Gain-probability diagrams as an alternative to significance testing in economics and finance

Trafimow, David; Wang, Ziyuan; Tong, Tingting; Wang, Tonghui Article Gain-probability diagrams as an alternative to significance testing in economics and finance Asian Journal of Economics and Banking (AJEB) Provided in Cooperation with: Ho Chi Minh University of Banking (HUB), Ho Chi Minh City Suggested Citation: Trafimow, David; Wang, Ziyuan; Tong, Tingting; Wang, Tonghui (2023) : Gainprobability diagrams as an alternative to significance testing in economics and finance, Asian Journal of Economics and Banking (AJEB), ISSN 2633-7991, Emerald, Leeds, Vol. 7, Iss. 3, pp. 333-357, https://doi.org/10.1108/AJEB-05-2023-0045 This Version is available at: https://hdl.handle.net/10419/334104 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Gain-probability diagrams as an alternative to significance testing in economics and finance David Trafimow Department of Psychology, New Mexico State University, Las Cruces, New Mexico, USA Ziyuan Wang University of Lynchburg, Lynchburg, Virginia, USA, and Tingting Tong and Tonghui Wang Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico, USA Abstract Purpose –The purpose of this article is to show the gains that can be made if researchers were to use gainprobability (G-P) diagrams. Design/methodology/approach –The authors present relevant mathematical equations, invented examples and real data examples. Findings –G-P diagrams provide a more nuanced understanding of the data than typical summary statistics, effect sizes or significance tests. Practical implications –Gain-probability diagrams provided a much better basis for making decisions than typical summary statistics, effect sizes or significance tests. Originality/value –G-P diagrams provide a completely new way to traverse the distance from data to decision-making implications. Keywords Mathematical and quantitative methods, Gain-probability diagrams, Skew normal distributions, Delta skew lognormal distributions, Decision-making, Probabilities Paper type Research Paper 1. Introduction Consider a medical scenario implying a government financing decision. Suppose researchers invent a medicine to treat a disease. Should the government invest to render the medicine more widely available to its citizens? At present, a widely accepted procedure would be to perform a significance test to determine if the medicine is effective at a statistically significant level. If so, that would constitute a reason to make theinvestment; if not, the government would not invest. However, significance testing is a poor way to evaluate the effectiveness of the medicine. It is quite possible for a medicine not to be effective, but nevertheless achieve statistical Gainprobability in economics and finance 333 © David Trafimow, Ziyuan Wang, Tingting Tong and Tonghui Wang. Published in Asian Journal of Economics and Banking. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http:// creativecommons.org/licences/by/4.0/legalcode It has come to the attention of the publisher that the article, Trafimow, D., Wang, Z., Tong, T. and Wang, T. “Gain-probability diagrams as an alternative to significance testing in economics and finance” published in Asian Journal of Economics and Banking https://doi.org/10.1108/AJEB-05-2023-0045 gave an incorrect location for Trafimow’s affiliation. This error was introduced in the editorial process and has now been corrected in the online version. The publisher sincerely apologises for this error. The current issue and full text archive of this journal is available on Emerald Insight at: https://www.emerald.com/insight/2615-9821.htm Received 23 May 2023 Revised 30 May 2023 Accepted 3 June 2023 Asian Journal of Economics and Banking Vol. 7 No. 3, 2023 pp. 333-357 Emerald Publishing Limited e-ISSN: 2633-7991 p-ISSN: 2615-9821 DOI 10.1108/AJEB-05-2023-0045 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025 significance, if the sample of cases is sufficiently large. This is because statistical significance depends on two items: the size of the effect and the sample size. It is possible for a sufficiently large sample size to compensate for even an extremely small effect. The p-values upon which significance testing depends confound the effect sizes researchers obtain with the sample sizes they collect. In fact, McQuitty (2004,2018) famously argued against too large sample sizes on the grounds that they ensure statistical significance even when there is not much of an effect. Of course, a problem with this argument is that statistical significance is not the only matter of concern. There is the much more important issue of the extent to which sample sizes are sufficiently large to engender confidence that the sample statistics provide good estimates of population parameters. The larger the sample size, the better