Module stikpetP.tests.test_freeman_tukey_gof

Functions

def ts_freeman_tukey_gof(data, expCounts=None, cc=None, modified=0)

Freeman-Tukey Test of Goodness-of-Fit

A test that can be used with a single nominal variable, to test if the probabilities in all the categories are equal (the null hypothesis). If the test has a p-value below a pre-defined threshold (usually 0.05) the assumption they are all equal in the population will be rejected.

There are quite a few tests that can do this. Perhaps the most commonly used is the Pearson chi-square test, but also an exact multinomial, G-test, Neyman, Mod-Log Likelihood, Cressie-Read, and Freeman-Tukey-Read test are possible.

The Freeman-Tukey attempts to make the distribution more like a normal distribution by using a square root transformation.

Lawal (1984) continued some work from Larntz (1978) and compared the modified Freeman-Tukey, G-test and the Pearson chi-square test, and concluded that for small samples the Pearson test is preferred, while for large samples either the Pearson or G-test. Making this Freeman-Tukey test perhaps somewhat redundant.

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

expCount : pandas dataframe, optional

the categories and expected counts

cc : {None, "yates", "yates2", "pearson", "williams"}, optional

which continuity correction to use. Default is None

modified : int, optional

indicate the use of the modified version. Either 0 (default), 1 or 2 (see notes)

Returns

pandas.DataFrame

A dataframe with the following columns:

• n, the sample size
• k, the number of categories
• statistic, the test statistic (chi-square value)
• df, degrees of freedom
• p-value, significance (p-value)
• minExp, the minimum expected count
• percBelow5, the percentage of categories with an expected count below 5
• test used, description of the test used

Notes

The formula used is (Ayinde & Abidoye, 2010, p. 21): $$\chi_{FT}^{2}=4\times\sum_{i=1}^{k}\left(\sqrt{F_{i}} - \sqrt{E_{i}}\right)^2$$ $$df = k - 1$$ $$sig. = 1 - \chi^2\left(\chi_{FT}^{2},df\right)$$

With: $$n = \sum_{i=1}^k F_i$$

If no expected counts provided: $$E_i = \frac{n}{k}$$ else: $$E_i = n\times\frac{E_{p_i}}{n_p}$$ $$n_p = \sum_{i=1}^k E_{p_i}$$

A modified version uses another possible smoothing (Bishop, 1969, p. 284; Larntz, 1978, p.253): $$\chi_{MFT}^{2} = \sum_{i=1}^{k}\left(\sqrt{F_{i}} + \sqrt{F_{i} + 1} - \sqrt{4\times E_{i} + 1}\right)^2$$

Or slightly different (Read & Cressie, 1988, p. 82): $$\chi_{MFT}^{2} = \sum_{i=1}^{k}\left(\sqrt{F_{i}} + \sqrt{F_{i} + 1} - \sqrt{4\times \left(E_{i} + 1\right)}\right)^2$$

Symbols used:

• $k$ the number of categories
• $F_i$ the (absolute) frequency of category i
• $E_i$ the expected frequency of category i
• $E_{p_i}$ the provided expected frequency of category i
• $n$ the sample size, i.e. the sum of all frequencies
• $n_p$ the sum of all provided expected counts
• $\chi^2\left(\dots\right)$ the chi-square cumulative density function

The test is attributed to Freeman and Tukey (1950), but couldn't really find it in there. Another source often mentioned is Bishop et al. (2007)

The Yates continuity correction (cc="yates") is calculated using (Yates, 1934, p. 222): $$F_i^\ast = \begin{cases} F_i - 0.5 & \text{ if } F_i > E_i \\ F_i + 0.5 & \text{ if } F_i < E_i \\ F_i & \text{ if } F_i = E_i \end{cases}$$

In some cases the Yates correction is slightly changed to (yates2) (Allen, 1990, p. 523): $$F_i^\ast = \begin{cases} F_i - 0.5 & \text{ if } F_i - 0.5 > E_i \\ F_i + 0.5 & \text{ if } F_i + 0.5 < E_i \\ F_i & \text{ else } \end{cases}$$

Note that the Yates correction is usually only considered if there are only two categories. Some also argue this correction is too conservative (see for details Haviland (1990)).

