Module stikpetP.tests.test_trinomial_os

Functions

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

One-Sample Trinomial Test

A test that could be used with ordinal data that includes ties

Similar as a sign-test but instead of ignoring scores that are tied with the hypothesized median they get included, hence instead of the binomial distribution, this will use the trinomial distribution.

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

Parameters

data : list or pandas data series

the data

levels : dictionary, optional

the categories and numeric value to use

mu : float, optional

hypothesized median. Default is the midrange of the data

Returns

pandas.DataFrame

A dataframe with the following columns:

• mu, the hypothesized median
• n-pos, the number scores above mu
• n-neg, the number scores below mu
• n-tied, the number of scores tied with mu
• p-value, significance (p-value)
• test, description of the test used

Notes

The test uses the trinomial probability mass function and can be found in Bian et al. (2009, p. 6).

The formula used is: $$p = 2\times \sum_{i=n_d}^n \sum_{j=0}^{\lfloor \frac{n - i}{2} \rfloor} \text{tri}\left(\left(j, j+i, n - i\right), \left(p_{pos}, p_{neg}, p_0\right) \right)$$

With: $$p_0 = \frac{n_0}{n}$$ $$p_{pos} = p_{neg} = \frac{1 - p_0}{n}$$ $$\left|n_{pos} - n_{neg}\right|$$

Symbols used:

• $n_0$, the number of scores equal to the hypothesized median
• $n_{pos}$, the number of scores above the hypothesized median
• $n_{neg}$, the number of scores below the hypothesized median
• $p_0$, the probability of the a score in the sample being equal to the hypothesized median
• $p_{pos}$, the population proportion of a score being above the hypothesized median
• $p_{neg}$, the population proportion of a score being below the hypothesized median
• $\text{tri}\left(…,… \right)$, the trinomial probability mass function

The paired version of the test is described in Bian et al. (1941), while Zaiontz (n.d.) mentions it can also be used for one-sample situations.

Before, After and Alternatives

Before this test you might want an impression using a frequency table or a visualisation: * tab_frequency for a frequency table * vi_bar_stacked_single for Single Stacked Bar-Chart * vi_bar_dual_axis for Dual-Axis Bar Chart

After this you might want to determine an effect size measure: * es_common_language_os for the Common Language Effect Size * es_dominance for the Dominance score * r_rank_biserial_os for the Rank-Biserial Correlation

Alternative tests: * ts_sign_os for One-Sample Sign Test * ts_wilcoxon_os for Wilcoxon Signed Rank Test (One-Sample)

The function makes use of: * di_mcdf for the Multinomial Cumulative Distribution Function

References

Bian, G., McAleer, M., & Wong, W.-K. (2009). A trinomial test for paired data when there are many ties. SSRN Electronic Journal. doi:10.2139/ssrn.1410589

Zaiontz, C. (n.d.). Trinomial test. Real Statistics Using Excel. Retrieved March 2, 2023, from https://real-statistics.com/non-parametric-tests/trinomial-test/

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

>>> pd.set_option('display.width',1000)
>>> pd.set_option('display.max_columns', 1000)

Example 1: pandas series

>>> student_df = pd.read_csv('https://peterstatistics.com/Packages/ExampleData/StudentStatistics.csv', sep=';', low_memory=False, storage_options={'User-Agent': 'Mozilla/5.0'})
>>> ex1 = student_df['Teach_Motivate']
>>> order = {"Fully Disagree":1, "Disagree":2, "Neither disagree nor agree":3, "Agree":4, "Fully agree":5}
>>> ts_trinomial_os(ex1, levels=order)
    mu  n-pos.  n-neg.  n-tied.   p-value                  test
0  3.0      13      29       12  0.016261  one-sample trinomial

Example 2: Numeric data

>>> ex2 = [1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5]
>>> ts_trinomial_os(ex2)
    mu  n-pos.  n-neg.  n-tied.   p-value                  test
0  3.0      10       6        2  0.385768  one-sample trinomial

Expand source code

def ts_trinomial_os(data, levels=None, mu = None):
    '''
    One-Sample Trinomial Test
    -------------------------
    A test that could be used with ordinal data that includes ties
    
    Similar as a sign-test but instead of ignoring scores that are tied with the hypothesized median they get included, hence instead of the binomial distribution, this will use the trinomial distribution.

    This function is shown in this [YouTube video](https://youtu.be/6kTZkgkEqHw) and the test is also described at [PeterStatistics.com](https://peterstatistics.com/Terms/Tests/WilcoxonSignedRankOneSample.html)
    
    Parameters
    ----------
    data : list or pandas data series 
        the data
    levels : dictionary, optional
        the categories and numeric value to use
    mu : float, optional 
        hypothesized median. Default is the midrange of the data
        
    Returns
    -------
    pandas.DataFrame
        A dataframe with the following columns:
    
        * *mu*, the hypothesized median
        * *n-pos*, the number scores above mu
        * *n-neg*, the number scores below mu
        * *n-tied*, the number of scores tied with mu
        * *p-value*, significance (p-value)
        * *test*, description of the test used
    
    Notes
    -----
    The test uses the trinomial probability mass function and can be found in Bian et al. (2009, p. 6).
    
