Module stikpetP.tests.test_wilcoxon_os

Functions

def ts_wilcoxon_os(data, levels=None, mu=None, ties=True, appr='wilcoxon', eqMed='wilcoxon', cc=False)

Wilcoxon Signed Rank Test (One-Sample)

The one-sample Wilcoxon signed rank test is often considered the non-parametric version of a one-sample t-test. It can be used to determine if the median is significantly different from an hypothesized value. It actually doesn't always tests this specifically, but more if the mean rank is significantly different.

If the p-value is the probability of a result as in the sample, or more extreme, if the assumption about the population would be true. If this is below a certain threshold (usually 0.05) the assumption about the population is rejected. For this test the assumed median for the population is then incorrect.

Results in software packages for this test can vary, since there are a few different approaches. Especially if there are so-called ties. See the notes for more information.

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

ties : boolean, optional

to use a tie correction. Default is True

appr : {"wilcoxon", "exact" "imanz", "imant"}, optional

method to use for approximation. Default is "wilcoxon"

eqMed : {"wilcoxon", "pratt", "zsplit"}, optional

method to deal with scores equal to mu. Default is "wilcoxon"

cc : boolean, optional

use a continuity correction. Default is False

Returns

pandas.DataFrame

A dataframe with the following columns:

• nr, the number of ranks used in calculation
• mu, the median according to the null hypothesis
• W, the Wilcoxon W value
• statistic, the test statistic (z-value, or t-value)
• df, degrees of freedom (only applicable for Iman t approximation)
• p-value, significance (p-value)
• test, description of the test used

Notes

This uses the ranks function from Pandas, the NormalDist function from Python's statistics, the t function from scipy.stats, comb function from Python's math library and two helper functions wcdf and srf for the exact Wilcoxon distribution.

The unadjusted test statistic is given by: $$W=\sum_{i=1}^{n_{r}^{+}}r_{i}^{+}$$

With: $$r=\text{rank}(|d|)$$ $$d_{i}=y_{i}-\theta$$

Symbols used: * $n_{r}^{+}$ is the number of ranks with a positive deviation from the hypothesized median * $r_{i}^{+}$ the i-th rank of the ranks with a positive deviation from the hypothesized median * $\theta$ is the median tested (the hypothesized median). * $y_i$ is the i-th score of the variable after removing scores that were equal to $\theta$

If there are no ties, an exact method can be used, using the Sign Rank Distribution. The exact test can be found in Zaiontz (n.d.)

Approximations

If the sample size is large enough, we can use a normal approximation. What is large enough varies quite per author. A few examples: n > 8 (slideplayer, 2015), n > 15 (SigMaxl, n.d.), n > 20 (Wikipedia, n.d.), n > 25 (Harris & Hardin, 2013), n > 30 (Winthrop, n.d.) .

The z-statistic is given by (appr="wilcoxon", ties=FALSE, cc=FALSE): $$Z = \frac{W - \mu_w}{\sigma_w}$$ or with a ties correction (appr="wilcoxon", ties=TRUE, cc=FALSE): $$Z_{adj} = \frac{W - \mu_w}{\sigma_w^*}$$

With: $$\mu_w = \frac{n_r\times\left(n_r + 1\right)}{4}$$ $$\sigma_w^2 = \frac{n_r\times\left(n_r + 1\right)\times\left(2\times n_r + 1\right)}{24}$$ $$\sigma_w^{*2} = \sigma_w^2 - A$$ $$A = \frac{\sum_{i=1}^k \left(t_i^3 - t_i\right)}{48}$$

Additional symbols used * $n_{r}$ is the number of ranks used * $k$ the number of unique ranks * $t_i$ the frequency of the i-th unique rank

A Yates continuity correction can simply be applied: In case of no ties (appr="wilcoxon", ties=FALSE, cc=TRUE): $$Z = \frac{\left|W - \mu_w\right| - 0.5}{\sigma_w}$$ In case of ties (appr="wilcoxon", ties=TRUE, cc=TRUE): $$Z_{adj} = \frac{\left|W - \mu_w\right| - 0.5}{\sigma_w^*}$$

An alternative approximation using the Student t distribution is given by Iman (1974, p. 799). The formula is (appr="imant", ties=FALSE, cc=FALSE): $$t = \frac{W - \mu_w}{\sqrt{\frac{\sigma_w^2\times n_r - \left(W - \mu_w\right)^2}{n_r - 1}}}$$ or with the ties correction (appr="imant", ties=TRUE, cc=FALSE): $$t = \frac{W - \mu_w}{\sqrt{\frac{\sigma_w^{*2}\times n_r - \left(W - \mu_w\right)^2}{n_r - 1}}}$$