The Pearson correction (cc="pearson") is calculated using (E.S. Pearson, 1947, p. 157): $$\chi_{PP}^2 = \chi_{FT}^{2}\times\frac{n - 1}{n}$$

The Williams correction (cc="williams") is calculated using (Williams, 1976, p. 36): $$\chi_{PW}^2 = \frac{\chi_{FT}^{2}}{q}$$

With: $$q = 1 + \frac{k^2 - 1}{6\times n\times df}$$

The formula is also used by McDonald (2014, p. 87)

Before, After and Alternatives

Before this an impression using a frequency table or a visualisation might be helpful: * tab_frequency * vi_bar_simple for Simple Bar Chart * vi_cleveland_dot_plot for Cleveland Dot Plot * vi_dot_plot for Dot Plot * vi_pareto_chart for Pareto Chart * vi_pie for Pie Chart

After this you might an effect size measure: * es_cohen_w for Cohen w * es_cramer_v_gof for Cramer's V for Goodness-of-Fit * es_fei for Fei * es_jbm_e for Johnston-Berry-Mielke E

or perform a post-hoc test: * ph_pairwise_bin for Pairwise Binary Tests * ph_pairwise_gof for Pairwise Goodness-of-Fit Tests * ph_residual_gof_bin for Residuals Tests using Binary tests * ph_residual_gof_gof for Residuals Using Goodness-of-Fit Tests

Alternative tests: * ts_pearson_gof for Pearson Chi-Square Goodness-of-Fit Test * ts_freeman_tukey_read for Freeman-Tukey-Read Test of Goodness-of-Fit * ts_g_gof for G (Likelihood Ratio) Goodness-of-Fit Test * ts_mod_log_likelihood_gof for Mod-Log Likelihood Test of Goodness-of-Fit * ts_multinomial_gof for Multinomial Goodness-of-Fit Test * ts_neyman_gof for Neyman Test of Goodness-of-Fit * ts_powerdivergence_gof for Power Divergence GoF Test

References

Allen, A. O. (1990). Probability, statistics, and queueing theory with computer science applications (2nd ed.). Academic Press.

Ayinde, K., & Abidoye, A. O. (2010). Simplified Freeman-Tukey test statistics for testing probabilities in contingency tables. Science World Journal, 2(2), 21–27. doi:10.4314/swj.v2i2.51730

Bishop, Y. M. M., Fienberg, S. E., & Holland, P. W. (2007). Discrete multivariate analysis. Springer.

Bishop, Y. M. M., Fienberg, S. E., & Holland, P. W. (2007). Discrete multivariate analysis. Springer.

Freeman, M. F., & Tukey, J. W. (1950). Transformations Related to the angular and the square root. The Annals of Mathematical Statistics, 21(4), 607–611. doi:10.1214/aoms/1177729756

Haviland, M. G. (1990). Yates’s correction for continuity and the analysis of 2 × 2 contingency tables. Statistics in Medicine, 9(4), 363–367. doi:10.1002/sim.4780090403

Larntz, K. (1978). Small-sample comparisons of exact levels for chi-squared goodness-of-fit statistics. Journal of the American Statistical Association, 73(362), 253–263. doi:10.1080/01621459.1978.10481567

Lawal, H. B. (1984). Comparisons of the X 2 , Y 2 , Freeman-Tukey and Williams’s improved G 2 test statistics in small samples of one-way multinomials. Biometrika, 71(2), 415–418. doi:10.2307/2336263

McDonald, J. H. (2014). Handbook of biological statistics (3rd ed.). Sparky House Publishing.

Pearson, E. S. (1947). The choice of statistical tests illustrated on the Interpretation of data classed in a 2 × 2 table. Biometrika, 34(1/2), 139–167. doi:10.2307/2332518

Read, T. R. C., & Cressie, N. A. C. (1988). Goodness-of-fit statistics for discrete multivariate data. Springer-Verlag.

