%%\VignetteIndexEntry{crmPack: Object-oriented implementation of CRM designs}
%%\VignetteKeywords{dose escalation, CRM, N-CRM, EWOC, Bayesian adaptive design}
%%\VignettePackage{crmPack}
%%\VignetteDepends{crmPack,ggmcmc,grid}

\documentclass[british]{scrartcl}

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\begin{document} 

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<<load, echo=FALSE, results=hide>>=
## load package already here to be able to reference the package version!!
library(crmPack)
crmPackVersion <- as.character(packageVersion("crmPack"))
@
\title{Using the package \texttt{crmPack}:\\ introductory examples}
\author{Daniel Saban\'es Bov\'e \\
Wai Yin Yeung}
\date{\today\\Package version \Sexpr{crmPackVersion}}
\maketitle

This short vignette shall introduce into the usage of the package
\texttt{crmPack}. Hopefully it makes it easy for you to set up your own CRM.

If you have any questions or feedback concerning the package, please write an
email to the package maintainer:
\href{mailto:sabanesd@roche.com}{\nolinkurl{sabanesd@roche.com}}
Thank you very much in advance!

<<setup, echo=FALSE, results=hide>>=
options(continue="  ")                  # use two blanks instead of "+" for
                                        # continued lines (for easy copying)
@

\section{Installation}
\label{sec:installation}

Many models in \texttt{crmPack} rely on \href{http://mcmc-jags.sourceforge.net/}{JAGS}
(please click on the link for going to the webpage of the project) for the
internal MCMC computations. WinBUGS is not required or supported anymore.

\section{Getting started}
\label{sec:getting-started}

Before being able to run anything, you have to load the package with
<<load>>=
library(crmPack)
@

For browsing the help pages for the package, it is easiest to start the web
browser interface with
<<webinterface, eval=FALSE>>=
crmPackHelp()
@
This gives you the list of all help pages available for the package. 
The whole R-package is built in a modular form, by using S4 classes and methods. Please have a look at the help page "Methods" to read an introduction into the S4 object framework of R, by typing \texttt{?Methods} in the R console. In the next sections we will therefore go one by one through the important building blocks (S4 classes and corresponding methods) of CRM designs with \texttt{crmPack}.

\section{Data}
\label{sec:data}

Figure \ref{fig:data} shows the structure of the data classes included in this package
\begin{figure} 
\centering
\begin{tikzpicture}[node distance=1cm, thick, >=stealth]
\node[block] (step1) {%
  GeneralData};
 \node[block, below=of step1] (step2) {%
 Data};
\node[block, below=of step2] (step3) {%
  DataDual};
\path[->] (step1) edge (step2)
           (step2) edge (step3)
;
\end{tikzpicture}
\caption{Data classes structure}\label{fig:data}
\end{figure}

We have three data classes for this package. The parent class is the \texttt{GeneralData} class. The \texttt{Data} class is inheriting from the \texttt{GeneralData} class and the \texttt{DataDual} class is inheriting from the \texttt{Data} class. Inheritance means that the subclass has all the slots (attributes) of the parent class, but can also have additional slots. Methods that work on the parent class also work the same way on the subclass, unless a specialized method for the subclass has been defined.

First, we will set up the data set. If you are at the beginning of a trial, no observations will be available. Then we can define an empty data set, for example:
<<emptydata>>=
emptydata <- Data(doseGrid=
                 c(0.1, 0.5, 1.5, 3, 6,
                   seq(from=10, to=80, by=2)))
@
The \texttt{R}-package \texttt{crmPack} uses the S4 class system for
implementation of the dose-escalation designs. There is the convention that
class initialization functions have the same name as the class, and all class
names are capitalized. Note that in order to create this \texttt{Data} object, we use the initialization function of the same name, and give it as parameters the contents of the object to be constructed.
At least the \texttt{doseGrid} parameter, which contains all possible dose levels to be potentially used in the trial, must be specified in a call of the \text{Data()} initialization function.
            
If you are in the middle of a trial and you would like to recommend the next
dose, then you have data from the previous patients for input into the model.
This data can also be captured in a \texttt{Data} object. For example:

<<data>>=
data <- Data(x=c(0.1, 0.5, 1.5, 3, 6, 10, 10, 10),
             y=c(0, 0, 0, 0, 0, 0, 1, 0),
             cohort=c(0, 1, 2, 3, 4, 5, 5, 5),
             doseGrid=
                 c(0.1, 0.5, 1.5, 3, 6,
                   seq(from=10, to=80, by=2)))
@

Most important are \texttt{x} (the doses) and \texttt{y} (the DLTs, 0 for no DLT
and 1 for DLT), and we are using the same dose grid \texttt{doseGrid} as before. 
All computations are using the dose grid specified in the \texttt{Data} object. So for example,
except for patient number 7, all patients were free of DLTs.

Again, you can find out the details in the help page \texttt{Data-class}. Note
that you have received a warning here, because you did not specify the patient
IDs -- however, automatic ones just indexing the patients have been created for
you:

<<ids>>=
data@ID
@

You can get a visual summary of the data by applying \texttt{plot} to the
object:\footnote{Note that for all \texttt{plot} calls in this vignette, you can
  leave away the wrapping \texttt{print} function call if you are working
  interactively with \texttt{R}. It is only because of the \texttt{Sweave}
  production of this vignette that the \texttt{print} statement is needed.}

<<plotdata, fig=TRUE>>=
print(plot(data))
@


\section{Structure of the model class}
\label{sec:modelstructure}

Figure \ref{fig:model} show the structure of the model class defined in this package. The \texttt{AllModels}
 is the parent class, from which all model classes inherit. There are two sub-classes: First, the \texttt{GeneralModel} class from which all models inherit that are using \texttt{JAGS} to specify the model and the prior distribution and will then be estimated by MCMC later on. Then, the second subclass is the \texttt{ModelPseudo} class for which the prior of the models are specified in terms of pseudo data and standard maximum likelihood routines from R will be used for 
computational purposes. All models included in this package will have a parent class of either the \texttt{GeneralModel} or the \texttt{ModelPseudo} classes. There are two further classes under \texttt{ModelPseudo} which are the \texttt{ModelTox} class include all DLT (the occurrence of a dose-limiting toxicity) class models, and class \texttt{ModelEff} which includes all efficacy class models.


\begin{figure}
\centering
\begin{tiny}
\begin{tikzpicture}[node distance=0.3cm, thick, >=stealth]
\node[block] (step1) {
  AllModels};
 \node[block, below left=0.3cm and 0.5cm of step1] (step2a) {
 GeneralModel};
  \node[block, below left=0.3cm of step2a] (step2aa) {
 Model};
 \node[block, below left=0.4cm and 0.7cm of step2aa] (step3aa) {
 LogisticNormal};
  \node[block, below left =1.5cm and 0.05 cm of step2aa] (step3bb) {
 LogisticLogNormalSub};
   \node[block, below right=1.9cm and 0.7 cm of step2aa] (step3cc) {
 LogisticKadane};
\node[block, below right=0.7cm and 0.5 cm of step1] (step2b) {
 ModelPseudo};
\node[block, below left=0.5cm and 0.8cm of step2b] (step3a) {
ModelTox};
\node[block, below right = 0.5cm and 1.5cm of step2b] (step3b){
ModelEff};
\node[block, below=2.2cm of step3a](step4a){
LogisticIndepBeta
};

\node[block, below left=0.5cm and 0.2cm of step3b](step4c){
Effloglog
};
\node[block, below= 0.5cm of step3b](step4d){
EffFlexi
};

  
\path[->] (step1) edge (step2a)
           (step1) edge (step2b)
           (step2a) edge (step2aa)
           (step2aa) edge (step3aa)
           (step2aa) edge (step3bb)
           (step2aa) edge (step3cc)
           (step2b) edge (step3a)
(step2b) edge (step3b)
(step3a) edge (step4a)

(step3b) edge (step4c)
(step3b) edge (step4d)


;
\end{tikzpicture}
\end{tiny}
\caption{Model classes structure}\label{fig:model}
\end{figure}


\section{Model setup}
\label{sec:model-setup}

\subsection{Logistic model with bivariate (log) normal prior}
\label{sec:logist-norm-model}

First, we will show an example of setting up of a model inheriting from the \texttt{Model} and \texttt{GeneralModel} classes, the logistic normal model. You can click on
the corresponding help page \texttt{LogisticLogNormal-class} as background
information for the next steps.

With the following command, we create a new model of class \texttt{LogisticLogNormal}, with certain mean and covariance prior parameters and reference dose:

<<model-setup>>=
model <- LogisticLogNormal(mean=c(-0.85, 1),
                           cov=
                               matrix(c(1, -0.5, -0.5, 1),
                                      nrow=2),
                           refDose=56)
@

We can query the class that an object belongs to with the
\texttt{class} function:

<<class>>=
class(model)
@

We can look in detail at the structure of \texttt{model} as follows:

<<str>>=
str(model)
@

We see that the object has \Sexpr{length(slotNames(model))} slots, and their
names. These can be accessed with the \verb|@| operator (similarly as for lists
the \verb|$| operator), for example we can extract the \texttt{dose} slot:

<<dose>>=
model@dose
@

This is the function that computes for given parameters \texttt{alpha0} and
\texttt{alpha1} the dose that gives the probability \texttt{prob} for a
dose-limiting toxicity (DLT). You can find out yourself about the other slots,
by looking at the help page for \texttt{Model-class} in the help browser,
because all univariate models with JAGS specification are just special cases (subclasses) of the \texttt{Model} class.
In the \texttt{Model-class} help page, you also find out that there are four
additional specific model classes that are sub-classes of the \texttt{Model}
class, namely \texttt{LogisticLogNormalSub}, \texttt{LogisticNormal},
\texttt{LogisticKadane} and \texttt{DualEndpoint}.

Next, we will show an example of setting up a model of the \texttt{ModelPseudo} class, the 
\texttt{LogisticIndepBeta} model. More specifically, this is also a model in \texttt{ModelTox} class.

The \texttt{LogisticIndepBeta} model is a two-parameter logistic regression model to describe the relationship between the probability of the occurrence of DLT and its corresponding log dose levels. The model parameters are $\phi_1$, for the intercept and $\phi_2$, the slope. 
This is also a model for which its prior is expressed in form of pseudo data. 

Here it is important that the data set has to be defined before defining any models from \texttt{ModelPseudo} class. This is to ensure we obtained the updated estimates for the model parameters using all currently available observations.
Either an empty data set or a data set that contains all currently available observations is needed.

Therefore, let's assume an empty data set is set up. For example, we will use 12 dose levels from 25 to 300 mg with increments of 25 mg. Then we have:
<<emptydata>>=
emptydata <- Data(doseGrid=
                   seq(from=25, to=300, by=25))
data1 <- emptydata                 
@
Then we express our prior in form of pseudo data. The idea is as follows. 
First fix two dose level $d_{(-1)}$ and $d_{(0)}$, which are usually the 
lowest and the highest dose level, so here we choose 25 and 300 mg. Then 
we elicit from experts or clinicians the probability of the occurrence of 
DLT, $p_{(-1)}$ and $p_{(0)}$ at these two dose levels. That is, assuming 
$n_{(l)}$ subjectsare treated at each of these two dose levels, $l=-1,0$, 
$t_{(l)}$ out of $n{(l)}$ subjects are expected to be observed with a DLT such 
that $p_{(l)} = t_{(l)}/n_{(l)}$. Let $\tilde{p}_{(l)}$ be the probability of 
the occurrence of a DLT at dose $l$ for dose $l=-1,0$. $\tilde{p}_{(l)}$ will 
follow independent Beta distributions and the joint probability density function 
of $p_{(l)}$ can be obtained. Therefore, this model is called 
\texttt{LogisticIndepBeta}.  We expressed the prior as if we have some data 
(pseudo data) before the trial start. The prior modal estimates of $\phi_1$ and 
$\phi_2$, which is also equivalent to the maximum likelihood estimators,
can be obtained with the R function \texttt{glm}.
Please refer to \cite{WhiteheadWilliamson1998} for details about the 
form of the prior and posterior density of the model parameters $\phi_1$ and 
$\phi_2$. 

