Concentration models

Let \(x\) denote a random variable from a lognormal distribution. Then, the log transformed variable \(y = ln(x)\) is normally distributed with \(\mu\) and variance \(\sigma\). The probability density function (p.d.f.) of y may be expressed as:

\[f_{y}(y, p_{0}, \mu_{y}, \sigma_{y}^2) = p_{0}I(y;0) + (1 - p_{0})(1 - I(y;0)) \cdot \frac{1}{\sqrt{2\pi\sigma_{y}}} \exp \frac{(y - \mu_{y})^2}{2\sigma_{y}^2}\]

where \(p_{0} = \mathit{Pr}(y< log(X_{lor})), x_{lor}\) is the limit of reporting and \(I(y;0)\) is an indicator function for \(y< log(X_{lor})\). For \(p_{0} = 0\) the p.d.f. of \(y\) reduces to the usual lognormal density. The left truncated density for \(y \geq \log(X_{lor})\) may be expressed as:

\[f_{y}(y; \mu_{y}, \sigma_{y}^2) = \frac{1}{\sqrt{2\pi\sigma_{y}}} \exp \frac{(y - \mu_{y})^2}{2\sigma_{y}^2} / (1 - \Phi(z))\]

with \(\Phi(\cdot)\) the standard normal c.d.f. and \(z = (\log(x_{lor}) - \mu_{y}) / \sigma_{z}\). Model parameters are estimated using maximum likelihood estimation based on the loglikelihood functions specified below. The loglikelihood functions are evaluated in R, using the optim algorithm to find estimates for \(\mu_{y}, \sigma_{y}^2\) and \(p_{0}\).

Mixture zero spike and censored lognormal

The loglikelihood may be expressed as:

\[\log L ( p_{0}, \mu_{y}, \sigma_{y}^2) = \sum_{i=1}^{n_{0}} \log(p_{0} + (1 - p_{0})\Phi(z_{i})) + n_{1} \log ( \frac{1 - p_{0}} {\sqrt{2\pi\sigma_{y}}}) - \sum_{i = n_{0} + 1}^n \frac{(y_{i} - \mu_{y})^2}{2\sigma_{y}^2}\]

where \(y_{i} = \log(x_{i})\), \(\Phi(\cdot)\) is the standard normal c.d.f., \(z = (\log(x_{i,lor}) - \mu_{y}) / \sigma_{y}\), \(z_{lor} = (\log(lor) - \mu_{y}) / \sigma_{y}\) with \(n_{0}\) number of censored values \((x_{i} < x_{i,lor}), n_{1}\) number of uncensored values \((x_{i} \geq x_{i,lor})\) and \(x_i{,} i = 1 \cdots n\).

Multiple values for LOR are allowed.

Censored lognormal

When \(p_{0} = 0\) the loglikelihood reduces to:

\[\log L ( \mu_{y}, \sigma_{y}^2) = \sum_{i=1}^{n_{0}} \log(\Phi(z)) + n_{1} \log ( \frac{1} {\sqrt{2\pi\sigma_{y}}}) - \sum_{i = n_{0} + 1}^n \frac{(y_{i} - \mu_{y})^2}{2\sigma_{y}^2}\]

Multiple values for LOR are allowed.

Mixture non-detect spike and truncated lognormal

Ignoring the \(n_{0}\) values below \(x_{lor}\), the loglikelihood may be expressed as:

\[\log L ( \mu_{y}, \sigma_{y}^2) = -n_{1} \log(1 - \Phi(z)) + n_{1} \log ( \frac{1} {\sqrt{2\pi\sigma_{y}}}) - \sum_{i = n_{0} + 1}^n \frac{(y_{i} - \mu_{y})^2}{2\sigma_{y}^2}\]

Only one value for LOR is allowed.

Mixture non-detect spike and lognormal

Ignoring the \(n_{0}\) values below \(x_{lor}\), the loglikelihood may be expressed as:

\[\log L ( \mu_{y}, \sigma_{y}^2) = n_{1} \log ( \frac{1} {\sqrt{2\pi\sigma_{y}}}) - \sum_{i = n_{0} + 1}^n \frac{(y_{i} - \mu_{y})^2}{2\sigma_{y}^2}\]

Only one value for LOR is allowed.