crmPack (version 1.0.6)

Effloglog-class: Class for the linear log-log efficacy model using pseudo data prior

Description

This is the efficacy model which describe the relationship of the continuous efficacy responses and the dose levels. More specifically, this is a model to describe the linear relationship between the continuous efficacy responses and its coressponding dose level in log-log scale. The efficacy log-log model is given as $$y_i=\theta_1 +theta_2 log(log(d_i))+\epsilon_i$$ where \(y_i\) is the efficacy responses for subject i, \(d_i\) is the dose level treated for subject i and \(\epsilon_i\) is the random error term of efficacy model at subject i such that \(\epsilon_i\) has a normal distribution of mean 0 and variance \(\sigma^2=\nu^{-1}\). This variance is assumed to be the same for all subjects.

Slots

Eff

the pseudo efficacy response, the scalar efficacy values. This must be a vector of at least length 2. Each element or value here must represents responses treated based on one subject. The order of its elements must corresponds to the values presented in vector Effdose (see details above)

Effdose

the pseudo efficacy dose level. This is the dose levels at which the pseudo efficacy responses are observed at. This must be a vector of at least length 2 and the orde of its elements must corresponds to values presented in vector Eff (see detial above)

nu

refers to the prior precision of pseudo efficacy responses. This is either a fixed value or a vector of elements a, a positive scalar for the shape parameter, and b, a positive scalar for the rate parameter for the gamma dsitribution. (see detail from above)

useFixed

a logical value if nu specified is a fixed value or not. This slot is needed for internal purposes and not to be touched by the user.

theta1

The intercept \(\theta_1\) parameter of 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 data and (if any) the observed data/ responses.

theta2

The slope \(theta_2\) parameter of the efficacy log-lgo model. This slot is used in output to display the resulting prior or posterior modal estimates obtained based on the pseudo data and (if any) the observed data/ responses.

Pcov

refers to the covariance matrix of the intercept (phi1) and slope (phi2) paramters of this model. This slot is used in output to display the covariance matrix obtained based on the pseudo data and (if any) the observed data/responses. This slot is needed for internal purposes.

vecmu

is the column vector of the prior or the posterior modal estimates of the intercept (phi1) and the slope (phi2). This slot is used in output to display as the mean of the prior or posterior bivariate normal distribtuion for phi1 and phi2. (see details from above)

matX

is the design matrix based on either the pseudo or all observed efficacy response. This is used in output to display the design matrix for the pseudo or the observed efficacy responses (see details from above)

matQ

is the square matrix of multiplying the the design matrix with its transponse. This is represented either using the only the pseudo efficay responses or only with the observed efficacy responses. This is display in the output (see details from above)

vecY

is the column vector either contains the pseudo efficay responses or all the observed efficacy responses. This is used in output to display the pseudo or observed efficacy responses (see detail from above)

c

is a constant value greater or equal to 0, with the default 0 leading to the model form described above. In general, the model has the form \(y_i=\theta_1 +theta_2 log(log(d_i + c))+\epsilon_i\), such that dose levels greater than \(1-c\) can be considered as described in Yeung et al. (2015).

Details

There are three parameters in this model which is to intercept \(\theta_1\), the slope \(\theta_2\) and the precision \(\nu\) of the efficay responses. It inherit all slots from ModelEff

The prior of this model is specified in form of pseudo data. First at least two dose levels are fixed. Then ask for experts' opinion about the efficacy values that can be obtained at each of the dose levels if one subject is treated at each of these dose levels. The prior modal estimates (same as the maximum likelihood estimates) can be obtained for the intercept and slope paramters in this model.

The Eff and Effdose are used to represent the prior in form of the pseudo data. The Eff represents the pseudo scalar efficacy values. The Effdose represents the dose levels at which these pseudo efficacy values are observed. These pseudo efficay values are always specified by assuming one subject are treated in each of the dose levels. Since at least 2 pseudo efficacy values are needed to obtain modal estimates of the intercept and slope parameters, both Eff and Effdose must be vector of at least length 2. The position of the values or elements specified in Eff or Effdose must be corresponds to the same elements or values in the other vector.

