{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,2]],"date-time":"2025-12-02T16:03:29Z","timestamp":1764691409489,"version":"3.46.0"},"reference-count":26,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T00:00:00Z","timestamp":1764547200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>When dealing with mathematical or statistical models involving one or more explanatory (independent) variables, one often wants to determine the effects of such variables on a response (independent) variable. In the case of linear regression models, one such effects measure is the so-called standardized regression coefficients used in various statistical software packages and discussed in regression textbooks. However, since strong reservations have been expressed against the use of standardized regression coefficients because of various limitations, the objective or initial working hypothesis behind the present research was that some alternative measure without such limitations ought to be explored. Consequently, elasticity coefficients are formulated and proposed as both relative and absolute such measures. While the standardized regression coefficients lack any convenient interpretation, the proposed elasticity coefficients have the particularly desirable property of having a logical and intuitively appealing interpretation in terms of the relative change in the value of the response variable as a consequence of a relative change (of 1 percent) in the value of one or more of the explanatory variables. Those elasticity measures have the flexibility of being applicable to individual or to all explanatory variables and to individual or to all observations or data sets. A numerical example is used to illustrate the use of these new measures. Comparison between values of the standardized regression coefficients and those of the corresponding elasticity coefficients based on reported data from various sources is provided. Also, the form of the elasticity coefficients for a variety of different types of models is presented. Statistical inferences are also discussed.<\/jats:p>","DOI":"10.3390\/computation13120279","type":"journal-article","created":{"date-parts":[[2025,12,2]],"date-time":"2025-12-02T15:31:17Z","timestamp":1764689477000},"page":"279","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Elasticity Coefficients as Effects Measures in Model Formulations"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6399-735X","authenticated-orcid":false,"given":"Tarald O.","family":"Kv\u00e5lseth","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA"},{"name":"Department of Industrial & Systems Engineering, University of Minnesota, Minneapolis, MN 55455, USA"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,1]]},"reference":[{"key":"ref_1","unstructured":"Draper, N.R., and Smith, H. (1981). Applied Regression Analysis, Wiley. [2nd ed.]."},{"key":"ref_2","unstructured":"Agresti, A., and Finlay, B. (2004). Statistical Methods for the Social Sciences, Pearson. [4th ed.]."},{"key":"ref_3","unstructured":"Montgomery, D.C., Peck, E.A., and Vining, G.G. (2021). Introduction to Linear Regression Analysis, Wiley. [6th ed.]."},{"key":"ref_4","unstructured":"Richard, D. (1985). Applied Linear Statistical Models, Irwin. [2nd ed.]."},{"key":"ref_5","unstructured":"Fox, J., and Weisberg, S. (2019). An R Companion to Applied Regression, Sage. [3rd ed.]."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Wolf, C., and Best, H. (2014). Linear regression. The Sage Handbook of Regression Analysis and Causal Inference, Sage.","DOI":"10.4135\/9781446288146.n4"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1093\/oxfordjournals.aje.a114229","article-title":"The fallacy of employing standardized regression coefficients and correlations as measures of effect","volume":"123","author":"Greenland","year":"1986","journal-title":"Am. J. Epidemiol."},{"key":"ref_8","unstructured":"Varian, H.R. (2010). Intermediate Microeconomics, W.W. Norton. [8th ed.]."},{"key":"ref_9","unstructured":"Hall, A.D. (1962). A Methodology for Systems Engineering, D. Van Nostrand Company."},{"key":"ref_10","unstructured":"Arfken, G. (1985). Mathematical Methods for Physicists, Academic Press."},{"key":"ref_11","unstructured":"Illowsky, B., Bergman, M., and Dean, S. (2024). Ch. 13 Multiple Linear Regression. Quantitative Analysis for Business, UW.PRESSBOOKS.PUB. Available online: https:\/\/uw.pressbooks.pub\/quantbusiness\/."},{"key":"ref_12","unstructured":"(2024, April 18). Interpretation of Regression Coefficients\u2014Elasticity and Logarithmic Transformation. Available online: https:\/\/biz.libretexts.org\/Workbench\/MGT_235\/10%3A_Linear_Regression_and_Correlation\/10.06%3A_Interpretation_of_Regression_Coefficients-_Elasticity_and_Logarithmic_Transformation."},{"key":"ref_13","unstructured":"Holmes, A., Illowsky, B., and Dean, S. (2017). Ch. 13.5. Interpretation of Regression Coefficients: Elasticity and Logarithmic Transformation. Introductory Business Statistics, OpenStax. Available online: https:\/\/openstax.org\/details\/books\/introductory-business-statistics."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"402","DOI":"10.1080\/02664763.2016.1174195","article-title":"Coefficient of variation: The second order alternative","volume":"44","year":"2017","journal-title":"J. Appl. Stat."},{"key":"ref_15","unstructured":"Ryan, T.P. (1997). Modern Regression Methods, Wiley."},{"key":"ref_16","unstructured":"Stone, J.V. (2022). Linear Regression: A Tutorial Introduction to the Mathematics of Regression Analysis, Sebtel Press."},{"key":"ref_17","unstructured":"Spiegel, M.R., and Stephens, L.J. (2018). Statistics, McGraw-Hill. [6th ed.]."},{"key":"ref_18","unstructured":"Marascuilo, L.A., and Serlin, R.C. (1988). Statistical Methods for the Social and Behavioral Sciences, W.H. Freeman."},{"key":"ref_19","unstructured":"McClave, J.T., and Benson, P.G. (1985). Statistics for Business and Economics, Dellen Publishing Company."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Walpole, R.E., and Meyers, R.H. (1978). Probability and Statistics for Engineers and Scientists, Macmillan. [2nd ed.].","DOI":"10.2307\/2530629"},{"key":"ref_21","unstructured":"Weisberg, S. (2014). Applied Linear Regression, Wiley. [4th ed.]."},{"key":"ref_22","unstructured":"Gujarati, D.N., Porter, D.C., and Gunasekar, S. (2012). Basic Econometrics, McGraw Hill Education. [5th ed.]."},{"key":"ref_23","first-page":"279","article-title":"Cautionary note about R2","volume":"39","year":"1985","journal-title":"Am. Stat."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Biecek, R., and Burzykowski, T. (2021). Explanatory Model Analysis: Explain, Explore, and Examine Predictive Models, Chapman and Hall\/CRC.","DOI":"10.1201\/9780429027192"},{"key":"ref_25","unstructured":"Fernihough, A. (2025, November 03). Marginal Effects for Generalized Linear Models: The Mfx Package for R. Available online: https:\/\/cran.r-project.org\/web\/packages\/mfx\/vignettes\/mfxarticle.pdf."},{"key":"ref_26","unstructured":"Molnar, C. (2025). Interpretable Machine Learning: A guide for Making Black Box Models Explainable, Shroff publisher. [3rd ed.]. Available online: https:\/\/christophm.github.io\/interpretable-ml-book\/."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/12\/279\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,2]],"date-time":"2025-12-02T16:00:39Z","timestamp":1764691239000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/12\/279"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,1]]},"references-count":26,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["computation13120279"],"URL":"https:\/\/doi.org\/10.3390\/computation13120279","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2025,12,1]]}}}