Module stikpetP.effect_sizes.eff_size_cohen_h_os

Functions

def es_cohen_h_os(data, p0=0.5, p0Cat=None, codes=None)

Cohen's h'

An adaptation of Cohen h for a one-sample case. It is an effect size measure that could be accompanying a one-sample binomial, score or Wald test.

See also https://youtu.be/ddWe94VKX_8, a video on Cohen h'.

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

Parameters

data : list or pandas data series

the data

p0 : float, optional

hypothesized proportion for the first category (default is 0.5)

p0Cat : optional

the category for which p0 was used

codes : list, optional

the two codes to use

Returns

pandas.DataFrame

A dataframe with the following columns:

• Cohen's h' : value of Cohen h'
• comment : description on which category for p0 was used.

Notes

To decide on which category is associated with p0 the following is used: * If codes are provided, the first code is assumed to be the category for the p0. * If p0Cat is specified that will be used for p0 and all other categories will be considered as category 2, this means if there are more than two categories the remaining two or more (besides p0Cat) will be merged as one large category. * If neither codes or p0Cat is specified and more than two categories are in the data a warning is printed and no results. * If neither codes or p0Cat is specified and there are two categories, p0 is assumed to be for the category closest matching the p0 value (i.e. if p0 is above 0.5 the category with the highest count is assumed to be used for p0)

Formula used (Cohen, 1988, p. 202): $$h'=\phi_{1}-\phi_{h_0}$$

With: $$\phi_{i}=2\times\text{arcsin}\sqrt{p_{i}}$$ $$p_i = \frac{F_i}{n}$$ $$n = \sum_{i=1}^k F_i$$

Symbols used:

• $F_i$, is the (absolute) frequency (count) of category i
• $n$, is the sample size, i.e. the sum of all frequencies
• $p_i$, the proportion of cases in category i
• $p_{h_0}$, the expected proportion (i.e. the proportion according to the null hypothesis)

Before, After and Alternatives

Before this effect size you might first want to perform a test: * ts_binomial_os for a One-Sample Binomial Test * ts_score_os for One-Sample Score Test * ts_wald_os for One-Sample Wald Test

After this, you might want a rule-of-thumb or first convert this to a 'regular' Cohen h: * es_convert to convert Cohen h' to Cohen h, use fr="cohenhos" and to=cohenh * th_cohen_h for rules-of-thumb for Cohen h

Alternatives could be: * es_cohen_g for Cohen g * es_alt_ratio for Alternative Ratio * r_rosenthal for Rosenthal Correlation if a z-value is available

References

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

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 list

>>> ex1 = [1, 1, 2, 1, 2, 1, 2, 1]
>>> es_cohen_h_os(ex1)
   Cohen h'            comment
0   0.25268  assuming p0 for 1
>>> es_cohen_h_os(ex1, p0=0.3)
   Cohen h'            comment
0  0.664197  assuming p0 for 1

Example 2: pandas Series

>>> import pandas as pd
>>> df1 = pd.read_csv('https://peterstatistics.com/Packages/ExampleData/GSS2012a.csv', sep=',', low_memory=False, storage_options={'User-Agent': 'Mozilla/5.0'})
>>> es_cohen_h_os(df1['sex'])
   Cohen h'                 comment
0   0.10251  assuming p0 for FEMALE
>>> es_cohen_h_os(df1['mar1'], codes=["DIVORCED", "NEVER MARRIED"])
   Cohen h'               comment
0 -0.114495  with p0 for DIVORCED

Expand source code

def es_cohen_h_os(data, p0=0.5, p0Cat=None, codes=None):
    '''
    Cohen's h'
    ----------
     
    An adaptation of Cohen h for a one-sample case. It is an effect size measure that could be accompanying a one-sample binomial, score or Wald test.

    See also https://youtu.be/ddWe94VKX_8, a video on Cohen h'.

    This function is shown in this [YouTube video](https://youtu.be/CHFPfThJ4aY) and the effect size is also described at [PeterStatistics.com](https://peterstatistics.com/Terms/EffectSizes/CohenH.html)
    
    Parameters
    ----------
    data : list or pandas data series 
        the data
    p0 : float, optional 
        hypothesized proportion for the first category (default is 0.5)
    p0Cat : optional
        the category for which p0 was used
    codes : list, optional 
        the two codes to use
        
    Returns
    -------
    pandas.DataFrame
        A dataframe with the following columns:
    
        - *Cohen's h'* : value of Cohen h' 
        - *comment* : description on which category for p0 was used.
   