The nu represents the prior presion \(\nu\) of the pseudo efficacy responses. It is also known as the inverse of the variance of the pseduo efficacy responses. The precision can be a fixed constant or having a gamma distribution. Therefore, single scalar value, a fixed value of the precision can be specified. If not, two positive scalar values must be specified as the shape and rate parameter of the gamma distribution. 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.

Given the variance of the pseudo efficacy responses, the joint prior distribution of the intercept \(\theta_1\) (theta1) and the slope \(\theta_2\) (theta2) of this model is a bivariate normal distribution. A conjugate posterior joint distribution is also used for theta1 and theta2. The joint prior bivariate normal distribution has mean \(\boldsymbol\mu_0\) and covariance matrix \((\nu \mathbf{Q}_0)^{-1}\). \(\boldsymbol\mu_0\) is a \(2 \times 1\) column vector contains the prior modal estimates of the intercept (theta1) and the slope (theta2). Based on \(r\) for \(r \geq 2\) pseudo efficacy responses specified, \(\mathbf{X}_0\) will be the \(r \times 2\) design matrix obtained for these pseudo efficacy responses. the matrix \(\mathbf{Q}_0\) will be calculated by \(\mathbf{Q}_0=\mathbf{X}_0 \mathbf{X}^T_0\) where \(\nu\) is the precision of the pseudo efficacy responses. For the joint posterior bivariate distribution, we have \(\boldsymbol{\mu}\) as the mean and \((\nu\mathbf{Q}_0)^{-1}\) as the covariance matrix. Here, \(\boldsymbol\mu\) is the column vector containing the posterior modal estimates of the intercept (theta1) and the slope (theta2). The design matrix \(\mathbf{X}\) obtained based only on observed efficacy responses will give \(\mathbf{Q}=\mathbf{X}\mathbf{X}^T\) with \(\nu\) as the precision of the observed efficay responses. If no observed efficay responses are availble (i.e only pseudo efficay responses are used), the vecmu, matX, matQ and vecY represents \(\boldsymbol\mu_0\), \(\mathbf{X}_0\), \(\mathbf{Q}_0\) and the column vector of pseudo efficay responses, respectively. If there are some observed efficacy responses, vecmu, matX, matQ and vecY will represent \(\boldsymbol\mu\), \(\mathbf{X}\), \(\mathbf{Q}\) and the column vector contains all observed efficacy responses, respectively. (see details in about the form of prior and posterior distribution)

Examples

##Obtain prior modal estimates for the Effloglog model (efficacy model) given the pseudo data.
##First define an empty data set by only define the dose levels used in the study,
## 12 dose levels are usesd from 25 to 300 mg with increments of 25.
emptydata<-DataDual(doseGrid=seq(25,300,25),placebo=FALSE)
data<-emptydata
## define the pseudo data as first fixed 2 dose levels 25 and 300 mg and specified in 
## (Effdose slot).
## Then the efficacy responses observed at these two dose levels are 1.223 and 2.513 and 
## specified in (Eff slot).
## The prior precision of the pseudo efficay responses. This can be either a fixed value of 
## specifying the shape (a) and the rate (b) parameters for the gamma distribution in (nu slot).
## Then specify all data currentl available in (data slot).

Effmodel<-Effloglog(Eff=c(1.223,2.513),Effdose=c(25,300),nu=c(a=1,b=0.025),data=data,c=0)

##Obtain posterior modal estimates and other estimates from the model given some observed responses
## If there is some observations available
## first specified the data
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=seq(25,300,25))

Effmodel<-Effloglog(Eff=c(1.223,2.513),Effdose=c(25,300),nu=c(a=1,b=0.025),data=data)