales in terms of number of scale items; interested readers should consult Bergkvist and Rossiter (2007), Drolet and Morrison (2001), Kardes et al. (1993), and Rossiter (2002) for further insights into that topic.

  4. In computing the summed scores, we assigned equal weight to each variable, as this is the common practice. However, as suggested by one of the reviewers, summed scores can also be obtained by weighting the variables by the loadings. For both the examples, we computed the loadings by constraining them to be equal across groups to meet the metric equivalency requirement and used the resulting loadings to also compute the weighted summed scores. Although the attenuated and disattenuated t values were slightly different when using the reviewer suggested procedure, there were no changes in the conclusions.

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Acknowledgments

The authors would like to thank Professor Terence A. Shimp, the Editor, and the two anonymous reviewers for their helpful comments on an earlier version of this manuscript.

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Authors and Affiliations

  1. College of Business Administration, Marquette University, Milwaukee, WI, 53233, USA

    Srinivas Durvasula

  2. Moore School of Business, University of South Carolina, Columbia, SC, 29208, USA

    Subhash Sharma

  3. Moore School of Business, University of South Carolina, Columbia, SC, 29208, USA

    Kealy Carter

Authors
  1. Srinivas Durvasula
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  2. Subhash Sharma
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  3. Kealy Carter
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Correspondence to Subhash Sharma.

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Durvasula, S., Sharma, S. & Carter, K. Correcting the t statistic for measurement error. Mark Lett 23, 671–682 (2012). https://doi.org/10.1007/s11002-012-9170-9

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Keywords

  • Methodology, testing of group means
  • Disattenuated t statistic
  • Composite scale scores
  • t statistic