    The formula used is:
    $$p = 2\\times \\sum_{i=n_d}^n \\sum_{j=0}^{\\lfloor \\frac{n - i}{2} \\rfloor} \\text{tri}\\left(\\left(j, j+i, n - i\\right), \\left(p_{pos}, p_{neg}, p_0\\right) \\right)$$
    
    With:
    $$p_0 = \\frac{n_0}{n}$$
    $$p_{pos} = p_{neg} = \\frac{1 - p_0}{n}$$
    $$\\left|n_{pos} - n_{neg}\\right|$$
    
    *Symbols used:*
    
    * $n_0$, the number of scores equal to the hypothesized median
    * $n_{pos}$, the number of scores above the hypothesized median
    * $n_{neg}$, the number of scores below the hypothesized median
    * $p_0$, the probability of the a score in the sample being equal to the hypothesized median
    * $p_{pos}$, the population proportion of a score being above the hypothesized median
    * $p_{neg}$, the population proportion of a score being below the hypothesized median
    * $\\text{tri}\\left(…,… \\right)$, the trinomial probability mass function
    
    The paired version of the test is described in Bian et al. (1941), while Zaiontz (n.d.) mentions it can also be used for one-sample situations.

    Before, After and Alternatives
    ------------------------------
    Before this test you might want an impression using a frequency table or a visualisation:
    * [tab_frequency](../other/table_frequency.html#tab_frequency) for a frequency table
    * [vi_bar_stacked_single](../visualisations/vis_bar_stacked_single.html#vi_bar_stacked_single) for Single Stacked Bar-Chart
    * [vi_bar_dual_axis](../visualisations/vis_bar_dual_axis.html#vi_bar_dual_axis) for Dual-Axis Bar Chart

    After this you might want to determine an 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_rank_biserial_os](../correlations/cor_rank_biserial_os.html#r_rank_biserial_os) for the Rank-Biserial Correlation
    
    Alternative tests:
    * [ts_sign_os](../tests/test_sign_os.html#ts_sign_os) for One-Sample Sign Test
    * [ts_wilcoxon_os](../tests/test_wilcoxon_os.html#ts_wilcoxon_os) for Wilcoxon Signed Rank Test (One-Sample)

    The function makes use of:
    * [di_mcdf](../distributions/dist_multinomial.html#di_mcdf) for the Multinomial Cumulative Distribution Function
    
    References
    ----------
    Bian, G., McAleer, M., & Wong, W.-K. (2009). A trinomial test for paired data when there are many ties. *SSRN Electronic Journal*. doi:10.2139/ssrn.1410589
    
    Zaiontz, C. (n.d.). Trinomial test. Real Statistics Using Excel. Retrieved March 2, 2023, from https://real-statistics.com/non-parametric-tests/trinomial-test/
    
    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
    ---------
    >>> pd.set_option('display.width',1000)
    >>> pd.set_option('display.max_columns', 1000)
    
    Example 1: pandas series
    >>> student_df = pd.read_csv('https://peterstatistics.com/Packages/ExampleData/StudentStatistics.csv', sep=';', low_memory=False, storage_options={'User-Agent': 'Mozilla/5.0'})
    >>> ex1 = student_df['Teach_Motivate']
    >>> order = {"Fully Disagree":1, "Disagree":2, "Neither disagree nor agree":3, "Agree":4, "Fully agree":5}
    >>> ts_trinomial_os(ex1, levels=order)
        mu  n-pos.  n-neg.  n-tied.   p-value                  test
    0  3.0      13      29       12  0.016261  one-sample trinomial
    
    Example 2: Numeric data
    >>> ex2 = [1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5]
    >>> ts_trinomial_os(ex2)
        mu  n-pos.  n-neg.  n-tied.   p-value                  test
    0  3.0      10       6        2  0.385768  one-sample trinomial
    
    '''
    
    if type(data) is list:
        data = pd.Series(data)
    
    #remove missing values
    data = data.dropna()
    if levels is not None:
        data = data.map(levels).astype('Int8')
    else:
        data = pd.to_numeric(data)
    
    #set hypothesized median to mid range if not provided
    if (mu is None):
        mu = (min(data) + max(data)) / 2
        
    nPos = sum(data>mu)
    nNeg = sum(data<mu)
    nNul = sum(data==mu)
    n = nPos + nNeg + nNul
    nd = abs(nPos-nNeg)

    pNul = nNul/n
    pPos = (1 - pNul)/2
    pNeg = pPos
    
    sig = 0
    for d in range(nd, n+1):
        for k in range(0, math.floor((n - d)/2)+1):
            pmf = di_mpmf([k, k + d, n-k-(k+d)], [pPos, pNeg, pNul])
            sig = sig + pmf
    
    sig = sig*2
    if sig>1:
        sig = 1
    
    testResults = pd.DataFrame([[mu, nPos, nNeg, nNul, sig, "one-sample trinomial"]], columns=["mu", "n-pos.", "n-neg.", "n-tied.", "p-value", "test"])
    pd.set_option('display.max_colwidth', None)
    
    return (testResults)