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 $y\ge \log (X_{lor})$ may be expressed as:

$$f_y(y;{\mu }_y,{\sigma }_y^2)=\frac{1}{\sqrt{2\pi {\sigma }_y}}\exp \frac{(y-{\mu }_y{)}^2}{2{\sigma }_y^2}/(1-\mathrm{\Phi }(z))$$

with $\mathrm{\Phi }(\cdot )$ 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:

$$\log L(p_0,{\mu }_y,{\sigma }_y^2)=\sum _{i=1}^{n_0}\log (p_0+(1-p_0)\mathrm{\Phi }(z_i))+n_1\log (\frac{1-p_0}{\sqrt{2\pi {\sigma }_y}})-\sum _{i=n_0+1}^n\frac{(y_i-{\mu }_y{)}^2}{2{\sigma }_y^2}$$

where $y_i=\log (x_i)$, $\mathrm{\Phi }(\cdot )$ 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 $(x_i\ge x_{i,lor})$ and $x_i,i=1\cdots n$.

Multiple values for LOR are allowed.

Censored lognormal

When $p_0=0$ the loglikelihood reduces to:

$$\log L({\mu }_y,{\sigma }_y^2)=\sum _{i=1}^{n_0}\log (\mathrm{\Phi }(z))+n_1\log (\frac{1}{\sqrt{2\pi {\sigma }_y}})-\sum _{i=n_0+1}^n\frac{(y_i-{\mu }_y{)}^2}{2{\sigma }_y^2}$$

Multiple values for LOR are allowed.

Mixture censored spike and truncated lognormal

Ignoring the $n_0$ values below $x_{lor}$, the loglikelihood may be expressed as:

$$\log L({\mu }_y,{\sigma }_y^2)=-n_1\log (1-\mathrm{\Phi }(z))+n_1\log (\frac{1}{\sqrt{2\pi {\sigma }_y}})-\sum _{i=n_0+1}^n\frac{(y_i-{\mu }_y{)}^2}{2{\sigma }_y^2}$$

Only one value for LOR is allowed.

Mixture censored 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.