With the following commands, we create the model of class \texttt{LogisticIndepBeta}, with the prior specified in form of pseudo data.

<<LogisticIndepBeta>>=
DLTmodel<-LogisticIndepBeta(binDLE=c(1.05,1.8),DLEweights=c(3,3),
                            DLEdose=c(25,300),data=data1)
@

(Note that in some functions including this initialization function, DLE instead of DLT is used. In this vignette we use the unified abbreviation DLT throughout the text and variable names.)

For the model specified, we have fixed two dose levels (25 and 300 mg) and represented them in the \texttt{DLEdose} slot. Then we assume that 3 subjects are treated at each of the dose levels, represented in the \texttt{DLEweights} slot. We have 1.05 subjects out of the 3 subjects treated at 25 mg observed with a DLT and 1.8 subjects out of the 3 subjects treated at 300 mg observed with a DLT and this is represented in the \texttt{binDLE} slot. Input to the \texttt{data} slot is also need to ensure the all currently available observations will be incorporated in the model to obtain updated modal estimates of the model parameters.
If an empty data set is used in the \texttt{data} slot, the prior modal estimates of the model parameters,
$\phi_1$ for the intercept and $\phi_2$ for the slope, can be obtained. If a data set with observations, e.g data1 in the DLTmodel above is used, we can obtaine the posterior modal estimates for the model parameters.
In addition, the pseudo data can be expressed by using more than 2 dose levels. On the other hand, at least two dose levels of pseudo information are needed to obtain modal estimates of the intercept and the slope parameter. Therefore, \texttt{binDLE},\texttt{DLEweights}, \texttt{DLEdose} must be vectors of at least length 2 and with their corresponding values specified at the same position in the other two vectors.

Since the imaginary nature of the pseudo data, the value $t_l$ for the number of subjects observed with DLT can be non-integer values. In principle, $n_l$ can also be non-integer values.

Then we can look at the structure of this model:

<<strLogisticIndepBeta>>=
str(DLTmodel)
@
 

There are in total 10 slots and their names are given. Remember that they can be accessed with the \verb|@| operator (similarly as for lists
the \verb|$| operator), for example we can extract the \texttt{phi1} slot:

<<phi1>>=
DLTmodel@phi1
@
This gives the updated modal estimate of the intercept parameter $\phi_1$. Please find out other slots using the \verb|@| operator and looking at the help page for \texttt{ModelPseudo}, \texttt{ModelTox} and \texttt{LogisticIndepBeta} classes. 



\subsection{Advanced model specification}
\label{sec:advanc-model-spec}

There are a few further, advanced ways to specify a model object in
\texttt{crmPack}.

First, a minimal informative prior \citep{Neuenschwander2008} can be computed
using the \texttt{MinimalInformative} function. The construction is based on the
input of a minimal and a maximal dose, where certain ranges of DLT probabilities
are deemed unlikely. A logistic function is then fitted through the
corresponding points on the dose-toxicity plane in order to derive Beta
distributions also for doses in-between. Finally these Beta distributions are
approximated by a common \texttt{LogisticNormal} (or \texttt{LogisticLogNormal})
model. So the minimal informative construction avoids explicit specification of
the prior parameters of the logistic regression model.

In our example, we could construct it as follows, assuming a minimal dose of 0.1
mg and a maximum dose of 100 mg:

<<min-inf>>=
coarseGrid <- c(0.1, 10, 30, 60, 100)
minInfModel <- MinimalInformative(dosegrid = coarseGrid,
                                  refDose=50,
                                  threshmin=0.2,
                                  threshmax=0.3,
                                  control=
                                 list(threshold.stop=0.03,
                                      maxit=200,
                                      max.time=0.01), # take this out for real application
                                  seed=432)
@

We use a few grid points between the minimum and the maximum to guide the
approximation routine, which is based on a stochastic optimization method (the
\texttt{control} argument is for this optimization routine, please see the help
page for \texttt{Quantiles2LogisticNormal} for details). Therefore we need to
set a random number generator seed beforehand to be able to reproduce the
results in the future. Please note that currently the reproducibility is under testing-- it is currently advised to save the approximation result in order to certainly be able to use the same model later on again. The \texttt{threshmin} and \texttt{threshmax} values specify the probability thresholds above and below, respectively, it is very
unlikely (only 5\% probability) to have the true probability of DLT at the minimum and
maximum dose, respectively.

The result \texttt{minInfModel} is a list, and we can use its contents to
illustrate the creation of the prior:

<<min-inf-res, fig=TRUE>>=
matplot(x=coarseGrid,
        y=minInfModel$required,
        type="b", pch=19, col="blue", lty=1,
        xlab="dose",
        ylab="prior probability of DLT")
matlines(x=coarseGrid,
         y=minInfModel$quantiles,
         type="b", pch=19, col="red", lty=1)
legend("right",
       legend=c("quantiles", "approximation"),
       col=c("blue", "red"),
       lty=1,
       bty="n")
@

In this plot we see in blue the quantiles (2.5\%, 50\%, and 97.5\%) of
the Beta distributions that we approximate with the red quantiles of the
logistic normal model. We see that the distance is still quite large, and the maximum
distance between any red and blue point is:

<<min-inf-dist>>=
minInfModel$distance
@

Therefore usually we would let the computations take longer (by removing the
\texttt{control} option from the \texttt{MinimalInformative} call) to obtain a
better approximation.
The final approximating model, which has produced the red points, is contained
in the \texttt{model} list element:

<<min-inf-model>>=
str(minInfModel$model)
@

Here we see in the slots \texttt{mean}, \texttt{cov} the parameters that have
been determined. At this point a slight warning: you cannot directly change
these parameters in the slots of the existing model object, because the
parameters have also been saved invisibly in other places in the model object.
Therefore, always use the class initialization function to create a new model
object, if new parameters are required. But if we want to further use the
approximation model, we can save it under a shorter name, e.g.:

<<min-inf-model-extract>>=
myModel <- minInfModel$model
@



\section{Obtaining the posterior}
\label{sec:obtaining-posterior}


As said before, models inheriting from the \texttt{GeneralModel} class rely on MCMC sampling for obtaining the posterior distribution of the model parameters, given the data. Most of the models, except the \texttt{EffFlexi} class model (please refer to \ref{subsec:dualresponses} for details), inheriting from the  \texttt{ModelPseudo} class do not necessarily require MCMC sampling to obtain posterior estimates. When no MCMC sampling is involved, the prior or posterior modal estimates of model estimates are used. But we can still obtain the full posterior distribution of the model parameters via MCMC for any models specified under \texttt{ModelPseudo} class. The MCMC sampling can be controlled with an object of class \texttt{McmcOptions}, created
for example as follows:

<<mcmc-opts>>=
options <- McmcOptions(burnin=10, # use larger burnin and samples for real application
                       step=2,
                       samples=200)
@

Now the object \texttt{options} specifies that you would like to have 200
parameter samples obtained from a Markov chain that starts with a ``burn-in''
phase of 10 iterations that are discarded, and then save a sample every 2
iterations. Note that these numbers are too low for actual production use and are
only used for illustrating purposes here; normally you would specify at least
the default parameters of the initialization function \texttt{McmcOptions}:
$10\,000$ burn-in iterations and $10\,000$ samples saved every 2nd iteration.
You can look these up in help browser under the link ``McmcOptions''.

After having set up the options, you can proceed to MCMC sampling by calling the
\texttt{mcmc} function:

<<mcmc-sampling>>=
set.seed(94)
samples <- mcmc(data, model, options)
@

The \texttt{mcmc} function takes the data object, the model and the MCMC
options. By default, JAGS is used for obtaining the samples. Use the option
\texttt{verbose=TRUE} to show a progress bar and detailed JAGS messages.

Finally, it is good practice to check graphically that the Markov chain has
really converged to the posterior distribution. To this end, \texttt{crmPack}
provides an interface to the convenient \texttt{R}-package \texttt{ggmcmc}. With
the function \texttt{get} you can extract the individual parameters from the
object of class \texttt{Samples}.

For example, we extract the $\alpha_{0}$ samples:
(please have a look at the help page for the \texttt{LogisticLogNormal} model
class for the interpretation of the parameters)

<<mcmc-extract>>=
## look at the structure of the samples object:
str(samples)
## now extract the alpha0 samples (intercept of the regression model)
alpha0samples <- get(samples, "alpha0")
@

\texttt{alpha0samples} now contains the $\alpha_{0}$ samples in a format
understood by \texttt{ggmcmc} and we can produce plots with it, e.g. a trace plot
and an autocorrelation plot:

<<ggmcmc, fig=TRUE>>=
library(ggmcmc)
print(ggs_traceplot(alpha0samples))
@

<<ggmcmc2, fig=TRUE>>=
print(ggs_autocorrelation(alpha0samples))
@

So here we see that we have some autocorrelation in the samples, and might
consider using a higher thinning parameter in order to decrease it.

You can find other useful plotting functions in the package information:

<<ggmcmc-help, eval=FALSE>>=
help(package="ggmcmc", help_type="html")
@

Similarly, using models from \texttt{ModelPseudo} class, we can also obtain the prior and posterior samples of the model parameters via MCMC. 

For example, using the DLTmodel, data1, the empty data set and options specified in earlier examples.
<<mcmc-DLTsamples, results=hide>>=
DLTsamples <- mcmc(data=data1,model=DLTmodel,options=options)
@

The prior samples of the model parameters are now saved in the variable \texttt{DLTsamples}. 

<<mcmc-postDLTsamples, results=hide>>==
data3 <-Data(x=c(25,50,50,75,100,100,225,300),
             y=c(0,0,0,0,1,1,1,1),
             doseGrid=seq(from=25,to=300,by=25))
DLTpostsamples <- mcmc(data=data3,model=DLTmodel,options=options)
@

Similarly, \texttt{DLTpostsamples} now contains the posterior samples of the model parameters. 
%Since an output message as seen from the above example will be display after each MCMC sampling when some observed data are involved, the \texttt{suppressMessages} function is used from now and throughout this document to avoid showing all of these message in this document 

This \texttt{mcmc} function also takes the data object, model and the MCMC options. This is not using JAGS but just R for the computations. 

Under this \texttt{DLTmodel}, we will obtain samples of $\phi_1$ and $\phi_2$. Using what has been described earlier in this section , we can also look at the structure using function \texttt{str}, extracting model parameters samples with \texttt{get} and produce plots with \texttt{ggs\_traceplot} and \texttt{ggs\_autocorrelation} 
for each of the model parameters.


When no MCMC sampling is involved, the posterior modal estimates of the model parameters can be obtained for models (except the \texttt{EffFlexi} class object) inheriting from the \texttt{ModelPseudo} class object. First you need to put together all currently available observations in form of a \texttt{Data} object (when only DLT responses are modelled) or \a texttt{DataDual} object (when both DLT and efficacy responses are modelled) class object. Then  using the \texttt{update} function to update your model, the posterior modal estimates of the model parameters will be display in the output of the model.