The two versions for with a continuity correction are: No ties correction, but continuity (appr="imant", ties=FALSE, cc=TRUE): $$t = \frac{\left|W - \mu_w\right| - 0.5}{\sqrt{\frac{\sigma_w^2\times n_r - \left(\left|W - \mu_w\right| - 0.5\right)^2}{n_r - 1}}}$$ Both corrections (appr="imant", ties=TRUE, cc=TRUE): $$t = \frac{\left|W - \mu_w\right| - 0.5}{\sqrt{\frac{\sigma_w^{*2}\times n_r - \left(\left|W - \mu_w\right| - 0.5\right)^2}{n_r - 1}}}$$

Iman (1974, p. 803) also provides a combination of the t-approximation and the regular z-approximation. The equation is given by (appr="imanz"): $$Z_{I} = \frac{Z}{2}\times\left(1 + \sqrt{\frac{n_r - 1}{n_r - Z^2}}\right)$$ The $Z$ is any of the previous methods.

Ties with mu

The default (eqMed="wilcoxon") removes first any scores that are equal to the hypothesized median. There are two alternative methods for this.Both re-define \eqn{d_i} to: $$d_i = x_i - \theta$$ Where $x_i$ is simply the i-th score.

For the z-split method we only need to re-define: $$W = \frac{\sum_{i=1}^{n_{d_0}}r_{i,0}}{2} + \sum_{i=1}^{n_{r}^{+}}r_{i}^{+}$$ Where $n_{d_0}$ is the number of scores that equal the hypothesized median, and $r_{i,0}$ is the rank of the i-th score that equals the hypothesized median.

In essence we added half the sum of the ranks that were equal to the hypothesized median.

For the z-split method all other calculations than go the same.

For the Pratt (1959) method we also re-define: $$\mu_w = \frac{n_r\times\left(n_r + 1\right) - n_{d_0}\times\left(n_{d_0} + 1\right)}{4}$$ $$\sigma_w^2 = \frac{n_r\times\left(n_r + 1\right)\times\left(2\times n_r + 1\right) - n_{d_0}\times\left(n_{d_0} + 1\right)\times\left(2\times n_{d_0} + 1\right)}{24}$$

For the Pratt method, the ties correction still excludes the ties for the scores that equal the hypothesized median, but for the z-split method it will include them.

For both methods now $n_r=n$, where n is the number of scores.

The Pratt (1959) method and z-split method were found in Python’s documentation for scipy’s Wilcoxon function (scipy, n.d.). They also refer to Cureton (1967) for the Pratt method.

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 * r_rosenthal for the Rosenthal Correlation if a normal approximation was used

Alternative tests: * ts_sign_os for One-Sample Sign Test * ts_trinomial_os for One-Sample Trinomial Test

The function makes use of: * di_wcdf for the Wilcoxon Cumulative Distribution Function

References

Cureton, E. E. (1967). The normal approximation to the signed-rank sampling distribution when zero differences are present. Journal of the American Statistical Association, 62(319), 1068–1069. doi:10.1080/01621459.1967.10500917

Harris, T., & Hardin, J. W. (2013). Exact Wilcoxon Signed-Rank and Wilcoxon Mann–Whitney Ranksum Tests. The Stata Journal, 13(2), 337–343. doi:10.1177/1536867X1301300208

Iman, R. L. (1974). Use of a t-statistic as an approximation to the exact distribution of the wildcoxon signed ranks test statistic. Communications in Statistics, 3(8), 795–806. doi:10.1080/03610927408827178

Pratt, J. W. (1959). Remarks on zeros and ties in the Wilcoxon signed rank procedures. Journal of the American Statistical Association, 54(287), 655–667. doi:10.1080/01621459.1959.10501526

scipy. (n.d.). Scipy.stats.wilcoxon. Scipy. https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.wilcoxon.html

SigMaxl. (n.d.). One Sample Wilcoxon Sign Test Exact. Retrieved August 30, 2020, from https://www.sigmaxl.com/OneSampleSignWilcoxonExact.shtml

slideplayer. (2015, June 13). Using statistics to make inferences 6.

