This function fits the distribution parameters, i.e. mean and sd, of a truncated
normal distribution from an arbitrary confidence interval and, optionally, the median.
paramtnormci_fit(
p,
ci,
median = mean(ci),
lowerTrunc = -Inf,
upperTrunc = Inf,
relativeTolerance = 0.05,
fitMethod = "Nelder-Mead",
...
)A list with elements mean and sd, i.e. the parameters of the underlying
normal distribution.
numeric 2-dimensional vector; probabilities of upper and lower bound of the
corresponding confidence interval.
numeric 2-dimensional vector; lower, i.e ci[[1]], and upper bound, i.e
ci[[2]], of the confidence interval.
if NULL: truncated normal is fitted only to lower and upper value of the
confidence interval; if numeric: truncated normal is fitted on the confidence interval
and the median simultaneously. For details cf. below.
numeric; lower truncation point of the distribution (>= -Inf).
numeric; upper truncation point of the distribution (<= Inf).
numeric; the relative tolerance level of deviation of the
generated probability levels from the specified confidence interval. If the relative deviation
is greater than relativeTolerance a warning is given.
optimization method used in constrOptim.
further parameters to be passed to constrOptim.
For details of the truncated normal distribution see tnorm.
The cumulative distribution of a truncated normal \(F_{\mu, \sigma}\)(x) gives the probability that a sampled value is less than \(x\). This is equivalent to saying that for the vector of quantiles \(q=(q(p_1), \ldots, q(p_k))\) at the corresponding probabilities \(p=(p_1, \ldots, p_k)\) it holds that $$p_i = F_{\mu, \sigma}(q_{p_i}),~i = 1, \ldots, k$$ In the case of arbitrary postulated quantiles this system of equations might not have a solution in \(\mu\) and \(\sigma\). A least squares fit leads to an approximate solution: $$\sum_{i=1}^k (p_i - F_{\mu, \sigma}(q_{p_i}))^2 = \min$$ defines the parameters \(\mu\) and \(\sigma\) of the underlying normal distribution. This method solves this minimization problem for two cases:
ci[[1]] < median < ci[[2]]: The parameters are fitted on the lower and upper value
of the confidence interval and the median, formally:
\(k=3\)
\(p_1\)=p[[1]], \(p_2\)=0.5 and \(p_3\)=p[[2]];
\(q(p_1)\)=ci[[1]],
\(q(0.5)\)=median and
\(q(p_3)\)=ci[[2]]
median=NULL: The parameters are fitted on the lower and upper value of the
confidence interval only, formally:
\(k=2\)
\(p_1\)=p[[1]], \(p_2\)=p[[2]];
\(q(p_1)\)=ci[[1]],
\(q(p_2)\)=ci[[2]]
The (p[[2]]-p[[1]]) - confidence interval must be symmetric in the sense that
p[[1]] + p[[2]] = 1.
tnorm, constrOptim