For example, we have some new observations specified in the data set \texttt{data3} and update the DLT model:

<<update-DLTmodel>>=
newDLTmodel <- update(object=DLTmodel,data=data3)
newDLTmodel@phi1
newDLTmodel@phi2
@

In the example, the \texttt{update} function is used to obtain the posterior modal estimates of the model parameters, $\phi_1$ and $\phi_2$, which can then be extracted using the \verb|@| operator on the updated result \texttt{newDLTmodel}. 

\section{Plotting the model fit}
\label{sec:plot-fit}

After having obtained the parameter samples, we can plot the model fit, by
supplying the samples, model and data to the generic plot function:

<<plot-model-fit, fig=TRUE>>=
print(plot(samples, model, data))
@

This plot shows the posterior mean curve and 95\% equi-tailed credible intervals
at each point of the dose grid from the \texttt{data} object.

Note that you can also produce a plot of the prior mean curve and credible
intervals, i.e. from the model without any data. This works in principle the
same way as with data, just that we use an empty data object:

<<empty-data, fig=TRUE>>=
## provide only the dose grid:
emptydata <- Data(doseGrid=data@doseGrid)
## obtain prior samples with this Data object
priorsamples <- mcmc(emptydata, model, options)
## then produce the plot
print(plot(priorsamples, model, emptydata))
@


This \texttt{plot} function can also apply to the \texttt{DLTmodel} when samples of the parameters have been generated:
<<plot-samplesdata1 , fig=TRUE>>=
print(plot(DLTsamples,DLTmodel,data1))
@


In addition, we can also plot the fitted dose-response curve using the prior or the posterior modal estimates of the model parameters when no MCMC sampling is used. For example, we have the \texttt{DLTmodel} specified earlier under the \texttt{ModelTox} class with the data set \texttt{data1} we specified earlier:

<<emptydatanoDLTsamples, fig=TRUE>>=
print(plot(data1,DLTmodel))
@

Since no samples are involved, only the curve using the prior or posterior modal estimates of the parameters are produced, without 95\% credibility intervals.


\section{Escalation Rules}
\label{sec:escalation-rules}

For the dose escalation, there are four kinds of rules:
\begin{enumerate}
\item \texttt{Increments}: For specifying maximum allowable increments between
  doses
\item \texttt{NextBest}: How to derive the next best dose
\item \texttt{CohortSize}: For specifying the cohort size
\item \texttt{Stopping}: Stopping rules for finishing the dose escalation
\end{enumerate}
We have listed here the classes of these rules, and there are multiple
subclasses for each of them, which you can find as links in the help pages
\texttt{Increments-class}, \texttt{NextBest-class}, \texttt{CohortSize-class}
and \texttt{Stopping-class}.

\subsection{Increments rules}
\label{sec:increments-rules}

Figure \ref{fig:increments} shows the structure of the \texttt{Increments} classes:

\begin{figure}
\centering
\begin{tikzpicture}[node distance=0.3cm, thick, >=stealth]
\node[block] (step1) {
  Increments};
 \node[block, below left=0.5 cm and 0.5cm of step1] (step2a) {
 IncrementsRelative};
\node[block, below right=0.5cm and  0.5cm of step1] (step2b) {
 IncrementsRelativeDLT};
 \node[block, below=0.5cm of step2a](step3){
 IncrementsRelativeParts
 };
\path[->] (step1) edge (step2a)
           (step1) edge (step2b)
           
           (step2a) edge (step3)

;
\end{tikzpicture}
\caption{Increments classes structure}\label{fig:increments}
\end{figure}

The \texttt{Increments} class is the basis for all maximum increments rule classes within this package. There are three subclasses, the \texttt{IncrementsRelative}, the \texttt{IncrementsRelativeParts} and the \texttt{IncrementsRelativeDLTs} classes.
Let us start with looking in detail at the increments rules. Currently two
specific rules are implemented: Maximum relative increments based on the current
dose (\texttt{IncrementsRelative} and \texttt{IncrementsRelativeParts}, which only works with \texttt{DataParts} objects), and maximum relative increments based on the
current cumulative number of DLTs that have happened
(\texttt{IncrementsRelativeDLT}).

For example, in order to specify a maximum increase of 100\% for doses up to 20 mg,
and a maximum of 33\% for doses above 20 mg, we can setup the following increments rule:

<<rel-incs>>=
myIncrements <- IncrementsRelative(intervals=c(0, 20),
                                   increments=c(1, 0.33))
@

Here the \texttt{intervals} slot specifies the left bounds of the intervals, in
which the maximum relative \texttt{increments} (note: decimal values here, no
percentages!) are valid.

The increments rule is used by the \texttt{maxDose} function to obtain the
maximum allowable dose given the current data:

<<max-dose>>=
nextMaxDose <- maxDose(myIncrements,
                       data=data)
nextMaxDose
@

So in this case, the next dose could not be larger than 20 mg.

In the following example the dose escalation will be restricted to a 3-fold (= 200\%) increase:
<<3fold-incs>>=
myIncrements1 <- IncrementsRelative(intervals=c(25),
                                   increments=c(2))
@
From all doses (since the dose grid starts at 25 mg) there is a maximum increase of 200\% here. 

The \texttt{IncrementsRelativeDLT} class works similarly, taking the number of DLTs in the whole trial so far as the basis for the maximum increments instead of the last dose.

\subsection{Rules for next best dose recommendation}
\label{sec:rules-next-best}

Figure \ref{fig:rules} show the structure of the next best dose recommendation rules currently implemented in \texttt{crmPack}.

\begin{figure}
\centering
\begin{tikzpicture}[node distance=0.3cm, thick, >=stealth]
\node[block] (step1) {
  NextBest};
 \node[block, below left=0.0001 cm and 4.5cm of step1] (step2a) {
 NextBestMTD};
\node[block, below left=1cm and 1.5cm of step1] (step2b) {
 NextBestNCRM};
\node[block, below left=3 cm and 0.05cm of step1] (step2c) {
  NextBestThreePlusThree};
\node[block, below left =5cm and 0.000001cm of step1] (step2d){
NextBestDualEndpoint};
\node[block, below right=5cm and 0.000001cm of step1](step2e){
NextBestTD
};
\node[block, below right=3cm and 0.1cm of step1](step2f){
NextBestTDSamples
};
\node[block, below right=1cm and 1.5cm of step1](step2g){
NextBestMaxGain
};
\node[block, below right=0.0001cm and 3.5cm of step1](step2h){
NextBestMaxGainSamples
};
  
  
\path[->] (step1) edge (step2a)
           (step1) edge (step2b)
           
           (step1) edge (step2c)
(step1) edge (step2d)
(step1) edge (step2e)
(step1) edge (step2f)
(step1) edge (step2g)
(step1) edge (step2h)
;
\end{tikzpicture}
\caption{Escalation classes structure}\label{fig:rules}
\end{figure}

All classes of escalation rules are contained in the \texttt{NextBest} class. There are two main types of escalation rules: either only the binary DLT responses are incorporated into the escalation process, or a binary DLT and a continuous efficacy/biomarker response are jointly incorporated into the escalation process.


There are two implemented rules for toxicity endpoint CRMs inheriting from the \texttt{GeneralModel} class: \texttt{NextBestMTD}
that uses the posterior distribution of the MTD estimate (given a target
toxicity probability defining the MTD), and \texttt{NextBestNCRM} that
implements the N-CRM, using posterior probabilities of target-dosing and
overdosing at the dose grid points to recommend a next best dose.

For example, in order to use the N-CRM with a target toxicity interval from 20\%
to 35\%, and a maximum overdosing probability of 25\%, we specify:

<<ncrm-spec>>=
myNextBest <- NextBestNCRM(target=c(0.2, 0.35),
                           overdose=c(0.35, 1),
                           maxOverdoseProb=0.25)
@

Alternatively, we could use an MTD driven recommendation rule. For example, with
a target toxicity rate of 33\%, and recommending the 25\% posterior quantile of
the MTD, we specify

<<mtd-spec>>=
mtdNextBest <- NextBestMTD(target=0.33,
                           derive=
                               function(mtdSamples){
                                   quantile(mtdSamples, probs=0.25)
                               })
@
Note that the \texttt{NextBestMTD} class is quite flexible, because you can
specify a function \texttt{derive} that derives the next best dose from the
posterior MTD samples.


There are also two further next best dose recommendation rules when the model is inheriting from the  \texttt{ModelTox} class. One rule is specified when no samples for the model parameters are involved and the other one is when samples of the model parameters are generated and are incorporated into the dose-escalation procedure.

The details about these rules are as follows. First, two probabilities of the occurrence of a DLT have to be fixed. The first one is called \texttt{targetDuringTrial} which is the target probability of the occurrence of a DLT to be used during the trial. The second probability is called \texttt{targetEndOfTrial} is the target probability of the occurrence of a DLT to be used at the end of a trial. The above two targets always have to be specified. For cases when samples are involved, an additional argument has to be used, which is a function to advise what we should recommend using the samples that we have. This will be elaborated in details in the example below.

<<TD-spec>>=
TDNextBest <- NextBestTD(targetDuringTrial=0.35,
                         targetEndOfTrial=0.3)
@

In this example, we fixed the target probability of the occurrence of a DLT to be used during the trial be 0.35. This means we will allow subjects to dose levels with probability of DLT closest and less than or equal 0.35 during the trial. At the end of the trial, we will therefore recommend a dose level which is closest and with probability of DLT less than or equal to 0.3. This \texttt{NextBestTD} rule class can be only used when no samples are involved in the escalation procedure. Next we will show an example of the \texttt{NextBestTDsamples} rule class when samples are involved in the escalation process.

<<TDsamples-spec>>=
TDsamplesNextBest <- NextBestTDsamples(targetDuringTrial=0.35,
                                       targetEndOfTrial=0.3,
                                       derive=function(TDsamples){
                                         quantile(TDsamples,probs=0.3)})

@

The slot for \texttt{targetDuringTrial} and \texttt{targetEndOfTrial} are specified in the same way as in the last example given the value of 0.35 and 0.3, respectively. The \texttt{derive} slot should always be specified with a function. 
% First, define the dose level for which its probability of the occurrence of a DLT during trial called the TDtargetDuringTrial and similarly, for TDtargetEndOfTrial. With this class and samples involved, we can obtain the TDtargetDuringTrial samples and TDtargetEndOfTrial samples. 
In this example, using the function specified in the \texttt{derive} slot says that we will recommend the 30\% posterior quantiles of the samples to be the estimates for the doses corresponding to the \texttt{targetDuringTrial} and \texttt{targetEndOfTrial} doses.

