Effect Sizes in Marketing Research: Why Cohen's Local f^2 Belongs in the Toolkit

Wolfgang Messner

arxiv: 2604.18363 · v1 · submitted 2026-04-20 · 📊 stat.ME

Effect Sizes in Marketing Research: Why Cohen's Local f² Belongs in the Toolkit

Wolfgang Messner This is my paper

Pith reviewed 2026-05-10 03:42 UTC · model grok-4.3

classification 📊 stat.ME

keywords effect sizesCohen's f-squaredmarketing researchregression modelssubstantive significancelocal effect sizeincremental validity

0 comments

The pith

Local Cohen's f-squared measures the unique contribution of individual predictors or blocks in marketing regression models and belongs in the effect-size toolkit.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper claims that recommendations for effect sizes in marketing research overlook the local form of Cohen's f-squared, known as f(B)^2. This measure assesses how much a specific predictor or group of predictors improves the explanation of an outcome after other variables are already in the model. Marketing studies often center on whether a focal idea adds real explanatory power beyond alternatives and controls, so a tool focused on that incremental contribution addresses a common need. The paper highlights that its foundation in R-squared makes it practical for the large data sets frequent in the field. It further suggests extending similar local effect sizes to multilevel models and machine-learning approaches.

Core claim

The paper claims that the omission of the local f(B)^2 from the editorial's framework is significant because marketing research frequently uses regression models to test whether a focal construct contributes explanatory power beyond competing predictors and controls. It positions the R-squared foundation of this local effect size as particularly advantageous in large-sample settings common to marketing. Furthermore, it proposes extending f-squared-type measures to multilevel models and, more tentatively, to neural networks and other machine-learning approaches.

What carries the argument

Cohen's local f(B)^2, the proportional increase in R-squared obtained by adding a focal predictor block B to a multivariable model that already contains other variables.

If this is right

Researchers gain a direct way to report the added explanatory power of a focal construct after controls and rivals are accounted for.
Statistical reporting in regression-based marketing papers can better match the theoretical emphasis on incremental validity rather than global fit alone.
The same local approach can be applied to multilevel models that handle nested marketing data structures.
Tentative extensions open the possibility of using analogous incremental measures inside neural networks and other machine-learning models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Widespread use could shift emphasis from overall model fit toward systematic comparison of what each added construct contributes.
It might help standardize how papers demonstrate that a new variable improves explanation beyond what is already known.
Testing the measure on both small and very large marketing data sets would clarify when its large-sample advantage appears most clearly.

Load-bearing premise

The local f-squared supplies information on substantive significance that is not already covered by other effect-size measures and that its R-squared foundation is especially useful for large marketing samples.

What would settle it

A direct comparison in published marketing regression studies showing whether local f-squared values produce different conclusions about which predictors matter substantively than alternatives such as standardized coefficients or partial R-squared.

read the original abstract

In an editorial in the Journal of Marketing, Steenkamp et al. (2026) make a valuable and timely intervention by urging marketing scholars to move beyond dichotomous significance testing and to report effect sizes that speak to substantive significance. Their editorial is especially strong in its insistence on exact p-values, richer statistical reporting, and closer alignment between rigor and relevance. Yet, their framework omits the local form of Cohen's f^2, that is f(B)^2 as an effect-size measure for the contribution of an individual predictor or predictor block B within a multivariable model. That omission matters because much of marketing research relies on regression-type models in which the central theoretical question is not merely whether a model fits globally, but whether a focal construct adds meaningful explanatory power beyond competing predictors and controls. This commentary argues that the R-squared foundation of local Cohen's f(B)^2 is a strength, especially in large-sample settings. Moreover, f-squared-type local effect sizes can be extended beyond ordinary least squares to multilevel models and, more tentatively, to neural networks and other machine-learning models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a short commentary correctly noting that local f² is a standard way to size a predictor's incremental contribution in regressions, but it offers no evidence that the rescaling changes any marketing conclusions compared with reporting ΔR² directly.

