binomcikit 0.0.3

Used to calculate Confidence Intervals

BINOMCIKIT:

Introduction:

In many statistical problems, we are interested in estimating the proportion of successes in a binomial process. For example, if you flip a coin 100 times and observe 55 heads, you might want to estimate the true proportion of heads for that coin. This is known as estimating a binomial proportion.

Estimating a single binomial proportion is a fundamental problem in statistics that applies to a wide range of real-world scenarios. In many fields, we encounter situations where we need to estimate the proportion of successes (or failures) in a fixed number of independent trials, with each trial having only two possible outcomes, such as success or failure, yes or no, pass or fail. This estimation problem is central to various industries, from healthcare to business to manufacturing.

Estimation Methods for Single Binomial Proportion:

There are several estimation procedures used to estimate a single binomial proportion:

1. Wald Interval
2. Wald-T Interval
3. Likelihood Interval (Exact Method)
4. Score Interval (Wilson Interval)
5. Logit-Wald Interval
6. ArcSine Interval

Each of these methods has its strengths and weaknesses, depending on the sample size, the observed proportion, and the desired accuracy. The choice of method depends on the specific characteristics of the data and the goals of the analysis.

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Summary Table with Additional Notes on Aberrations and Continuity Corrections:

Method	Formula	Key Issues/Considerations
Wald Interval	$\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$	Issues with $\hat{p} = 0$ or $\hat{p} = 1$; continuity correction helps in small $n$
Wald-T Interval	$\hat{p} \pm t_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$	Better for small $n$; still struggles with extreme $\hat{p}$; continuity correction can help
Likelihood Interval	Based on likelihood ratio test	Exact method, no issues with boundary values (0 or 1), no need for continuity correction
Score Interval	$\hat{p} \pm \frac{z_{\alpha/2}}{2n} \left( 1 \pm \sqrt{1 + \frac{4 \hat{p}(1-\hat{p})}{n z_{\alpha/2}^2}} \right)$	Robust for small $n$, less affected by $\hat{p} = 0$ or $\hat{p} = 1$; no continuity correction needed
Logit-Wald	Logit transform followed by Wald method	Helps with extreme $\hat{p}$; no continuity correction required, but check for small sample sizes
ArcSine	$\hat{p} = \sin^2\left(\frac{\text{ArcSine}(\hat{p})}{2}\right)$	Helps stabilize variance at extremes (0 or 1); no need for continuity correction for most cases

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Naming convention for Methods:

We use "ci" in the start of each function name to indicate that the "Confidence Interval" is being called.

We then call the method used by first calling "ci", followed by the Naming Convention used. (ci is the prefix).

Example Usage:

"cias" - Confidence Interval called using the ArcSine Method.

The names used to call each of the methods are listed below:

Method Name	Naming Convention Used
1. Wald Interval	wd
2. Wald-T Interval	tw
3. Likelihood Interval (Exact Method)	lr
4. Score Interval (Wilson Interval)	sc
5. Logit-Wald Interval	lt
6. ArcSine Interval	as
7. Exact	ex
8. All	all

Naming Convention for Submethods:

We use the submethod name to specify what sub-type of method is being called.

We then call the method used by first calling "ci{method name}", followed by the Naming Convention used. (ci{method name} is the prefix).

Example Usage:

"ciasx" - Adjusted Score Method of Confidence Interval called using the ArcSine Method.

The names used to call each of the sub-methods/type of method are listed below:

Type of Method	Naming Convention Used	Where it is called	Example
1. Base Method	x	postfix	ciwdx
2. Adjusted Method	a	prefix	ciawd
3. Adjusted X Method	a_x	prefix and postfix	ciawdx
4. Continuity Corrected	c	prefix	cicwd
3. Continuity Corrected X	c_x	prefix and postfix	cicwdx

Naming convention for plots:

Just add "plot" prefix before ci{methodname} to print the plot.

Example Usage:

"plotciwdx" - plots the Based Method of Confidence Interval called using the Wald Method.