During the study, in order to derive the next best dose, we supply the generic
\texttt{nextBest} function with the rule, the maximum dose, the posterior
samples, the model and the data:

<<next-best-run>>=
doseRecommendation <- nextBest(myNextBest,
                               doselimit=nextMaxDose,
                               samples=samples, model=model, data=data)
@

The result is a list with two elements: \texttt{value} contains the numeric
value of the recommended next best dose, and \texttt{plot} contains a plot that
illustrates how the next best dose was computed. In this case we used the N-CRM
rule, therefore the plot gives the target-dosing and overdosing probabilities
together with the safety bar of 25\%, the maximum dose and the final
recommendation (the red triangle):

<<next-best-results, fig=TRUE>>=
doseRecommendation$value
print(doseRecommendation$plot)
@

Similarly, we can use the the generic \texttt{nextBest} function for the\texttt{NextBestTD} and \texttt{NextBestTDsamples} rules. In the example below we will use the data set \texttt{data3} with DLT observations. We can compute the next best dose to be given to the next cohort using the posterior modal estimates of the DLT model (i.e., no MCMC sampling involved here):

<<next-best-TD-run>>=
doseRecDLT <- nextBest(TDNextBest,doselimit=300,model=newDLTmodel,data=data3)
@

A list of numerical values and a plot showing how the next best dose was computed will be given. This list of results will provide the numerical values for the next dose level, \texttt{nextdose}; the target probability of DLT used during the trial, \texttt{targetDuringTrial}; the estimated dose level for which its probability of DLT equals the target probability used during the trial, \texttt{TDtargetDuringTrial}; the target probability of DLT used at the end of a trial, \texttt{targetEndOfTrial}; the estimated dose level for which its probability of DLT equals the target probability of DLT used at the end of a trial. \texttt{TDtargetEndOfTrial}; and the dose level at dose grid closest and less than the \texttt{TDtargetEndOfTrial}, \texttt{TDtargetEndOfTrialdoseGrid}. We can use the \verb|$| operator to obtain these values and the plot from the list. For example,

<<next-bestTD-results,fig=TRUE>>=
doseRecDLT$nextdose
doseRecDLT$targetDuringTrial
doseRecDLT$TDtargetDuringTrial
print(doseRecDLT$plot)
@


We can see that the next dose suggested to be given to the next cohort of subjects is 50 mg. The target probability of DLT during the trial is 0.35 and the TD35 (the tolerated dose with probability of DLT equal to 0.35) is estimated to be 52.28 mg. As we are using 12 dose levels or dose grids from 25 mg to 300 mg with increments of 25 mg for this data set, \texttt{data3}, we can see that what is suggested for the next dose 50 mg is also the dose level closest below 52.28 mg, the estimated \texttt{TDtargetDuringTrial}. Similarly, at the end of a trial we could also obtain all "End Of Trial" estimates by using the \verb|$| operator. In addition, we also have a plot to show next dose allocation. The red curve shows the estimated DLT curve obtained using the posterior modal estimates of the model parameters. We also assumed the maximum allowable dose be 300 mg which was specified as the \texttt{doselimit} parameter of the \texttt{nextBest} function call and the red vertical line denoted with "Max" shows the maximum dose level (at x-axis) that is allowed in this case. The vertical purple line denoted with "Next" marks the dose level to be allocated to the next cohort of subjects. In this example, the target probability of DLT used during trial and at the end of a trial were 0.35 and 0.3, respectively. The circle and the square on the DLT curve show  where the probability of DLT is estimated to be equal to 0.3 and 0.35, respectively. Hence, the value of the estimated TD30 and TD35 can be checked at the x-axis vertically below these symbols.

When MCMC sampling is involved, we will use the samples of model parameters to choose the next best dose. For example, in the following code chunk we use the data set, \texttt{data3}, with some DLT observations and the posterior samples of the model parameters, \texttt{DLTpostsamples} to compute the next best dose:
<<next-bestTDsamples-run>>=
doseRecDLTSamples <- nextBest(TDsamplesNextBest,doselimit=300,
                              samples=DLTpostsamples,model=newDLTmodel,
                              data=data3)
@

The same list of results will be produced as in the example before: The values of the \texttt{nextdose}, \texttt{targetDuringTrial}, \texttt{TDtargetDuringTrial}, \texttt{targetEndOfTrial}, \texttt{TDtargetEndOfTrial} and \texttt{TDtargetEndOfTrialdoseGrid} can be obtained using the \verb|$| operator. The only difference is that the plot in this example will look slightly different than in the previous example:

<<next-bestTDsamples-results, fig=TRUE>>=
print(doseRecDLTSamples$plot)
@

In the plot, vertical lines are given to show the value for the next dose, the TD30 estimate, the TD35 estimate and the maximum allowable dose level. Since samples of model parameters were utilized, the density curves of the TD30 (pink) and the TD35 (grey) are plotted.
% , the area bounded under the grey curves shows the density of the TD35 samples while the area under the pink curve shows the density of the TD30 samples.



\subsection{Cohort size rules}
\label{sec:cohort-rules}

Figure \ref{fig:CohortSize} shows the inheritance structure of the classes inheriting from \texttt{CohortSize}, which are used for implementing the cohort size rules of the dose escalation designs:

\begin{figure}
\centering
\begin{tikzpicture}[node distance=0.3cm, thick, >=stealth]
\node[block] (step1) {
  CohortSize};
 \node[block, below left=0.0001 cm and 3.5cm of step1] (step2a) {
 CohortSizeRange};
\node[block, below left=1cm and  2cm of step1] (step2b) {
 CohortSizeDLT};
 \node[block, below left=2cm and 0.5cm of step1](step2c){
 CohortSizeConst
 };
  \node[block, below right=2cm and 0.5cm of step1](step2d){
 CohortSizeParts
 };
  \node[block, below right=1cm and 2cm of step1](step2e){
 CohortSizeMax
 };
  \node[block, below right=0.0001cm and 3.5cm of step1](step2f){
 CohortSizeMin
 };
\path[->] (step1) edge (step2a)
           (step1) edge (step2b)
           
           (step1) edge (step2c)
           (step1) edge (step2d)
           (step1) edge (step2e)
           (step1) edge (step2f)

;
\end{tikzpicture}
\caption{CohortSize classes structure}\label{fig:CohortSize}
\end{figure}

% All classes related to cohort size in this package are contains within \texttt{CohortSize} class.

Similarly to the increments rules, you can define intervals in the dose space
and/or the DLT space to define the size of the cohorts. For example, let's
assume we want to have one patient only in the cohorts until we reach 30 mg or
the first DLT is encountered, and then proceed with three patients per cohort.

We start by creating the two separate rules, first for the dose range:

<<size-range>>=
mySize1 <- CohortSizeRange(intervals=c(0, 30),
                           cohortSize=c(1, 3))
@

Then for the DLT range:

<<size-dlt>>=
mySize2 <- CohortSizeDLT(DLTintervals=c(0, 1),
                         cohortSize=c(1, 3))
@

Finally we combine the two rules by taking the maximum number of patients of
both rules:

<<size-combined>>=
mySize <- maxSize(mySize1, mySize2)
@

The \texttt{CohortSize} rule is used by the \texttt{size} function, together
with the next dose and the current data, in order to determine the size of the
next cohort:

<<size-eval>>=
size(mySize,
     dose=doseRecommendation$value,
     data=data)
@

Here, because we have one DLT already, we would go for 3 patients for the next cohort.

Moreover, if you would like to have a constant cohort size, you can use the
following \texttt{CohortSizeConst} class, which we will use (with three
patients) for simplicity for the remainder of this vignette:

<<size-const>>=
mySize <- CohortSizeConst(size=3)
@


\subsection{Stopping rules}
\label{sec:stopping-rules}

%Figure \ref{fig:stopping} will show the structure of the \texttt{Stopping} rules classes

All of the stopping rules classes inherit directly from the \texttt{Stopping} class. There are in total 11 stopping rules, listed as follows:
% \begin{minipage}{0.45\textwidth}
\begin{itemize}
\item \texttt{StoppingCohortNearDose}
\item \texttt{StoppingPatientsNearDose}
\item \texttt{StoppingMinCohorts}
\item \texttt{StoppingMinPatients}
\item \texttt{StoppingTargetProb}
\item \texttt{StoppingMTDdistribution}
% \end{itemize}
% \end{minipage}
% \begin{minipage}{0.45\textwidth}
% \begin{itemize}
\item \texttt{StoppingTargetBiomarker}
\item \texttt{StoppingTDCIRatio}

\item \texttt{StoppingGstarCIRatio}

\end{itemize}
% \end{minipage}

From the names of these stopping rules, we can have an idea of what criteria have been used for stopping decisions and we will explain briefly here what are these criteria. For further details please refer to examples presented later in this vignette or examples given in the help pages. You can find a link to all implemented stopping rule parts in the help page \texttt{Stopping-class}.


For example, \texttt{StoppingCohortNearDose} class objects can be used to stop the dose escalation based on the numbers of cohorts treated near to the next best dose (where the required proximity is given as the percentage of relative deviation from the next best dose). Similarly, for \texttt{StoppingPatientsNearDose}, stopping is based on the number of patients treated near the next best dose. \texttt{StoppingMinCohorts} and \texttt{StoppingMinPatients} rules can be used to stop the dose escalation if a minimum overall number of patients or cohorts have been enrolled. We have also other stopping rules such that a trial will be stopped either based on the MTD distribution (\texttt{StoppingMTDdistribution}), or reached a pre-specified probability of the next dose being in the target toxicity interval (\texttt{StoppingTargetProb}) or or target biomarker interval (\texttt{StoppingTargetBiomarker}) or when the current estimate of the quantity of interest is 'accurate' enough (\texttt{StoppingTDCIRatio} and  \texttt{StoppingGstarCIRatio})

Stopping rules are often quite complex, because they are built from "and/or" combinations of multiple parts. Therefore the \texttt{crmPack} implementation mirrors this, and
multiple atomic stopping rules can be combined easily. For example, let's assume
we would like to stop the trial if there are at least 3 cohorts and at least
50\% probability in the target toxicity interval $(20\%, 35\%)$, or the maximum
sample size of 20 patients has been reached. 

Then we start by creating the three
pieces the rule is composed of:

<<rules-bits>>=
myStopping1 <- StoppingMinCohorts(nCohorts=3)
myStopping2 <- StoppingTargetProb(target=c(0.2, 0.35),
                                  prob=0.5)
myStopping3 <- StoppingMinPatients(nPatients=20)
@

Finally we combine these with the ``and'' operator \texttt{\&} and the ``or'' operator \verb-|-:

<<rules-compose>>=
myStopping <- (myStopping1 & myStopping2) | myStopping3
@

We can also stop the trial when the current estimate of the quantity of interest, such as the TD30 given in earlier examples, is 'accurate' enough.
The accuracy of the current estimate of TD30 is quantified by the width of the associated 95\% credibility interval. The wider the interval, the less accurate the estimate is. In particular, the ratio of the upper to the lower limit of this 95\% credibility interval is used. The smaller the ratio, the more accurate is the estimate. 

For example, we will stop our trial if we obtain a ratio of less than 5 for the 95\% credibility interval of the TD30 estimate in this case, deciding that we have obtained an estimate which is 'accurate' enough. 
The \texttt{StoppingTDCIRatio} function can be  used in both cases when no DLT samples or DLT samples are involved:
<<rules2-bits>>=
myStopping4 <- StoppingTDCIRatio(targetRatio=5,
                                        targetEndOfTrial=0.3)

@

% In the above two examples, the \texttt{targetRatio} and \texttt{targetEndOfTrial} has to be specified. 

During the dose escalation study, any (atomic or combined) stopping rule can be used by the function \texttt{stopTrial} to determine if the rule has already been fulfilled. For example in our case:

<<rules-try>>=
stopTrial(stopping=myStopping, dose=doseRecommendation$value,
          samples=samples, model=model, data=data)
@


We receive here \texttt{FALSE}, which means that the stopping rule criteria have
not been met. The attribute \texttt{message} contains the textual results of the
atomic parts of the stopping rule. Here we can read that the probability for
target toxicity was just 30\% for the recommended dose 20 mg and therefore too
low, and also the maximum sample size has not been reached, therefore the trial
shall continue.

