Module stikpetP.effect_sizes.convert_es

Functions

def es_convert(es, fr, to, ex1=None, ex2=None)

Convert Effect Sizes

Function to convert various effect sizes to other effect sizes.

Parameters

es : float

the effect size value to convert

fr : string

name of the original effect size (see details)

to : string

name of the effect size to convert to (see details)

ex1 : float or string, optional

extra for some conversions (see details)

ex2 : float or string, optional

extra for some conversions (see details)

Returns

res : float

the converted effect size value

Notes

COHEN D

Cohen d to Odds Ratio

fr="cohend", to="or", ex1="chinn"

This uses (Chinn, 2000, p. 3129): $$OR = e^{d\times 1.81}$$

fr="cohend", to="or", ex1="borenstein"

This uses (Borenstein et. al, 2009, p. 3): $$OR = e^{\frac{d\times\pi}{\sqrt{3}}}$$

Cohen d to Rank Biserial (Cliff delta)

fr = "cohend", to = "rb"

This uses (Marfo & Okyere, 2019, p. 4): $$r_b = \frac{2\times\Phi\left(\frac{d}{2}\right)-1}{\Phi\left(\frac{d}{2}\right)}$$

COHEN D'

Convert a Cohen d' to Cohen d

fr="cohendos" to="cohend"

This uses (Cohen , 1988, p. 46): $$d = d'\times\sqrt{2}$$

COHEN F

Cohen f to Eta-squared

fr="cohenf" to="etasq"

This uses (Cohen, 1988, p. 284): $$\eta^2 = \frac{f^2}{1 + f^2}$$

COHEN H'

Cohen h' to Cohen h

fr= "cohenhos", to = "cohenh"

This uses (Cohen, 1988, p. 203): $$h = h' imes\sqrt{2}$$

COHEN U

Cohen U to Cohen d fr="cohenu1", to="cohend" fr="cohenu2", to="cohend" fr="cohenu3", to="cohend"

This uses (Cohen, 1988, p. 23): $$d = \Phi^{-1}\left(U_3\right)$$ $$d = 2\times\Phi^{-1}\left(U_2\right)$$ $$d = 2\times\Phi^{-1}\left(\frac{1}{2 - U_1}\right)$$

COHEN w

Cohen w to Contingency Coefficient

fr="cohenw", to="cc"

Cohen w to Cramér V GoF

fr="cohenw", to="cramervgof", ex1=k

This uses (Cohen, 1988, p. 223): $$v = \frac{w}{\sqrt{k - 1}}$$

Cohen w to Cramér V ind.

fr="cohenw", to="cramervind", ex1=r, ex2=c

This uses: $$v = \frac{w}{\sqrt{\min\left(r - 1, c - 1\right)}}

Cohen w to Fei

fr="cohenw", to="fei", ex1=minExp/n

This uses: $$Fei = \frac{w}{\sqrt{\frac{1}{p_E}-1}}

CRAMER V GoF

Cramer's v for Goodness-of-Fit to Cohen w

fr="cramervgof", to = "cohenw", ex1 = k

This uses (Cohen, 1988, p. 223): $$w = v\times\sqrt{k - 1}$$

CRAMER V Ind

Cramer's v for Independence to Cohen w

fr="cramervgof", to = "cohenw", ex1 = r, ex2 = c

This uses: $$w = v\times\sqrt{\min\left(r - 1, c - 1\right)}$$

EPSILON SQUARED

Epsilon Squared to Eta Squared

fr="epsilonsq", to="etasq", ex1 = n, ex2 = k

This uses: $$\eta^2 = 1 - \frac{\left(1 - \epsilon^2\right)\times\left(n - k\right)}{n - 1}$$

Epsilon Squared to Omega Squared

fr="epsilonsq", to="omegasq", ex1=MS_w, ex2 = SS_t

This uses: $$\hat{\omega}^2 = \epsilon^2\times\left(1 - \frac{MS_w}{SS_t + MS_w}\right)$$

ETA SQUARED

Eta squared to Cohen f

fr="etasq", to="cohenf"

