Abstract
Cohen’s d is the most popular effect size index for traditional experimental designs, and it is desirable to know the minimum sample size necessary to obtain a sample value that is a good estimate of the population value. The present work addresses that lack with an application to independent samples under the umbrella of skew normal distributions. In addition to derivations of relevant equations, there is a link to a free and user-friendly computer program. Finally, we present computer simulations and a worked example to support the equations and program.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Azzalini, A., Dalla Valle, A.: The multivariate skew-normal distribution. Biometrica 83(4), 715–726 (1996)
Blanca, M.J., Arnau, J., López-Montiel, D., Bono, R., Bendayan, R.: Skewness and kurtosis in real data samples. Methodol. Eur. J. Res. Meth. Behav. Soc. Sci. 9(2), 78–84 (2013). https://doi.org/10.1027/1614-2241/a000057
Chen, X., Trafimow, D., Wang, T., Tong, T., Wang, C.: The APP procedure for estimating Cohen’s effect size. Asian J. Econ. Bank. 5(3), 289–306 (2021). https://doi.org/10.1108/AJEB-08-2021-0095
Cohen, J.: Statistical Power Analysis for the Behavioral Sciences, 2nd edn. Erlbaum, Hillsdale (1988)
Cao, L., Wang, C., Chen, X., Trafimow, D., Wang, T.: The a priori procedure (APP) for estimating the Cohen’s effect size for matched pairs under skew normal settings. Int. J. Uncert. Fuzziness Knowl. Based Syst. (2023, accepted)
González-Farías, G., Domínguez-Molina, A., Gupta, A.K.: Additive properties of skew normal random vectors. J. Stat. Plan. Infer. 126(2004), 521–534 (2004)
Ho, A.D., Yu, C.C.: Descriptive statistics for modern test score distributions: skewness, kurtosis, discreteness, and ceiling effects. Educ. Psychol. Measur. 75, 365–388 (2015). https://doi.org/10.1177/0013164414548576
Micceri, T.: The unicorn, the normal curve, and other improbable creatures. Psychol. Bull. 105, 156–166 (1989). https://doi.org/10.1037/0033-2909.105.1.156
New Mexico State University (2018/19) Budget estimate. [Salaries] (2018/19), Las Cruces, N.M. The University 1994/95
Trafimow, D.: Using the coefficient of confidence to make the philosophical switch from a posteriori to a priori inferential statistics. Educ. Psychol. Measur. 77, 831–854 (2017). https://doi.org/10.1177/0013164416667977
Trafimow, D.: A frequentist alternative to significance testing, p-values, and confidence intervals. Econometrics 7(2), 1–14 (2019). https://www.mdpi.com/2225-1146/7/2/26
Trafimow, D., MacDonald, J.A.: Performing inferential statistics prior to data collection. Educ. Psychol. Measur. 77, 204–219 (2017). https://doi.org/10.1177/0013164416659745
Trafimow, D., Wang, T., Wang, C.: From a sampling precision perspective, skewness is a friend and not an enemy! Educ. Psychol. Measure. 79, 129–150 (2018). https://doi.org/10.1177/0013164418764801
Wang, C., Wang, T., Trafimow, D., Chen, J.: Extending a priori procedure to two independent samples under skew normal setting. Asian J. Econ. Bank. 03(02), 29–40 (2019)
Wang, C., Wang, T., Trafimow, D., Myüz, H.A.: Desired sample size for estimating the skewness under skew normal settings. In: Kreinovich, V., Sriboonchitta, S. (eds.) TES 2019. SCI, vol. 808, pp. 152–162. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-04263-9_11
Wang, C., Wang, T., Trafimow, T., Zhang, X.: Necessary sample size for estimating the scale parameter with specified closeness and confidence. Int. J. Intell. Technol. Appl. Stat. 12(1), 17–29 (2019). https://doi.org/10.6148/IJITAS.201903_12(1).0002
Wang, C., Wang, T., Trafimow, D., Li, H., Hu, L., Rodriguez, A.: Extending the a priori procedure (APP) to address correlation coefficients. In: Ngoc Thach, N., Kreinovich, V., Trung, N.D. (eds.) Data Science for Financial Econometrics. SCI, vol. 898, pp. 141–149. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-48853-6_10
Wang, Z., Wang, C., Wang, T.: Estimation of location parameter on the skew normal setting with known coefficient of variation and skewness. Int. J. Intell. Technol. Appl. Stat. 9(3), 45–63 (2016)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Appendix