read the letter

The core point is straightforward: Steenkamp et al. left local Cohen's f² out of their effect-size recommendations, and this note argues it belongs there because marketing regressions often center on whether one construct adds explanatory power beyond controls. The paper recalls the definition f(B)² = (R²_full − R²_reduced) / (1 − R²_full) and says the R² base is helpful in large samples. That part is accurate and uncontroversial; anyone who has worked with hierarchical regression knows the formula and its interpretation as a local effect size. It also flags possible extensions to multilevel models and, more cautiously, to neural nets, which keeps the door open for future work without overclaiming. Those are the useful reminders here. The limitation is that the note never shows a side-by-side case where f² produces a different substantive ranking, a different threshold decision, or better alignment with theory than the incremental R² or partial R² that software already outputs. Because the two are strictly monotonic, any preference for one over the other is a matter of convention rather than new information. Without even a small marketing dataset or simulation illustrating the difference, the claim that the omission “matters” stays at the level of preference. The paper is therefore best read as a targeted suggestion for the Journal of Marketing’s reporting guidelines rather than a methodological advance. Readers who track effect-size debates in applied fields will find it relevant and easy to absorb; it does not require deep new technical knowledge. I would send it to peer review as a short commentary if the journal wants to keep the conversation going, but it does not need heavy refereeing.

Referee Report

2 major / 2 minor

Summary. The manuscript is a commentary on the Steenkamp et al. (2026) editorial in the Journal of Marketing. It argues that the editorial omits the local form of Cohen's f² (denoted f(B)²) as an effect-size measure for the incremental contribution of an individual predictor or block B in multivariable regression models. The paper claims this omission is consequential for marketing research because such models typically ask whether a focal construct adds explanatory power beyond controls, asserts that the R² foundation of local f² is advantageous especially in large-sample settings, and suggests that f²-type measures can be extended to multilevel models and (tentatively) to neural networks and other ML models.

Significance. If the argument holds and is substantiated with evidence, the commentary would usefully supplement the editorial by reminding readers of a standard, R²-based local effect size already available in regression software. It correctly identifies that marketing regressions often focus on incremental explanatory power rather than global fit alone. The paper draws on well-established properties of R² and Cohen's f² without introducing new derivations or data, so its primary value is advocacy and potential extension rather than methodological innovation.

major comments (2)

[Abstract] Abstract and the paragraph asserting the R² foundation as a strength: the claim that local f² is advantageous in large-sample marketing data because it speaks to substantive significance is not supported by any side-by-side comparison, simulation, or empirical example showing that f(B)² produces materially different predictor rankings, substantive conclusions, or alignment with theory than the incremental R² (or partial R²) that standard software already reports. Since local f² is defined exactly as (R²_full − R²_reduced) / (1 − R²_full), the manuscript must demonstrate an informational gap rather than assert a preference for the rescaled form.
[Discussion of extensions] Section discussing extensions beyond OLS: the statement that f-squared-type local effect sizes 'can be extended' to neural networks and other machine-learning models is presented without any concrete formula, computational procedure, or reference for how a local incremental contribution would be defined or estimated in non-linear, non-parametric models; the extension therefore remains speculative and does not yet meet the load-bearing standard for the paper's broader recommendation.

minor comments (2)

[Throughout] Clarify the notation f(B)² versus the global f² throughout to prevent reader confusion, especially when the manuscript contrasts local and global measures.
[Introduction] Verify the citation details and year for Steenkamp et al. (2026); if the editorial is forthcoming, note its status explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our commentary. We address each major comment point by point below, indicating where revisions have been made to the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract and the paragraph asserting the R² foundation as a strength: the claim that local f² is advantageous in large-sample marketing data because it speaks to substantive significance is not supported by any side-by-side comparison, simulation, or empirical example showing that f(B)² produces materially different predictor rankings, substantive conclusions, or alignment with theory than the incremental R² (or partial R²) that standard software already reports. Since local f² is defined exactly as (R²_full − R²_reduced) / (1 − R²_full), the manuscript must demonstrate an informational gap rather than assert a preference for the rescaled form.

