Module stikpetP.correlations.cor_rank_biserial_os

Functions

def r_rank_biserial_os(data, levels=None, mu=None)

Rank biserial correlation coefficient (one-sample)

This function will calculate Rank biserial correlation coefficient (one-sample)

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

Parameters

data : list or pandas data series

The data to analyse.

levels : dictionary, optional

with the categories and numeric value to use

mu : float, optional

parameter to set the hypothesized median. If not used the midrange is used

Returns

results : pandas dataframe

with the hypothesized median and effect size.

Notes

The formula used (Kerby, 2014, p. 5): $$r_{rb} = \frac{\left|R_{pos} - R_{neg}\right|}{R}$$

This is actually the same as (King & Minium, 2008, p. 403): $$r_{rb} = \frac{4\times\left|R_{min} - \frac{R_{pos} + R_{min}}{2}\right|}{n\times\left(n + 1\right)}$$

Symbols used:

• $R_{pos}$ the sum of the ranks with a positive deviation from the hypothesized median
• $R_{neg}$ the sum of the ranks with a positive deviation from the hypothesized median
• $R_{min}$ the minimum of $R_{pos}$, $R_{neg}$
• $n$ the number of ranks with a non-zero difference with the hypothesized median
• $R$ the sum of all ranks, i.e. $R_{pos} + R_{neg}$

If no hypothesized median is provided, the midrange is used, defined as: $$\frac{x_{max} - x_{min}}{2}$$

Where $x_{max}$ is the maximum value of the scores, and $x_{min}$ the minimum

Classification

Use the th_rank_biserial() function for a classification of the value.

Before, After and Alternatives

Before this measure you might want to perform the test: * ts_sign_os for One-Sample Sign Test * ts_trinomial_os for One-Sample Trinomial Test * ts_wilcoxon_os for Wilcoxon Signed Rank Test (One-Sample)

After this you might want a rule-of-thumb: * th_rank_biserial for Rank Biserial Correlation rule-of-thumb

Alternative effect size measure: * es_common_language_os for the Common Language Effect Size * es_dominance for the Dominance score * r_rosenthal for the Rosenthal Correlation if a z-value is available

References

Kerby, D. S. (2014). The simple difference formula: An approach to teaching nonparametric correlation. Comprehensive Psychology, 3, 1–9. doi:10.2466/11.IT.3.1

King, B. M., & Minium, E. W. (2008). Statistical reasoning in the behavioral sciences (5th ed.). John Wiley & Sons, Inc.

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

Load example data

>>> df2 = pd.read_csv('https://peterstatistics.com/Packages/ExampleData/StudentStatistics.csv', sep=';', low_memory=False, storage_options={'User-Agent': 'Mozilla/5.0'})

Example with pandas series

>>> ex1 = df2['Teach_Motivate']
>>> order = {"Fully Disagree":1, "Disagree":2, "Neither disagree nor agree":3, "Agree":4, "Fully agree":5}
>>> r_rank_biserial_os(ex1, levels=order)
    mu       rb
0  3.0 -0.47619

Example with a list

>>> ex2 = [1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5]
>>> r_rank_biserial_os(ex2)
    mu        rb
0  3.0  0.338235

Expand source code

def r_rank_biserial_os(data, levels=None, mu=None):
    '''
    Rank biserial correlation coefficient (one-sample)
    --------------------------------------------------
     
    This function will calculate Rank biserial correlation coefficient (one-sample)

    This function is shown in this [YouTube video](https://youtu.be/0PEwwuhgqdw) and the measure is also described at [PeterStatistics.com](https://peterstatistics.com/Terms/Correlations/RankBiserialCorrelation.html)
    
    Parameters
    ----------
    data : list or pandas data series
            The data to analyse.
    levels : dictionary, optional
            with the categories and numeric value to use
    mu : float, optional
        parameter to set the hypothesized median. If not used the midrange is used
        
    Returns
    -------
    results : pandas dataframe 
        with the hypothesized median and effect size.
    