Wikipedia. (n.d.). Wilcoxon signed-rank test. In Wikipedia. Retrieved August 30, 2020, from https://en.wikipedia.org/w/index.php?title=Wilcoxon_signed-rank_test&oldid=974561084

Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics Bulletin, 1(6), 80. doi:10.2307/3001968

Winthrop. (n.d.). The Wilcoxon signed rank test for one sample. Winthrop Univerisy Hospital. https://nyuwinthrop.org/wp-content/uploads/2019/08/wilcoxon-sign-rank-test-one-sample.pdf

Zaiontz, C. (n.d.). Wilcoxon signed ranks exact test. Real Statistics Using Excel. Retrieved January 25, 2023, from https://real-statistics.com/non-parametric-tests/wilcoxon-signed-ranks-test/wilcoxon-signed-ranks-exact-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

>>> df2 = pd.read_csv('https://peterstatistics.com/Packages/ExampleData/StudentStatistics.csv', sep=';', low_memory=False, storage_options={'User-Agent': 'Mozilla/5.0'})
>>> ex1 = df2['Teach_Motivate']
>>> order = {"Fully Disagree":1, "Disagree":2, "Neither disagree nor agree":3, "Agree":4, "Fully agree":5}
>>> ts_wilcoxon_os(ex1, levels=order)
   nr   mu      W  statistic    df   p-value                                                        test
0  42  3.0  236.5   2.788301  n.a.  0.005299  one-sample Wilcoxon signed rank test, with ties correction

Example 2: Numeric data

>>> ex2 = [1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5]
>>> ts_wilcoxon_os(ex2)
   nr   mu     W  statistic    df   p-value                                                        test
0  16  3.0  91.0   1.231162  n.a.  0.218262  one-sample Wilcoxon signed rank test, with ties correction

Expand source code

def ts_wilcoxon_os(data, levels=None, mu = None, ties = True, 
               appr = "wilcoxon", eqMed = "wilcoxon", cc = False):
    '''
    Wilcoxon Signed Rank Test (One-Sample)
    --------------------------------------
     
    The one-sample Wilcoxon signed rank test is often considered the non-parametric version of a one-sample t-test. It can be used to determine if the median is significantly different from an hypothesized value. It actually doesn't always tests this specifically, but more if the mean rank is significantly different.
    
    If the p-value is the probability of a result as in the sample, or more extreme, if the assumption about the population would be true. If this is below a certain threshold (usually 0.05) the assumption about the population is rejected. For this test the assumed median for the population is then incorrect.
    
    Results in software packages for this test can vary, since there are a few different approaches. Especially if there are so-called ties. See the notes for more information.

    This function is shown in this [YouTube video](https://youtu.be/zChr4HhWMhY) 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
    ties : boolean, optional 
        to use a tie correction. Default is True
    appr : {"wilcoxon", "exact" "imanz", "imant"}, optional
        method to use for approximation. Default is "wilcoxon"
    eqMed : {"wilcoxon", "pratt", "zsplit"}, optional 
        method to deal with scores equal to mu. Default is "wilcoxon"
    cc : boolean, optional 
        use a continuity correction. Default is False
        
    Returns
    -------
    pandas.DataFrame
        A dataframe with the following columns:
    
        * *nr*, the number of ranks used in calculation
        * *mu*, the median according to the null hypothesis
        * *W*, the Wilcoxon W value
        * *statistic*, the test statistic (z-value, or t-value)
        * *df*, degrees of freedom (only applicable for Iman t approximation)
        * *p-value*, significance (p-value)
        * *test*, description of the test used
   
    Notes
    -----
    This uses the ranks function from Pandas, the NormalDist function from Python's statistics, the t function from scipy.stats, comb function from Python's math library and two helper functions wcdf and srf for the exact Wilcoxon distribution.
    
    The unadjusted test statistic is given by:
    $$W=\\sum_{i=1}^{n_{r}^{+}}r_{i}^{+}$$
    
    With:
    $$r=\\text{rank}(|d|)$$
    $$d_{i}=y_{i}-\\theta$$
    
    *Symbols used:*
    * $n_{r}^{+}$ is the number of ranks with a positive deviation from the hypothesized median
    * $r_{i}^{+}$ the i-th rank of the ranks with a positive deviation from the hypothesized median
    * $\\theta$ is the median tested (the hypothesized median).
    * $y_i$ is the i-th score of the variable after removing scores that were equal to $\\theta$
    
    If there are no ties, an exact method can be used, using the Sign Rank Distribution. The exact test can be found in Zaiontz (n.d.)
    
