Module stikpetP.effect_sizes.eff_size_hedges_g_os

Functions

def es_hedges_g_os(data, mu=None, appr=None)

Hedges g (one-sample)

This function will calculate Hedges g (one-sample). An effect size measure that can be used with a test for a single mean (for example a one-sample Student t-test).

Hedges g is a correction for Cohen's d'. Actually Hedges (1981) didn't seem to have a one-sample version for Hedges g, and this correction is the one for Hedges g used for the independent samples.

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

Parameters

data : list

numeric list with numbers

mu : float

the hypothesized mean

appr : {None, 'hedges', 'durlak', 'xue'}, optional

approximation to use. Default is None

Returns

pandas.DataFrame

A dataframe with the following columns:

• mu, the hypothesized mean used
• g, Hedges g for a one-sample
• version, description of version used.

Notes

The formula used for the exact method (appr=NULL) (Hedges, 1981, p. 111): $$g = d' \times\frac{\Gamma\left(m\right)}{\Gamma\left(m - \frac{1}{2}\right)\times\sqrt{m}}$$

With: $$m = \frac{df}{2}$$ $$df = n - 1$$

Symbols used:

• $d'$ Cohen’s d for one-sample
• $df$ the degrees of freedom
• $n$ the sample size (i.e. the number of scores)
• $\Gamma\left(\dots\right)$ the gamma function

The formula used for the approximation from Hedges (1981, p. 114) (appr="hedges"): $$g = d' \times\left(1 - \frac{3}{4\times df - 1}\right)$$

The formula used for the approximation from Durlak (2009, p. 927) (appr="durlak"): $$g = d' \times\frac{n - 3}{n - 2.25} \times\sqrt{\frac{n - 2}{n}}$$

The formula used for the approximation from Xue (2020, p. 3) (appr="xue"): $$g = d' \times \sqrt[12]{1 - \frac{9}{df} + \frac{69}{2\times df^2} - \frac{72}{df^3} + \frac{687}{8\times df^4} - \frac{441}{8\times df^5} + \frac{247}{16\times df^6}}$$

Since Hedges g is a correction for Cohen d', it can be converted to a 'regular' Cohen d and then rules of thumb for the interpertation could be used.

Before, After and Alternatives

Before this you might want to perform a test: * ts_student_t_os for One-Sample Student t-Test * ts_trimmed_mean_os for One-Sample Trimmed (Yuen or Yuen-Welch) Mean Test * ts_z_os for One-Sample Z Test

After this you might want a rule-of-thumb for the effect size, first convert to regular Cohen d: * es_convert to convert Hedges g to Cohen d, use fr = "cohendos" and to = "cohend" * th_cohen_d for rules-of-thumb for Cohen d

Alternative Effect Sizes: * es_cohen_d_os for Cohen d' * es_common_language_os for the Common Language Effect Size

References

Durlak, J. A. (2009). How to select, calculate, and interpret effect sizes. Journal of Pediatric Psychology, 34(9), 917–928. doi:10.1093/jpepsy/jsp004

Hedges, L. V. (1981). Distribution Theory for Glass’s Estimator of Effect Size and Related Estimators. Journal of Educational Statistics, 6(2), 107–128. doi:10.2307/1164588

Xue, X. (2020). Improved approximations of Hedges’ g*. doi:10.48550/arXiv.2003.06675

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

Example 1: Numeric Pandas Series

>>> import pandas as pd
>>> df2 = pd.read_csv('https://peterstatistics.com/Packages/ExampleData/StudentStatistics.csv', sep=';', low_memory=False, storage_options={'User-Agent': 'Mozilla/5.0'})
>>> ex1 = df2['Gen_Age']
>>> es_hedges_g_os(ex1)
     mu         g version
0  68.5 -2.857185   exact

Example 2: Numeric list

>>> ex2 = [1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5]
>>> es_hedges_g_os(ex2)
    mu         g version
0  3.0  0.268649   exact

Expand source code

def es_hedges_g_os(data, mu=None, appr=None):
    '''
    Hedges g (one-sample)
    ---------------------
    
    This function will calculate Hedges g (one-sample). An effect size measure that can be used with a test for a single mean (for example a one-sample Student t-test).
    
    Hedges g is a correction for Cohen's d'. Actually Hedges (1981) didn't seem to have a one-sample version for Hedges g, and this correction is the one for Hedges g used for the independent samples.

    This function is shown in this [YouTube video](https://youtu.be/k4qt9iMcse8) and the measure is also described at [PeterStatistics.com](https://peterstatistics.com/Terms/EffectSizes/HedgesG.html)
    
    Parameters
    ----------
    data : list
        numeric list with numbers
    mu : float
        the hypothesized mean
    appr : {None, 'hedges', 'durlak', 'xue'}, optional
        approximation to use. Default is None 
    
    Returns
    -------
    pandas.DataFrame
        A dataframe with the following columns:
    
        * *mu*, the hypothesized mean used
        * *g*, Hedges g for a one-sample
        * *version*, description of version used.
    