In the same way the stopping rule \texttt{myStopping4} (no samples and with samples) can be evaluated:
<<rules2-try>>=
stopTrial(stopping= myStopping4, dose=doseRecDLTSamples$nextdose,
          samples=DLTpostsamples,model=newDLTmodel,data=data3)

stopTrial(stopping= myStopping4, dose=doseRecDLT$nextdose,
          model=newDLTmodel,data=data3)
@
% when DLT samples or no DLT samples are involved. 
Note that at the moment the "and" operator \texttt{\&} and the ``or''
operator \verb-|- cannot be used together with \texttt{StoppingTDCIRatio} class objects. This is still under development.




\section{Simulations}
\label{sec:simulations}

In order to run simulations, we first have to build a specific design, that
comprises a model, the escalation rules, starting data, a cohort size and a starting dose.


The structure of the design classes in this package is shown in figure \ref{fig:Design}.

\begin{figure}
\centering
\begin{tikzpicture}[node distance=0.3cm, thick, >=stealth]
\node[block] (step1) {
  RuleDesign};
 \node[block, below left=0.5 cm and 2cm of step1] (step2a) {
 Design};
 \node[block, below =1cm of step2a] (step3a) {
 DualDesign};
\node[block, below =1cm of step1] (step2b) {
 TDDesign};
 \node[block, below=1cm of step2b](step3b){
 DualResponsesDesign
 };
  \node[block, below right=0.5cm and 2cm of step1](step2c){
 TDsamplesDesign
 };
  \node[block, below=1cm of step2c](step3c){
 DualResponsesSamplesDesign
 };

\path[->] (step1) edge (step2a)
           (step1) edge (step2b)
           
           (step1) edge (step2c)
           (step2a) edge (step3a)
           (step2b) edge (step3b)
           (step2c) edge (step3c)

;
\end{tikzpicture}
\caption{Design classes structure}\label{fig:Design}
\end{figure}


It might seem strange at first
sight that we have to supply starting data to the design, but we will show below
that this makes sense. First, we use our \texttt{emptydata} object that only
contains the dose grid, and a cohorts of 3 patients, starting from 0.1 mg:

<<design-setup>>=
design <- Design(model=model,
                 nextBest=myNextBest,
                 stopping=myStopping,
                 increments=myIncrements,
                 cohortSize=mySize,
                 data=emptydata,
                 startingDose=3)
@


Another example will be given when the \texttt{TDDesign} class is used. The empty data set, \texttt{data1} will be used, and the starting dose will be 25 mg. The code below will be a design defined when no MCMC sampling is involved. The \texttt{nextBest} slot under this \texttt{TDDesign} class function has to be defined with the \texttt{TDNextBest} class object to ensure we will pick the next best dose using rules as defined when no MCMC sampling is involved. In addition, we define here with \texttt{myStopping4} that the trial will only stop when the ratio of the 95\% credibility interval limits of the current estimate of TD30 (TDtargetEndOfTrial) is less than or equal to 5. In addition, we also use \texttt{myIncrements1}, \texttt{mySize} and \texttt{data1} defined in earlier examples for the \texttt{increments}, \texttt{cohortSize} and  \texttt{data} slots in defining the \texttt{TDDesign} object:

<<TDDesign-setup>>=
DLTdesign <-TDDesign(model=DLTmodel,
                     nextBest=TDNextBest,
                     stopping=myStopping4,
                     increments=myIncrements1,
                     cohortSize=mySize,
                     data=data1,
                     startingDose=25)
@

When MCMC samples are involved, we also have to specify a design to ensure our package will run the simulations using the MCMC samples of the model parameters for models specified under the \texttt{ModelPseudo} class object. In the example, the \texttt{TDsamplesDesign} class object has to be used with the \texttt{TDsamplesNextBest} class object in the \texttt{nextBest} slot to ensure MCMC sampling is involved for this design. We also apply the stopping rule \texttt{myStopping4} or \texttt{myStopping3} such that the trial will stop either when the ratio of the 95\% credibility interval limits of the current estimate of TD30 (TDtargetEndOfTrial) is less than or equal to 5 (\texttt{myStopping4}) or when a maximum of 30 patients has been enrolled in the trial (\texttt{myStopping3}):

<<TDsamplesDesign-set-up>>=
DLTsamplesDesign <- TDsamplesDesign(model=DLTmodel,
                                    nextBest=TDsamplesNextBest,
                                    stopping=(myStopping4|myStopping3),
                                    increments = myIncrements1,
                                    cohortSize=mySize,
                                    data=data1,
                                    startingDose=25)
@

\subsection{Examining single trial behavior}
\label{sec:examine-single-trial}

Before looking at the ``many trials'' operating characteristics, it is important
to look at the ``single trial'' operating characteristics of the dose escalation
design. For this, \texttt{crmPack} provides the function \texttt{examine}, which
generates a dataframe showing the beginning of several hypothetical trial
courses under the design. Assuming no DLTs have been seen until a certain dose,
then the consequences of different number of DLTs being observed at this dose
are shown. In the current example we have

<<design-examine>>=
set.seed(23)
examine(design)
@

Note that it is important to set a seed, since minor changes might occur due to
sampling variations. However, the \texttt{mcmcOptions} parameter should be
chosen in order to minimize such variation. The default setting, used implicitly
in the above call, should normally be sufficient, but checking this (by running the function twice with different seeds and comparing the results) is important.

The resulting data frame gives the dose of the cohort until which no DLTs are
observed, the number of DLTs, the resulting next dose recommendation, whether
the design would stop, and the relative increment of the next dose compared to
the current dose in percentage. Note that cohort size rules are taken into
account by \texttt{examine}. \texttt{NA} entries mean that the design would stop
without a valid dose, since all doses are considered too toxic after observing
the number of DLTs at that dose.

\subsection{Simulating from a true scenario}
\label{sec:simulating-from-true}

For the ``many trials'' operating characteristics, we have to define a true
scenario, from which the data should arise. In this case, this only requires a
function that computes the probability of DLT given a dose. Here we use a
specific case of the function contained in the model space:

<<true-def, fig=TRUE>>=
## define the true function
myTruth <- function(dose)
{
    model@prob(dose, alpha0=7, alpha1=8)
}

## plot it in the range of the dose grid
curve(myTruth(x), from=0, to=80, ylim=c(0, 1))
@


In a similar way, we can also simulate trials based on a true DLT scenario using the \texttt{TDDesign} and the \texttt{TDsamplesDesign}.
First, we will specified the true DLT scenario such that
<<trueDLT, fig=TRUE>>=
## define the true function
TrueDLT <- function(dose)
{
    DLTmodel@prob(dose, phi1=-53.66584, phi2=10.50499)
}

## plot it in the range of the dose grid
curve(TrueDLT, from=25, to=300, ylim=c(0, 1))
@

This true DLT scenario will be used for both the \texttt{TDDesign} and the \texttt{TDsamplesDesign}

Now we can proceed to the simulations. We only generate 10 trial outcomes here
for illustration, for the actual study this should be increased of course to at
least 500:

<<run-sims>>=
time <- system.time(mySims <- simulate(design,
                                       args=NULL,
                                       truth=myTruth,
                                       nsim=10, # increase this in real application
                                       seed=819,
                                       mcmcOptions=options,
                                       parallel=FALSE))[3]
time
@

We have wrapped the call to \texttt{simulate} in a \texttt{system.time} to
obtain the required time for the simulations (about \Sexpr{round(time)}~seconds
in this case). The argument \texttt{args} could contain additional arguments for
the \texttt{truth} function, which we did not require here and therefore let it
at the default \texttt{NULL}. We specify the number of simulations with
\texttt{nsim} and the random number generator seed with \texttt{seed}. Note that
we also pass again the MCMC options object, because during the trial simulations
the MCMC routines are used. Finally, the argument \texttt{parallel} can be used
to enable the use of all processors of the computer for running the simulations
in parallel. This can yield a meaningful speedup, especially for larger number
of simulations.

As (almost) always, the result of this call is again an object with a class, in
this case \texttt{Simulations}:

<<sim-class>>=
class(mySims)
@

From the help page

<<sim-help>>=
help("Simulations-class", help="html")
@

we see that this class is a subclass of the ``GeneralSimulations'' class. By
looking at the help pages for ``Simulations'' and the parent class
``GeneralSimulations'', we can find the description of all slots of
\texttt{mySims}. In particular, the \texttt{data} slot contains the list of
produced \texttt{Data} objects of the simulated trials. Therefore, we can plot
the course of e.g. the third simulated trial as follows:

<<third-trial, fig=TRUE>>=
print(plot(mySims@data[[3]]))
@

The final dose for this trial was

<<third-dose>>=
mySims@doses[3]
@

and the stopping reason was

<<third-stop>>=
mySims@stopReasons[[3]]
@

Furthermore, with this object, we can apply two methods. First, we can plot it,
i.e. we can apply the plot method:

<<sim-plot, fig=TRUE>>=
print(plot(mySims))
@

The resulting plot shows on the top panel a summary of the trial trajectories.
On the bottom, the proportions of doses tried, averaged over the simulated
trials, are shown. Note that you can select the plots by changing the
\texttt{type} argument of \texttt{plot}, which by default is \texttt{type =
  c("trajectory", "dosesTried")}.

Second, we can summarize the simulation results. Here again we have to supply a
true dose-toxicity function. We take the same (\texttt{myTruth}) as above:

<<sim-summary>>=
summary(mySims,
        truth=myTruth)
@

Note that sometimes the observed toxicity rate at the dose most often selected
(here 20 mg) is not available, because it can happen that no patients were
actually treated that dose during the simulations. (Here it is available.) This
illustrates that the MTD can be selected based on the evidence from the data at
other dose levels -- which is an advantage of model-based dose-escalation
designs.

Now we can also produce a plot of the summary results, which gives a bit more
detail than the textual summary we have just seen:

<<sim-sum-plot, fig=TRUE>>=
simSum <- summary(mySims,
                  truth=myTruth)
print(plot(simSum))
@

The top left panel shows the distribution of the sample size across the
simulated trials. In this case the trials had between 15 and 21 patients. The
top right panel shows the distribution of the final MTD estimate / recommended
dose across the simulated trials. The middle left panel shows the distribution
across the simulations of the DLT proportions observed in the patients dosed.
Here in most trials between 20 and 30\% of the patients had DLTs.
The middle right panel shows the distribution across simulations of the number
of patients treated above the target toxicity window (here we used the default
from 20\% to 35\%). Finally, in the bottom panel we see a comparison
of the true dose-toxicity curve (black) with the estimated dose-toxicity curves,
averaged (continuous red line) across the trials and with 95\% credible interval
across the trials. Here we see that the steep true dose-toxicity curve is not
recovered by the model fit.

If we find that e.g. the top right plot with the distribution of the final
selected doses is too small and shows not the right x-axis window, we can only
plot this one and add x-axis customization on top: (see the \texttt{ggplot2}
documentation for more information on customizing the plots)

<<sim-sum-plot2, fig=TRUE>>=
dosePlot <- plot(simSum, type="doseSelected") +
      scale_x_continuous(breaks=10:30, limits=c(10, 30))
print(dosePlot)
@

Some further examples will be given for simulations using the \texttt{TDDesign} and the \texttt{TDsamplesDesign} classes. For illustration purpose, we will generate only 10 trial outcomes.

