quantCI

For the quantCI function, there are two methods that can be specified to calculate the confidence interval specified. The “exact” method solves for the percentiles numerically, while the “approximate” method uses an approximation that may be faster with large sets of data.

If the “approximate” method is specified, let \(n\) be the number of non-missing values for a variable, and let \(x_{1},x_{2},...,x_{n}\) represent the ordered values of the variable. Let the \(t^{th}\) percentile be \(y\), \(p = \frac{t}{100}\), and let \((n+1)p = j + g\), where \(j\) is the integer part of \(n(p+1)\), and \(g\) is the fractional part of \(n(p+1)\). Then:

\[y = (1-g)x_{j} + gx_{j+1}\]

If the “exact” method is specified, let \(u_{1}\) be the lower percentile, \(u_{2}\) be the upper percentile, \(0 < u_{1} < u_{2} < 1\), and \(n^{'} = n + 1\). \(I_{u}(a,b)\) is the incomplete beta function. Then:

\[I_{u}[n^{'}u_{1},n^{'}(1-u_{1})] = 1 - \alpha/2\]

\[I_{u}[n^{'}u_{2},n^{'}(1-u_{2})] = \alpha/2\]

\[y = (1-g)x_{j} + gx_{j+1}\]

The function returns a list of 5 values: the lower/upper confidence limit of the quantile, the estimated data value at the quantile and its lower/upper bound of the confidence interval.