I have to simulate (in javascript) the likelihood that members of a certain population would have a disease, given a contextual value. All information I have comes from papers that show odds ratios with $95\%$ confidence interval.
I found Box-Muller method and other approaches to normal distribution, but I cannot realize the way to generate random numbers with OR, CI percentage and the interval.
It appears you're asking how to generate bivariate binary data with a pre-specified odds ratio. Here I will describe how you can do this, as long as you can generate a discrete random variables (as described here), for example.
If you want to generate data with a particular odds ratio, you're talking about binary that comes from a $2 \times 2$ table, so the normal distribution is not relevant. Let $X,Y$ be the two binary outcomes; the $2 \times 2$ table can be parameterized in terms of the cell probabilities $p_{ij} = P(Y = i, X = j)$. The parameters $p_{11}, p_{01}, p_{10}$ will suffice, since $p_{00} = 1 - p_{11} - p_{01} - p_{10}$.
It can be shown that there is a 1-to-1 invertible mapping $\{ p_{11}, p_{01}, p_{10} \} \longrightarrow \{ M_{X}, M_{Y}, OR \}$ where $M_{X} = p_{11} + p_{01}, M_{Y} = p_{11} + p_{10}$ are the marginal probabilities and $OR$ is the odds ratio.
That is, we can map back and forth at will between the $\{$cell probabilities $\}$ and $\{$ the marginal probabilities & Odds ratio$\}$. Using this fact, you can generate bivariate binary data with a pre-specified odds ratio. This rest of this answer will walk one through that process and supply some crude R code to carry it out
The '$\longrightarrow$' is simple enough; to generate data with a particular odds ratio you have to invert this mapping. For a fixed value of $M_{X}, M_{Y}$, we have
\begin{equation} \log( OR ) = \log(p_{11}) + \log \left(1 - M_{Y} - M_{X} + p_{11}\right) - \log \left(M_{Y}-p_{11}\right) - \log \left(M_{X}-p_{11}\right). \end{equation}
It is a fact that
\begin{equation} {\rm max}\Big(0, M_X + M_Y-1\Big) \le p_{11}\le {\rm min}\Big(M_X, M_Y\Big). \end{equation}
As $p_{11}$ moves through this range, $OR$ increases monotonically from 0 to $\infty$, thus there is a unique root of
\begin{equation} \log(p_{11}) + \log \left(1 - M_{Y} - M_{X} + p_{11}\right) - \log \left(M_{Y}-p_{11}\right) - \log \left(M_{X}-p_{11}\right) - \log(OR) \end{equation}
as a function of $p_{11}$. After solving for this root, $p_{10} = M_{Y} - p_{11}$ and $p_{01} = M_{X} - p_{11}$ and $p_{00} = 1 - p_{11} - p_{01} - p_{10}$, at which point we have the cell probabilities and the problem reduces to simply generating discrete random variables.
The width of the confidence interval will be a function of the cell counts so more information is needed to precisely reproduce the results.
Here is some crude R code to generate data as specified above.
# return a 2x2 table of n outcomes with row marginal prob M1, column marginal prob
# M2, and odds ratio OR
f = function(n, M1, M2, OR)
{
# find p11
g = function(p) log(p) + log(1-M1-M2+p) - log(M1-p) - log(M2-p) - log(OR)
br = c( max(0,M1+M2-1), min(M1,M2) )
p11 = uniroot(g, br)$root
# fill in other cell probabilities
p10 = M1 - p11
p01 = M2 - p11
p00 = 1-p11-p10-p01
# generate random numbers with those cell probabilities
x = runif(n)
n11 = sum(x < p11)
n10 = sum(x < (p11+p10)) - n11
n01 = sum(x < (p11+p10+p01)) - n11 - n10
n00 = n - (n11+n10+n01)
z = matrix(0,2,2)
z[1,] = c(n11,n10)
z[2,] = c(n01,n00)
return(z)
}
