Module `stikpetP.effect_sizes.eff_size_jbm_e`

Expand source code

from math import log
import pandas as pd

def es_jbm_e(chi2, n, minExp, test='chi'):
    '''
    Johnston-Berry-Mielke E
    -----------------------
     
    An effect size measure that could be used with a chi-square test or g-test. 

    Johnston, Berry and Mielke (2006) proposed to simply divide the found chi-square value, by the maximum possible chi-square value. Something Cohen w (an alternative effect size measure) does not. The advantage of this measure is that it will therefor alway be between 0 and 1. Ben-Shachar et al. (2023) expanded on this idea and also took the square root out of the result, and called this Fei.
    
    The function is shown in this [YouTube video](https://youtu.be/5a8Sv_wBoxI) and the measure is also described at [PeterStatistics.com](https://peterstatistics.com/Terms/EffectSizes/JohnstonBerryMielke-E.html)
    
    Parameters
    ----------
    chi2 : float
        the chi-square test statistic
    n : int
        the sample size
    minExp: float
        the minimum expected count
    test : {"chi", "g"}, optional
        either "chi" (default) for chi-square tests, or "g" for likelihood ratio
        
    Returns
    -------
    E : float
        the value of JBM E
   
    Notes
    -----
    Two versions of this effect size. The formula for a chi-square test is:
    $$E_{\\chi^2} = \\frac{q}{1-q} \\times \\left( \\sum_{i=1}^{k} \\frac{p_{i}^{2}}{q_{i}} - 1 \\right) = \\frac{\\chi_{GoF}^2 \\times E_{min}}{n \\times \\left(n - E_{min} \\right)}$$
    For a Likelihood Ratio (G) test:
    $$E_{L}=-\\frac{1}{\\text{ln}(q)}\\times\\sum_{i=1}^{k}\\left(p_{i}\\times\\text{ln}\\left(\\frac{p_{i}}{q_{i}}\\right)\\right) = -\\frac{1}{\\text{ln}\\left(q\\right)\\times\\frac{\\chi_L^2}{2\\times n}}$$
    
    *Symbols used:*
    
    * $q$ the minimum of all $q_i$
    * $q_i$ the expected proportion in category i
    * $p_i$ the observed proportion in category i
    * $n$ the total sample size
    * $E_{min}$ the minimum expected count
    * $\\chi_{GoF}^2$ the chi-square test statistic of a Pearson chi-square test of goodness-of-fit
    * $\\chi_L^2$ the chi-square test statistic of a likelihood ratio test of goodness-of-fit 
    
    Both formulas are from Johnston et al. (2006, p. 413)
    
    *Classification*
    
    A qualification rule-of-thumb could be obtained by converting this to Cohen's w (use **es_convert(e, fr="jbme", to="cohenw", ex1=minExp/n)**)
    
    Before, After and Alternatives
    ------------------------------
    Before this you will need a chi-square value. From either:
    * [ts_pearson_gof](../tests/test_pearson_gof.html#ts_pearson_gof) for Pearson Chi-Square Goodness-of-Fit Test
    * [ts_freeman_tukey_gof](../tests/test_freeman_tukey_gof.html#ts_freeman_tukey_gof) for Freeman-Tukey Test of Goodness-of-Fit
    * [ts_freeman_tukey_read](../tests/test_freeman_tukey_read.html#ts_freeman_tukey_read) for Freeman-Tukey-Read Test of Goodness-of-Fit
    * [ts_g_gof](../tests/test_g_gof.html#ts_g_gof) for G (Likelihood Ratio) Goodness-of-Fit Test
    * [ts_mod_log_likelihood_gof](../tests/test_mod_log_likelihood_gof.html#ts_mod_log_likelihood_gof) for Mod-Log Likelihood Test of Goodness-of-Fit
    * [ts_neyman_gof](../tests/test_neyman_gof.html#ts_neyman_gof) for Neyman Test of Goodness-of-Fit
    * [ts_powerdivergence_gof](../tests/test_powerdivergence_gof.html#ts_powerdivergence_gof) for Power Divergence GoF Test
    * [ph_pairwise_gof](../other/poho_pairwise_gof.html#ph_pairwise_gof) for Pairwise Goodness-of-Fit Tests
    * [ph_residual_gof_gof](../other/poho_residual_gof_gof.html#ph_residual_gof_gof) for Residuals Using Goodness-of-Fit Tests