Williams, D. A. (1976). Improved likelihood ratio tests for complete contingency tables. Biometrika, 63(1), 33–37. doi:10.2307/2335081

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

>>> df1 = pd.read_csv('https://peterstatistics.com/Packages/ExampleData/GSS2012a.csv', sep=',', low_memory=False, storage_options={'User-Agent': 'Mozilla/5.0'})
>>> ex1 = df1['mar1']
>>> ts_freeman_tukey_gof(ex1)
      n  k    statistic  df        p-value  minExp  percBelow5                              test used
0  1941  5  1166.495639   4  2.919359e-251   388.2         0.0  Freeman-Tukey test of goodness-of-fit

Example 2: pandas series with various settings

>>> ex2 = df1['mar1']
>>> eCounts = pd.DataFrame({'category' : ["MARRIED", "DIVORCED", "NEVER MARRIED", "SEPARATED"], 'count' : [5,5,5,5]})
>>> ts_freeman_tukey_gof(ex2, expCounts=eCounts, cc="yates")
      n  k    statistic  df        p-value  minExp  percBelow5                                                    test used
0  1760  4  1044.117361   3  4.836441e-226   440.0         0.0  Freeman-Tukey test of goodness-of-fit, and Yates correction

>>> ts_freeman_tukey_gof(ex2, expCounts=eCounts, cc="pearson")
      n  k    statistic  df        p-value  minExp  percBelow5                                                      test used
0  1760  4  1047.365448   3  9.547419e-227   440.0         0.0  Freeman-Tukey test of goodness-of-fit, and Pearson correction
>>> ts_freeman_tukey_gof(ex2, expCounts=eCounts, cc="williams")
      n  k    statistic  df        p-value  minExp  percBelow5                                                       test used
0  1760  4  1047.464921   3  9.084607e-227   440.0         0.0  Freeman-Tukey test of goodness-of-fit, and Williams correction

Example 3: a list

>>> ex3 = ["MARRIED", "DIVORCED", "MARRIED", "SEPARATED", "DIVORCED", "NEVER MARRIED", "DIVORCED", "DIVORCED", "NEVER MARRIED", "MARRIED", "MARRIED", "MARRIED", "SEPARATED", "DIVORCED", "NEVER MARRIED", "NEVER MARRIED", "DIVORCED", "DIVORCED", "MARRIED"]
>>> ts_freeman_tukey_gof(ex3)
    n  k  statistic  df   p-value  minExp  percBelow5                              test used
0  19  4   3.632589   3  0.303969    4.75       100.0  Freeman-Tukey test of goodness-of-fit

Expand source code

def ts_freeman_tukey_gof(data, expCounts=None, cc=None, modified=0):
    '''
    Freeman-Tukey Test of Goodness-of-Fit
    -------------------------------------
    
    A test that can be used with a single nominal variable, to test if the probabilities in all the categories are equal (the null hypothesis). If the test has a p-value below a pre-defined threshold (usually 0.05) the assumption they are all equal in the population will be rejected.
    
    There are quite a few tests that can do this. Perhaps the most commonly used is the Pearson chi-square test, but also an exact multinomial, G-test, Neyman, Mod-Log Likelihood, Cressie-Read, and Freeman-Tukey-Read test are possible.
    
    The Freeman-Tukey attempts to make the distribution more like a normal distribution by using a square root transformation.
    
    Lawal (1984) continued some work from Larntz (1978) and compared the modified Freeman-Tukey, G-test and the Pearson chi-square test, and concluded that for small samples the Pearson test is preferred, while for large samples either the Pearson or G-test. Making this Freeman-Tukey test perhaps somewhat redundant.