the estimate. Thus, from the point of view of estimation, McQuitty’s famous advice is contraindicated. In addition, the fact that the advice was so well received by business and economics researchers and cited over 1,000 times and received an accolade by Journal ofGlobal Scholarsof Marketing Science, points to one of many problems with significance testing thinking (Trafimow et al., 2021). There have been many criticisms of significance testing, including several reviews (Hubbard, 2016;Ziliak and McCloskey, 2016;Trafimow, 2019) and 43 articles in the 2019 special issue of The American Statistician, the highly respected journal of The American Statistical Association. Because the disadvantages of significance testing have been covered so extensively, there is no point in rehashing them here. Rather, the present goal is to present a more useful alternative. In the medical scenario, there are many factors to consider. These include the seriousness of the disease, the cost of the medicine and many others. In addition, and crucially, there is the issue of how well the medicine works. Put in the form of a question, what is the probability that a randomly selected person who takes the medicine will be better off, or worse off, and by how much, with the medicine than without it? Nor must we restrict ourselves to economics issues pertaining to medicine. If an economic intervention is proposed to increase people’s living space, what is the probability that a randomly selected person will be better off, or worse off, and by how much living space, with the intervention than without it? Or if a government wishes to institute a policy to increase incomein a poor area, what isthe probability that a person will bebetter off, or worse off, and by how much income, with the new policy than without it? The issue of probabilities of being better off, or worse off, by varying degrees, is relevant to many potential economics applications. To estimate the probability of being better off, or worse off, by varying degrees, with the intervention or policy change than without it, it is necessary to look carefully at the data to determine the distribution. We will consider two families of distributions: skew normal distributions, which include normal distributions, and delta log-skew-normal (LSN) distributions, which include lognormal distributions. The subsequent section includes crucial equations, with subsections for skew normal and delta LSN distributions. Following that, we provide examples of how to use the equations to construct gain-probability (G-P) diagrams to draw conclusions far beyond that which significance tests allow (Tong et al., 2022;Trafimow et al., 2022;Wang et al., 2022). 2. Probability of being better off or worse off by how much? Skew normal distributions Whereas normal distributions have two parameters, mean μ and standard deviation σ ; skew normal distributions have three parameters. The location ξreplaces the mean, the scale ω replaces the standard deviation, and there is a shape (or skewness) parameter α . When the shape parameter equals 0 the distribution is normal, and the location equals the mean and the scale equals the standard deviation. But when the shape parameter does not equal 0, the location does not equal the mean and the scale does not equal the standard deviation. In symbols, when α 50, ξ5 μ and ω 5 σ ; but when α ≠0, ξ≠ μ and ω ≠ σ . The family of normal AJEB 7,3 334 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025 distributions is a subset of the family of skew normal distributions. Thus, the family of skew normal distributions is more generally applicable than the family of normal distributions. Let us now derive the probability that a randomly chosen person from one skew normal population has a higher score on the dependent variable than a randomly chosen person from another skew normal population. We will assume dependent populations at any level of correlation between 1andþ1, with independent populations (correlation 50) as a special case. D2.1. Azzalini and Valle (1996) A random vector X¼ðX1;...;XkÞ0is said to have an kdimensional multivariate skew normal distribution with the vector of location parameter μ ¼ð μ 1;...; μ kÞ0∈Rk,the scale parameter of the positive definite matrix Σ,and the vector of skewness (shape) parameters α ¼ð α 1;...; α kÞ0,denoted as X∼SN k ( μ ,Σ, α ), if its density function (pdf) is given by fXðxÞ¼2 f kðx; μ ;ΣÞΦ α 0Σ−1=2ðx μ Þ  ;(2.1) where f k (x; μ ,Σ)is the density of the k-dimensional multivariate normal distribution N k ( μ ,Σ) with mean μ and covariance matrix Σ,and Φ(z)is the cumulative distribution function (cdf) of the standard normal random variable Z ∼N(0, 1). Here x0is the transpose of the vector x∈Rk. 2.1 Calculation of δ ab 5P(Z> 0) in independent distributions As we mentioned above, Let X∼SN(ξ 1 , ω 1 , α 1 ), Y∼SN(ξ 2 , ω 2 , α 2 ) and we assume that Xand Y are independent. We want to obtain the distribution of the linear combination, Z5XþaY þb,ofXand Yfor specified a∈Rand b∈Rand then calculate δ ab 5P(Z> 0). We need to find the distribution of Zfirst, which is given below. There are two advantages of using Z5XþaY þb. Both advantages stem from the possibility that a researcher might be interested in more than simply a probabilistic advantage for one group over another. A researcher might be interested in the probabilities of being better off, or worse off, to varying degrees. The equation provides two ways to assess this. One way is by setting a51 and varying b, to obtain the probability that a randomly selected person from one condition will score higher than a randomly selected person from another condition, by the amount the researcher specifies by setting bto that value. This is the most straightforward advantage. A second advantage is that the equation provides a way to assess the probability of being better off, or worse off, by varying degrees, in terms of multiples. For example, what is the probability that a randomly selected person from one group will score twice as much, thrice as much and so on, as a randomly selected person from the other group? The equation facilitates such an assessment. The researcher merely sets b50andletsavary at multiples of interest. Theorem 2.1. Let X ∼SNðξ1; ω 2 1; α 1Þ,Y∼SNðξ2; ω 2 2; α 2Þ.First we assume that two skew normal populations are independent. Then the probability density function (pdf) of Z 5XþaY þbis fZðzÞ¼c f z; ν ; τ 2Φ2½Bðz ν Þ;02;Δ;(2.2) where c¼Φ−1 202;02;Δþ τ 2BB0; ν ¼ξ1þaξ2þb; τ 2¼ ω 2 1þa2 ω 2 2;B¼d0 τ 2; 02¼0 0  ;Δ¼1þ α 2 10 01þ α 2 2 ! dd0 τ 2;d¼ α 1 ω 1 a α 2 ω 2  : Gainprobability in economics and finance 335 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025 Here f (z; ν , τ 2 )is the pdf of the normal distribution N( ν , τ 2 )with mean ν and variance τ 2 ,and Φ 2 (u;0 2 ,Δ)is the cumulative distribution function (cdf) of the bivariate normal distribution with mean vector 0 2 and covariance Δ. The proof of Theorem 2.1 is given in Appendix. Note that the pdf of Zgiven in Equation (2.2) is the special case of the closed skew normal density. For details of closed skew normal distributions, see Gupta et al. and Zhu et al. Now we can compute the probability P(Z> 0), which is given by δab ¼PrðZ>0Þ¼Z∞ 0 c f z; ν ; τ 2Φ2½Bðz ν Þ;02;Δdz:(2.3) Density curves of Zfor different parameters and their corresponding P(Z> 0) are given in Figures 1 and 2, respectively. These values can be instantiated into the equations presented earlier, but there is an easier way too. We provide a freely available online calculator at https://probab.shinyapps.io/inde_prob/ 2.2 Calculation of δ ρ ab 5P(U> 0) in dependent distributions Now we consider the bivariate skew normal random vector X¼ðX1;X2Þ0∼SN2ðξ;Σ; α Þ, with location parameter ξ, scale parameter Σand skewness parameter α given by ξ¼ξ1 ξ2  ;Σ¼ ω 2 1 ρω 1 ω 2 ρω 1 ω 2 ω 2 2 ! ; α ¼ α 1 α 2  : Then we have the following result and its proof. Theorem 2.2. Let X∼SN 2 (ξ,Σ, α )given above and consider U 5X 1 þaX 2 þb with a;b∈R.Then the pdf of U is Figure 1. The density of Zwhen location parameters of both populations equal 0, the scale parameters for first and second population equal 1 and 2 respectively AJEB 7,3 336 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025 fUðuÞ¼2 f u;ξ1þaξ2þb; ω 2 1þa2 ω 2 2þ2a ρω 1 ω 2 3Φ α *ðuξ1aξ2bÞ  ω 2 1þa2 ω 2 2þ2a ρω 1 ω 21=2 "# ;(2.4) where α *¼ α 1d1þ α 2d2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 α 2 1 α 2 2Þk2 σ 2ð α 1d1þ α 2d2Þ2 q with d1¼ ω 2 1þ ω 1 ω 2ðffiffiffiffiffiffiffiffiffiffiffiffi 1− ρ 2 pþa ρ Þ,d2¼a ω 2 2þ ω 1 ω 2ðaffiffiffiffiffiffiffiffiffiffiffiffi 1− ρ 2 p− ρ Þ, k¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω 2 1þ ω 2 2þ2ffiffiffiffiffiffiffiffiffiffiffiffi 1− ρ 2 p ω 1 ω 2 qand σ 2¼ ω 2 1þa2 ω 2 2þ2a ρω 1 ω 2. The proof of Theorem 2.2 is given in Appendix. The probability δ ρ ab 5P(U> 0) is given by δ ρ ab ¼PðU>0Þ¼Z∞ 0 fUðuÞdu:(2.5) Density curves of Ufor different parameters and their corresponding probability P(U> 0) are given in Figures 3 and 4, respectively. For example, in Figure 3, we can see that the density curves are affected by the location parameter ξ 2 50, 2, 4, 2 with other parameters specified and ρ 50.25, together with their corresponding probabilities P(U>0)50.433816, 0.566184, 0.226627, 0.10565, respectively. We provide a different online calculator, still freely available online at https://probab.shinyapps.io/ProbU/ Figure 2. The density of Z when location parameters for first and second population equal 2 and 0 respectively, the scale parameters for first and second population equal 1 and 2 respectively Gainprobability in economics and finance 337 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025 Figure 3. The density of Uwhen