    After this you might want to use some rule-of-thumb for the interpretation by converting it to Cohen w:
    * [es_convert](../effect_sizes/convert_es.html#es_convert) to convert JBM-E to Cohen w (using fr="jbme", to="cohenw", ex1=minExp/n)
    * [th_cohen_w](../other/thumb_cohen_w.html#th_cohen_w) for various rules-of-thumb for Cohen w

    Alternative effect sizes that use a chi-square value:
    * [es_cramer_v_gof](../effect_sizes/eff_size_cramer_v_gof.html#es_cramer_v_gof) for Cramer's V for Goodness-of-Fit
    * [es_cohen_w](../effect_sizes/eff_size_cohen_w.html#es_cohen_w) for Cohen's w
    * [es_fei](../effect_sizes/eff_size_fei.html#es_fei) for Fei
    
    References
    ----------
    Ben-Shachar, M. S., Patil, I., Thériault, R., Wiernik, B. M., & Lüdecke, D. (2023). Phi, fei, fo, fum: Effect sizes for categorical data that use the chi-squared statistic. *Mathematics, 11*(1982), 1–10. doi:10.3390/math11091982
    
    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. doi: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
    
    Examples
    --------
    >>> chi2Value = 3.106
    >>> n = 19
    >>> minExp = 3
    >>> es_jbm_e(chi2Value, n, minExp)
    0.030651315789473683
    >>> es_jbm_e(chi2Value, n, minExp, test="likelihood")
    0.04428196998451468
    
    '''
    if test=='chi':
        E = chi2 * minExp / (n * (n - minExp))       
    else:
        E = -1/log(minExp/n) * chi2/(2*n)
    return E

Functions

def es_jbm_e(chi2, n, minExp, test='chi')

Johnston-Berry-Mielke E

An effect size measure that could be used with a chi-square test or g-test.

Johnston, Berry and Mielke (2006) proposed to simply divide the found chi-square value, by the maximum possible chi-square value. Something Cohen w (an alternative effect size measure) does not. The advantage of this measure is that it will therefor alway be between 0 and 1. Ben-Shachar et al. (2023) expanded on this idea and also took the square root out of the result, and called this Fei.

The function is shown in this YouTube video and the measure is also described at PeterStatistics.com

Parameters

chi2 : float: the chi-square test statistic
n : int: the sample size
minExp : float: the minimum expected count
test : {"chi", "g"}, optional: either "chi" (default) for chi-square tests, or "g" for likelihood ratio

Returns

E : float: the value of JBM E

Notes

Two versions of this effect size. The formula for a chi-square test is: $E_{\chi^2} = \frac{q}{1-q} \times \left( \sum_{i=1}^{k} \frac{p_{i}^{2}}{q_{i}} - 1 \right) = \frac{\chi_{GoF}^2 \times E_{min}}{n \times \left(n - E_{min} \right)}$ For a Likelihood Ratio (G) test: $E_{L}=-\frac{1}{\text{ln}(q)}\times\sum_{i=1}^{k}\left(p_{i}\times\text{ln}\left(\frac{p_{i}}{q_{i}}\right)\right) = -\frac{1}{\text{ln}\left(q\right)\times\frac{\chi_L^2}{2\times n}}$

Symbols used:

$q$ the minimum of all $q_i$
$q_i$ the expected proportion in category i
$p_i$ the observed proportion in category i
$n$ the total sample size
$E_{min}$ the minimum expected count
$\chi_{GoF}^2$ the chi-square test statistic of a Pearson chi-square test of goodness-of-fit
$\chi_L^2$ the chi-square test statistic of a likelihood ratio test of goodness-of-fit