    This function is shown in this [YouTube video](https://youtu.be/x_dzouxszKI) and the test is also described at [PeterStatistics.com](https://peterstatistics.com/Terms/Tests/Freeman-Tukey.html)
    
    Parameters
    ----------
    data :  list or pandas data series 
        the data
    
    expCount : pandas dataframe, optional 
        the categories and expected counts
    
    cc : {None, "yates", "yates2", "pearson", "williams"}, optional 
        which continuity correction to use. Default is None
    
    modified : int, optional
        indicate the use of the modified version. Either 0 (default), 1 or 2 (see notes)
    
    Returns
    -------
    pandas.DataFrame
        A dataframe with the following columns:
    
        * *n*, the sample size
        * *k*, the number of categories
        * *statistic*, the test statistic (chi-square value)
        * *df*, degrees of freedom
        * *p-value*, significance (p-value)
        * *minExp*, the minimum expected count
        * *percBelow5*, the percentage of categories with an expected count below 5
        * *test used*, description of the test used
    
    Notes
    -----
    The formula used is (Ayinde & Abidoye, 2010, p. 21):
    $$\\chi_{FT}^{2}=4\\times\\sum_{i=1}^{k}\\left(\\sqrt{F_{i}} - \\sqrt{E_{i}}\\right)^2$$
    $$df = k - 1$$
    $$sig. = 1 - \\chi^2\\left(\\chi_{FT}^{2},df\\right)$$
    
    With:
    $$n = \\sum_{i=1}^k F_i$$
    
    If no expected counts provided:
    $$E_i = \\frac{n}{k}$$
    else:
    $$E_i = n\\times\\frac{E_{p_i}}{n_p}$$
    $$n_p = \\sum_{i=1}^k E_{p_i}$$
    
    A modified version uses another possible smoothing (Bishop, 1969, p. 284; Larntz, 1978, p.253):
    $$\\chi_{MFT}^{2} = \\sum_{i=1}^{k}\\left(\\sqrt{F_{i}} + \\sqrt{F_{i} + 1} - \\sqrt{4\\times E_{i} + 1}\\right)^2$$

    Or slightly different (Read & Cressie, 1988, p. 82):
    $$\\chi_{MFT}^{2} = \\sum_{i=1}^{k}\\left(\\sqrt{F_{i}} + \\sqrt{F_{i} + 1} - \\sqrt{4\\times \\left(E_{i} + 1\\right)}\\right)^2$$

    
    *Symbols used:*
    
    * $k$ the number of categories
    * $F_i$ the (absolute) frequency of category i
    * $E_i$ the expected frequency of category i
    * $E_{p_i}$ the provided expected frequency of category i
    * $n$ the sample size, i.e. the sum of all frequencies
    * $n_p$ the sum of all provided expected counts
    * $\\chi^2\\left(\\dots\\right)$ the chi-square cumulative density function
    
    The test is attributed to Freeman and Tukey (1950), but couldn't really find it in there. Another source often mentioned is Bishop et al. (2007)
    
    The Yates continuity correction (cc="yates") is calculated using (Yates, 1934, p. 222):
    $$F_i^\\ast  = \\begin{cases} F_i - 0.5 & \\text{ if } F_i > E_i \\\\ F_i + 0.5 & \\text{ if } F_i < E_i \\\\ F_i & \\text{ if } F_i = E_i \\end{cases}$$
    
    In some cases the Yates correction is slightly changed to (yates2) (Allen, 1990, p. 523):
    $$F_i^\\ast  = \\begin{cases} F_i - 0.5 & \\text{ if } F_i - 0.5 > E_i \\\\ F_i + 0.5 & \\text{ if } F_i + 0.5 < E_i \\\\ F_i & \\text{ else } \\end{cases}$$
    
    Note that the Yates correction is usually only considered if there are only two categories. Some also argue this correction is too conservative (see for details Haviland (1990)).
    
    The Pearson correction (cc="pearson") is calculated using (E.S. Pearson, 1947, p. 157):
    $$\\chi_{PP}^2 = \\chi_{FT}^{2}\\times\\frac{n - 1}{n}$$
    
    The Williams correction (cc="williams") is calculated using (Williams, 1976, p. 36):
    $$\\chi_{PW}^2 = \\frac{\\chi_{FT}^{2}}{q}$$
    
    With:
    $$q = 1 + \\frac{k^2 - 1}{6\\times n\\times df}$$
    
    The formula is also used by McDonald (2014, p. 87)
    