    Notes
    -----
    The formula used for the exact method (appr=NULL) (Hedges, 1981, p. 111):
    $$g = d' \\times\\frac{\\Gamma\\left(m\\right)}{\\Gamma\\left(m - \\frac{1}{2}\\right)\\times\\sqrt{m}}$$
    
    With:
    $$m = \\frac{df}{2}$$
    $$df = n - 1$$
    
    *Symbols used:*
    
    * $d'$ Cohen’s d for one-sample
    * $df$ the degrees of freedom
    * $n$ the sample size (i.e. the number of scores)
    * $\\Gamma\\left(\\dots\\right)$ the gamma function
    
    The formula used for the approximation from Hedges (1981, p. 114) (appr="hedges"):
    $$g = d' \\times\\left(1 - \\frac{3}{4\\times df - 1}\\right)$$
    
    The formula used for the approximation from Durlak (2009, p. 927) (appr="durlak"):
    $$g = d' \\times\\frac{n - 3}{n - 2.25} \\times\\sqrt{\\frac{n - 2}{n}}$$
    
    The formula used for the approximation from Xue (2020, p. 3) (appr="xue"):
    $$g = d' \\times \\sqrt[12]{1 - \\frac{9}{df} + \\frac{69}{2\\times df^2} - \\frac{72}{df^3} + \\frac{687}{8\\times df^4} - \\frac{441}{8\\times df^5} + \\frac{247}{16\\times df^6}}$$
    
    Since Hedges g is a correction for Cohen d', it can be converted to a 'regular' Cohen d and then rules of thumb for the interpertation could be used. 
    
    Before, After and Alternatives
    ------------------------------
    Before this you might want to perform a test:
    * [ts_student_t_os](../tests/test_student_t_os.html#ts_student_t_os) for One-Sample Student t-Test
    * [ts_trimmed_mean_os](../tests/test_trimmed_mean_os.html#ts_trimmed_mean_os) for One-Sample Trimmed (Yuen or Yuen-Welch) Mean Test
    * [ts_z_os](../tests/test_z_os.html#ts_z_os) for One-Sample Z Test

    After this you might want a rule-of-thumb for the effect size, first convert to regular Cohen d:
    * [es_convert](../effect_sizes/convert_es.html#es_convert) to convert Hedges g to Cohen d, use *fr = "cohendos"* and *to = "cohend"*
    * [th_cohen_d](../other/thumb_cohen_d.html#th_cohen_d) for rules-of-thumb for Cohen d
    
    Alternative Effect Sizes:
    * [es_cohen_d_os](../effect_sizes/eff_size_cohen_d_os.html#es_cohen_d_os) for Cohen d'
    * [es_common_language_os](../eff_size_common_language_os/meas_variation.html#es_common_language_os) for the Common Language Effect Size
    
    References
    ----------
    Durlak, J. A. (2009). How to select, calculate, and interpret effect sizes. *Journal of Pediatric Psychology, 34*(9), 917–928. doi:10.1093/jpepsy/jsp004
    
    Hedges, L. V. (1981). Distribution Theory for Glass’s Estimator of Effect Size and Related Estimators. *Journal of Educational Statistics, 6*(2), 107–128. doi:10.2307/1164588
    
    Xue, X. (2020). Improved approximations of Hedges’ g*. doi:10.48550/arXiv.2003.06675
    
    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
    --------
    Example 1: Numeric Pandas Series
    >>> import pandas as pd
    >>> df2 = pd.read_csv('https://peterstatistics.com/Packages/ExampleData/StudentStatistics.csv', sep=';', low_memory=False, storage_options={'User-Agent': 'Mozilla/5.0'})
    >>> ex1 = df2['Gen_Age']
    >>> es_hedges_g_os(ex1)
         mu         g version
    0  68.5 -2.857185   exact
    
    Example 2: Numeric list
    >>> ex2 = [1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5]
    >>> es_hedges_g_os(ex2)
        mu         g version
    0  3.0  0.268649   exact
    
    '''
    #data as a pandas series
    if type(data) is list:
        data = pd.Series(data)
        
    #set hypothesized median to mid range if not provided
    if (mu is None):
        mu = (min(data) + max(data)) / 2
    
    #remove missing values
    data = data.dropna()
    
    # sample size and degrees of freedom
    n = len(data)
    df = n - 1
    
    # sample mean and standard deviation
    avg = sum(data)/n
    s = (sum((data-avg)**2)/df)**(1/2)
    
    # Cohen's d for one-sample
    d = (avg - mu) / s
    
    # helper variable m
    m = df/2
    
    if appr is None and m < 172:
        g = d*math.gamma(m)/(math.gamma(m-0.5)*m**(1/2))
        comment = "exact"
    elif appr=="hedges":
        g = d*(1 - 3/(4*df-1))
        comment = "Hedges approximation"
    elif appr=="durlak":
        g = d*(n-3)/(n-2.25)*((n-2)/n)**0.5
        comment = "Durlak approximation"
    else:
        g = d*(1-9/df+69/(2*df**2)-72/(df**3)+687/(8*df**4)-441/(8*df**5)+247/(16*df**6))**(1/12)
        comment = "Xue approximation"
    
    # print warning if exact was asked but couldn't be used
    if appr is None and m >= 172:
        print("WARNING: exact method could not be computed due to large sample size, Xue approximation used instead")
        comment = "Xue approximation"
    
    
    results = pd.DataFrame([[mu, g, comment]], columns=["mu", "g", "version"])
        
    pd.set_option('display.max_colwidth', None)
    
    return results