Effloglog — Effloglog-class • crmPack

Effloglog is the class for the linear log-log efficacy model using pseudo data prior. It describes the relationship between continuous efficacy responses and corresponding dose levels in log-log scale. This efficacy log-log model is given as $$y_i = theta1 + theta2 * log(log(x_i)) + epsilon_i,$$ where $y_i$ is the efficacy response for subject $i$, $x_i$ is the dose level treated for subject $i$ and $epsilon_i$ is the random error term of efficacy model at subject $i$. The error term $epsilon_i$ is a random variable that follows normal distribution with mean $0$ and variance $nu^{-1}$, which is assumed to be the same for all subjects. There are three parameters in this model, the intercept $theta1$, the slope $theta2$ and the precision $nu$ of the efficacy responses, also known as the inverse of the variance of the pseudo efficacy responses. It can be a fixed constant or having a gamma distribution. Therefore, a single scalar value or a vector with two positive numbers values must be specified for nu slot. If there are some observed efficacy responses available, in the output, nu will display the updated value of the precision or the updated values for the parameters of the gamma distribution. The Effloglog inherits all slots from ModelEff class.

Usage

Effloglog(eff, eff_dose, nu, data, const = 0)

.DefaultEffloglog()

Arguments

eff: (numeric)
the pseudo efficacy responses. Elements of eff must correspond to the elements of eff_dose.
eff_dose: (numeric)
dose levels that correspond to pseudo efficacy responses in eff.
nu: (numeric)
the precision (inverse of the variance) of the efficacy responses. This is either a fixed value or a named vector with two positive numbers, the shape (a), and the rate (b) parameters for the gamma distribution.
data: (DataDual)
observed data to update estimates of the model parameters.
const: (number)
the constant value added to the dose level when the dose level value is less than or equal to 1 and a special form of the linear log-log has to applied (Yeung et al. (2015).).

Details

The prior of this model is specified in form of pseudo data. First, at least two dose levels are fixed. Then, using e.g. experts' opinion, the efficacy values that correspond to these dose levels can be obtained, The eff and eff_dose arguments represent the prior in form of the pseudo data. The eff represents the pseudo efficacy values. The eff_dose represents the dose levels at which these pseudo efficacy values are observed. Hence, the positions of the elements specified in eff and eff_dose must correspond to each other between these vectors. Since at least 2 pseudo efficacy values are needed to obtain modal estimates of the intercept and slope parameters, both eff and eff_dose must be vectors of length at least 2.

The joint prior distribution of the intercept $theta1$ and the slope $theta2$ of this model follows bivariate normal distribution with mean $mu$ and covariance matrix $(nu * Q)^{-1}$. The mean $mu$ is a $2 x 1$ column vector that contains the prior modal estimates of the intercept and the slope. Scalar $nu$ is the precision of the pseudo efficacy responses and $Q$ is the prior or posterior (given that observed, no DLT data is available) precision matrix. It is specified as $Q = X0^T * X0 + X^T * X$, where $X0$ is a design matrix that is based on pseudo dose levels only, and $X$ is a design matrix that is based on dose levels corresponding to the no DLT efficacy responses observed only (if any). Hence, the $X0$ (or $X$) will be of size $r x 2$, if there are $r >= 2$ pseudo efficacy responses specified (or if there are $r$ no DLT efficacy responses observed in the data).

Slots

eff: (numeric)
the pseudo efficacy responses. Each element here must represent responses treated based on one subject. It must be a vector of length at least 2 and the order of its elements must correspond to values specified in eff_dose.
eff_dose: (numeric)
the pseudo efficacy dose levels at which the pseudo efficacy responses are observed. It must be a vector of length at least 2 and the order of its elements must correspond to values specified in eff.
nu: (numeric)
parameter of the prior precision of pseudo efficacy responses. This is either a fixed value or a named vector with two positive numbers, the shape (a), and the rate (b) parameters for the gamma distribution.
use_fixed: (flag)
indicates whether nu specified is a fixed value or a vector with two parameters for gamma distribution. This slot is for internal purposes only and must not be used by the user.
theta1: (number)
the intercept in this efficacy log-log model. This slot is used in output to display the resulting prior or posterior modal estimates obtained based on the pseudo and observed (if any) data.
theta2: (number)
the slope in this efficacy log-log model. This slot is used in output to display the resulting prior or posterior modal estimates obtained based on the pseudo and observed (if any) data.
Pcov: (matrix)
refers to the $2 x 2$ covariance matrix of the estimators of the intercept $theta1$ and the slope $theta2$ parameters in this model. This is used in output to display the resulting prior and posterior covariance matrix of $theta1$ and $theta2$ obtained, based on the pseudo and observed (if any) data. This slot is needed for internal purposes.
X: (matrix)
is the design matrix that is based on either the pseudo dose levels or observed dose levels (without DLT). This is used in the output to display the design matrix for the pseudo or the observed efficacy responses.
Y: (numeric)
is a vector that either contains the pseudo efficacy responses or observed efficacy responses (without DLT).
mu: (numeric)
a vector of the prior or the posterior modal estimates of the intercept ($theta1$) and the slope ($theta2$). This slot is used in output to display as the mean of the prior or posterior bivariate normal distribution for $theta1$ and $theta2$.
Q: (matrix)
is the prior or posterior (given that observed, no DLT data is available) precision matrix. It is specified as $Q = X0^T * X0 + X^T * X$, where $X0$ is a design matrix that is based on pseudo dose levels only, and $X$ is a design matrix that is based on dose levels corresponding to the observed, no DLT efficacy values only (if any).
const: (number)
a non-negative number (default to 0), leading to the model form described above. In general, the model has the form $y_i = theta1 + theta2 * log(log(x_i + const)) + epsilon_i$, such that dose levels greater than $1 - const$ can be considered as described in Yeung et al. (2015).

Note

Typically, end users will not use the .DefaultEffloglog() function.

Examples

# Obtain prior modal estimates given the pseudo data.
# First we use an empty data set such that only the dose levels under
# investigations are given. In total, 12 dose levels are under investigation
# ranging from 25 to 300 mg with increments of 25 (i.e 25, 50, 75, ..., 300).
emptydata <- DataDual(doseGrid = seq(25, 300, 25), placebo = FALSE)

# Define the pseudo data as first by fixing two dose levels 25 and 300 mg (`eff_dose`).
# Then, the efficacy responses observed at these two dose levels are 1.223 and 2.513 (`eff`).
# We specify the prior precision of the pseudo efficacy responses (`nu`) as a vector
# with the shape (a) and the rate (b) parameters for the gamma distribution.
# Obtain modal estimates and other estimates from the model (no observations,
# only pseudo data).
my_model1 <- Effloglog(
  eff = c(1.223, 2.513),
  eff_dose = c(25, 300),
  nu = c(a = 1, b = 0.025),
  data = emptydata
)

# Observed data.
my_data <- DataDual(
  x = c(25, 50, 50, 75, 100, 100, 225, 300),
  y = c(0, 0, 0, 0, 1, 1, 1, 1),
  w = c(0.31, 0.42, 0.59, 0.45, 0.6, 0.7, 0.6, 0.52),
  doseGrid = emptydata@doseGrid
)
#> Used default patient IDs!
#> Used best guess cohort indices!

# Obtain posterior modal estimates and other estimates from the model given some
# observed data.
my_model2 <- Effloglog(
  eff = c(1.223, 2.513),
  eff_dose = c(25, 300),
  nu = c(a = 1, b = 0.025),
  data = my_data
)