    Before, After and Alternatives
    ------------------------------
    Before this an impression using a frequency table or a visualisation might be helpful:
    * [tab_frequency](../other/table_frequency.html#tab_frequency)
    * [vi_bar_simple](../visualisations/vis_bar_simple.html#vi_bar_simple) for Simple Bar Chart
    * [vi_cleveland_dot_plot](../visualisations/vis_cleveland_dot_plot.html#vi_cleveland_dot_plot) for Cleveland Dot Plot
    * [vi_dot_plot](../visualisations/vis_dot_plot.html#vi_dot_plot) for Dot Plot
    * [vi_pareto_chart](../visualisations/vis_pareto_chart.html#vi_pareto_chart) for Pareto Chart
    * [vi_pie](../visualisations/vis_pie.html#vi_pie) for Pie Chart
    
    After this you might an effect size measure:
    * [es_cohen_w](../effect_sizes/eff_size_cohen_w.html#es_cohen_w) for Cohen w
    * [es_cramer_v_gof](../effect_sizes/eff_size_cramer_v_gof.html#es_cramer_v_gof) for Cramer's V for Goodness-of-Fit
    * [es_fei](../effect_sizes/eff_size_fei.html#es_fei) for Fei
    * [es_jbm_e](../effect_sizes/eff_size_jbm_e.html#es_jbm_e) for Johnston-Berry-Mielke E

    or perform a post-hoc test:
    * [ph_pairwise_bin](../other/poho_pairwise_bin.html#ph_pairwise_bin) for Pairwise Binary Tests
    * [ph_pairwise_gof](../other/poho_pairwise_gof.html#ph_pairwise_gof) for Pairwise Goodness-of-Fit Tests
    * [ph_residual_gof_bin](../other/poho_residual_gof_bin.html#ph_residual_gof_bin) for Residuals Tests  using Binary tests
    * [ph_residual_gof_gof](../other/poho_residual_gof_gof.html#ph_residual_gof_gof) for Residuals Using Goodness-of-Fit Tests

    Alternative tests:
    * [ts_pearson_gof](../tests/test_pearson_gof.html#ts_pearson_gof) for Pearson Chi-Square Goodness-of-Fit Test
    * [ts_freeman_tukey_read](../tests/test_freeman_tukey_read.html#ts_freeman_tukey_read) for Freeman-Tukey-Read Test of Goodness-of-Fit
    * [ts_g_gof](../tests/test_g_gof.html#ts_g_gof) for G (Likelihood Ratio) Goodness-of-Fit Test
    * [ts_mod_log_likelihood_gof](../tests/test_mod_log_likelihood_gof.html#ts_mod_log_likelihood_gof) for Mod-Log Likelihood Test of Goodness-of-Fit
    * [ts_multinomial_gof](../tests/test_multinomial_gof.html#ts_multinomial_gof) for Multinomial Goodness-of-Fit Test
    * [ts_neyman_gof](../tests/test_neyman_gof.html#ts_neyman_gof) for Neyman Test of Goodness-of-Fit
    * [ts_powerdivergence_gof](../tests/test_powerdivergence_gof.html#ts_powerdivergence_gof) for Power Divergence GoF Test
    
    References
    ----------
    Allen, A. O. (1990). *Probability, statistics, and queueing theory with computer science applications* (2nd ed.). Academic Press.
    
    Ayinde, K., & Abidoye, A. O. (2010). Simplified Freeman-Tukey test statistics for testing probabilities in contingency tables. *Science World Journal, 2*(2), 21–27. doi:10.4314/swj.v2i2.51730

    Bishop, Y. M. M., Fienberg, S. E., & Holland, P. W. (2007). *Discrete multivariate analysis*. Springer.
    
    Bishop, Y. M. M., Fienberg, S. E., & Holland, P. W. (2007). *Discrete multivariate analysis*. Springer.
    
