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https://github.com/kukuster/ci_methods_analyser

Analyse efficacy of your own methods for calculating confidence interval
https://github.com/kukuster/ci_methods_analyser

analytical-solution ci-methods confidence-interval confidence-interval-plot confidence-intervals efficacy proportions statistical-models statistics statistics-toolbox wald-interval wilson-score z-test

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Analyse efficacy of your own methods for calculating confidence interval

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README 

        
# CI methods analyser

A toolkit for measuring the efficacy of various methods for calculating a confidence interval.

Currently provides with a toolkit for measuring the efficacy of methods for a confidence interval for the following statistics:



 - proportion

 - difference between two proportions



This library was mainly inspired by the library:

["Five Confidence Intervals for Proportions That You Should Know About" by Dr. Dennis Robert](https://towardsdatascience.com/five-confidence-intervals-for-proportions-that-you-should-know-about-7ff5484c024f)



## Installation



https://pypi.org/project/CI-methods-analyser/



## Usage



### Testing Wald Interval - a popular method for calculating confidence interval for proportion



Wald Interval is defined as so:





    $$ (w^-, w^+) = p\,\pm\,z\sqrt{\frac{p(1-p)}{n}} $$




```python

from CI_methods_analyser import CImethodForProportion_efficacyToolkit as toolkit, methods_for_CI_for_proportion



# take an already implemented method for calculating CI for proportions

wald_interval = methods_for_CI_for_proportion.wald_interval



# initialize the toolkit

wald_interval_test_toolkit = toolkit(

    method=wald_interval, method_name="Wald Interval")



# calculate the real coverage that the method produces

# for each case of a true population proportion (taken from the list `proportions`)

wald_interval_test_toolkit.calculate_coverage_analytically(

    sample_size=100, proportions=('0.001', '0.999', '0.01'), confidence=0.95)

# now you can access the calculated coverage and a few statistics:

# wald_interval_test_toolkit.coverage  # 1-d array of 0-100, the same shape as passed `proportions`

# wald_interval_test_toolkit.average_coverage  # np.longdouble 0-100, avg of `coverage`

# wald_interval_test_toolkit.average_deviation  # np.longdouble 0-100, avg abs diff w/ `confidence`



# plots the calculated coverage in a matplotlib.pyplot figure

wald_interval_test_toolkit.plot_coverage(

    plt_figure_title="Wald Interval coverage")

# you can access the figure here:

# wald_interval_test_toolkit.figure



# shows the figure (non-blocking)

wald_interval_test_toolkit.show_plot()



# because show_plot() is non-blocking,

# you have to pause the execution in order for the figure to be rendered completely

input('press Enter to exit')

```



This will output the image:



![Wald Interval - real coverage](https://github.com/Kukuster/CI_methods_analyser/raw/master/docs/wald_interval_p_0.001_0.999_0.01_n100_conf95.png)



The plot indicates overall bad performance of the method and particularly poor performance for extreme proportions. 






*You really might want to use a different method. Check out this wonderful medium.com article by **Dr. Dennis Robert**:*

 - ***[Five Confidence Intervals for Proportions That You Should Know About](https://towardsdatascience.com/five-confidence-intervals-for-proportions-that-you-should-know-about-7ff5484c024f)** [code in R]*










The shortcut function `calculate_coverage_and_show_plot` will yield the equivalent calculation and render the same picture:



```python

from CI_methods_analyser import CImethodForProportion_efficacyToolkit as toolkit, methods_for_CI_for_proportion



toolkit(

    method=methods_for_CI_for_proportion.wald_interval, method_name="Wald Interval"

).calculate_coverage_and_show_plot(

    sample_size=100, proportions=('0.001', '0.999', '0.01'), confidence=0.95,

    plt_figure_title="Wald Interval coverage"

)



input('press Enter to exit')

```



I personally prefer **night light-friendly** styling:



```python

from CI_methods_analyser import CImethodForProportion_efficacyToolkit as toolkit, methods_for_CI_for_proportion



toolkit(

    method=methods_for_CI_for_proportion.wald_interval, method_name="Wald Interval"

).calculate_coverage_and_show_plot(

    sample_size=100, proportions=('0.001', '0.999', '0.01'), confidence=0.95,

    plt_figure_title="Wald Interval coverage",

    theme='dark_background', plot_color="green", line_color="orange"

)



input('press Enter to exit')