    Notes
    -----
    The formula used (Kerby, 2014, p. 5):
    $$r_{rb} = \\frac{\\left|R_{pos} - R_{neg}\\right|}{R}$$
    
    This is actually the same as (King & Minium, 2008, p. 403):
    $$r_{rb} = \\frac{4\\times\\left|R_{min} - \\frac{R_{pos} + R_{min}}{2}\\right|}{n\\times\\left(n + 1\\right)}$$
    
    *Symbols used:*
    
    * $R_{pos}$ the sum of the ranks with a positive deviation from the hypothesized median 
    * $R_{neg}$ the sum of the ranks with a positive deviation from the hypothesized median 
    * $R_{min}$ the minimum of $R_{pos}$, $R_{neg}$
    * $n$ the number of ranks with a non-zero difference with the hypothesized median 
    * $R$ the sum of all ranks, i.e. $R_{pos} + R_{neg}$
    
    If no hypothesized median is provided, the midrange is used, defined as:
    $$\\frac{x_{max} - x_{min}}{2}$$
    
    Where $x_{max}$ is the maximum value of the scores, and $x_{min}$ the minimum
    
    *Classification*
    
    Use the **th_rank_biserial()** function for a classification of the value.

    Before, After and Alternatives
    ------------------------------
    Before this measure you might want to perform the test:
    * [ts_sign_os](../tests/test_sign_os.html#ts_sign_os) for One-Sample Sign Test
    * [ts_trinomial_os](../tests/test_trinomial_os.html#ts_trinomial_os) for One-Sample Trinomial Test
    * [ts_wilcoxon_os](../tests/test_wilcoxon_os.html#ts_wilcoxon_os) for Wilcoxon Signed Rank Test (One-Sample)

    After this you might want a rule-of-thumb:
    * [th_rank_biserial](../other/thumb_rank_biserial.html#th_rank_biserial) for Rank Biserial Correlation rule-of-thumb
    
    Alternative effect size measure:
    * [es_common_language_os](../effect_sizes/eff_size_common_language_os.html#es_common_language_os) for the Common Language Effect Size
    * [es_dominance](../effect_sizes/eff_size_dominance.html#es_dominance) for the Dominance score
    * [r_rosenthal](../correlations/cor_rosenthal.html#r_rosenthal) for the Rosenthal Correlation if a z-value is available
    
    References 
    ----------
    Kerby, D. S. (2014). The simple difference formula: An approach to teaching nonparametric correlation. *Comprehensive Psychology*, 3, 1–9. doi:10.2466/11.IT.3.1
    
    King, B. M., & Minium, E. W. (2008). *Statistical reasoning in the behavioral sciences* (5th ed.). John Wiley & Sons, Inc.
    
    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
    --------
    Load example data
    >>> df2 = pd.read_csv('https://peterstatistics.com/Packages/ExampleData/StudentStatistics.csv', sep=';', low_memory=False, storage_options={'User-Agent': 'Mozilla/5.0'})
    
    Example with pandas series
    >>> ex1 = df2['Teach_Motivate']
    >>> order = {"Fully Disagree":1, "Disagree":2, "Neither disagree nor agree":3, "Agree":4, "Fully agree":5}
    >>> r_rank_biserial_os(ex1, levels=order)
        mu       rb
    0  3.0 -0.47619
    
    Example with a list
    >>> ex2 = [1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5]
    >>> r_rank_biserial_os(ex2)
        mu        rb
    0  3.0  0.338235
    
    '''
    #data as a pandas series
    if type(data) is list:
        data = pd.Series(data)
        
    #remove missing values
    data = data.dropna()
    if levels is not None:
        pd.set_option('future.no_silent_downcasting', True)
        data = data.map(levels).astype('Int8')
    else:
        data = pd.to_numeric(data)
    
    data = data.sort_values()
    
    #set hypothesized median to mid range if not provided
    if (mu is None):
        mu = (min(data) + max(data)) / 2
        
    #remove scores equal to hypothesized median
    rankDf = pd.DataFrame(data[data-mu != 0])
    
    #determine differences with mu and absolute values of those
    rankDf['diff'] = rankDf.iloc[:,0] - mu    
    rankDf['abs. diff.'] = abs(rankDf['diff'])
    
    #rank the absolute differences
    rankDf['ranks'] = rankDf['abs. diff.'].rank()
    
    #determine the sum of the positive and the negative ranks
    Rplus = rankDf.query("diff > 0", engine="python")["ranks"].sum()
    Rmin = rankDf.query("diff < 0", engine="python")["ranks"].sum()
    
    #calculate the rank biserial correlation
    rb = (Rplus - Rmin)/(Rplus + Rmin)
    
    #prepare results
    results = pd.DataFrame([[mu, rb]], columns=["mu", "rb"])        
    pd.set_option('display.max_colwidth', None)
    
    return results