    **Approximations**
    
    If the sample size is large enough, we can use a normal approximation. What is large enough varies quite per author. A few examples: n > 8 (slideplayer, 2015), n > 15 (SigMaxl, n.d.), n > 20 (Wikipedia, n.d.), n > 25 (Harris & Hardin, 2013), n > 30 (Winthrop, n.d.) . 
    
    The z-statistic is given by (appr="wilcoxon", ties=FALSE, cc=FALSE):
    $$Z = \\frac{W - \\mu_w}{\\sigma_w}$$
    or with a ties correction (appr="wilcoxon", ties=TRUE, cc=FALSE):
    $$Z_{adj} = \\frac{W - \\mu_w}{\\sigma_w^*}$$
    
    With:
    $$\\mu_w = \\frac{n_r\\times\\left(n_r + 1\\right)}{4}$$
    $$\\sigma_w^2 = \\frac{n_r\\times\\left(n_r + 1\\right)\\times\\left(2\\times n_r + 1\\right)}{24}$$
    $$\\sigma_w^{*2} = \\sigma_w^2 - A$$
    $$A = \\frac{\\sum_{i=1}^k \\left(t_i^3 - t_i\\right)}{48}$$
    
    *Additional symbols used*
    * $n_{r}$ is the number of ranks used
    * $k$ the number of unique ranks
    * $t_i$ the frequency of the i-th unique rank
    
    A Yates continuity correction can simply be applied: 
    In case of no ties (appr="wilcoxon", ties=FALSE, cc=TRUE):
    $$Z = \\frac{\\left|W - \\mu_w\\right| - 0.5}{\\sigma_w}$$
    In case of ties (appr="wilcoxon", ties=TRUE, cc=TRUE):
    $$Z_{adj} = \\frac{\\left|W - \\mu_w\\right| - 0.5}{\\sigma_w^*}$$
    
    An alternative approximation using the Student t distribution is given by Iman (1974, p. 799). The formula is (appr="imant", ties=FALSE, cc=FALSE):
    $$t = \\frac{W - \\mu_w}{\\sqrt{\\frac{\\sigma_w^2\\times n_r - \\left(W - \\mu_w\\right)^2}{n_r - 1}}}$$
    or with the ties correction (appr="imant", ties=TRUE, cc=FALSE):
    $$t = \\frac{W - \\mu_w}{\\sqrt{\\frac{\\sigma_w^{*2}\\times n_r - \\left(W - \\mu_w\\right)^2}{n_r - 1}}}$$
    
    The two versions for with a continuity correction are:
    No ties correction, but continuity (appr="imant", ties=FALSE, cc=TRUE):
    $$t = \\frac{\\left|W - \\mu_w\\right| - 0.5}{\\sqrt{\\frac{\\sigma_w^2\\times n_r - \\left(\\left|W - \\mu_w\\right| - 0.5\\right)^2}{n_r - 1}}}$$
    Both corrections (appr="imant", ties=TRUE, cc=TRUE):
    $$t = \\frac{\\left|W - \\mu_w\\right| - 0.5}{\\sqrt{\\frac{\\sigma_w^{*2}\\times n_r - \\left(\\left|W - \\mu_w\\right| - 0.5\\right)^2}{n_r - 1}}}$$
    
    Iman (1974, p. 803) also provides a combination of the t-approximation and the regular z-approximation. The equation is given by (appr="imanz"):
    $$Z_{I} = \\frac{Z}{2}\\times\\left(1 + \\sqrt{\\frac{n_r - 1}{n_r - Z^2}}\\right)$$
    The $Z$ is any of the previous methods.
    
    **Ties with mu**
    
    The default (eqMed="wilcoxon") removes first any scores that are equal to the hypothesized median. There are two alternative methods for this.Both re-define \\eqn{d_i} to:
    $$d_i = x_i - \\theta$$
    Where $x_i$ is simply the i-th score.
    
    For the z-split method we only need to re-define:
    $$W = \\frac{\\sum_{i=1}^{n_{d_0}}r_{i,0}}{2} + \\sum_{i=1}^{n_{r}^{+}}r_{i}^{+}$$
    Where $n_{d_0}$ is the number of scores that equal the hypothesized median, and $r_{i,0}$ is the rank of the i-th score that equals the hypothesized median.
    