location parameters of the first population equal 0, the scale parameters for first and second population equal 1 and 2 respectively Figure 4. The density of Uwhen location parameters of the first and second population equal 0 and 4, the scale parameters for first and second population equal 1 and 2 respectively AJEB 7,3 338 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025 2.3 Estimation of δ ab 5P(Z> 0) and δ ρ ab 5P(U>0) The above two theorems imply that if the location, scale and shape parameters are known in two populations, one can calculate the probabilities δ ab 5P(Z>0)andδ ρ ab 5P(U>0). But in most situations in real life the population parameters are not known. Thus estimating δ ab and δ ρ ab is necessary to be studied. The parametric estimation of P(X>Y)for normal distributions in the context of probabilistic environmental risk assignment was discussed by Jacobs et al. Here, we consider the method of moment estimator (MME) and maximum likelihood estimator (MLE). The estimators obtained in this two ways are denoted by bδMME ab and bδMLE ab respectively. Method of moment estimator (MME) Equation (2.6), below, relates location to mean and scale to standard deviation. ξ¼ μ ffiffiffi2 π rδ ω and ω 2¼ σ 212 π δ2  −1 ;(2.6) where δ¼ α =ffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ α 2 p. To obtain parameter estimators from samples, it is necessary to obtain an estimate of delta δ, which is the moment estimate bδin Equation (2.7) below. jbδj¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π 2bγ1j2 3 bγ1j2 3þðð4 π Þ=2Þ2 3 v u u u u t(2.7) wherebγ1is the sample skewness and the sign ofbδis the same as the sign ofbγ1. In turn, it is easy to obtain an estimate of the shape parameter α using Equation (2.8): b α ¼bδ ffiffiffiffiffiffiffiffiffiffiffiffi 1bδ2 q:(2.8) Rewriting Equation (2.6) in terms of sample estimates, as opposed to population parameters, renders Equation (2.9). bξ¼Xffiffiffi2 π rbδb ω and b ω 2¼S212 π bδ2  −1 :(2.9) Then we can obtain the bδMME ab by substituting the MMEs of ξ 1 ,ξ 2 , ω 1 , ω 2 , α 1 and α 2 in Equation (2.3). Maximum likelihood estimator (MLE) The log-likelihood function for the skew normal distribution is given by logðLÞ¼nlog 2 ω n 2logð2 π Þ1 2X n i¼1 xiξ ω  2 þX n i¼1 log Φ α xiξ ω  (2.10) To derive the MLEs of the parameters in skew normal distribution, we take the partial derivatives of the lnL functions with respect to parameters of interest and equal them to zero. Then the corresponding MLEs are obtained by solving the following equations: Gainprobability in economics and finance 339 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025 logL ξ¼X n i¼1 xiξ ω   α X n i¼1 f α xiξ ω  Φ α xiξ ω  ¼0; logL ω ¼−nþX n i¼1 xiξ ω  2  α X n i¼1 f α xiξ ω  Φ α xiξ ω  xiξ ω  ¼0; logL α ¼X n i¼1 f α xiξ ω  Φ α xiξ ω  xiξ ω  ¼0: (2.11) By the invariance property of MLEs, we obtain bδMLE ab by substituting the MLEs of ξ 1 ,ξ 2 , ω 1 , ω 2 , α 1 and α 2 in Equation (2.3). To derive bδMME ρ ab and bδMLE ρ ab under matched data setting, instead of estimating ξ 1 ,ξ 2 , ω 1 , ω 2 , α 1 and α 2 we estimate ξ * 5ξ 1 þaξ 2 þb, σ 2¼c0Σc¼ ω 2 1þa2 ω 2 2þ2a ρω 1 ω 2and α *¼δ* ffiffiffiffiffiffiffiffiffiffiffiffiffi 1δ2 * q ¼ α 1d1þ α 2d2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 α 2 1 α 2 2k2 σ 2ð α 1d1þ α 2d2Þ2 q (2.12) by using the same methods as in the independent setting. Then by substituting the MMEs and MLEs of ξ * , σ 2 and α * in Equation (2.5), we can get bδMME ρ ab and bδMLE ρ ab . 3. Probability of being better off or worse off by how much? Delta log-skewnormal distributions The log-normal distributions is very popular in both theory and applications of probability and statistics. However, when modeling complex random phenomena in many applied areas, there are a real need of more flexible models which extend the log-normal. Lin and Stoyanov introduced the LSN distribution which is an extension for modeling positive data using log-normal models. A positive random variable Xis LSNly distributed (i.e. X∼ LSN( μ , σ , α )) if the logarithm of Xis skew normally distributed with location parameter μ , scale parameter σ and slant parameter α .Letw(.) and Φ(.) be respectively probability density function (pdf) and cumulative distribution function (cdf) of the standard normal distribution, then we have that fXðxÞ¼ 2 x σ wlog x; μ ; σ 2Φ α log x μ σ  ;x>0:(3.1) AJEB 7,3 340 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025 However, let us now consider lognormal statistics. The log transformed mean and standard deviation for Business A is 2.4 and 0.25 (variance is 0.0625), respectively. These values are 2.0 and 0.9 (variance is 0.81) for Business B. Thus, the probability that a randomly selected person in Business A will waste more time than a randomly selected person in Business B is approximately 0.67, despite the miniscule conventional effect