Authors: We acknowledge that the manuscript does not contain new simulations or empirical examples comparing local f² directly to incremental or partial R². As a commentary focused on advocacy rather than methodological innovation, the argument centers on the established role of local f² within Cohen's framework as a standardized, bounded measure of incremental contribution. To respond to the concern, we have revised the abstract and the relevant discussion paragraph to emphasize the interpretive standardization (normalization by residual variance) without asserting unshown empirical superiority, and we have added a concise numerical illustration using typical regression output to clarify how the rescaled form aids substantive interpretation in practice. revision: partial
Referee: [Discussion of extensions] Section discussing extensions beyond OLS: the statement that f-squared-type local effect sizes 'can be extended' to neural networks and other machine-learning models is presented without any concrete formula, computational procedure, or reference for how a local incremental contribution would be defined or estimated in non-linear, non-parametric models; the extension therefore remains speculative and does not yet meet the load-bearing standard for the paper's broader recommendation.

Authors: We agree that the original phrasing regarding extensions to neural networks and other machine-learning models was tentative and lacked concrete details, rendering it speculative. In the revised manuscript we have removed the specific reference to neural networks and ML models. The discussion of extensions is now confined to multilevel models, for which established pseudo-R² and local effect-size approaches are available in the literature, and we note only in passing that further adaptation to complex ML settings remains an open methodological question. revision: yes

Circularity Check

0 steps flagged

No circularity: commentary advocates established measure without derivations or self-referential reductions

full rationale

The paper is a commentary arguing that an editorial omits local Cohen's f² as an effect-size measure for individual predictor contributions in regression models. It contains no derivation chain, equations, fitted parameters, or predictions that reduce to the paper's own inputs by construction. References to R² foundations and extensions to multilevel or ML models cite established statistical concepts without self-citation loops or ansatzes smuggled from prior author work. The claim that the omission matters rests on substantive discussion of marketing research needs rather than any mathematical reduction or renaming of known results. This is a self-contained opinion piece on reporting practices with no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The commentary relies on domain assumptions from statistics without introducing new free parameters, axioms beyond standard ones, or invented entities.

axioms (1)

domain assumption Local Cohen's f^2 is a valid and useful effect size based on changes in R-squared for assessing predictor importance.
This is drawn from standard statistical literature on effect sizes, assumed to hold in the context of marketing regression models.

pith-pipeline@v0.9.0 · 5485 in / 1315 out tokens · 49035 ms · 2026-05-10T03:42:45.817456+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

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work page doi:10.3758/bf03207705 1999
[2]

Cohen, J. (1988). Statistical power analysis for the behavioral science (2nd ed.). Lawrence Erlbaum

work page 1988
[3]

Cohen, J., & Cohen, P. (1983). Applied multiple regression/correlation analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum. De La Rosa, W., Silverman, J., Sussman, A. B., Rino, G., Dorie, V .,

work page 1983
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Hell, M., Giannella, E., & Dillman, L. (2025). Using Expenditure Reframes to Increase Interest in Claiming Government Benefits. Journal of Marketing , 1 –22. https://doi.org/10.1177/00222429251356992

work page doi:10.1177/00222429251356992 2025
[5]

Dedecker, J., Guedj, O., & Taupin, M. L. (2025). Asymptotic confidence interval for R2 in multiple linear regression. Statistics, 59(1), 1 –36. https://doi.org/10.1080/02331888.2024.2428978

work page doi:10.1080/02331888.2024.2428978 2025
[6]

Muluk, H. (2018). Hypocrisy and culture: Failing to practice what you preach receives harsher interpersonal reactions in independent (vs. interdependent) cultures. Journal of Experimental Social Psychology , 76, 371 –384. https://doi.org/10.1016/j.jesp.2017.12.009

work page doi:10.1016/j.jesp.2017.12.009 2018
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Fisher, R. A. (1928). The general sampling distribution of the multiple correlation coefficient. Proceedings of the Royal Society of London. Series A , 121(788), 654 –673. https://doi.org/10.1098/rspa.1928.0224

work page doi:10.1098/rspa.1928.0224 1928
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Kelley, K. (2008). Sample size planning for the squared multiple correlation coefficient: Accuracy in parameter estimation via narrow confidence intervals. Multivariate Behavioral Research, 43(4), 524 –555. https://doi.org/10.1080/00273170802490632

work page doi:10.1080/00273170802490632 2008
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Khalilzadeh, J., & Tasci, A. D. A. (2017). Large sample size, significance level, and the effect size: Solutions to perils of using big data for academic research. Tourism Management, 62, 89–96. https://doi.org/10.1016/j.tourman.2017.03.026

work page doi:10.1016/j.tourman.2017.03.026 2017
[10]