This uses:

$$f = \sqrt{\frac{\eta^2}{1 - \eta^2}}$$

Eta squared to Epsilon Squared

fr="etasq", to="epsilonsq", ex1=n, ex2=k

This uses: $$\epsilon^2 = \frac{n\times\eta^2 - k + \left(1 - \eta^2\right)}{n - l}$$

FEI

Fei to Cohen w

fr="fei", to="cohenw", ex1=minExp/n

This uses: $$w = Fei\times\sqrt{\frac{1}{p_E}-1}$$

Fei to Johnston-Berry-Mielke E

fr="fei", to="jbme"

This uses: $$E = Fei^2$$

JOHNSTON-BERRY-MIELKE

Johnston-Berry-Mielke E to Cohen w

fr="jbme", to="cohenw", ex1=minExp/n

This uses (Johnston et al., 2006, p. 413): $$w = \sqrt{\frac{E\times\left(1 - \right)}{q}}$$

Johnston-Berry-Mielke E to Cohen w

fr="jbme", to="fei"

This uses: $$Fei = \sqrt(E)$$

ODDS RATIO

Odds Ratio to Cohen d

fr="or", to="cohend", ex1="chinn"

This uses (Chinn, 2000, p. 3129): $$d = \frac{\ln{\left(OR\right)}}{1.81}$$

fr="or", to="cohend", ex1="borenstein"

This uses (Borenstein et. al, 2009, p. 3): $$d = \ln\left(OR\right)\times\frac{\sqrt{3}}{\pi}$$

Odds Ratio to Yule Q

fr="or", to="yuleq"

This uses: $$Q = \frac{OR - 1}{OR + 1}$$

Odds Ratio to Yule Y

This uses $$Y = \frac{\sqrt{OR} - 1}{\sqrt{OR} + 1}$$

OMEGA SQUARED

Omega Squared to Epsilon Squared

fr="omegasq", to="epsilonsq", ex1=MS_w, ex2 = SS_t

This uses: $$\epsilon^2 = \frac{\hat{\omega}^2}{1 - \frac{MS_w}{SS_t + MS_w}}$$

RANK BISERIAL (CLIFF DELTA)

Rank Biserial (Cliff delta) to Cohen d

fr = "rb", to = "cohend"

This uses (Marfo & Okyere, 2019, p. 4): $$d =2 \times \Phi^{-1}\left(-\frac{1}{r_b - 2}\right)$$

Rank Biserial (Cliff delta) to Vargha-Delaney A

fr = "rb", to = "vda"

This uses: $$r_b = 2\times A - 1$$

VARGHA-DELANEY A

Vargha-Delaney A to Rank Biserial (Cliff delta)

fr = "cle", to = "rb"

This uses: $$A = \frac{r_b + 1}{2}$$

YULE Q

Yule Q to Odds Ratio

fr="yuleq", to="or"

This uses:

$$OR = \frac{1 + Q}{1 - Q}$$

Yule Q to Yule Y

fr="yuleq", to="yuley"

This uses: $$Y = \frac{1 - sqrt{1 - Q^2}}{Q}$$

YULE Y

Yule Y to Yule Q

fr="yuley", to=="yuleq"

This uses: $$Q = \frac{2\times Y}{1 + Y^2}$$

Yule Y to Odds Ratio

fr="yuley", to=="or"

This uses $$OR = \left(\frac{1 + Y}{1 - Y}\right)^2$$

References

Borenstein, M., Hedges, L. V., Higgins, J. P. T., & Rothstein, H. R. (2009). Converting Among Effect Sizes. In Introduction to Meta-Analysis. John Wiley & Sons, Ltd. https://doi.org/10.1002/9780470743386

Chinn, S. (2000). A simple method for converting an odds ratio to effect size for use in meta-analysis. Statistics in Medicine, 19(22), 3127–3131. <https://doi.org/10.1002/1097-0258(20001130)19:22<3127::aid-sim784>>3.0.co;2-m

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). L. Erlbaum Associates.

Johnston, J. E., Berry, K. J., & Mielke, P. W. (2006). Measures of effect size for chi-squared and likelihood-ratio goodness-of-fit tests. Perceptual and Motor Skills, 103(2), 412–414. https://doi.org/10.2466/pms.103.2.412-414

Author

Made by P. Stikker

Companion website: https://PeterStatistics.com
YouTube channel: https://www.youtube.com/stikpet
Donations: https://www.patreon.com/bePatron?u=19398076

Expand source code

def es_convert(es, fr, to, ex1=None, ex2=None):
    '''
    Convert Effect Sizes
    --------------------
    
    Function to convert various effect sizes to other effect sizes.
    