size in the previous paragraph. In addition, it is possible to construct a G-P diagram with more nuanced information. One capability of G-P diagrams is that effects can be reported as multiples. For example, it is possible to concern ourselves with the probability that time-wasting is twice, thrice, etc. that in one business relative to the other. The G-P diagram renders clear that which would be obscured otherwise; time-wasting is much more of a problem for Business A than for Business B (see Figure 8). Imagine two large cities, City A and City B and an economist is concerned with comparing commuting times. Suppose commuting times are lognormally distributed and the mean commuting time for City A is 41.264 min, the standard deviation is 74.054 and the variance is 5484.041. For City B, these values are 30.114, 33.641 and 1131.691, respectively. There seems an obvious advantage for City B over City A in that the mean commute is shorter for City B than for City A by 11.15 min. The mean of the logarithmically transformed distribution for City A is 3.0, the standard deviation is 1.2 and the variance is 1.44. For City B, these values are 3.0, 0.90 and 0.81, respectively. The probability that a randomly selected person from City A would have a shorter commuting time than a randomly selected person from City B is 0.50; despite appearances based on the previous paragraph, there is no probabilistic advantage for City A over City B. Moreover, the G-P diagramissymmetric;aseachblackbarinFigure 9 isthesamesizeasitscorrespondinggraybar, thereisnoprobabilisticadvantageforeithercity, no matter the multiple under consideration. Figure 8. The probability of being better off or worse off, by differing multiples Gainprobability in economics and finance 347 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025 Consider another example, using the same log normal parameter values as the previous example, but with the addition of considering shape. Suppose that the shape parameter for City A is 0.10 and the shape parameter for City B is 0.10. In that case, the probability that a randomly selected person in City A would have a longer commuting tie than a randomly selected person in City B is 0.54, not 0.50. And the probability that a randomly selected person in City B would have a longer commuting time than a randomly selected person in City A is 0.46. Thus, in contrast to the previous example, the slight difference in shape parameters leads to a clear probabilistic disadvantage for City A relative to City B. In addition, in contrast to Figure 9 that is symmetrical, Figure 10 that illustrates the present example is asymmetric. 5. Application: real data In the previous section, we examined invented examples to demonstrate the lessons that can be learned by constructing G-P diagrams. In this section, we present analyses of real data. 5.1 NBA guards (2022–2023 regular season) We downloaded data from the following website: https://www.espn.com/nba/stats/player/_/ season/2023/seasontype/2/table/defensive/sort/avgSteals/dir/desc. Among other types of information, it contains data for minutes played by point guards and shooting guards. Let us compare them using traditional normal statistics versus skew normal statistics. The means for point guards and shooting guards are 24.0 and 19.8, respectively. The standard deviations are 8.8 and 9.4, respectively. In addition, the skews are 0.3 and 0.2, respectively. Thus, point guards average 4.2 more minutes per game than shooting guards. Figure 9. The probability of being better off by varying multiples, with respect to commuting times, depending on city AJEB 7,3 348 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025 We might ask about how likely a randomly chosen point guard would average more minutes than a randomly chosen shooting guard. Based on the difference in averages, we might guess that the probability that randomly chosen point guard would average more minutes than a randomly chosen shooting guard would be greater than 0.50, but perhaps not too much more than that. But rather than guess, let us run out the calculations. As usual, the first step is to find the skew normal values for location, scale and shape. For point guards these are 31.4, 11.5 (scale squared is 131) and 1.4, respectively. For shooting guards, these are 13.1, 11.6 (scale squared is 134) and 1.1, respectively. Based on these values, the probability that a randomly selected point guard would play more minutes per game than a randomly selected shooting guard is 0.63. Figure 11 provides a G-P diagram. An interesting aspect of the figure is that the middle two bars are not very far apart, thereby indicating that at small amounts of difference in minutes played, there is only a slight probabilistic advantage for point guards over shooting guards. However, comparing black against gray bars at greater extremes indicates a strong probabilistic advantage for point guards over shooting guards. The G-P diagram suggests a more subtle substantive story than does a mere difference in means or difference in locations. 