Both formulas are from Johnston et al. (2006, p. 413)

Classification

A qualification rule-of-thumb could be obtained by converting this to Cohen's w (use es_convert(e, fr="jbme", to="cohenw", ex1=minExp/n))

Before, After and Alternatives

Before this you will need a chi-square value. From either: * ts_pearson_gof for Pearson Chi-Square Goodness-of-Fit Test * ts_freeman_tukey_gof for Freeman-Tukey Test of Goodness-of-Fit * ts_freeman_tukey_read for Freeman-Tukey-Read Test of Goodness-of-Fit * ts_g_gof for G (Likelihood Ratio) Goodness-of-Fit Test * ts_mod_log_likelihood_gof for Mod-Log Likelihood Test of Goodness-of-Fit * ts_neyman_gof for Neyman Test of Goodness-of-Fit * ts_powerdivergence_gof for Power Divergence GoF Test * ph_pairwise_gof for Pairwise Goodness-of-Fit Tests * ph_residual_gof_gof for Residuals Using Goodness-of-Fit Tests

After this you might want to use some rule-of-thumb for the interpretation by converting it to Cohen w: * es_convert to convert JBM-E to Cohen w (using fr="jbme", to="cohenw", ex1=minExp/n) * th_cohen_w for various rules-of-thumb for Cohen w

Alternative effect sizes that use a chi-square value: * es_cramer_v_gof for Cramer's V for Goodness-of-Fit * es_cohen_w for Cohen's w * es_fei for Fei

References

Ben-Shachar, M. S., Patil, I., Thériault, R., Wiernik, B. M., & Lüdecke, D. (2023). Phi, fei, fo, fum: Effect sizes for categorical data that use the chi-squared statistic. Mathematics, 11(1982), 1–10. doi:10.3390/math11091982

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. doi: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

Examples

>>> chi2Value = 3.106
>>> n = 19
>>> minExp = 3
>>> es_jbm_e(chi2Value, n, minExp)
0.030651315789473683
>>> es_jbm_e(chi2Value, n, minExp, test="likelihood")
0.04428196998451468

Expand source code

def es_jbm_e(chi2, n, minExp, test='chi'):
    '''
    Johnston-Berry-Mielke E
    -----------------------
     
    An effect size measure that could be used with a chi-square test or g-test. 

    Johnston, Berry and Mielke (2006) proposed to simply divide the found chi-square value, by the maximum possible chi-square value. Something Cohen w (an alternative effect size measure) does not. The advantage of this measure is that it will therefor alway be between 0 and 1. Ben-Shachar et al. (2023) expanded on this idea and also took the square root out of the result, and called this Fei.
    
    The function is shown in this [YouTube video](https://youtu.be/5a8Sv_wBoxI) and the measure is also described at [PeterStatistics.com](https://peterstatistics.com/Terms/EffectSizes/JohnstonBerryMielke-E.html)
    
    Parameters
    ----------
    chi2 : float
        the chi-square test statistic
    n : int
        the sample size
    minExp: float
        the minimum expected count
    test : {"chi", "g"}, optional
        either "chi" (default) for chi-square tests, or "g" for likelihood ratio
        
    Returns
    -------
    E : float
        the value of JBM E
   
    Notes
    -----
    Two versions of this effect size. The formula for a chi-square test is:
    $$E_{\\chi^2} = \\frac{q}{1-q} \\times \\left( \\sum_{i=1}^{k} \\frac{p_{i}^{2}}{q_{i}} - 1 \\right) = \\frac{\\chi_{GoF}^2 \\times E_{min}}{n \\times \\left(n - E_{min} \\right)}$$
    For a Likelihood Ratio (G) test:
    $$E_{L}=-\\frac{1}{\\text{ln}(q)}\\times\\sum_{i=1}^{k}\\left(p_{i}\\times\\text{ln}\\left(\\frac{p_{i}}{q_{i}}\\right)\\right) = -\\frac{1}{\\text{ln}\\left(q\\right)\\times\\frac{\\chi_L^2}{2\\times n}}$$
    