    In essence we added half the sum of the ranks that were equal to the hypothesized median.
    
    For the z-split method all other calculations than go the same.
    
    For the Pratt (1959) method we also re-define:
    $$\\mu_w = \\frac{n_r\\times\\left(n_r + 1\\right) - n_{d_0}\\times\\left(n_{d_0} + 1\\right)}{4}$$
    $$\\sigma_w^2 = \\frac{n_r\\times\\left(n_r + 1\\right)\\times\\left(2\\times n_r + 1\\right) - n_{d_0}\\times\\left(n_{d_0} + 1\\right)\\times\\left(2\\times n_{d_0} + 1\\right)}{24}$$
    
    For the Pratt method, the ties correction still excludes the ties for the scores that equal the hypothesized median, but for the z-split method it will include them.
    
    For both methods now $n_r=n$, where n is the number of scores.
    
    The Pratt (1959) method and z-split method were found in Python’s documentation for scipy’s Wilcoxon function (scipy, n.d.). They also refer to Cureton (1967) for the Pratt method.

    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
    * [r_rosenthal](../correlations/cor_rosenthal.html#r_rosenthal) for the Rosenthal Correlation if a normal approximation was used
    
    Alternative tests:
    * [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

    The function makes use of:
    * [di_wcdf](../distributions/dist_wilcoxon.html#di_wcdf) for the Wilcoxon Cumulative Distribution Function
    
    References
    -----------
    Cureton, E. E. (1967). The normal approximation to the signed-rank sampling distribution when zero differences are present. *Journal of the American Statistical Association, 62*(319), 1068–1069. doi:10.1080/01621459.1967.10500917
    
    Harris, T., & Hardin, J. W. (2013). Exact Wilcoxon Signed-Rank and Wilcoxon Mann–Whitney Ranksum Tests. *The Stata Journal, 13*(2), 337–343. doi:10.1177/1536867X1301300208
    
    Iman, R. L. (1974). Use of a t-statistic as an approximation to the exact distribution of the wildcoxon signed ranks test statistic. *Communications in Statistics, 3*(8), 795–806. doi:10.1080/03610927408827178
    
    Pratt, J. W. (1959). Remarks on zeros and ties in the Wilcoxon signed rank procedures. *Journal of the American Statistical Association, 54*(287), 655–667. doi:10.1080/01621459.1959.10501526
    
    scipy. (n.d.). Scipy.stats.wilcoxon. Scipy. https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.wilcoxon.html
    
    SigMaxl. (n.d.). One Sample Wilcoxon Sign Test Exact. Retrieved August 30, 2020, from https://www.sigmaxl.com/OneSampleSignWilcoxonExact.shtml
    
    slideplayer. (2015, June 13). Using statistics to make inferences 6.
    
    Wikipedia. (n.d.). Wilcoxon signed-rank test. In Wikipedia. Retrieved August 30, 2020, from https://en.wikipedia.org/w/index.php?title=Wilcoxon_signed-rank_test&oldid=974561084
    
    Wilcoxon, F. (1945). Individual comparisons by ranking methods. *Biometrics Bulletin, 1*(6), 80. doi:10.2307/3001968
    
    Winthrop. (n.d.). The Wilcoxon signed rank test for one sample. Winthrop Univerisy Hospital. https://nyuwinthrop.org/wp-content/uploads/2019/08/wilcoxon-sign-rank-test-one-sample.pdf
    
    Zaiontz, C. (n.d.). Wilcoxon signed ranks exact test. Real Statistics Using Excel. Retrieved January 25, 2023, from https://real-statistics.com/non-parametric-tests/wilcoxon-signed-ranks-test/wilcoxon-signed-ranks-exact-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
    >>> df2 = pd.read_csv('https://peterstatistics.com/Packages/ExampleData/StudentStatistics.csv', sep=';', low_memory=False, storage_options={'User-Agent': 'Mozilla/5.0'})
    >>> ex1 = df2['Teach_Motivate']
    >>> order = {"Fully Disagree":1, "Disagree":2, "Neither disagree nor agree":3, "Agree":4, "Fully agree":5}
    >>> ts_wilcoxon_os(ex1, levels=order)
       nr   mu      W  statistic    df   p-value                                                        test
    0  42  3.0  236.5   2.788301  n.a.  0.005299  one-sample Wilcoxon signed rank test, with ties correction