5.2 Living space for males versus females Hyman et al. (2002) obtained data comparing living space for their male and female participants, in square feet. The mean amount of living space for men is 2300 and the standard deviation is 705. The mean amount of living space for women is 2186 and the standard deviation is 671. The effect size is miniscule and not statistically significant: Cohen’s d50.04, t(564) < 1. However, the data are lognormally distributed and a G-P diagram Figure 10. Probability of being better off or worse off, by varying multiples, with respect to commuting times, depending on city and taking shape parameters into account Gainprobability in economics and finance 349 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025 suggests a very different conclusion. To commence, let us find the means and standard deviations after a logarithmic transformation. The transformed mean and standard deviation for males is 7.69 and 0.31 (variance 50.0961), respectively. The transformed mean and standard deviation for females is 7.64 and 0.32 (variance 50.1024), respectively. Thus, the probability that a randomly selected male will have more living space than a randomly selected female is 0.54 whereas the probability that a randomly selected female will have more living space than a randomly selected male is only 0.46, a more impressive difference than is suggested by the Cohen’sdor the nonsignificant p-value. For a more fine-grained analysis, Figure 12 provides the G-P diagram. Because it is convenient to express gender effects with respect to living space in multiples, Figure 12 is in terms of multiples. That is, the black bars provide the probabilities that a randomly selected male will have 1.0 to 1.5 times as much living space as a randomly selected female, 1.5 to 2.0 times as much living space and so on. The gray bars provide analogous probabilistic advantages for females. Each black bar is larger than its corresponding gray bar thereby indicating that although the probabilistic advantage for males over females is not large, it is not trivial either. Again, we see that a G-P diagram provides both more valid and more nuanced conclusions than typical differences between means, Cohen’sd, or significance tests. Consider precipitation in Buffalo for 306 days starting in June in 2019 versus 2020 [https://www.ncdc.noaa.gov/cdo-web/]. Because there are days when there is no precipitation, the data fall under the category of delta skew lognormal distributions. Thus, there are the following bullet-listed summary parameter estimates for each year: locations are 2.8712 for 2019 and 2.8327 for 2020, squared scales are 3.884841 for 2019 and 3.806401 for 2020, Figure 11. Probability of varying degrees of advantage (or disadvantage) for shooting guards or point guards with respect to minutes played AJEB 7,3 350 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025 shapes are 2.929 for 2019 and -3.019 for 2020 and probabilities of zeroes (lack of precipitation for a day) are 0.49346 for 2019 and 0.59150 for 2020. This is an especially interesting example because (a) it engages all the parameters of delta skew lognormal distributions and (b) ties are possible. A tie can occur if there is no precipitation on a randomly selected day in 2019 or 2020. The probability of more precipitation on a randomly selected day in 2019 than on a randomly selected day in 2020 is 0.404. The probability of more precipitation on a randomly selected day in 2020than on a randomly selected day in 2019 is 0.304 and the probability of a tie is 0.292. Thus, there is a general probabilistic advantage for 2019 over 2020, assuming precipitation is positive. (If precipitation is negative, then there is a probabilistic disadvantage for 2019 relative to 2020. However, we will assume that precipitation is positive.) Figure 13 shows a G-P diagram. However, the diagram differs from the others because ties are possible. The black bar in the middle shows the probability of ties. The dark gray bars show the probability of more precipitation in 2019 than in 2020, by varying millimeters. And the light gray bars show the probability of more precipitation in 2020 than in 2019, by varying millimeters. The figure shows that the probabilistic advantage for 2019 over 2020 is clear both at small amounts and at large ones. 6. Discussion The G-P diagrams based on both invented and real data indicate important lessons for economics researchers. One lesson is that both summary statistics and significance tests can Figure 12. Probability that males or females are better off, by varying multiples, with respect to living space Gainprobability in economics