Lee, Y.-S. (1971). Some results on the sampling distribution of the multiple correlation coefficient. Journal of the Royal Statistical Society: Series B , 33(1), 117 –130. https://doi.org/10.1111/j.2517-6161.1971.tb00863.x

work page doi:10.1111/j.2517-6161.1971.tb00863.x 1971
[11]

Lorah, J. (2018). Effect size measures for multilevel models: definition, interpretation, and TIMSS example. Large-Scale Assessments in Education , 6(1), 1 –11. https://doi.org/10.1186/s40536-018-0061-2

work page doi:10.1186/s40536-018-0061-2 2018
[12]

Messner, W. (2023). From black box to clear box: A hypothesis testing framework for scalar regression problems using deep artificial neural networks. Applied Soft Computing, 146, 1–14. https://doi.org/10.1016/j.asoc.2023.110729

work page doi:10.1016/j.asoc.2023.110729 2023
[13]

Messner, W. (2024). Exploring multilevel data with deep learning and XAI: The effect of personal -care advertising spending on subjective happiness. International Business Review, 33(1), 1–22. https://doi.org/10.1016/j.ibusrev.2023.102203

work page doi:10.1016/j.ibusrev.2023.102203 2024
[14]

Messner, W., Greene, T., & Matalone, J. (2025). From bytes to biases: Investigating the cultural self -perception of large language models. Journal of Public Policy & Marketing, 44(3), 370–391. https://doi.org/10.1177/07439156251319788

work page doi:10.1177/07439156251319788 2025
[15]

Ogasawara, H. (2006). Asymptotic expansion and conditional robustness for the sample multiple correlation coefficient under nonnormality. Communications in Statistics - Simulation and Computation , 35(1), 177 –199. https://doi.org/10.1080/03610910500416207

work page doi:10.1080/03610910500416207 2006
[16]

Olkin, I., & Pratt, J. W. (1958). Unbiased estimation of certain correlation coefficients. The Annals of Mathematical Statistics, 29(1), 201–211

work page 1958
[17]

Mermelstein, R. J. (2012). A practical guide to calculating Cohen's f2, a measure of local effect size, from PROC MIXED. Frontiers in Psychology , 3, 1 –6. https://doi.org/10.3389/fpsyg.2012.00111

work page doi:10.3389/fpsyg.2012.00111 2012
[18]

Soper, H. E. (1929). The general sampling distribution of the multiple correlation coefficient. Journal of the Royal Statistical Society, 92(3), 445–447. https://doi.org/10.2307/2342239

work page doi:10.2307/2342239 1929
[19]

Slotegraaf, R. J. (2026). Cementing JM’s impact on the marketing ecosystem: Empirical Execution. Journal of Marketing, 90(3), 1 –12. https://doi.org/10.1177/00222429261426047

work page doi:10.1177/00222429261426047 2026

[1] [1]

Bakeman, R., & McArthur, D. (1999). Determining the power of multiple regression analyses both with and without repeated measures. Behavior Research Methods, Instruments, and Computers, 31(1), 150 –154. https://doi.org/10.3758/BF03207705

work page doi:10.3758/bf03207705 1999

[2] [2]

Cohen, J. (1988). Statistical power analysis for the behavioral science (2nd ed.). Lawrence Erlbaum

work page 1988

[3] [3]

Cohen, J., & Cohen, P. (1983). Applied multiple regression/correlation analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum. De La Rosa, W., Silverman, J., Sussman, A. B., Rino, G., Dorie, V .,

work page 1983

[4] [4]

Hell, M., Giannella, E., & Dillman, L. (2025). Using Expenditure Reframes to Increase Interest in Claiming Government Benefits. Journal of Marketing , 1 –22. https://doi.org/10.1177/00222429251356992

work page doi:10.1177/00222429251356992 2025

[5] [5]

Dedecker, J., Guedj, O., & Taupin, M. L. (2025). Asymptotic confidence interval for R2 in multiple linear regression. Statistics, 59(1), 1 –36. https://doi.org/10.1080/02331888.2024.2428978

work page doi:10.1080/02331888.2024.2428978 2025

[6] [6]