    Example 2: Numeric data
    >>> ex2 = [1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5]
    >>> ts_wilcoxon_os(ex2)
       nr   mu     W  statistic    df   p-value                                                        test
    0  16  3.0  91.0   1.231162  n.a.  0.218262  one-sample Wilcoxon signed rank test, with ties correction

    '''
    
    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)
    
    data = data.sort_values()
    
    #set hypothesized median to mid range if not provided
    if (mu is None):
        mu = (min(data) + max(data)) / 2
        
    #sample size (n)
    n = len(data)
    
    #adjust sample size if wilcoxon method is used for equal distance
    if (eqMed == "wilcoxon"):
        nr = n - len(data[data == mu])
    else:
        nr = n
        
    absDiffs = 0
    #remove scores equal to mu if eqMed is wilcoxon
    if (eqMed == "wilcoxon" or appr=="exact"):
        data = data[data != mu]
        
    #determine the absolute deviations from the mu
    diffs = data - mu
    absDiffs = abs(diffs)
    ranks = absDiffs.rank()
    W = sum(ranks[diffs > 0])
    
    if appr=="exact":
        #check if ties exist
        if max(ranks.value_counts()) > 1:
            return "ties exist, cannot compute exact method"
        else:
            Wmin = sum(ranks[diffs < 0])
            statistic = min(W, Wmin)
            pVal = di_wcdf(int(statistic), len(ranks))*2
            testUsed = "one-sample Wilcoxon signed rank exact test"
            df = "n.a."
            
    else:
        #add half the equal to median ranks if zsplit is used
        nD0 = sum(diffs==0)
        if (eqMed == "zsplit"):
            W = W + sum(ranks[diffs == 0])/2

        rAvg = nr*(nr + 1)/4
        s2 = nr * (nr+1) * (2*nr + 1)/24
        #adjust if Pratt method is used
        if (eqMed == "pratt"):
            #normal approximation adjustment based on Cureton (1967)
            s2 = s2 - nD0 * (nD0 + 1) * (2 * nD0 + 1) / 24
            rAvg = (nr * (nr + 1) - nD0 * (nD0 + 1)) / 4

        #the ties correction
        tCorr = 0
        if (ties):
            #remove ranks of scores equal to mu for Pratt (wilcoxon already removed)
            if (eqMed == "pratt"):
                ranks = ranks[absDiffs != 0]
            for i in ranks.value_counts():
                tCorr = tCorr + (i**3 - i)
            tCorr = tCorr/48
            s2 = s2 - tCorr

        se = (s2)**0.5
        num = abs(W - rAvg)
        #apply continuity correction if needed
        if (cc):
            num = num - 0.5

        df = "n.a."

        if (appr=="imant"):
            if cc:
                statistic = num / ((s2 * nr - (abs(W - rAvg)-0.5)**2) / (nr - 1))**0.5

            else:
                statistic = num / ((s2 * nr - (W - rAvg)**2) / (nr - 1))**0.5
            df = nr - 1
            pVal = 2 * (1 - t.cdf(abs(statistic), df))
        else:
            statistic = num / se

            if (appr == "imanz"):
                statistic = statistic / 2 * (1 + ((nr - 1) / (nr - statistic**2))**0.5)

            pVal = 2 * (1 - NormalDist().cdf(abs(statistic))) 

        testUsed = "one-sample Wilcoxon signed rank test"
        if (ties and cc):
            testUsed = ", ".join([testUsed, "with ties and continuity correction"])
        elif (ties):
            testUsed = ", ".join([testUsed, "with ties correction"])
        elif (cc):
            testUsed = ", ".join([testUsed, "with continuity correction"])

        if (appr == "imant"):
            testUsed = ", ".join([testUsed, "using Iman (1974) t approximation"])
        elif (appr == "imanz"):
            testUsed = ", ".join([testUsed, "using Iman (1974) z approximation"])

        if (eqMed == "pratt"):
            testUsed = ", ".join([testUsed, " Pratt method for equal to hyp. med. (inc. Cureton adjustment for normal approximation)"])
        elif (eqMed == "zsplit"):
            testUsed = ", ".join([testUsed, "z-split method for equal to hyp. med."])

    testResults = pd.DataFrame([[nr, mu, W, statistic, df, pVal, testUsed]], columns=["nr", "mu", "W", "statistic", "df", "p-value", "test"])
    pd.set_option('display.max_colwidth', None)
    
    return testResults