    Freeman, M. F., & Tukey, J. W. (1950). Transformations Related to the angular and the square root. *The Annals of Mathematical Statistics, 21*(4), 607–611. doi:10.1214/aoms/1177729756
    
    Haviland, M. G. (1990). Yates’s correction for continuity and the analysis of 2 × 2 contingency tables. *Statistics in Medicine, 9*(4), 363–367. doi:10.1002/sim.4780090403
    
    Larntz, K. (1978). Small-sample comparisons of exact levels for chi-squared goodness-of-fit statistics. *Journal of the American Statistical Association, 73*(362), 253–263. doi:10.1080/01621459.1978.10481567
    
    Lawal, H. B. (1984). Comparisons of the X 2 , Y 2 , Freeman-Tukey and Williams’s improved G 2 test statistics in small samples of one-way multinomials. *Biometrika, 71*(2), 415–418. doi:10.2307/2336263
    
    McDonald, J. H. (2014). *Handbook of biological statistics* (3rd ed.). Sparky House Publishing.
    
    Pearson, E. S. (1947). The choice of statistical tests illustrated on the Interpretation of data classed in a 2 × 2 table. *Biometrika, 34*(1/2), 139–167. doi:10.2307/2332518

    Read, T. R. C., & Cressie, N. A. C. (1988). Goodness-of-fit statistics for discrete multivariate data. Springer-Verlag.
    
    Williams, D. A. (1976). Improved likelihood ratio tests for complete contingency tables. *Biometrika, 63*(1), 33–37. doi:10.2307/2335081
    
    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
    >>> df1 = pd.read_csv('https://peterstatistics.com/Packages/ExampleData/GSS2012a.csv', sep=',', low_memory=False, storage_options={'User-Agent': 'Mozilla/5.0'})
    >>> ex1 = df1['mar1']
    >>> ts_freeman_tukey_gof(ex1)
          n  k    statistic  df        p-value  minExp  percBelow5                              test used
    0  1941  5  1166.495639   4  2.919359e-251   388.2         0.0  Freeman-Tukey test of goodness-of-fit
    
    Example 2: pandas series with various settings
    >>> ex2 = df1['mar1']
    >>> eCounts = pd.DataFrame({'category' : ["MARRIED", "DIVORCED", "NEVER MARRIED", "SEPARATED"], 'count' : [5,5,5,5]})
    >>> ts_freeman_tukey_gof(ex2, expCounts=eCounts, cc="yates")
          n  k    statistic  df        p-value  minExp  percBelow5                                                    test used
    0  1760  4  1044.117361   3  4.836441e-226   440.0         0.0  Freeman-Tukey test of goodness-of-fit, and Yates correction
    
    >>> ts_freeman_tukey_gof(ex2, expCounts=eCounts, cc="pearson")
          n  k    statistic  df        p-value  minExp  percBelow5                                                      test used
    0  1760  4  1047.365448   3  9.547419e-227   440.0         0.0  Freeman-Tukey test of goodness-of-fit, and Pearson correction
    >>> ts_freeman_tukey_gof(ex2, expCounts=eCounts, cc="williams")
          n  k    statistic  df        p-value  minExp  percBelow5                                                       test used
    0  1760  4  1047.464921   3  9.084607e-227   440.0         0.0  Freeman-Tukey test of goodness-of-fit, and Williams correction
    
    Example 3: a list
    >>> ex3 = ["MARRIED", "DIVORCED", "MARRIED", "SEPARATED", "DIVORCED", "NEVER MARRIED", "DIVORCED", "DIVORCED", "NEVER MARRIED", "MARRIED", "MARRIED", "MARRIED", "SEPARATED", "DIVORCED", "NEVER MARRIED", "NEVER MARRIED", "DIVORCED", "DIVORCED", "MARRIED"]
    >>> ts_freeman_tukey_gof(ex3)
        n  k  statistic  df   p-value  minExp  percBelow5                              test used
    0  19  4   3.632589   3  0.303969    4.75       100.0  Freeman-Tukey test of goodness-of-fit
    
    
    '''
    
    if type(data) == list:
        data = pd.Series(data)
        
    #Set correction factor to 1 (no correction)
    corFactor = 1
    if modified:
        testUsed = "modified Freeman-Tukey test of goodness-of-fit"
    else:
        testUsed = "Freeman-Tukey test of goodness-of-fit"
    