<<DLTSim-run>>=
DLTSim <- simulate(DLTdesign,
                   args=NULL,
                   truth=TrueDLT,
                   nsim=10,
                   seed=819,
                   parallel=FALSE)
@

The above is an example when no MCMC sampling is involved and we have another example below for simulation when MCMC sampling is involved:

<<DLTsampSim-run,results=hide>>=
DLTsampSim <- simulate(DLTsamplesDesign,
                       args=NULL,
                       truth=TrueDLT,
                       nsim=10,
                       seed=819, 
                       mcmcOptions=options,
                       parallel=FALSE)
@

The meaning of these arguments are the same as those defined and explained above in the \texttt{simulate} example for the \texttt{Design} class. 
% With slots \texttt{args} to specify any additional arguments for the \texttt{truth} function, \texttt{truth} for the real DLT scenario to simulate responses from, \texttt{nsim} for the number of simulations, \texttt{seed}, the random generator seed and \texttt{parallel} to specify whether parallel computing is to be used for running the simulations. 

Similarly, the results of individual simulations can be obtained graphically using the \texttt{plot} function. 
%' Again for example we want to look at the results of the third trial. Since the \texttt{data} slot of the simulations contains a list of data obtained in the simulated trial, the plots of the simulated results of the third trial for these two simulations are given as 
%' <<plotDLTSimdata,fig=TRUE>>=
%' print(plot(DLTSim@data[[3]]))
%' @
%' 
%' <<plotDLTsampSimdata,fig=TRUE>>=
%' print(plot(DLTsampSim@data[[3]]))
%' @
%' 
The dose level for recommendation that is the dose levels closest below the final estimated TD30 (the
final estimates of the dose level with probability of DLT equals to the target end of trial) was

<<DLTSim-dose>>=
DLTSim@doses[3]
@

<<DLTsampSim-dose>>=
DLTsampSim@doses[3]
@


The overall results of the 10 trials for these two simulations can also be plotted as
<<plotDLTSim,fig=TRUE>>=
print(plot(DLTSim))
@

<<plotDLTsampSim,fig=TRUE>>=
print(plot(DLTsampSim))
@
which show the trial trajectories and the proportion of doses level tried.

These simulation results can also be summarized using the \texttt{summary} function given the truth:
<<DLTSim-summary>>=
summary(DLTSim,
        truth=TrueDLT)
@

<<DLTsampSim-summary>>=
summary(DLTsampSim,
        truth=TrueDLT)
@

Then we can also plot the summary of these two simulations using the \texttt{plot} function:
<<DLTSim-plotsummary,fig=TRUE>>=
DLTsimSum <- summary(DLTSim,
             truth=TrueDLT)
print(plot(DLTsimSum))
@

and 
<<DLTsampSim-plotsummary,fig=TRUE>>=
DLTsimsampSum <- summary(DLTsampSim,
                truth=TrueDLT)
print(plot(DLTsimsampSum))
@


\subsection{Predicting the future course of the trial}
\label{sec:pred-future-course}

By simulating parameters from their current posterior distribution instead of an
assumed true scenario, it is possible to generate trial simulations from the posterior
predictive distribution at any time point during the trial. This means that we
can predict the future course of the trial, given the current data. In our
illustrating example, this would work as follows.

The rationale of the \texttt{simulate} call is now that we specify as the
\texttt{truth} argument the \texttt{prob} function from our assumed model, which
has additional arguments (in our case \texttt{alpha0} and \texttt{alpha1}) on
top of the first argument \texttt{dose}:

<<explain-fut>>=
model@prob
@

For the simulations, these arguments are internally given by the values
contained in the data frame given to \texttt{simulate} as the \texttt{args}
argument. In our case, we want to supply the posterior samples of
\texttt{alpha0} and \texttt{alpha1} in this data frame. We take only 10 out of
the posterior samples in order to reduce the runtime for this example:

<<fut-samples>>=
postSamples <- as.data.frame(samples@data)[(1:2)*5, ]
postSamples
@

Therefore, each simulated trial will come from a posterior sample of our
estimated model, given all data so far.

Furthermore we have to make a new \texttt{Design} object that contains the
current data to start from, and the current recommended dose as the starting dose:

<<design-future>>=
nowDesign <- Design(model=model,
                    nextBest=myNextBest,
                    stopping=myStopping,
                    increments=myIncrements,
                    cohortSize=mySize,
                    ## use the current data:
                    data=data,
                    ## and the recommended dose as the starting dose:
                    startingDose=doseRecommendation$value)
@

Finally we can execute the simulations:

<<sim-future>>=
time <- system.time(futureSims <- simulate(
    ## supply the new design here
    nowDesign,
    ## the truth is the assumed prob function
    truth=model@prob,
    ## further arguments are the
    ## posterior samples
    args=postSamples,
    ## do exactly so many simulations as
    ## we have samples
    nsim=nrow(postSamples),
    seed=918,
    ## this remains the same:
    mcmcOptions=options,
    parallel=FALSE))[3]
time
@

And now, exactly in the same way as above for the operating characteristics
simulations, we can summarize the resulting predictive simulations, for example
show the predicted trajectories of doses:

<<sim-future-plot, fig=TRUE>>=
print(plot(futureSims))
@

In the summary, we do not need to look at the characteristics involving the true
dose-toxicity function, because in this case we are not intending to compare the
performance of our CRM relative to a truth:

<<sim-future-summary>>=
summary(futureSims,
        truth=myTruth)
@

We see here e.g. that the estimated number of patients overall is 19, so 11 more
than the current 8 patients are expected to be needed before finishing the
trial.

\section{Simulating 3+3 design outcomes}
\label{sec:simul-3+3-design}

While \texttt{crmPack} focuses on model-based dose-escalation designs, it also 
includes the 3+3 design in order to allow for convenient comparisons. Note that
actually no simulations would be required for the 3+3 design, because all
possible outcomes can be enumerated, however we still rely here on simulations
for consistency with the overall \texttt{crmPack} design.

The easiest way to setup a 3+3 design is the function
\texttt{ThreePlusThreeDesign}:

<<three-plus-three-setup>>=
threeDesign <- ThreePlusThreeDesign(doseGrid=c(5, 10, 15, 25, 35, 50, 80))
class(threeDesign)
@

We have used here a much coarser dose grid than for the model-based design
before, because the 3+3 design cannot jump over doses. The starting dose is
automatically chosen as the first dose from the grid. The outcome is a
\texttt{RuleDesign} object, and you have more setup options if you directly use
the \texttt{RuleDesign()} initialization function. We can then simulate trials,
again assuming that the \texttt{myTruth} function gives the true dose-toxicity
relationship:

<<three-sims>>=
threeSims <- simulate(threeDesign,
                      nsim=100,
                      seed=35,
                      truth=myTruth,
                      parallel=FALSE)
@

As before for the model-based design, we can summarize the simulations:

<<three-sims-summary>>=
threeSimsSum <- summary(threeSims,
                        truth=myTruth)
threeSimsSum
@

Here we see that \Sexpr{threeSimsSum@doseMostSelected} mg was the dose most
often selected as MTD, and this is actually too low when comparing with the
narrow target dose interval going from
\Sexpr{round(threeSimsSum@targetDoseInterval[1],1)} to
\Sexpr{round(threeSimsSum@targetDoseInterval[2],1)} mg. This is an inherent problem of
dose-escalation designs where the dose grid has to be coarse: you might not know
before starting the trial which is the range where you need a more refined dose
grid. In this case we obtain doses that are too low, as one can see from the
average true toxicity of
\Sexpr{round(mean(threeSimsSum@toxAtDosesSelected)*100)}~\% at doses selected.
Graphical summaries are again obtained by calling ``plot'' on the summary
object:

<<three-sims-plot, fig=TRUE>>=
print(plot(threeSimsSum))
@

\section{Dual-endpoint dose escalation designs}
\label{sec:dual-end}

In this section, we will look into dose-escalation procedures included in this package where two end points are incorporated into the study.
% Assume that two endpoints/responses of observations will be collected from each subject. 
The first endpoint is the binary DLT response that we discussed already in the last sections. The second endpoint is the continuous biomarker/efficacy response. In this package, we can either model these two responses jointly (using a single model class, assuming correlation) or separately (using two separate model classes, assuming no correlation).
% For the former, a correlation between the two responses can be assumed where the latter no correlation between the two responses are assumed. 
Now we will first describe how we model the two responses jointly.

\subsection{Dual-endpoint designs with a joint model}
\label{subsec:dual-endp-designs}

As a disclaimer, please note that the designs in this section are
still under development, and so far we have not yet been published. Therefore please consider them as experimental.

In the help page ``DualEndpoint-class'' the general joint model structure is
described. Basically the idea is that a (single)
biomarker variable is the second endpoint of the dose-escalation design, with
the aim to maximize the biomarker response while controlling toxicity in a safe
range. This is useful when it can not be assumed that just increasing the dose
will always lead to better efficacy.

Let's look at the data structure. Here is an example:
<<dual-data-struct>>=
data <- DataDual(
    x=
        c(0.1, 0.5, 1.5, 3, 6, 10, 10, 10,
          20, 20, 20, 40, 40, 40, 50, 50, 50),
    y=
        c(0, 0, 0, 0, 0, 0, 1, 0,
          0, 1, 1, 0, 0, 1, 0, 1, 1),
    w=
        c(0.31, 0.42, 0.59, 0.45, 0.6, 0.7, 0.55, 0.6,
          0.52, 0.54, 0.56, 0.43, 0.41, 0.39, 0.34, 0.38, 0.21),
    doseGrid=
        c(0.1, 0.5, 1.5, 3, 6,
          seq(from=10, to=80, by=2)))
@

The corresponding plot can again be obtained with:

<<dual-data-plot, fig=TRUE>>=
print(plot(data))
@

Here we see that there seems to be a maximum biomarker response at around 10~mg
already. In order to model this data, we consider a dual-endpoint model with a first-order random-walk (RW1)
structure for the dose-biomarker relationship:

<<dual-rw1-model>>=
model <- DualEndpointRW(mu=c(0, 1),
                        Sigma=matrix(c(1, 0, 0, 1), nrow=2),
                        sigma2betaW=
                        0.01,
                        sigma2W=
                        c(a=0.1, b=0.1),
                        rho=
                        c(a=1, b=1),
                        smooth="RW1")
@

We use a smoothing parameter $\sigma^{2}_{\beta_{W}} = 0.01$, an inverse-gamma
prior $\mathrm{IG}(0.1, 0.1)$ on the biomarker variance $\sigma^{2}_{W}$ and a
uniform prior (or $\mathrm{Beta}(1, 1)$ prior) on the correlation $\rho$ between
the latent DLT and the biomarker variable.

As the dual-endpoint models are more complex, it is advisable to use a
sufficiently long Markov chain for fitting them. Here we just use for illustration
purposes a quite small Markov chain -- again, for the real application, this would need to be at least 100 times longer!