    *Symbols used:*
    
    * $q$ the minimum of all $q_i$
    * $q_i$ the expected proportion in category i
    * $p_i$ the observed proportion in category i
    * $n$ the total sample size
    * $E_{min}$ the minimum expected count
    * $\\chi_{GoF}^2$ the chi-square test statistic of a Pearson chi-square test of goodness-of-fit
    * $\\chi_L^2$ the chi-square test statistic of a likelihood ratio test of goodness-of-fit 
    
    Both formulas are from Johnston et al. (2006, p. 413)
    
    *Classification*
    
    A qualification rule-of-thumb could be obtained by converting this to Cohen's w (use **es_convert(e, fr="jbme", to="cohenw", ex1=minExp/n)**)
    
    Before, After and Alternatives
    ------------------------------
    Before this you will need a chi-square value. From either:
    * [ts_pearson_gof](../tests/test_pearson_gof.html#ts_pearson_gof) for Pearson Chi-Square Goodness-of-Fit Test
    * [ts_freeman_tukey_gof](../tests/test_freeman_tukey_gof.html#ts_freeman_tukey_gof) for Freeman-Tukey Test of Goodness-of-Fit
    * [ts_freeman_tukey_read](../tests/test_freeman_tukey_read.html#ts_freeman_tukey_read) for Freeman-Tukey-Read Test of Goodness-of-Fit
    * [ts_g_gof](../tests/test_g_gof.html#ts_g_gof) for G (Likelihood Ratio) Goodness-of-Fit Test
    * [ts_mod_log_likelihood_gof](../tests/test_mod_log_likelihood_gof.html#ts_mod_log_likelihood_gof) for Mod-Log Likelihood Test of Goodness-of-Fit
    * [ts_neyman_gof](../tests/test_neyman_gof.html#ts_neyman_gof) for Neyman Test of Goodness-of-Fit
    * [ts_powerdivergence_gof](../tests/test_powerdivergence_gof.html#ts_powerdivergence_gof) for Power Divergence GoF Test
    * [ph_pairwise_gof](../other/poho_pairwise_gof.html#ph_pairwise_gof) for Pairwise Goodness-of-Fit Tests
    * [ph_residual_gof_gof](../other/poho_residual_gof_gof.html#ph_residual_gof_gof) for Residuals Using Goodness-of-Fit Tests

    After this you might want to use some rule-of-thumb for the interpretation by converting it to Cohen w:
    * [es_convert](../effect_sizes/convert_es.html#es_convert) to convert JBM-E to Cohen w (using fr="jbme", to="cohenw", ex1=minExp/n)
    * [th_cohen_w](../other/thumb_cohen_w.html#th_cohen_w) for various rules-of-thumb for Cohen w

    Alternative effect sizes that use a chi-square value:
    * [es_cramer_v_gof](../effect_sizes/eff_size_cramer_v_gof.html#es_cramer_v_gof) for Cramer's V for Goodness-of-Fit
    * [es_cohen_w](../effect_sizes/eff_size_cohen_w.html#es_cohen_w) for Cohen's w
    * [es_fei](../effect_sizes/eff_size_fei.html#es_fei) for Fei
    
    References
    ----------
    Ben-Shachar, M. S., Patil, I., Thériault, R., Wiernik, B. M., & Lüdecke, D. (2023). Phi, fei, fo, fum: Effect sizes for categorical data that use the chi-squared statistic. *Mathematics, 11*(1982), 1–10. doi:10.3390/math11091982
    
    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. doi: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
    
    Examples
    --------
    >>> chi2Value = 3.106
    >>> n = 19
    >>> minExp = 3
    >>> es_jbm_e(chi2Value, n, minExp)
    0.030651315789473683
    >>> es_jbm_e(chi2Value, n, minExp, test="likelihood")
    0.04428196998451468
    
    '''
    if test=='chi':
        E = chi2 * minExp / (n * (n - minExp))       
    else:
        E = -1/log(minExp/n) * chi2/(2*n)
    return E