Module stikpetP.tests.test_g_gof

Functions

def ts_g_gof(data, expCounts=None, cc=None)

G (Likelihood Ratio) Goodness-of-Fit Test

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, Freeman-Tukey, Neyman, Mod-Log Likelihood and Cressie-Read test are possible.

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

expCounts : pandas dataframe, optional

the categories and expected counts

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

which continuity correction to use. Default is None

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

It uses chi2 from scipy's stats library for the chi-square distribution

The formula used (Wilks, 1938, p. 62): $$G=2\times\sum_{i=1}^{k}\left(F_{i}\times ln\left(\frac{F_{i}}{E_{i}}\right)\right)$$ $$df = k - 1$$ $$sig. = 1 - \chi^2\left(G,df\right)$$

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

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

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 term ‘Likelihood Ratio Goodness-of-Fit’ can for example be found in an article from Quine and Robinson (1985), the term ‘Wilks’s likelihood ratio test’ can also be found in Li and Babu (2019, p. 331), while the term G-test is found in Hoey (2012, p. 4)

The Yates continuity correction (cc="yates") is calculated using (Yates, 1934, p. 222): $$F_i^\ast = \begin{cases} F_i - 0.5 & \text{ if } F_i > E_i \\ F_i + 0.5 & \text{ if } F_i < E_i \\ F_i & \text{ if } F_i = E_i \end{cases}$$ $$G_Y=2\times\sum_{i=1}^{k}\left(F_i^\ast\times ln\left(\frac{F_i^\ast}{E_{i}}\right)\right)$$

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)).

Where if $F_i^\ast = 0$ then $F_i^\ast\times \ln\left(\frac{F_i^\ast}{E_{i}}\right) = 0$

The Pearson correction (cc="pearson") is calculated using (E.S. Pearson, 1947, p. 157): $$G_{P} = G\times\frac{n - 1}{n}$$

The Williams correction (cc="williams") is calculated using (Williams, 1976, p. 36): $$G_{W} = \frac{G}{q}$$ 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 Test * ph_pairwise_gof for Pairwise Goodness-of-Fit Tests * ph_residual_gof_bin for Residuals 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_gof for Freeman-Tukey Test of Goodness-of-Fit * ts_freeman_tukey_read for Freeman-Tukey-Read Test of Goodness-of-Fit * 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.

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

Hoey, J. (2012). The two-way likelihood ratio (G) test and comparison to two-way chi squared test. 1–6. doi:10.48550/ARXIV.1206.4881

Li, B., & Babu, G. J. (2019). A graduate course on statistical inference. Springer.

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

Quine, M. P., & Robinson, J. (1985). Efficiencies of chi-square and likelihood Ratio goodness-of-fit tests. The Annals of Statistics, 13(2), 727–742. doi:10.1214/aos/1176349550

Wilks, S. S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypotheses. The Annals of Mathematical Statistics, 9(1), 60–62. doi:10.1214/aoms/1177732360

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

Yates, F. (1934). Contingency tables involving small numbers and the chi square test. Supplement to the Journal of the Royal Statistical Society, 1(2), 217–235. doi:10.2307/2983604

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_g_gof(ex1)
      n  k    statistic  df        p-value  minExp  percBelow5                  test used
0  1941  5  1137.011676   4  7.187038e-245   388.2         0.0  G 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_g_gof(ex2, expCounts=eCounts, cc="yates")
      n  k  statistic  df        p-value  minExp  percBelow5                                        test used
0  1760  4  971.38162   3  2.905646e-210   440.0         0.0  G test of goodness-of-fit, and Yates correction
>>> ts_g_gof(ex2, expCounts=eCounts, cc="pearson")
      n  k   statistic  df        p-value  minExp  percBelow5                                          test used
0  1760  4  971.779494   3  2.381959e-210   440.0         0.0  G test of goodness-of-fit, and Pearson correction
>>> ts_g_gof(ex2, expCounts=eCounts, cc="williams")
      n  k   statistic  df        p-value  minExp  percBelow5                                           test used
0  1760  4  971.871789   3  2.274643e-210   440.0         0.0  G 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_g_gof(ex3)
    n  k  statistic  df   p-value  minExp  percBelow5                  test used
0  19  4   3.397304   3  0.334328    4.75       100.0  G test of goodness-of-fit

Expand source code

def ts_g_gof(data, expCounts=None, cc=None):
    '''
    G (Likelihood Ratio) Goodness-of-Fit Test
    ------------------------------------------
     
    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, Freeman-Tukey, Neyman, Mod-Log Likelihood and Cressie-Read test are possible.