and finance 351 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025 be very misleading. It is possible for a difference in means to hide that there is no probabilistic advantage for either group over the other. However, it is possible, too, for a lack of difference in means to hide a very sizable probabilistic advantage for one group over the other. One solution is to use more appropriate summary statistics. For example, if the data are obtained from a skew-normal distribution, locations, scales and shapes can be better than means and standard deviations. However, even distribution-appropriate sample statistics can be misleading. For instance, we have seen an example where the locations and scales of two groups are the same and the difference in skews very small. Nevertheless, the G-P diagram (Figure 7) indicated a very large probabilistic difference. Although it is dramatic to show that traditional statistics and G-P diagrams can come to opposing conclusions, G-P diagrams have another advantage. Specifically, G-P diagrams can support subtle conclusions that cannot be addressed by either summary statistics or significance tests. Figure 11 provides a nice case in point. It shows that at relatively small differences in minutes played, there is only a miniscule probabilistic advantage for point guards over shooting guards. But at more extreme differences in minutes played, the probabilistic advantage for point guards increases substantially. In conclusion, there is no need for economics researchers to constrain themselves by depending solely on summary statistics and null hypothesis significance tests. As we have seen, G-P diagrams provide opportunities to contradict the seeming implications of summary statistics or null hypothesis significance tests. In addition, even in cases where there is no contradiction, such as the data pertaining to minutes played by National Basketball Association (NBA) point guards and shooting guards, G-P diagrams provide much more nuanced information than do summary statistics and null hypothesis significance tests. We Figure 13. Probability that there is a precipitation advantage, by varying degrees, for 2019 or 2020, or that there is no advantage for either year (black bar) AJEB 7,3 352 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025 hope and expect that future economics researchers will avail themselves of the potential advantages to be enjoyed by exploiting the capabilities of G-P diagrams for providing subtle probabilistic information about comparisons of interest. References Azzalini, A. and Valle, A.D. (1996), “The multivariate skew normal distribution”,Biometrika, Vol. 83 No. 4, pp. 715-726, doi: 10.1093/biomet/83.4.715. Hubbard, R. (2016), Corrupt Research: the Case for Reconceptualizing Empirical Management and Social Science, Sage Publications, Los Angeles, CA. Hyman, M.R., Ganesh, G. and McQuitty, S. (2002), “Augmenting the household affluence construct”, Journal of Marketing Theory and Practice, Vol. 10 No. 3, pp. 13-31, doi: 10.1080/10696679.2002. 11501917. McQuitty, S. (2004), “Statistical power and structural equation models in business research”,Journal of Business Research, Vol. 57 No. 2, pp. 175-183, doi: 10.1016/S0148-2963(01)00301-0. McQuitty, S. (2018), “Reflections on ‘Statistical power and structural equation models in business research’”,Journal of Global Scholars of Marketing Science, Vol. 28 No. 3, pp. 272-277, doi: 10. 1080/21639159.2018.1434806. Tong, T., Wang, T., Trafimow, D. and Wang, C. (2022), “The probability of being better or worse off, and by how much, depending on experimental conditions with skew normal populations”,in SriboonchittaKreinovich, W.Y.E.S.V. (Ed.), Credible Asset allocation, Optimal Transport Methods, and Related Topics (141-149), Springer-Verlag. Trafimow, D. (2019), “A frequentist alternative to significance testing, p-values, and confidence intervals”, Econometrics,Vol.7No.2,pp.1-14,availableat:https://www.mdpi.com/2225-1146/7/2/26 Trafimow, D., Hyman, M.R., Kostyk, A., Wang, C. and Wang, T. (2021), “The harmful effect of null hypothesis significance testing on marketing research: an example”,Journal of Business Research, Vol. 125, pp. 39-44, doi: 10.1016/j.jbusres.2020.11.069. Trafimow, D., Hyman, M.R., Kostyk, A., Wang, Z., Tong, T., Wang, T. and Wang, C. (2022), “Gainprobability diagrams in consumer research”,International Journal of Market Research, Vol. 64 No. 4, pp. 470-483, doi: 10.1177/14707853221085509. Wang, Z., Wang, T., Trafimow, D. and Xu, Z. (2022), “A different kind of effect size based on samples from two populations with delta log-skew-normal distributions”, in Ngoc Thach, N., Ha, D.T., Trung, N.D. and Kreinovich, V. (Eds), Prediction and Causality in Econometrics and Related Topics. ECONVN 2021. Studies in Computational Intelligence, Springer, Cham, Vol. 983, doi: 10. 