Analysis of proportions over time

Question

My knowledge of statistics is limited and I am looking for resources to read on the matter if possible.

Anyways, I am currently trying to estimate a confidence interval for a proportion over time. The specific example pertains to a trader's Win % as he make more trades. My current idea is to take the first $n$ sample data to calculate a proportion, then see how that changes over the amount of trades. I want to ultimately calculate the variability of the win %. My confidence interval would be calculated as such:

$$p \pm \sqrt[2]{\frac{1}{n} * p(1-p)}$$

I know this is clearly the wrong approach to look into but I would appreciate it if I am pointed to in the right direction.

Answer 1

When a proportion is transformed using the Anscombe transform $A$, its sampling distribution is normal (for $n$ larger than 20) and its standard error is theoretically given to be (Anscombe, 1948) :

$$SE_A = 1/\sqrt{4(n+1/2)}.$$

The Anscombe transform is a variation of the arcsine transform given by

$$A(s, n) = \sin^{-1}\left(\sqrt{\frac{s+3/8}{n+3/4}}\right)$$

where $s$ is the number of wins, and $n$ is the total number of play.

This transform returns scores between $0$ and $\pi/2 \approx 1.57$ (instead of between 0 and 1 for the proportion $s/n$).

Consequently, the confidence interval of $A(s,n)$ is obtained with a $z$ critical value,

$$\left[ A - z_{1/2-\gamma/2}, \;\;A + z_{1/2+\gamma/2} \right]$$

where $\gamma$ (typically .95) is the confidence level desired (Laurencelle & Cousineau, 2023).

In a final step, you can reverse the function $A$ on the confidence limits to obtain the confidence interval of your proportion, or if used in a plot, use an arcsine scale. This last step is automatized if you are using R's library of mine ANOPA and its function anopaPlot() (see this vignette here).