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.