    Parameters
    -----------
    es : float
        the effect size value to convert
    fr : string
        name of the original effect size (see details)
    to : string
        name of the effect size to convert to (see details)
    ex1 : float or string, optional
        extra for some conversions (see details)
    ex2 : float or string, optional
        extra for some conversions (see details)
    
    Returns
    -------
    res : float
        the converted effect size value
    
    Notes
    -----
    **COHEN D**
    
    **Cohen d to Odds Ratio**
    
    fr="cohend", to="or", ex1="chinn"
    
    This uses (Chinn, 2000, p. 3129):
    $$OR = e^{d\\times 1.81}$$
    
    fr="cohend", to="or", ex1="borenstein"
    
    This uses (Borenstein et. al, 2009, p. 3):
    $$OR = e^{\\frac{d\\times\\pi}{\\sqrt{3}}}$$
    
    **Cohen d to Rank Biserial (Cliff delta)**
    
    fr = "cohend", to = "rb"
    
    This uses (Marfo & Okyere, 2019, p. 4):
    $$r_b = \\frac{2\\times\\Phi\\left(\\frac{d}{2}\\right)-1}{\\Phi\\left(\\frac{d}{2}\\right)}$$
    
    **COHEN D'**
    
    *Convert a Cohen d' to Cohen d*
    
    fr="cohendos" to="cohend"
    
    This uses (Cohen , 1988, p. 46):
    $$d = d'\\times\\sqrt{2}$$
    
    **COHEN F**
    
    *Cohen f to Eta-squared*
    
    fr="cohenf" to="etasq"
    
    This uses (Cohen, 1988, p. 284):
    $$\\eta^2 = \\frac{f^2}{1 + f^2}$$
    
    **COHEN H'**
    
    *Cohen h' to Cohen h*
    
    fr= "cohenhos", to = "cohenh"
    
    This uses (Cohen, 1988, p. 203):
    $$h = h'\times\\sqrt{2}$$

    **COHEN U**
    
    *Cohen U to Cohen d*
    fr="cohenu1", to="cohend"
    fr="cohenu2", to="cohend"
    fr="cohenu3", to="cohend"
    
    This uses (Cohen, 1988, p. 23):
    $$d = \\Phi^{-1}\\left(U_3\\right)$$
    $$d = 2\\times\\Phi^{-1}\\left(U_2\\right)$$
    $$d = 2\\times\\Phi^{-1}\\left(\\frac{1}{2 - U_1}\\right)$$
    
    **COHEN w**
    
    *Cohen w to Contingency Coefficient*
    
    fr="cohenw", to="cc"
    
    *Cohen w to Cramér V GoF*
    
    fr="cohenw", to="cramervgof", ex1=k
    
    This uses (Cohen, 1988, p. 223):
    $$v = \\frac{w}{\\sqrt{k - 1}}$$
    
    *Cohen w to Cramér V ind.*
    
    fr="cohenw", to="cramervind", ex1=r, ex2=c
    
    This uses:
    $$v = \\frac{w}{\\sqrt{\\min\\left(r - 1, c - 1\\right)}}
    
    *Cohen w to Fei*
    
    fr="cohenw", to="fei", ex1=minExp/n
    
    This uses:
    $$Fei = \\frac{w}{\\sqrt{\\frac{1}{p_E}-1}}
    
    **CRAMER V GoF**
    
    *Cramer's v for Goodness-of-Fit to Cohen w*
    
    fr="cramervgof", to = "cohenw", ex1 = k
    
    This uses (Cohen, 1988, p. 223):
    $$w = v\\times\\sqrt{k - 1}$$
    
    **CRAMER V Ind**
    
    *Cramer's v for Independence to Cohen w*
    
    fr="cramervgof", to = "cohenw", ex1 = r, ex2 = c
    
    This uses:
    $$w = v\\times\\sqrt{\\min\\left(r - 1, c - 1\\right)}$$
    
    **EPSILON SQUARED**
    
    *Epsilon Squared to Eta Squared*
    
    fr="epsilonsq", to="etasq", ex1 = n, ex2 = k
    
    This uses:
    $$\\eta^2 = 1 - \\frac{\\left(1 - \\epsilon^2\\right)\\times\\left(n - k\\right)}{n - 1}$$
    