Appendix
A. Proof of Theorem 1
Proof
-
(a)
By Proposition 1 with \(\omega _1=\omega _2=\omega \), we know that \(\bar{X}-\bar{Y}\) is closed skew normal distributed and its pdf is given in Eq. (3) with \(\tau =\frac{m+n}{mn}\) and \(\boldsymbol{b}'=\omega (\lambda _1, -\lambda _2)\). From Lemma 1, we know that
$$ \frac{(n-1)S^2_X}{\omega ^2}\sim \chi ^2_{n-1}\qquad \text{ and }\qquad \frac{(m-1)S^2_Y}{\omega ^2}\sim \chi ^2_{m-1} $$are independent so that
$$ U\equiv \frac{(m+n-2)S_p}{\omega ^2} \sim \chi ^2_{\nu } $$and the pdf of U
$$ f_U(u)=\frac{1}{2^{\nu /2-1}\varGamma (\nu /2)}u^{\nu /2-1}e^{-u/2}, $$where \(\nu =m+n-2\). Since \(Z=\bar{X}-\bar{Y}\) and \(S_p^2\) are independent, we can define \(T=\frac{Z}{S_p}=\frac{Z/\omega }{\sqrt{U/\nu }}\), and the density function is
$$ g_T(t)=\int _{0}^{\infty }f_Z(\omega tu)f_U(\sqrt{\nu }u) \omega \sqrt{\nu }u du, $$from which, let \(v=\sqrt{\nu }u\),
$$\begin{aligned} \begin{aligned} g_{_T}(t)= \frac{4}{2^{\nu /2-1}\sqrt{\nu }\varGamma (\nu /2)}\int _0^\infty v^{\nu }e^{-v^2/2}\phi \left( \frac{v}{\sqrt{\nu }}t;\ \theta ,\ \ \tau ^2\right) \\ \varPhi _2\left[ B\left( \frac{v}{\sqrt{\nu }}t-\theta \right) ;\ \boldsymbol{0}_2,\ \varDelta \right] dv, \end{aligned} \end{aligned}$$where \(\tau ^2=\frac{m+n}{mn}\), \(\mathbf {b_*}=(\lambda _1,\ -\lambda _2)'\), \(B=\frac{1}{\tau ^2}\mathbf {b_*}\) and \(\varDelta =I_2 + diag \,(n\lambda _1^2,\ m\lambda ^2_2)-\mathbf {b_*b_*'}/\tau ^2\).
-
(b)
Note that both \(\bar{X}\) and \(\bar{Y}\) are skew normal distributed and they are independent, the mean and variance of Z are
$$ E(Z)=\xi _1-\xi _2 +\sqrt{\frac{2}{\pi }}(\delta _1 -\delta _2)\omega \quad \text{ and }\quad Var(Z)=\omega ^2 \left[ \frac{n+m}{mn} -\frac{2}{\pi }(\delta ^2_1+\delta _2^2)\right] , $$respectively. Also it is easy to obtain
$$\begin{aligned} E\left( \frac{1}{S_p}\right) =\frac{\sqrt{\nu }}{\omega }E(\frac{1}{\sqrt{U}}) =\sqrt{\frac{\nu }{2\omega }}\frac{\varGamma [(\nu -1)/2]}{\varGamma (\nu /2)} \quad \text{ and }\quad E\left( \frac{1}{S_p^2}\right) =\frac{\nu }{(\nu -2)\omega ^2}. \end{aligned}$$(8)Let \(a=\sqrt{\frac{\nu }{2}}\frac{\varGamma [(\nu -1)/2]}{\varGamma (\nu /2)}\). Now the mean of T is
$$ \mu _{_T}=E(\bar{X}-\bar{Y})E\left( \frac{1}{S_p}\right) =a\left[ \theta +\sqrt{\frac{2}{\pi }}(\delta _1-\delta _2)\right] . $$For the variance of T, we have
$$ E(T^2)=E(\bar{X}-\bar{Y})^2E\left( \frac{1}{S_p^2}\right) =\{Var(\bar{X})+Var(\bar{Y})+[E(\bar{X}-\bar{Y})]^2\}E\left( \frac{1}{S_p^2}\right) $$so that \(\sigma ^2_T=E(T^2)-\mu _{_T}^2\), the desired result follows after simplification.
B. Proof of Theorem 2
Proof
Let \(\hat{T}=\frac{T}{a}-\sqrt{\frac{2}{\pi }}(\delta _1-\delta _2)\), the unbiased estimator of \(\theta \). Then Eq. (6) can be rewritten to be
then
where a is a constant related to n given in (ii) of Theorem 1. If we denote \(Z_*=\frac{T-E(T)}{\sigma _{_T}}\), then Eq. (7) holds in which \(g_{_{Z_*}}(z)\) is the density function of \(Z_*\), and \(g_{_{Z_*}}(z)=\sigma _{_T}g_{_T}(z+\mu _{_T})\), where \(g_{_T}(t)\) is the density function of T given in the Theorem 1. If we have had an estimate \(\hat{\theta }\) of \(\theta \) by the previous information, then the required sample size n can be obtained by solving Eq. (7).
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Wang, C. et al. (2023). The a Priori Procedure for Estimating the Cohen’s Effect Size Under Independent Skew Normal Settings. In: Huynh, VN., Le, B., Honda, K., Inuiguchi, M., Kohda, Y. (eds) Integrated Uncertainty in Knowledge Modelling and Decision Making. IUKM 2023. Lecture Notes in Computer Science(), vol 14375. Springer, Cham. https://doi.org/10.1007/978-3-031-46775-2_10
Download citation
DOI: https://doi.org/10.1007/978-3-031-46775-2_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-46774-5
Online ISBN: 978-3-031-46775-2
eBook Packages: Computer ScienceComputer Science (R0)