```



![Wald Interval - real coverage (dark theme)](https://github.com/Kukuster/CI_methods_analyser/raw/master/docs/wald_interval_p_0.001_0.999_0.01_n100_conf95_dark.png)







### Testing custom method for CI for proportion



You can implement your own methods and test them:



```python

from CI_methods_analyser import CImethodForProportion_efficacyToolkit as toolkit

from CI_methods_analyser.math_functions import normal_z_score_two_tailed

from functools import lru_cache



# not a particularly good method for calculating CI for proportion

@lru_cache(100_000)

def im_telling_ya_test(x: int, n: int, conflevel: float = 0.95):

    z = normal_z_score_two_tailed(conflevel)



    p = float(x)/n

    return (

        p - 0.02*z,

        p + 0.02*z

    )



toolkit(

    method=im_telling_ya_test, method_name='"I\'m telling ya" test'

).calculate_coverage_and_show_plot(

    sample_size=100, proportions=('0.001', '0.999', '0.01'), confidence=0.95,

    plt_figure_title='"I\'m telling ya" coverage',

    theme='dark_background', plot_color="green", line_color="orange"

)



input('press Enter to exit')



```

!["I'm telling ya" test - real coverage](https://github.com/Kukuster/CI_methods_analyser/raw/master/docs/im_telling_ya_test_p_0.001_0.999_0.01_n100_conf95_dark.png)



```python

from CI_methods_analyser import CImethodForProportion_efficacyToolkit as toolkit

from CI_methods_analyser.math_functions import normal_z_score_two_tailed

from functools import lru_cache



# you could say, this method is "too good"

@lru_cache(100_000)

def God_is_my_witness_score(x: int, n: int, conflevel: float = 0.95):

    z = normal_z_score_two_tailed(conflevel)



    p = float(x)/n

    return (

        (0 + p)/2 - 0.005*z,

        (1 + p)/2 + 0.005*z

    )



toolkit(

    method=God_is_my_witness_score, method_name='"God is my witness" score'

).calculate_coverage_and_show_plot(

    sample_size=100, proportions=('0.001', '0.999', '0.01'), confidence=0.95,

    plt_figure_title='"God is my witness" score coverage', theme='dark_background'

)



input('press Enter to exit')

```



!["God is my witness" score - real coverage](https://github.com/Kukuster/CI_methods_analyser/raw/master/docs/God_is_my_witness_score_p_0.001_0.999_0.01_n100_conf95_dark.png)



### Testing methods for CI for the difference between two proportions



Let's use the implemented **Pooled Z test**:





    $$ (\delta^-, \delta^+) = \hat{p}_T - \hat{p}_C \pm z_{\alpha}\sqrt{\bar{p}(1-\bar{p})(\frac{1}{n_T}+\frac{1}{n_C})} $$



, where:



    $$ \bar{p} = \frac{n_T\hat{p}_T + n_C\hat{p}_C}{n_T + n_C} $$




```python

from CI_methods_analyser import CImethodForDiffBetwTwoProportions_efficacyToolkit as toolkit_d, methods_for_CI_for_diff_betw_two_proportions as methods



toolkit_d(

    method=methods.Z_test_pooled, method_name='Z test pooled'

).calculate_coverage_and_show_plot(

    sample_size1=100, sample_size2=100, proportions=('0.001', '0.999', '0.01'), confidence=0.95,

    plt_figure_title='Z test pooled', theme='dark_background',

)



input('press Enter to exit')

```



![Z test (unpooled) - real coverage](https://github.com/Kukuster/CI_methods_analyser/raw/master/docs/z_test_pooled_p_0.001_0.999_0.01_n1_100_n2_100_conf95.png)



As you can see, this test is generally very good for close proportions, unless proportions have extreme values [purple]



Also, this test is extremely concervative for the high and extreme differences between two proportions, i.e. for proportions which values a far apart [green]