    *Epsilon Squared to Omega Squared*
    
    fr="epsilonsq", to="omegasq", ex1=MS_w, ex2 = SS_t
    
    This uses:
    $$\\hat{\\omega}^2 = \\epsilon^2\\times\\left(1 - \\frac{MS_w}{SS_t + MS_w}\\right)$$
    
    **ETA SQUARED**
    
    *Eta squared to Cohen f*
    
    fr="etasq", to="cohenf"
    
    This uses:
    
    $$f = \\sqrt{\\frac{\\eta^2}{1 - \\eta^2}}$$
    
    *Eta squared to Epsilon Squared*
    
    fr="etasq", to="epsilonsq", ex1=n, ex2=k
    
    This uses:
    $$\\epsilon^2 = \\frac{n\\times\\eta^2 - k + \\left(1 - \\eta^2\\right)}{n - l}$$
    
    **FEI**
    
    *Fei to Cohen w*
    
    fr="fei", to="cohenw", ex1=minExp/n
    
    This uses:
    $$w = Fei\\times\\sqrt{\\frac{1}{p_E}-1}$$
    
    *Fei to Johnston-Berry-Mielke E*
    
    fr="fei", to="jbme"
    
    This uses:
    $$E = Fei^2$$
    
    **JOHNSTON-BERRY-MIELKE**
    
    *Johnston-Berry-Mielke E to Cohen w*
    
    fr="jbme", to="cohenw", ex1=minExp/n
    
    This uses (Johnston et al., 2006, p. 413):
    $$w = \\sqrt{\\frac{E\\times\\left(1 - \\right)}{q}}$$
    
    *Johnston-Berry-Mielke E to Cohen w*
    
    fr="jbme", to="fei"
    
    This uses:
    $$Fei = \\sqrt(E)$$
    
    
    **ODDS RATIO**
    
    *Odds Ratio to Cohen d*
    
    fr="or", to="cohend", ex1="chinn"
    
    This uses (Chinn, 2000, p. 3129):
    $$d = \\frac{\\ln{\\left(OR\\right)}}{1.81}$$
    
    fr="or", to="cohend", ex1="borenstein"
    
    This uses (Borenstein et. al, 2009, p. 3):
    $$d = \\ln\\left(OR\\right)\\times\\frac{\\sqrt{3}}{\\pi}$$
    
    *Odds Ratio to Yule Q*
    
    fr="or", to="yuleq"
    
    This uses:
    $$Q = \\frac{OR - 1}{OR + 1}$$
    
    *Odds Ratio to Yule Y*
    
    This uses
    $$Y = \\frac{\\sqrt{OR} - 1}{\\sqrt{OR} + 1}$$
    
    **OMEGA SQUARED**
    
    *Omega Squared to Epsilon Squared*
    
    fr="omegasq", to="epsilonsq", ex1=MS_w, ex2 = SS_t
    
    This uses:
    $$\\epsilon^2 = \\frac{\\hat{\\omega}^2}{1 - \\frac{MS_w}{SS_t + MS_w}}$$
    
    **RANK BISERIAL (CLIFF DELTA)**
    
    **Rank Biserial (Cliff delta) to Cohen d**
    
    fr = "rb", to = "cohend"
    
    This uses (Marfo & Okyere, 2019, p. 4):
    $$d =2 \\times \\Phi^{-1}\\left(-\\frac{1}{r_b - 2}\\right)$$
    
    **Rank Biserial (Cliff delta) to Vargha-Delaney A**
    
    fr = "rb", to = "vda"
    
    This uses:
    $$r_b = 2\\times A - 1$$
    
    **VARGHA-DELANEY A**
    
    **Vargha-Delaney A to Rank Biserial (Cliff delta)**
    
    fr = "cle", to = "rb"
    