<<dual-options>>=
options <- McmcOptions(burnin=10,
                       step=2,
                       samples=50)
@

Then we can obtain the MCMC samples:

<<dual-mcmc>>=
samples <- mcmc(data, model, options)
@

And we check the convergence by picking a few of the fitted biomarker means and
plotting their traceplots:

<<dual-conv, fig=TRUE>>=
data@nGrid
betaWpicks <- get(samples, "betaW", c(1, 5, 10, 25))
ggs_traceplot(betaWpicks)
@

Here all 4 $\beta_{W,j}$ ($j=1, 5, 10, 25$) means, which are the biomarker means
at the first, 5th, 10th and 25th gridpoint, respectively, seem to have
converged, as the traceplots show. (Remember that \verb|data@nGrid| gives the
number of gridpoints.) So we can plot the model fit:

\setkeys{Gin}{width=\textwidth}
<<dual-modelfit, fig=TRUE>>=
print(plot(samples, model, data, extrapolate=FALSE))
@
\setkeys{Gin}{width=0.65\textwidth}

We specify \texttt{extrapolate = FALSE} to focus the biomarker plot in the right
panel on the observed dose range, so we don't want to extrapolate the
biomarker fit to higher dose levels. We can also look at the estimated biomarker
precision $1 / \sigma^{2}_{W}$. For that we extract the precision ``precW'' and
then use another \texttt{ggmcmc} function to create the histogram:

<<dual-variance, fig=TRUE>>=
ggs_histogram(get(samples, "precW"))
@

For the selection of the next best dose, a special class
``NextBestDualEndpoint'' has been implemented. It tries to maximize the
biomarker response, under an NCRM-type safety constraint. If we want to have at
least 90\% of the maximum biomarker response, and a 25\% maximum overdose
probability for the next dose, we specify:

<<dual-nextbest>>=
myNextBest <- NextBestDualEndpoint(target=c(0.9,1),
                                   overdose=c(0.35, 1),
                                   maxOverdoseProb=0.25)
@

In our example, and assuming a dose limit of 50~mg given by the maximum
allowable increments, the next dose can then be found as follows:

<<dual-nextdose-eval>>=
nextDose <- nextBest(myNextBest,
                     doselimit=50,
                     samples=samples,
                     model=model,
                     data=data)
nextDose$value
@

A corresponding plot can be produced by printing the ``plot'' element of the
returned list:

<<dual-nextdose-plot>>=
print(nextDose$plot)
@

Here the bottom panel shows (as for the NCRM) the overdose probability, and we
see that doses above 6~mg are too toxic. In the top panel, we see the
probability for each dose to reach at least 90\% of the maximum biomarker
response in the dose grid --- this is here our target probability. While the
numbers are low, we clearly see that there is a local maximum at 10~mg of the
target probability, confirming what we have seen in the previous data and
model fit plots.

A corresponding stopping rule exists. When we have a certain probability to be
above a relative biomarker target, then the ``StoppingTargetBiomarker'' rule
gives back \texttt{TRUE} when queried if it has been fulfilled by the
\texttt{stopTrial} function. For example, if we require at least 50\%
probability to be above 90\% biomarker response, we specify:

<<dual-stop>>=
myStopping6 <- StoppingTargetBiomarker(target=c(0.9,1),
                                       prob=0.5)
@

In this case, the rule has not been fulfilled yet, as we see here:

<<dual-stop-try>>=
stopTrial(myStopping6, dose=nextDose$value,
          samples, model, data)
@

Again, this dual-endpoint specific rule can be combined as required with any
other stopping rule. For example, we could combine it with a maximum sample size
of 40~patients:

<<dual-stop-whole>>=
myStopping <- myStopping6 | StoppingMinPatients(40)
@

If one or both of the stopping rules are fulfilled, then the trial is stopped.

Let's try to build a corresponding dual-endpoint design. We start with an empty
data set, and use the relative increments rule defined in a previous section and
use a constant cohort size of 3 patients:

<<dual-design>>=
emptydata <- DataDual(doseGrid=data@doseGrid)
design <- DualDesign(model=model,
                     data=emptydata,
                     nextBest=myNextBest,
                     stopping=myStopping,
                     increments=myIncrements,
                     cohortSize=CohortSizeConst(3),
                     startingDose=6)
@

In order to study operating characteristics, we need to determine true biomarker
and DLT probability functions. Here we are going to use a biomarker function
from the beta family. Note that there is a corresponding ``DualEndpointBeta''
model class, that allows to have dual-endpoint designs with the beta biomarker
response function. Have a look at the corresponding help page for more
information on that. But let's come back to our scenario definition:

<<dual-scenario>>=
betaMod <- function (dose, e0, eMax, delta1, delta2, scal)
{
    maxDens <- (delta1^delta1) * (delta2^delta2)/((delta1 + delta2)^(delta1 + delta2))
    dose <- dose/scal
    e0 + eMax/maxDens * (dose^delta1) * (1 - dose)^delta2
}
trueBiomarker <- function(dose)
{
    betaMod(dose, e0=0.2, eMax=0.6, delta1=5, delta2=5 * 0.5 / 0.5, scal=100)
}
trueTox <- function(dose)
{
    pnorm((dose-60)/10)
}
@

We can draw the corresponding curves:

<<dual-sc-plot, fig=TRUE>>=
par(mfrow=c(1, 2))
curve(trueTox(x), from=0, to=80)
curve(trueBiomarker(x), from=0, to=80)
@

So the biomarker response peaks at 50~mg, where the toxicity is still low. After
deciding for a true correlation of $\rho=0$ and a true biomarker variance of
$\sigma^{2}_{W} = 0.01$ (giving a high signal-to-noise ratio), we can start
simulating trials (starting each with 6~mg):

<<dual-sims>>=
mySims <- simulate(design,
                   trueTox=trueTox,
                   trueBiomarker=trueBiomarker,
                   sigma2W=0.01,
                   rho=0,
                   nsim=10,
                   parallel=FALSE,
                   seed=3,
                   startingDose=6,
                   mcmcOptions =
                       McmcOptions(burnin=1000,
                                   step=1,
                                   samples=3000))
@

Note that we are having a ``small'' MCMC option set here, in order to reduce
simulation time --- for the real application, this should be ``larger''.

Plotting the result gives not only an overview of the final dose recommendations
and trial trajectories, but also a summary of the biomarker variance and
correlation estimates in the simulations:

<<dual-sims-plot, fig=TRUE>>=
print(plot(mySims))
@

Finally, a summary of the simulations can be obtained with the corresponding function:

<<dual-sims-sum>>=
sumOut <- summary(mySims,
                  trueTox=trueTox,
                  trueBiomarker=trueBiomarker)
sumOut
@

We see here that all trials proceeded until the maximum sample size of 40
patients (reaching 42 because of the cohort size 3). The doses selected are
lower than the toxicity target range, because here we were aiming for a
biomarker target instead, and the true biomarker response peaked at 50~mg.

The corresponding plot looks as follows:

<<dual-sim-sum-plot, fig=TRUE>>=
print(plot(sumOut))
@

We see that the average biomarker fit is not too bad in the range up to 50~mg,
but the toxicity curve fit is bad --- probably a result of the very low
frequency of DLTs. 

Again the warning here: the dual-endpoint designs are still
experimental!


% Next we will talk about dose escalation designs where the two endpoints are modelled separately. 

\subsection{Dual-endpoint designs with separate models}
\label{subsec:dualresponses}

In this subsection, we will look into the dose-escalation designs where we model the binary DLT responses and the continuous biomarker/efficacy responses separately. Here we hence assume that there is no correlation between the binary DLT and continuous efficacy responses.

First, we have to define the data sets for the dual responses using the \texttt{DataDual} function just like in the example given in the last subsection.

<<dualdata>>=
data2<-DataDual(doseGrid=seq(25,300,25))

data4<-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))

@
In this example, \texttt{data2} is the empty data set where 12 dose levels, from 25 to 300 mg with increments of 25 mg are used. The variable \texttt{data4} contains the data set with both the binary DLT and continuous efficacy responses observations. The elements in the slot \texttt{x} are the dose levels where 8 subjects are treated. The elements in slot \texttt{y} represent the corresponding binary DLT responses observed for these 8 subjects and the elements in slot \texttt{w} represent the continuous efficacy responses obtained for the 8 subjects.

Similarly, we can also obtain a plot of the data sets using the \texttt{plot} function as described in the last subsection.

As described, we will model the two responses separately. In order to do so, we will only use models inheriting from the \texttt{ModelPseudo} class. 

For the binary DLT responses, we can use any of the models inheriting from the \texttt{ModelTox} class. In the example below we will use models inheriting from the \texttt{LogisticIndepBeta} class which is the variable \texttt{DLTmodel} (or \texttt{newDLTmodel} with observations) given in previous examples.

For the continuous efficacy responses, we can use any models inheriting from the \texttt{ModelEff} class. In the current version of the package, there are two model classes, the \texttt{Effloglog} and the \texttt{EffFlexi} model, inheriting from the \texttt{ModelEff} class. Since \texttt{ModelEff} is also inheriting from the \texttt{ModelPseudo} class, the prior for this efficacy model also needs to be specified in the form of pseudo data. (Please refer to \ref{fig:model} for the structure of model classes defined in this package.)


The following commands show how we set up the \texttt{Effloglog} model. This is an efficacy model to describe the relationship between the efficacy responses to their corresponding dose levels on a double logarithmic ("log-log") scale. This refers to a linear model with three unknown parameters: the intercept $\theta_1$, the slope $\theta_2$ and the precision $\nu$ (inverse of the variance) of the efficacy responses. 
Similarly to other pseudo models, the data set has to be specified before setting up the model:

<<modelEff>>=
Effmodel<-Effloglog(Eff=c(1.223,2.513),Effdose=c(25,300),nu=c(a=1,b=0.025),data=data2)
@

For the specification of the prior pseudo data, two dose levels 25 and 300 mg are fixed and specified in the \texttt{Effdose} slot. After eliciting the prior expected efficacy values at these two dose levels (e.g. by asking for experts'), these are specified in the \texttt{Eff}  slot. Here for example, 1.223 is the expected efficacy value for subjects treated at dose 25 mg and 2.513 is the expected efficacy value for subjects treated at 300 mg. The slot \texttt{nu} represents the prior precision of the efficacy responses. In this example, two positive scalars for $a$ and $b$ are specified suggesting the prior distribution of the precision is gamma with shape parameter $a=1$ and rate parameter $b=0.025$. Note here, since a gamma distribution is used as the prior distribution of $\nu$, the posterior distribution will again be a gamma distribution, because the gamma prior on the precision is conjugate to the normal likelihood. 
If a fixed value of the precision is preferred, a single positive scalar can also be specified in \texttt{nu} slot. Finally the \texttt{data} slot is specified either with an empty data set or a data set with all currently available observations.

Similarly, we can also look at the structure of the Effmodel by applying the \texttt{str} function:
<<strEff>>=
str(Effmodel)
@

There are 15 slots, which can be accessed with the \verb|@| operator. From this efficacy model, we can obtain the prior (if using an empty data set) or the posterior modal estimates of model parameters $\theta_1$ (intercept) and $\theta_2$ (slope). In addition, if a gamma prior distribution is used for $\nu$ and we have some observations (data) available, then we can obtain the updated values of the shape $a$ and the rate $b$ parameters for the gamma distribution, via the model. The joint prior and posterior density functions of $\theta_1$ and $\theta_2$ are described in details in \citep{wy15}.  


Next, we will describe an example when a flexible semi-parametric function is used to describe the relationship between the efficacy values and their corresponding dose levels. The differences of the mean efficacy responses of neighboring dose levels are modeled by the either first or second order random walk models. This flexible model aims to capture different shapes of the dose-efficacy curve. We will estimate the mean efficacy responses obtained at each of the dose levels by MCMC sampling.
% will be involved to obtain the prior and posterior distribution of all these mean efficacy responses. 

