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The a Priori Procedure for Estimating the Cohen’s Effect Size Under Independent Skew Normal Settings

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Integrated Uncertainty in Knowledge Modelling and Decision Making (IUKM 2023)

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.

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Correspondence to Tonghui Wang .

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Appendix

Appendix

A. Proof of Theorem 1

Proof

  1. (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 \(\varDelta =I_2 + diag \,(n\lambda _1^2,\ m\lambda ^2_2)-\mathbf {b_*b_*'}/\tau ^2\).

  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

$$ P\left[ f_1 \sigma _{_{T_{11}}}\le (T-\mu _{_T})/a\le f_2\sigma _{_{T_{11}}}\right] , $$

then

$$ P\left[ f_1 a\frac{\sigma _{_{T_{11}}}}{\sigma _{_T}} \le Z_* \le f_2 a\frac{\sigma _{_{T_{11}}}}{\sigma _{_T}}\right] , $$

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).

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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

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  • DOI: https://doi.org/10.1007/978-3-031-46775-2_10

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