You may want to change the color palette (although I wouldn't):



```python

from CI_methods_analyser import CImethodForDiffBetwTwoProportions_efficacyToolkit as toolkit_d, methods_for_CI_for_diff_betw_two_proportions as methods



toolkit_d(

    method=methods.Z_test_pooled, method_name='Z test pooled'

).calculate_coverage_and_show_plot(

    sample_size1=100, sample_size2=100, proportions=('0.001', '0.999', '0.01'), confidence=0.95,

    plt_figure_title='Z test pooled', theme='dark_background',

    colors=("gray", "purple", "white", "orange", "#d62728")

)



input('press Enter to exit')

```



![Z test (unpooled) - real coverage](https://github.com/Kukuster/CI_methods_analyser/raw/master/docs/z_test_pooled_p_0.001_0.999_0.01_n1_100_n2_100_conf95_hotcolors.png)










## NOTES



### Methods for measuring efficacy of CI methods

Two ways can be used to calculate the efficacy of CI methods:

 - approximately, with random simulation (as implemented in R by Dr. Dennis Robert, see link above). Here: `calculate_coverage_randomly`

 - precisely, with the analytical solution. Here: `calculate_coverage_analytically`



Both methods are implemented for CI for both statistics: *proportion*, and *difference between two proportions*. For the precise analytical solution, an optimization was made. Theoretically, it is lossy, but practically the error is always negligible (as proven by `test_z_precision_difference.py`). Optimization is regulated with the parameter `z_precision` and it is automatically estimated by default.







## Various links

**1. Equivalence and Noninferiority Testing (as I understand, are fancy terms for 2-sided and 1-sided p tests for the difference between two proportions)**

 - **[https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/PASS/Confidence_Intervals_for_the_Difference_Between_Two_Proportions.pdf](https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/PASS/Confidence_Intervals_for_the_Difference_Between_Two_Proportions.pdf)**

 - **[https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/PASS/Non-Inferiority_Tests_for_the_Difference_Between_Two_Proportions.pdf](https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/PASS/Non-Inferiority_Tests_for_the_Difference_Between_Two_Proportions.pdf)**

 - [https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Two_Proportions-Non-Inferiority,_Superiority,_Equivalence,_and_Two-Sided_Tests_vs_a_Margin.pdf](https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Two_Proportions-Non-Inferiority,_Superiority,_Equivalence,_and_Two-Sided_Tests_vs_a_Margin.pdf) 

 - [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3019319/](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3019319/)

 - [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2701110/](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2701110/)

 - [https://pubmed.ncbi.nlm.nih.gov/9595617/](https://pubmed.ncbi.nlm.nih.gov/9595617/)

 - [http://thescipub.com/pdf/10.3844/amjbsp.2010.23.31](http://thescipub.com/pdf/10.3844/amjbsp.2010.23.31) 



**2. Biostatistics course (Dr. Nicolas Padilla Raygoza, et al.)**

 - [https://docs.google.com/presentation/d/1t1DowyVDDRFYGHDlJgmYMRN4JCrvFl3q/edit#slide=id.p1](https://docs.google.com/presentation/d/1t1DowyVDDRFYGHDlJgmYMRN4JCrvFl3q/edit#slide=id.p1) 

 - [https://www.google.com/search?q=Dr.+Sc.+Nicolas+Padilla+Raygoza+Biostatistics+course+Part+10&oq=Dr.+Sc.+Nicolas+Padilla+Raygoza+Biostatistics+course+Part+10&aqs=chrome..69i57.3448j0j7&sourceid=chrome&ie=UTF-8](https://www.google.com/search?q=Dr.+Sc.+Nicolas+Padilla+Raygoza+Biostatistics+course+Part+10&oq=Dr.+Sc.+Nicolas+Padilla+Raygoza+Biostatistics+course+Part+10&aqs=chrome..69i57.3448j0j7&sourceid=chrome&ie=UTF-8) 

 - [https://slideplayer.com/slide/9837395/](https://slideplayer.com/slide/9837395/)



**3. Using z-test instead of a binomial test:**

 - When can use [https://stats.stackexchange.com/questions/424446/when-can-we-use-a-z-test-instead-of-a-binomial-test](https://stats.stackexchange.com/questions/424446/when-can-we-use-a-z-test-instead-of-a-binomial-test) 

 - How to use [https://cogsci.ucsd.edu/~dgroppe/STATZ/binomial_ztest.pdf](https://cogsci.ucsd.edu/~dgroppe/STATZ/binomial_ztest.pdf)