1007/978-3-030-77094-5_10. Ziliak, S.T. and McCloskey, D.N. (2016), The Cult of Statistical Significance: How the Standard Error Costs Us Jobs, Justice, and Lives, The University of Michigan Press, Ann Arbor, MI. Appendix Source(s): Appendix by authors In this section, we provide detailed proofs of Theorem 2.1,2.2,3.1,Theorem 3.2. A.1 . Proof of Theorem 2.1 The joint pdf of X∼SN(ξ 1 , ω 1 , α 1 ) and Y∼SN(ξ 2 , ω 2 , α 2 )is fðx;yÞ¼4 f x;ξ1; ω 2 1 f y;ξ2; ω 2 2Φ α 1 xξ1 ω 1  Φ α 2 yξ2 ω 2  : Let Z5XþaY þb,Y5Y, then X5ZaY b,Y5Yand it is easy to see that the Jacobian Jof this transformation is 1. Thus, we obtain the pdf of Zas Gainprobability in economics and finance 353 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025 fZðzÞ¼Z∞ −∞ fXðzay bÞfYðyÞdy; which is fZðzÞ¼4Z∞ −∞ f zay b;ξ1; ω 2 1 f y;ξ2; ω 2 2Φ α 1 zay bξ1 ω 1  Φ α 2 yξ2 ω 2  ¼2 πω 1 ω 2Z∞ −∞ e −ðzaybξ1Þ2 2 ω 2 1þðyξ2Þ2 2 ω 2 2  Φ2 α 1 zay bξ1 ω 1 ; α 2 yξ2 ω 2  0;0;I2  dy ¼4 ω * f z;ξ1þaξ2þb; ω 2 1þa2 ω 2 2Z∞ −∞ f y;c; ω 2 * 3Φ2 α 1 zay bξ1 ω 1 ; α 2 yξ2 ω 2  0;0;I2  dy; (A.1) where ω *¼ ω 1 ω 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω 2 1þa2 ω 2 2 pand c¼a ω 2 2ðzbξ1Þþ ω 2 1ξ2 2 ω 2 1þa2 ω 2 2 : Let T¼y−c ω *. Then Equation(A.1) will be fZðzÞ¼4 f z;ξ1þaξ2þb; ω 2 1þa2 ω 2 23 ETΦ2 α 1 zat ω *þcξ1b ω 1 ; α 2 t ω *þaþcξ2 ω 2 ! 0 ;0;I2 "#() (A.2) with T∼N(0, 1). If we denote X1¼ α 1 z−aðt ω *þcÞ−ξ1−b ω 1and X2¼ α 2 t ω *þaþc−ξ2 ω 2, then ET_in Equation (A.2) can be simplified as ETΦ2ðX1;X2Þ0;0;I2  ¼ET½PðU1≤X1;U2≤X2jX1;X2Þ ¼PðU1≤X1;U2≤X2Þ ¼PðU1X1≤0;U2X2≤0Þ; (A.3) where U 1 ,U 2 and Tare independent standard normal random variables. Note that E(U i X i )5E(X i ) and Var(U i X i )51þVar(X i ) for i51, 2. Additionally, CovðU1X1;U2X2Þ¼CovðX1;X2Þ¼−a α 1 α 2 ω 2 * ω 1 ω 2 : Therefore, the joint distribution of ðU1−X1;U2−X2Þ0is ðU1X1;U2X2Þ0∼N2ð μ ;ΣJÞ; where AJEB 7,3 354 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025 μ ¼ α 1 zac ξ1b ω 1  α 2 cξ2 ω 2 0 B B B @1 C C C Aand ΣJ¼ 1þ α 2 1 a2 ω 2 * ω 2 1a α 1 α 2 ω 2 * ω 1 ω 2 a α 1 α 2 ω 2 * ω 1 ω 2 1þ α 2 2 ω 2 * ω 2 2 0 B B B B @ 1 C C C C A: Then Equation(A.3) can be written to be ETΦ2ðX1;X2Þ0;0;I2  ¼Φ2ð− μ ;0;ΣJÞ; and therefore, Equation (A.1) will be f1;2ðzÞ¼4 f z;ξ1þaξ2þb; ω 2 1þa2 ω 2 2Φ2½Bðzðξ1þaξ2þbÞÞ;02;ΣJ;(A.4) where B¼ α 1 ω 1 ω 2 1þa2 ω 2 2 ;a α 2 ω 2 ω 2 1þa2 ω 2 2  0. A.2 . Proof of Theorem 2.2 Let c5(1,a)0, we are trying to find the distribution of U5X 1 þaX 2 þb5c0Xþb. First we derive the moment generating function (mgf) of U: MUðtÞ¼E½expðtc0XþtbÞ ¼2 exp tðc0ξþbÞþ1 2t2c0Σc  Φδ0Σ1=2ct:(A.5) It is easy to obtain that c0ξþb5ξ 1 þaξ 2 þband σ 2¼c0Σc¼ ω 2 1þa2 ω 2 2þ2a ρω 1 ω 2. From Wang et al., Σ1=2¼ ω 2 1þffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ρ 2 q ω 1 ω 2 k ρω 1 ω 2 k ρω 1 ω 2 k ω 2 2þffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ρ 2 q ω 1 ω 2 k 2 6 6 6 6 6 4 3 7 7 7 7 7 5 ; where k¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω 2 1þ ω 2 2þ2ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ρ 2 p ω 1 ω 2 q: Thus, δ * 5δ0Σ 1/2 c(c0Σc) 1/2 .Byequation (2.1), we obtain that U∼SN(ξ 1 þaξ 2 þb, σ 2 , α * ) with α *¼δ* ffiffiffiffiffiffiffiffiffiffiffiffiffi 1δ2 * q ¼ α 1d1þ α 2d2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 α 2 1 α 2 2k2 σ 2ð α 1d1þ α 2d2Þ2 q; (A.6) where d1¼ ω 2 1þ ω 1 ω 2ðffiffiffiffiffiffiffiffiffiffiffiffi 1− ρ 2 pþa ρ Þand d2¼a ω 2 2þ ω 1 ω 2ðaffiffiffiffiffiffiffiffiffiffiffiffi 1− ρ 2 pþ ρ Þ. Then the density of Uis given by Gainprobability in economics and finance 355 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025 fUðuÞ¼2 f ðu;c0ξþb; σ 2Φ α *ub0ξb σ (A.7) ¼2 f u;ξ1þaξ2þb; ω 2 1þa2 ω 2 2þ2a ρω 1 ω 2 3Φ α *ðuξ1aξ2bÞ  ω 2 1þa2 ω 2 2þ2a ρω 1 ω 21=2 "# :(A.8) A.3 . Proof of Theorem 3.1 The joint pdf of X∼Δðδ1; μ 1; σ 2 1Þand Y∼Δðδ2; μ 2; σ 2 2Þis gðx;yÞ¼ δ1δ2ifx¼0;y¼0 ð1δ1Þð1δ2ÞfXðxÞfYðyÞifx>0;y>0 δ1ð1δ2ÞfYðyÞifx¼0;y>0 δ2ð1δ1ÞfXðxÞifx>0;y¼0 8 > > < > > : (A.9) where f X (x), f Y (y) are the probability distribution functions of Xwhen x> 0 and Ywhen y>0, respectively. (1) For x>0,y> 0, let Z5XþaY þb,Y5Yso that X5ZaY b,Y5Y. It is easy to see that the Jacobian Jof this transformation is 1. Thus, the pdf of Zas fZðzÞ¼ZfXðzay bÞfYðyÞdy; (2) For x50, y> 0, we have x50, Z5aY þband the pdf of Zis fZðzÞ¼δ1ð1δ2ÞfY zb a  1 jaj: (3) Similarly, for x>0,y50, the pdf of Zis fZðzÞ¼δ2ð1δ1ÞfXðzbÞ: Note that the sign of adetermines the range of Z.Ifa< 0, then the pdf of Z5XþaY þbis fZðzÞ¼ Z∞ z−b a fXðzay bÞfYðyÞdy 1 afY zb a ifz<b δ1δ2ifz¼b Z∞ 0 fXðzay bÞfYðyÞdy þfXðzbÞifz>b: 8 > > > > > > < > > > > > > : (A.10) Also if a> 0, the pdf of Zis given by fZðzÞ¼ δ1δ2ifz¼b Zz−b a 0 fXðzay bÞfYðyÞdy þ1 afY zb a  þfXðzbÞifz>b 8 > < > : (A.11) so that desired result follows. AJEB 7,3 356 Downloaded from http://www.emerald.com/ajeb/article-pdf/7/3/333/358346/ajeb-05-2023-0045.pdf by ZBW German National Library of Economics user on 16 December 2025