    #The test itself        
    freqs = data.value_counts()
    k = len(freqs)

    #Determine expected counts if not provided
    if expCounts is None:
        expCounts = [sum(freqs)/len(freqs)]* k
        expCounts = pd.Series(expCounts, index=list(freqs.index.values))
    
    else:
        #if expected counts are provided
        ne = 0
        k = len(expCounts)
        #determine sample size of expected counts
        for i in range(0,k):
            ne = ne + expCounts.iloc[i,1]

        #remove categories not provided from observed counts
        for i in freqs.index:
            if i not in list(expCounts.iloc[:,0]):
                freqs = freqs.drop(i)
        # and sort based on the index
        freqs = freqs.sort_index()

        #set the column names
        expCounts.columns = ["category", "count"]
        #sort the expected counts
        expCounts.sort_values(by="category", inplace=True)
        #adjust based on observed count total
        expCounts['count'] = expCounts['count'].astype('float64')
        n = sum(freqs)
        for i in range(0,k):
            expCounts.at[i, 'count'] = float(expCounts.at[i, 'count'] / ne * n)
        
        expCounts = pd.Series(expCounts.iloc[:, 1])
        
    n = sum(freqs)
    df = k - 1

    #set williams correction factor
    if cc=="williams":
        corFactor = 1/(1 + (k**2 - 1)/(6*n*df))
        testUsed = testUsed + ", and Williams correction"
    
    #adjust frequencies if Yates correction is requested
    if cc=="yates":
        k = len(freqs)
        adjFreq = list(freqs).copy()
        for i in range(0, k):
            if adjFreq[i] > expCounts.iloc[i]:
                adjFreq[i] = adjFreq[i] - 0.5
            elif adjFreq[i] < expCounts.iloc[i]:
                adjFreq[i] = adjFreq[i] + 0.5

        freqs = pd.Series(adjFreq, index=list(freqs.index.values))
        testUsed = testUsed + ", and Yates correction"
    
    if cc=="yates2":
        k = len(freqs)
        adjFreq = list(freqs).copy()
        for i in range(0, k):
            if adjFreq[i] - 0.5 > expCounts.iloc[i]:
                adjFreq[i] = adjFreq[i] - 0.5
            elif adjFreq[i] + 0.5 < expCounts.iloc[i]:
                adjFreq[i] = adjFreq[i] + 0.5

        freqs = pd.Series(adjFreq, index=list(freqs.index.values))
        testUsed = testUsed + ", and Yates correction"
        
    #determine the test statistic
    if modified==1:
        ts = sum([(freqs.iloc[i]**0.5 + (freqs.iloc[i]+1)**0.5 - (4*expCounts.iloc[i] + 1)**0.5)**2 for i in range(0,k)]) 
    if modified==2:
        ts = sum([(freqs.iloc[i]**0.5 + (freqs.iloc[i]+1)**0.5 - (4*(expCounts.iloc[i] + 1))**0.5)**2 for i in range(0,k)])
    elif modified==0:
        ts = 4*sum([(freqs.iloc[i]**0.5 - expCounts.iloc[i]**0.5)**2 for i in range(0,k)])
    
    #set E.S. Pearson correction
    if cc=="pearson":
        corFactor = (n - 1)/n
        testUsed = testUsed + ", and Pearson correction"
    
    #Adjust test statistic
    ts = ts*corFactor
    
    #Determine p-value
    pVal = chi2.sf(ts, df)
    
    #Check minimum expected counts
    #Cells with expected count less than 5
    nbelow = len([x for x in expCounts if x < 5])
    #Number of cells
    ncells = len(expCounts)
    #As proportion
    pBelow = nbelow/ncells
    #the minimum expected count
    minExp = min(expCounts)
    
    #prepare results
    testResults = pd.DataFrame([[n, k, ts, df, pVal, minExp, pBelow*100, testUsed]], columns=["n", "k","statistic", "df", "p-value", "minExp", "percBelow5", "test used"])        
    pd.set_option('display.max_colwidth', None)
    
    return testResults