    This uses:
    $$A = \\frac{r_b + 1}{2}$$
    
    **YULE Q**
    
    *Yule Q to Odds Ratio*
    
    fr="yuleq", to="or"
    
    This uses:
    
    $$OR = \\frac{1 + Q}{1 - Q}$$
    
    *Yule Q to Yule Y*
    
    fr="yuleq", to="yuley"
    
    This uses:
    $$Y = \\frac{1 - sqrt{1 - Q^2}}{Q}$$
    
    **YULE Y**
    
    *Yule Y to Yule Q*
    
    fr="yuley", to=="yuleq"
    
    This uses:
    $$Q = \\frac{2\\times Y}{1 + Y^2}$$
    
    *Yule Y to Odds Ratio*
    
    fr="yuley", to=="or"
    
    This uses
    $$OR = \\left(\\frac{1 + Y}{1 - Y}\\right)^2$$
    
    References
    ----------
    Borenstein, M., Hedges, L. V., Higgins, J. P. T., & Rothstein, H. R. (2009). Converting Among Effect Sizes. In *Introduction to Meta-Analysis*. John Wiley & Sons, Ltd. https://doi.org/10.1002/9780470743386
    
    Chinn, S. (2000). A simple method for converting an odds ratio to effect size for use in meta-analysis. *Statistics in Medicine, 19*(22), 3127–3131. https://doi.org/10.1002/1097-0258(20001130)19:22<3127::aid-sim784>3.0.co;2-m
    
    Cohen, J. (1988). *Statistical power analysis for the behavioral sciences* (2nd ed.). L. Erlbaum Associates.
    
    Johnston, J. E., Berry, K. J., & Mielke, P. W. (2006). Measures of effect size for chi-squared and likelihood-ratio goodness-of-fit tests. *Perceptual and Motor Skills, 103*(2), 412–414. https://doi.org/10.2466/pms.103.2.412-414
    
    
    Author
    ------
    Made by P. Stikker
    
    Companion website: https://PeterStatistics.com  
    YouTube channel: https://www.youtube.com/stikpet  
    Donations: https://www.patreon.com/bePatron?u=19398076
    
    '''
    
    #"or", "cohend" , "yuleq", "yuley", "vda", "rb"
    #fr="cohenh2", to = "cohenh"
    #fr="cramervgof", to = "cohenw", ex1 = df


    #COHEN d
    #Cohen d one-sample to Cohen d
    if(fr=="cohendos" and to=="cohend"):
        res = es*(2)**0.5
      #Cohen d to Odds Ratio
    elif(fr=="cohend" and to=="or"):
        #Chinn (2000, p. 3129)
        if(ex1=="chinn"):
            res = math.exp(1.81*es)
        else:
            #Borenstein et. al (2009, p. 3)
            res = math.exp(es*math.pi/(3**0.5))
    
    #Cohen d to Rank Biserial (Cliff delta)
    elif (fr=="cohend" and to=="rb"):
        pf= NormalDist().cdf(es/(2))
        res = (2*pf - 1)/pf
    
    #COHEN F
    #Cohen f to eta squared
    elif(fr=="cohenf" and to=="etasq"):
        res = es**2/(1 + es**2)
        
    #COHEN h'
    #Cohen h' to Cohen h
    elif(fr=="cohenhos" and to=="cohenh"):
        res = es*(2)**0.5

    #COHEN U
    #Cohen U1 to Cohen d
    elif(fr=="cohenu1" and to=="cohend"):
        res = NormalDist().inv_cdf(1/(2 - es))*2
    elif(fr=="cohenu2" and to=="cohend"):
        res = NormalDist().inv_cdf(es)*2
    elif(fr=="cohenu3" and to=="cohend"):
        res = NormalDist().inv_cdf(es)
    
    #COHEN w
    #Cohen w to Contingency Coefficient
    elif(fr=="cohenw" and to=="cc"):
        res = (es**2 / (1 + es**2))**0.5
    
    #Cohen w to Cramer V
    elif(fr=="cohenw" and to =="cramervgof"):
        res = es/(ex1 - 1)**0.5
    elif(fr=="cohenw" and to =="cramervind"):
        res = es/(min(ex1 - 1, ex2 -1))**0.5
        