    Notes
    -----
    To decide on which category is associated with p0 the following is used:
    * If codes are provided, the first code is assumed to be the category for the p0.
    * If p0Cat is specified that will be used for p0 and all other categories will be considered as category 2, this means if there are more than two categories the remaining two or more (besides p0Cat) will be merged as one large category.
    * If neither codes or p0Cat is specified and more than two categories are in the data a warning is printed and no results.
    * If neither codes or p0Cat is specified and there are two categories, p0 is assumed to be for the category closest matching the p0 value (i.e. if p0 is above 0.5 the category with the highest count is assumed to be used for p0)
    
    Formula used (Cohen, 1988, p. 202):
    $$h'=\\phi_{1}-\\phi_{h_0}$$
    
    With:
    $$\\phi_{i}=2\\times\\text{arcsin}\\sqrt{p_{i}}$$
    $$p_i = \\frac{F_i}{n}$$
    $$n = \\sum_{i=1}^k F_i$$
    
    *Symbols used*:
    
    * $F_i$, is the (absolute) frequency (count) of category i
    * $n$, is the sample size, i.e. the sum of all frequencies
    * $p_i$, the proportion of cases in category i
    * $p_{h_0}$, the expected proportion (i.e. the proportion according to the null hypothesis)
    
    Before, After and Alternatives
    ------------------------------
    Before this effect size you might first want to perform a test:
    * [ts_binomial_os](../tests/test_binomial_os.html#ts_binomial_os) for a One-Sample Binomial Test
    * [ts_score_os](../tests/test_score_os.html#ts_score_os) for One-Sample Score Test
    * [ts_wald_os](../tests/test_wald_os.html#ts_wald_os) for One-Sample Wald Test

    After this, you might want a rule-of-thumb or first convert this to a 'regular' Cohen h:
    * [es_convert](../effect_sizes/convert_es.html#es_convert) to convert Cohen h' to Cohen h, use fr="cohenhos" and to=cohenh
    * [th_cohen_h](../other.thumb/cohen_h.html#th_cohen_h) for rules-of-thumb for Cohen h

    Alternatives could be:
    * [es_cohen_g](../effect_sizes/eff_size_cohen_g.html#es_cohen_g) for Cohen g
    * [es_alt_ratio](../effect_sizes/eff_size_alt_ratio.html#es_alt_ratio) for Alternative Ratio
    * [r_rosenthal](../correlations/cor_rosenthal.html#r_rosenthal) for Rosenthal Correlation if a z-value is available
    
    References
    ----------
    Cohen, J. (1988). *Statistical power analysis for the behavioral sciences* (2nd ed.). L. Erlbaum Associates.
    
    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 list
    >>> ex1 = [1, 1, 2, 1, 2, 1, 2, 1]
    >>> es_cohen_h_os(ex1)
       Cohen h'            comment
    0   0.25268  assuming p0 for 1
    >>> es_cohen_h_os(ex1, p0=0.3)
       Cohen h'            comment
    0  0.664197  assuming p0 for 1
    
    Example 2: pandas Series
    >>> import pandas as pd
    >>> df1 = pd.read_csv('https://peterstatistics.com/Packages/ExampleData/GSS2012a.csv', sep=',', low_memory=False, storage_options={'User-Agent': 'Mozilla/5.0'})
    >>> es_cohen_h_os(df1['sex'])
       Cohen h'                 comment
    0   0.10251  assuming p0 for FEMALE
    >>> es_cohen_h_os(df1['mar1'], codes=["DIVORCED", "NEVER MARRIED"])
       Cohen h'               comment
    0 -0.114495  with p0 for DIVORCED
    
    '''
    
    if type(data) is list:
        data = pd.Series(data)

    #remove missing values
    data = data.dropna()
        
    #Determine number of successes, failures, and total sample size
    if codes is None:
        #create a frequency table
        freq = data.value_counts()

        if p0Cat is None:
            #check if there were exactly two categories or not
            if len(freq) != 2:
                # unable to determine which category p0 would belong to, so print warning and end
                print("WARNING: data does not have two unique categories, please specify two categories using codes parameter")
                return
            else:
                #simply select the two categories as cat1 and cat2
                n1 = freq.values[0]
                n2 = freq.values[1]
                n = n1 + n2
                #determine p0 was for which category
                p0_cat = freq.index[0]
                if p0 > 0.5 and n1 < n2:
                    n3=n2
                    n2 = n1
                    n1 = n3
                    p0_cat = freq.index[1]
                cat_used =  "assuming p0 for " + str(p0_cat)
        else:
            n = sum(freq.values)
            n1 = sum(data==p0Cat)
            n2 = n - n1
            p0_cat = p0Cat
            cat_used = "with p0 for " + str(p0Cat)
    else:        
        n1 = sum(data==codes[0])
        n2 = sum(data==codes[1])
        n = n1 + n2
        cat_used =  "with p0 for " + str(codes[0])
        
    p1 = n1/n
    
    phi1 = 2 * math.asin(p1**0.5)
    phic = 2 * math.asin(p0**0.5)
    
    h2 = phi1 - phic

    results = pd.DataFrame([[h2, cat_used]], columns=["Cohen h'", "comment"])
    
    return (results)