Updated difference_test, ttest, and anova documentation

Updated difference_test, ttest, and anova documentation.

Also trying to get the citations to only show those which were cited.

source/anova_documentation.rst

@@ -30,6 +30,7 @@ anova methods
 ^^^^^^^^^^^^^
 
   * **results(return_type = "Dataframe", decimals = 4, pretty_format = True)**
+
       * **return_type** : The type of data structure the results should be returned as. Supported options
       are 'Dataframe' which will return a Pandas DataFrame or 'Dictionary' which will return a dictionary.
       * **decimals** : The number of decimal places the data should be rounded too.

source/difference_test_documentation.rst

@@ -10,19 +10,19 @@ This method is similar to researchpy.ttest(), except it allows the user to use
 the formula syntax.
 
 This method can perform the following tests:
-  * Independent sample t-test :cite:`scipy_ttest_ind`
+  * Independent sample t-test :cite:`2018:scipy_ttest_ind`
 
       * `psudo-code: difference_test(formula_like, data, equal_variances = True, independent_samples = True)`
 
-  * Paired sample t-test :cite:`scipy_ttest_rel`
+  * Paired sample t-test :cite:`2018:scipy_ttest_rel`
 
       * `psudo-code: difference_test(formula_like, data, equal_variances = True, independent_samples = False)`
 
-  * Welch's t-test :cite:`scipy_ttest_ind`
+  * Welch's t-test :cite:`2018:scipy_ttest_ind`
 
       * `psudo-code: difference_test(formula_like, data, equal_variances = False, independent_samples = True)`
 
-  * Wilcoxon ranked-sign test :cite:`scipy_wilcoxon`
+  * Wilcoxon ranked-sign test :cite:`2018:scipy_wilcoxon`
 
       * By default, discards all zero-differences; this is known as the 'wilcox' method.
       * `psudo-code: difference_test(formula_like, data, equal_variances = False, independent_samples = False)`
@@ -37,14 +37,15 @@ will include the effect size measures if indicated.
 
 Arguments
 -----------------
-**difference_test(formula_like, data = {}, conf_level = 0.95, equal_variances = True, independent_samples = True, wilcox_parameters = {"zero_method" : "wilcox", "correction" : False, "mode" : "auto"}, **keywords)**
+**difference_test(formula_like, data = {}, conf_level = 0.95, equal_variances = True, independent_samples = True, wilcox_parameters = {"zero_method" : "wilcox", "correction" : False, "mode" : "auto"}, welch_dof = "satterthwaite", **keywords)**
 
   * **formula_like**: A valid `formula <https://patsy.readthedocs.io/en/latest/formulas.html>`_ ; for example, "DV ~ IV".
   * **data**: data to perform the analysis on - contains the dependent and independent variables.
   * **conf_level**: Specify the confidence interval to be calculated.
   * **equal_variances**: Boolean to indicate if equal variances are assumed.
   * **independent_samples**: Boolean to indicate if groups and independent of each other.
   * **wilcox_parameters**: A dictionary with optional methods for calculating the Wilcoxon signed-rank test. For more information, see `scipy.stats.wilcoxon <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.wilcoxon.html#scipy.stats.wilcoxon>`_.
+  * **welch_dof** : A string which indicates how to calculate the degrees of freedom for the Welch's t-test; options are "satterthwaite" (default) and "welch".
 
 **conduct(return_type = "Dataframe", effect_size = None)**
 
@@ -84,6 +85,25 @@ Arguments
 
 
 
+Welch Degrees of freedom
+^^^^^^^^^^^^^^^^^^^^^^^^
+There are two degrees of freedom options available when calculating the Welch's t-test. The default is to use
+the Satterthwaite :cite:`1946:Satterthwaite` calculation with the option to use the Welch :cite:`1947:Welch` calculation.
+
+Satterthwaite (1946)
+""""""""""""""""""""
+
+.. math::
+
+  \frac{(\frac{s^2_x}{n_x} + \frac{s^2_y}{n_y})^2}{\frac{(\frac{s^2_x}{n_x})^2}{n_x-1} + \frac{(\frac{s^2_y}{n_y})^2}{n_y-1} }
+
+
+Welch (1947)
+""""""""""""
+
+.. math::
+
+  -2 + \frac{(\frac{s^2_x}{n_x} + \frac{s^2_y}{n_y})^2}{\frac{(\frac{s^2_x}{n_x})^2}{n_x+1} + \frac{(\frac{s^2_y}{n_y})^2}{n_y+1
 