    #Cohen w to Fei
    elif(fr=="cohenw" and to =="fei"):
        res = es/(1/ex1 - 1)**0.5
        
    #CONTINGENCY COEFFICIENT
    #Contingency Coefficient to Cohen w
    elif(fr=="cc" and to=="cohenw"):
        res = (es**2 / (1 - es**2))**0.5
        
    #CRAMÉR V
    #Cramer's v GoF to Cohen w
    elif(fr=="cramervgof" and to=="cohenw"):
        res = es*(ex1 - 1)**0.5
    
    #Cramer's v Ind. to Cohen w
    elif(fr=="cramervind" and to=="cohenw"):
        res = es*(min(ex1 - 1, ex2 -1))**0.5
        
    #EPSILON SQUARED
    #Epsilon squared to Eta squared
    elif(fr=="epsilonsq" and to=="etasq"):
        res = 1 - (1 - es)*(ex1 - ex2)/(ex1 - 1)
        
    #Epsilon squared to Omega squared
    elif(fr=="epsilonsq" and to=="omegasq"):
        res = es*(1 - ex1/(ex2 + ex1))
        
        
    #ETA SQUARED
    #Eta squared to Cohen f
    elif(fr=="etasq" and to=="cohenf"):
        res = (es/(1-es))**0.5
        
    #Eta squared to Epsilon Squared
    elif(fr=="etasq" and to=="epsilonsq"):
        res = (ex1*es - ex2 + (1 - es))/(ex1 - ex2)
        
    #FEI    
    #Fei to Cohen w
    elif(fr=="fei" and to =="cohenw"):
        res = es*(1/ex1 - 1)**0.5
    
    #Fei to Johnston-Berry-Mielke E
    elif(fr=="fei" and to =="jbme"):
        res = es**2
    
    #JOHNSTON-BERRY-MIELKE E
    #Johnston-Berry-Mielke E to Cohen w
    elif(fr=="jbme" and to=="cohenw"):
        res = (es*(1 - ex1)/(ex1))**0.5
 
    #Johnston-Berry-Mielke E to Fei
    elif(fr=="jbme" and to=="fei"):
        res = (es)**0.5
    
    #ODDS RATIO
    #Odds Ratio to Cohen d (Chinn, 2000, p. 3129)
    elif(fr=="or" and to=="cohend"):
        if(ex1=="chinn"):
            res = math.log(es)/1.81
        else:
            #Borenstein et. al (2009, p. 3)
            res = math.log(es)*(3)**0.5/math.pi

    #Odds Ratio to Yule Q
    elif(fr=="or" and to=="yuleq"):
        res = (es - 1)/(es + 1)
        
    #Odds Ratio to Yule Y
    elif(fr=="or" and to=="yuley"):
        res = ((es)**0.5 - 1)/((es)**0.5 + 1)

        
    #OMEGA SQUARED
    #Omega squared to Epsilon squared
    elif(fr=="omegasq" and to=="epsilonsq"):
        res = es/(1 - ex1/(ex2 + ex1))
        
        
    #RANK BISERIAL
    #Rank Biserial (Cliff delta) to Cohen d
    elif (fr=="rb" and to=="cohend"):        
        res = 2*NormalDist().inv_cdf(-1/(es-2))
        
    #Rank Biserial to CLE / Vargha and Delaney A
    elif(fr=="rb" and to=="cle"):
        res = (es + 1)/2
        
    #VARGHA AND DELANEY A
    #Vargha and Delaney A to Rank Biserial
    
    elif(fr=="cle" and to=="rb"):
        res = 2*es - 1
        
        
    #YULE Q
    #Yule Q to Odds Ratio
    elif(fr=="yuleq" and to=="or"):
        res = (1 + es)/(1 - es)
        
    #Yule Q to Yule Y
    elif(fr=="yuleq" and to=="yuley"):
        res = (1 - (1 - es**2)**0.5)/es
        
        
    #YULE Y
    #Yule Y to Odds Ratio
    elif(fr=="yuley" and to=="or"):
        res = ((1 + es)/(1 - es))**2
        
    #Yule Y to Yule Q
    elif(fr=="yuley" and to=="yuleq"):
        res = (2*es)/(1 + es**2)
        
    return(res)