    This function is shown in this [YouTube video](https://youtu.be/8RZgl7rGZZE) and the test is also described at [PeterStatistics.com](https://peterstatistics.com/Terms/Tests/Gtest.html)
    
    Parameters
    ----------
    data :  list or pandas data series 
        the data
        
    expCounts : pandas dataframe, optional 
        the categories and expected counts
        
    cc : {None, "yates", "yates2", "pearson", "williams"}, optional 
        which continuity correction to use. Default is None
        
    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
    -----
    It uses chi2 from scipy's stats library for the chi-square distribution
    
    The formula used (Wilks, 1938, p. 62):
    $$G=2\\times\\sum_{i=1}^{k}\\left(F_{i}\\times ln\\left(\\frac{F_{i}}{E_{i}}\\right)\\right)$$
    $$df = k - 1$$
    $$sig. = 1 - \\chi^2\\left(G,df\\right)$$
    
    With:
    $$n = \\sum_{i=1}^k F_i$$
    
    If no expected counts provided:
    $$E_i = \\frac{n}{k}$$
    else:
    $$E_i = n\\times\\frac{E_{p_i}}{n_p}$$
    $$n_p = \\sum_{i=1}^k E_{p_i}$$
    
    *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 term ‘Likelihood Ratio Goodness-of-Fit’ can for example be found in an article from Quine and Robinson (1985), the term ‘Wilks’s likelihood ratio test’ can also be found in Li and Babu (2019, p. 331), while the term G-test is found in Hoey (2012, p. 4)
    
    The Yates continuity correction (cc="yates") is calculated using (Yates, 1934, p. 222):
    $$F_i^\\ast  = \\begin{cases} F_i - 0.5 & \\text{ if } F_i > E_i \\\\ F_i + 0.5 & \\text{ if } F_i < E_i \\\\ F_i & \\text{ if } F_i = E_i \\end{cases}$$
    $$G_Y=2\\times\\sum_{i=1}^{k}\\left(F_i^\\ast\\times ln\\left(\\frac{F_i^\\ast}{E_{i}}\\right)\\right)$$

    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)).
    
    Where if $F_i^\\ast = 0$ then $F_i^\\ast\\times \\ln\\left(\\frac{F_i^\\ast}{E_{i}}\\right) = 0$
    
    The Pearson correction (cc="pearson") is calculated using (E.S. Pearson, 1947, p. 157):
    $$G_{P} = G\\times\\frac{n - 1}{n}$$
    
    The Williams correction (cc="williams") is calculated using (Williams, 1976, p. 36):
    $$G_{W} = \\frac{G}{q}$$
    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 Test
    * [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
    * [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_gof](../tests/test_freeman_tukey_gof.html#ts_freeman_tukey_gof) for Freeman-Tukey Test of Goodness-of-Fit
    * [ts_freeman_tukey_read](../tests/test_freeman_tukey_read.html#ts_freeman_tukey_read) for Freeman-Tukey-Read Test of Goodness-of-Fit
    * [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.
    
    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
    
    Hoey, J. (2012). The two-way likelihood ratio (G) test and comparison to two-way chi squared test. 1–6. doi:10.48550/ARXIV.1206.4881
    
    Li, B., & Babu, G. J. (2019). *A graduate course on statistical inference*. Springer.
    
    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
    
    Quine, M. P., & Robinson, J. (1985). Efficiencies of chi-square and likelihood Ratio goodness-of-fit tests. *The Annals of Statistics, 13*(2), 727–742. doi:10.1214/aos/1176349550
    
    Wilks, S. S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypotheses. *The Annals of Mathematical Statistics, 9*(1), 60–62. doi:10.1214/aoms/1177732360
    
    Williams, D. A. (1976). Improved likelihood ratio tests for complete contingency tables. *Biometrika, 63*(1), 33–37. doi:10.2307/2335081
    
    Yates, F. (1934). Contingency tables involving small numbers and the chi square test. *Supplement to the Journal of the Royal Statistical Society, 1*(2), 217–235. doi:10.2307/2983604
    
    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_g_gof(ex1)
          n  k    statistic  df        p-value  minExp  percBelow5                  test used
    0  1941  5  1137.011676   4  7.187038e-245   388.2         0.0  G 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_g_gof(ex2, expCounts=eCounts, cc="yates")
          n  k  statistic  df        p-value  minExp  percBelow5                                        test used
    0  1760  4  971.38162   3  2.905646e-210   440.0         0.0  G test of goodness-of-fit, and Yates correction
    >>> ts_g_gof(ex2, expCounts=eCounts, cc="pearson")
          n  k   statistic  df        p-value  minExp  percBelow5                                          test used
    0  1760  4  971.779494   3  2.381959e-210   440.0         0.0  G test of goodness-of-fit, and Pearson correction
    >>> ts_g_gof(ex2, expCounts=eCounts, cc="williams")
          n  k   statistic  df        p-value  minExp  percBelow5                                           test used
    0  1760  4  971.871789   3  2.274643e-210   440.0         0.0  G 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_g_gof(ex3)
        n  k  statistic  df   p-value  minExp  percBelow5                  test used
    0  19  4   3.397304   3  0.334328    4.75       100.0  G test of goodness-of-fit
    
    '''
    
    if type(data) == list:
        data = pd.Series(data)
        
    #Set correction factor to 1 (no correction)
    corFactor = 1
    testUsed = "G 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
    ts = 2*sum(list(freqs)*np.log(list(freqs)/expCounts))
    
    #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