 
 Effect size measures formulas
@@ -445,4 +465,3 @@ References
 ----------
 .. bibliography:: refs.bib
    :list: bullet
-   :cited:

source/refs.bib

@@ -1,5 +1,32 @@
 % Encoding: UTF-8
 
+@Article{Satterthwaite,
+ ISSN = {00994987},
+ URL = {http://www.jstor.org/stable/3002019},
+ author = {F. E. Satterthwaite},
+ journal = {Biometrics Bulletin},
+ number = {6},
+ pages = {110--114},
+ publisher = {[International Biometric Society, Wiley]},
+ title = {An Approximate Distribution of Estimates of Variance Components},
+ volume = {2},
+ year = {1946}
+}
+
+@Article{Welch,
+ ISSN = {00063444},
+ URL = {http://www.jstor.org/stable/2332510},
+ author = {B. L. Welch},
+ journal = {Biometrika},
+ number = {1/2},
+ pages = {28--35},
+ publisher = {[Oxford University Press, Biometrika Trust]},
+ title = {The Generalization of `Student's' Problem when Several Different Population Variances are Involved},
+ volume = {34},
+ year = {1947}
+}
+
+
 @Book{cohen1988,
   title     = {Statistical Power Analysis for the Behavioral Sciences},
   publisher = {Lawrence Erlbaum Associates},

source/ttest_documentation.rst

@@ -39,7 +39,7 @@ Parameters
 
 Input
 -----
-**ttest(group1, group2, group1_name= None, group2_name= None, equal_variances= True, paired= False, wilcox_parameters = {"zero_method" : "pratt", "correction" : False, "mode" : "auto"})**
+**ttest(group1, group2, group1_name= None, group2_name= None, equal_variances= True, paired= False, wilcox_parameters = {"zero_method" : "pratt", "correction" : False, "mode" : "auto"}, welch_dof = "satterthwaite")**
 
   * **group1** and **group2** : requires the data to be a Pandas Series
   * **group1_name** and **group2_name** : will override the series name
@@ -50,6 +50,7 @@ Input
       variance is assumed, a paired sample t-test is conducted, otherwise a Wilcoxon
       ranked-sign test is conducted. The default is False.
   * **wilcox_parameters** : A dictionary which contains the testing specifications for the Wilcoxon signed-rank test.
+  * **welch_dof** : A string to indicate which calculation is to be used when calculating the degrees of freedom. Can either be "welch" or "satterthwaite" (default).
 
 
 Returns
@@ -82,6 +83,28 @@ effect size measures of Pearson r and Rank-Biserial r.
 
 
 
+Welch Degrees of freedom
+========================
+There are two degrees of freedom options available when calculating the Welch's t-test. The default is to use
+the Satterthwaite (1946) calculation with the option to use the Welch (1947) calculation.
+
+Satterthwaite (1946)
+^^^^^^^^^^^^^^^^^^^^
+
+.. math::
+
+  \frac{(\frac{s^2_x}{n_x} + \frac{s^2_y}{n_y})^2}{\frac{(\frac{s^2_x}{n_x})^2}{n_x-1} + \frac{(\frac{s^2_y}{n_y})^2}{n_y-1} }
+
+
+Welch (1947)
+^^^^^^^^^^^^
+
+.. math::
+
+  -2 + \frac{(\frac{s^2_x}{n_x} + \frac{s^2_y}{n_y})^2}{\frac{(\frac{s^2_x}{n_x})^2}{n_x+1} + \frac{(\frac{s^2_y}{n_y})^2}{n_y+1} }
+
+
+
 Effect size measures formulas
 =============================