Muluk, H. (2018). Hypocrisy and culture: Failing to practice what you preach receives harsher interpersonal reactions in independent (vs. interdependent) cultures. Journal of Experimental Social Psychology , 76, 371 –384. https://doi.org/10.1016/j.jesp.2017.12.009

work page doi:10.1016/j.jesp.2017.12.009 2018

[7] [7]

Fisher, R. A. (1928). The general sampling distribution of the multiple correlation coefficient. Proceedings of the Royal Society of London. Series A , 121(788), 654 –673. https://doi.org/10.1098/rspa.1928.0224

work page doi:10.1098/rspa.1928.0224 1928

[8] [8]

Kelley, K. (2008). Sample size planning for the squared multiple correlation coefficient: Accuracy in parameter estimation via narrow confidence intervals. Multivariate Behavioral Research, 43(4), 524 –555. https://doi.org/10.1080/00273170802490632

work page doi:10.1080/00273170802490632 2008

[9] [9]

Khalilzadeh, J., & Tasci, A. D. A. (2017). Large sample size, significance level, and the effect size: Solutions to perils of using big data for academic research. Tourism Management, 62, 89–96. https://doi.org/10.1016/j.tourman.2017.03.026

work page doi:10.1016/j.tourman.2017.03.026 2017

[10] [10]

Lee, Y.-S. (1971). Some results on the sampling distribution of the multiple correlation coefficient. Journal of the Royal Statistical Society: Series B , 33(1), 117 –130. https://doi.org/10.1111/j.2517-6161.1971.tb00863.x

work page doi:10.1111/j.2517-6161.1971.tb00863.x 1971

[11] [11]

Lorah, J. (2018). Effect size measures for multilevel models: definition, interpretation, and TIMSS example. Large-Scale Assessments in Education , 6(1), 1 –11. https://doi.org/10.1186/s40536-018-0061-2

work page doi:10.1186/s40536-018-0061-2 2018

[12] [12]

Messner, W. (2023). From black box to clear box: A hypothesis testing framework for scalar regression problems using deep artificial neural networks. Applied Soft Computing, 146, 1–14. https://doi.org/10.1016/j.asoc.2023.110729

work page doi:10.1016/j.asoc.2023.110729 2023

[13] [13]

Messner, W. (2024). Exploring multilevel data with deep learning and XAI: The effect of personal -care advertising spending on subjective happiness. International Business Review, 33(1), 1–22. https://doi.org/10.1016/j.ibusrev.2023.102203

work page doi:10.1016/j.ibusrev.2023.102203 2024

[14] [14]

Messner, W., Greene, T., & Matalone, J. (2025). From bytes to biases: Investigating the cultural self -perception of large language models. Journal of Public Policy & Marketing, 44(3), 370–391. https://doi.org/10.1177/07439156251319788

work page doi:10.1177/07439156251319788 2025

[15] [15]

Ogasawara, H. (2006). Asymptotic expansion and conditional robustness for the sample multiple correlation coefficient under nonnormality. Communications in Statistics - Simulation and Computation , 35(1), 177 –199. https://doi.org/10.1080/03610910500416207

work page doi:10.1080/03610910500416207 2006

[16] [16]

Olkin, I., & Pratt, J. W. (1958). Unbiased estimation of certain correlation coefficients. The Annals of Mathematical Statistics, 29(1), 201–211

work page 1958

[17] [17]

Mermelstein, R. J. (2012). A practical guide to calculating Cohen's f2, a measure of local effect size, from PROC MIXED. Frontiers in Psychology , 3, 1 –6. https://doi.org/10.3389/fpsyg.2012.00111

work page doi:10.3389/fpsyg.2012.00111 2012

[18] [18]

Soper, H. E. (1929). The general sampling distribution of the multiple correlation coefficient. Journal of the Royal Statistical Society, 92(3), 445–447. https://doi.org/10.2307/2342239

work page doi:10.2307/2342239 1929

[19] [19]

Slotegraaf, R. J. (2026). Cementing JM’s impact on the marketing ecosystem: Empirical Execution. Journal of Marketing, 90(3), 1 –12. https://doi.org/10.1177/00222429261426047

work page doi:10.1177/00222429261426047 2026