diff --git a/vignettes/statistical-specification.Rmd b/vignettes/statistical-specification.Rmd index b0f4acae..37066411 100644 --- a/vignettes/statistical-specification.Rmd +++ b/vignettes/statistical-specification.Rmd @@ -76,6 +76,8 @@ Where: - $\lambda > 0$ is the rate parameter - $\gamma > 0$ is the shape parameter Note that with $\gamma = 1$ we obtain the exponential distribution as a special case. + + ## Log-Logistic Distribution $$ @@ -166,6 +168,7 @@ Where: If using the non-centred parameterisation then the following alternative formulation is used: + $$ \begin{align*} b_i &= exp(\mu_{bl(i)} + \omega_{b l(i)} * \eta_{b i}) \\ @@ -179,6 +182,7 @@ g_i &= exp(\mu_{gk(i)} + \omega_{g k(i)} * \eta_{g i}) \\ $$ Where: + * $\eta_{\theta i}$ is a random effects offset on parameter $\theta$ for subject $i$ @@ -291,12 +295,13 @@ Where: If using the non-centred parameterisation then the following alternative formulation is used: + $$ \begin{align*} b_i &= exp(\mu_{bl(i)} + \omega_{b l(i)} * \eta_{b i}) \\ s_i &= exp(\mu_{sk(i)} + \omega_{s k(i)} * \eta_{s i}) \\ g_i &= exp(\mu_{gk(i)} + \omega_{g k(i)} * \eta_{g i}) \\ -\phi_i &= \text{logistic}(\mu_{gk(i)} + \omega_{\phi k(i)} * \eta_{\phi i}) \\ +\phi_i &= \text{logistic}(\mu_{\phi k(i)} + \omega_{\phi k(i)} * \eta_{\phi i}) \\ \\ \eta_{b i} &\sim N(0, 1)\\ \eta_{s i} &\sim N(0, 1) \\ @@ -306,6 +311,7 @@ g_i &= exp(\mu_{gk(i)} + \omega_{g k(i)} * \eta_{g i}) \\ $$ Where: + * $\eta_{\theta i}$ is a random effects offset on parameter $\theta$ for subject $i$ @@ -418,6 +424,7 @@ Where: If using the non-centred parameterisation then the following alternative formulation is used: + $$ \begin{align*} b_i &= exp(\mu_{b l(i)} + \omega_{b l(i)} * \eta_{b i}) \\