This flexible form can be specified by using the \texttt{EffFlexi} class object. The \texttt{EffFlexi} 
class is inheriting from the \texttt{ModelEff} class and its prior is also specified with pseudo data:

<<modelEffFlexi>>=
Effmodel2<- EffFlexi(Eff=c(1.223, 2.513),
                    Effdose=c(25,300),sigma2=c(a=0.1,b=0.1),
                    sigma2betaW=c(a=20,b=50),smooth="RW2",data=data2)
@

Here, similarly to above, we also fixed two dose levels 25 and 300 mg and supplied the prior expected efficacy responses 1.223 and 2.513. The variance of the efficacy responses $\sigma^2$ under this model can be specified with a single positive scalar value or two positive scalar values for the shape $a$ and the scale $b$ parameters of the inverse gamma distribution in the slot \texttt{sigma2}. In here, we specified the variance of the efficacy responses with the inverse gamma distribution with shape parameter $a=0.1$ and scale parameter $b=0.1$. Then, the variance of the random walk model $\sigma^2_{\beta_W}$ can also be specified either with a single positive scalar or two positive scalar for the parameters of the inverse gamma distribution in the slot \texttt{sigma2betaW}. In here, we specified the variance of the random walk model with the inverse gamma distribution with shape parameter $a=20$ and scale parameter $b=50$. In addition, we can also specify how we would like to smooth the mean efficacy response function. Either the first order (with \texttt{RW1}) or the second order (with \texttt{RW2}) random walk model can be used to describe the relationship between the neighbouring mean efficacy responses and is specified in the the slot \texttt{smooth}. As seen in our example, \texttt{RW2}, the second order random walk model is used. Finally, we also have to specify the data set in \texttt{data} to be used for the model which is \texttt{data2} in this example.

The structure of the \texttt{EffFlexi} model object is as follows:
<<strEffmodel2>>=
str(Effmodel2)
@

The slot and the names are shown which can be accessed with the \verb|@| operator. The value 'FALSE' of the slot \texttt{useFixed} shows that both the variance of the efficacy response \texttt{sigma2} and the variance of the random walk model \texttt{sigma2betaW} are not fixed, but estimated and assigned an inverse gamma prior distribution in this model. The slot \texttt{useRW1} also gives a 'FALSE' value which means that the second order random walk model has been used to model the smooth dose-reponse function. In addition, the (only internally required) random walk difference matrix and the rank of this matrix are also shown in the slot \texttt{RWmat} and \texttt{RWmatRank}, respectively.

As discussed, the posterior estimates for model parameters specified under \texttt{ModelPseudo} class (except the \texttt{EffFlexi} model class) can be obtained as the modal estimates or via MCMC sampling. In here, we will first show how we obtain the estimates of the parameters via MCMC sampling. (Similarly, we can also use the \texttt{mcmc} function to obtain prior and posterior samples of the \texttt{Effloglog} and the \texttt{EffFlexi} models.)
% using Effmodel or Effmodel2, respectively with data2 and options specified earlier

<<mcmc-Effsamples>>=
Effsamples <- mcmc(data=data2,model=Effmodel,options)
Effsamples2 <- mcmc(data=data2, model=Effmodel2, options)
@

<<mcmc-Effpostsamples>>=
Effpostsamples <- mcmc(data=data2,model=Effmodel,options)
Effpostsamples2 <- mcmc(data=data2, model=Effmodel2, options)
@

Under the \texttt{Effloglog} (Effmodel) model, samples of the intercept $\theta_1$, the slope $\theta_2$ of the efficacy linear log-log model and the precision $\nu$ of the efficacy responses can be obtained. For the \texttt{EffFlexi} (Effmodel2) model, the samples of the mean efficacy responses at all dose levels, the variance $\sigma^2$ (sigma2) of the efficacy responses and the variance $\sigma^2_{\beta_W}$ (sigma2betaW) of the random walk model are obtained. It is also again possible to look at the structure (\texttt{str}) and extract (\texttt{get}) and obtain plots (\texttt{\text{ggs\_traceplot}} and \texttt{\text{ggs\_autocorrelation}}) of the samples of the parameters.

If no MCMC sampling is involved, the prior or the posterior modal estimates can be obtained in the output of the models. If some observations for both responses are available, they can be put in a \texttt{DataDual} data set, and given to the \texttt{data} slot under each model. We can also do so by updating the current model with the new observations using the \texttt{update} function. Then the prior or the posterior modal estimates of the model parameters can be obtained using the \verb|@| operator of the model. For example, for the \texttt{Effloglog} class model:

<<update-Effmodel>>=
newEffmodel <- update(object=Effmodel,data=data4)
newEffmodel@theta1
newEffmodel@theta2
newEffmodel@nu
@
The posterior modal estimates of $\theta_1$ and $\theta_2$ and the updated values of parameters of the gamma distribution of $\nu$ can be read now from the output above.

Similarly we can update with new data for the \texttt{EffFlexi} class model:
<<update-Effmodel2>>=
newEffmodel2 <- update(object=Effmodel2,data=data4)
newEffmodel2@RWmat
@
% such that the updated random walk difference matrix can be obtained.

The \texttt{plot} function can also be applied to the \texttt{Effloglog} model class or the \texttt{EffFlexi} model class objects, when samples of the parameters are generated under all these models:

<<plot-samplesdata2loglog, fig=TRUE>>=
print(plot(Effpostsamples, newEffmodel,data4))
@

<<plot-samplesdata2Flexi, fig=TRUE>>=
print(plot(Effpostsamples2,newEffmodel2,data4))
@

In addition, we can also plot the fitted dose-efficacy curve using the prior or the posterior modal estimates of the model parameters when no MCMC sampling is used. For example, using \texttt{Effmodel} and data set \texttt{data2} specified earlier:

<<plot-nosamplesEffmodel, fig=TRUE>>=
print(plot(data2,Effmodel))
@


Since no samples are involved, only the curves using the prior or posterior modal estimates of the parameters are produced, and no 95\% credibility intervals are provided.


Furthermore, we can also plot the estimated DLT probability and efficacy curve side by side using the \texttt{plotDualResponses} function.
For example, using the \texttt{DLTmodel}, \texttt{Effmodel} and \texttt{data2} specified in earlier examples:

<<plotDualResponseNoSamples, fig=TRUE>>=
plotDualResponses(DLEmodel=DLTmodel,
                  Effmodel=Effmodel,data=data2)
@

When the MCMC samples are used, we have:
<<plotDualResponseSamples, fig=TRUE>>=
plotDualResponses(DLEmodel=DLTmodel,DLEsamples=DLTsamples,
                  Effmodel=Effmodel,Effsamples=Effsamples,data=data2)
@


Next we will talk about the dose escalation rules when two separate models are used for the dual responses. All \texttt{Increments}, \texttt{CohortSize} and the \texttt{Stopping} rules classes described earlier can be applied here. We will now look into additional \texttt{NextBest} and \texttt{Stopping} classes rules that we can use in this situation.

In here, the decision of choosing the next best dose for administration is based on a gain function we defined \citep{wy15}. This gain function represents a trade-off between the DLT and efficacy responses such that we will allocate the dose which gives the best trade-off between these responses. In other words, the dose which gives the maximum gain value will be the dose allocated to the next cohort of subjects. The basic ideas of this rules are as follows. 
% old text: Assume that no DLT is observed in a subject, the gain value observed at a particular dose level is obtained by multiplying the probability of no DLT observed at this dose levels and all expected efficacy responses when no DLT is observed at this particular dose level.
%new:
The gain value at a particular dose level is obtained by multiplying the probability of no DLT at this dose level and the expected efficacy response at this dose level. As the data accumulates in the trial, the estimate of the gain function will improve.
This gain function consists of two components, one part is about the DLT responses and the other about the efficacy response. It depends on the values obtained in each of the components which will affect the values of the gain.

For example, the most ideal case is that both the probability of no DLT and the expected value of the efficacy response are high. The gain value obtained will then be very high. This is the reason why the dose which gives the maximum gain value should be allocated to the next cohort of subjects.

We can plot the gain function given a DLT model specified under the \texttt{ModelTox} class and an efficacy model specified under the \texttt{ModelEff} class using the \texttt{plotGain} function. For example, using the variables \texttt{newDLTmodel}, \texttt{newEffmodel} and the data set with observations, \texttt{data4}, specified in earlier examples, we have:

<<plotGain, fig=TRUE>>=
plotGain(DLEmodel=newDLTmodel,Effmodel=newEffmodel,data=data4)
@

This is a case where no MCMC sampling is involved such that the prior and posterior modal estimates of the model parameters are used.

There are two implemented \texttt{NextBest} rules for the dual responses using the gain function: the \texttt{NextBestMaxGain} and the \texttt{NextBestMaxGainSamples} class object. The \texttt{NextBestMaxGain} is used when no MCMC sampling are involved and we will use the prior or the posterior modal estimates of the model parameters to obtain the gain values at each of the dose levels, while the \texttt{NextBestMaxGainSamples} is used when MCMC sampling is involved to obtain the posterior estimates. For example, when no MCMC sampling is involved:

<<nextbestmaxgain>>=
GainNextBest <-NextBestMaxGain(DLEDuringTrialtarget=0.35,
                               DLEEndOfTrialtarget=0.3)
@

To use the \texttt{NextBestMaxGain}, we have to specify the target probability of the occurrence of a DLT to be used during the trial or at the end of the trial. In this example, the target probability of DLT to be used during the trial and at the end of the trial are 0.35 and 0.3, respectively. Therefore, under this rule we will suggest the dose level which gives the maximum gain value and has a probability of DLT less than or equal to 0.35 to administer to the next cohort of subjects. At the end of the trial, we will recommend the dose with maximum gain value and probability of DLT less than or equal to 0.3. In order to derive the next best dose for administration, we have to use the \texttt{nextBest} function with this \texttt{NextBestMaxGain} object given the doselimit, both the DLT and the efficacy models and the data set, which includes all currently available observations:
<<nextbest-maxgain>>=
doseRecGain <- nextBest(GainNextBest,
                        doselimit=max(data4@doseGrid),
                        model=newDLTmodel,
                        Effmodel=newEffmodel,
                        data=data4)
@
The results will be a list of numerical values with a plot illustrating how the next best dose was computed. The list of numerical values include the next best dose suggested, the values of the target probabilities of DLT used during and at the end of a trial. Furthermore, the estimated doses for these two targets, as well as the "Gstar" estimated dose (the dose with gives the maximum gain value) are provided along with the corresponding dose level in the dose grid for the above three estimates. We can also get to see the plot about the next best dose recommendation using the \verb|$| operator.

<<nextbestplot-maxgain,fig=TRUE>>=
doseRecGain$plot
@


As usual, we have the solid red, blue and green lines as the curves to represent the relationship between the probability of DLT, the mean efficacy response and gain values, respectively, to their corresponding dose levels. The vertical line in purple shows the next best dose suggested for administration and the vertical brown line shows the maximum allowable dose level to be administered to the next cohort of subjects. Furthermore, the circle and the square on the DLT curve also show the current estimate of the estimated TD30 and TD35.

Next we will look at the \texttt{NextBestMaxGainSamples} class object when MCMC sampling is involved. In the following code, we specify the target probabilities of DLT used during or at the end of a trial to be 0.35 and 0.3 again, and we specify that the 30\% posterior quantile will be used as the estimate for the TD35 and TD30, while we specify the 50\% posterior quantile for the Gstar estimate:

<<nextbestmaxgainsamples>>=
GainsamplesNextBest <- NextBestMaxGainSamples(DLEDuringTrialtarget=0.35,
                                   DLEEndOfTrialtarget=0.3,
                                   TDderive=function(TDsamples){
                                     quantile(TDsamples,prob=0.3)},
                                   Gstarderive=function(Gstarsamples){
                                     quantile(Gstarsamples,prob=0.5)})
@ 
Note that the two functions, \texttt{TDderive} and \texttt{Gstarderive} have to be specified to derive the corresponding estimates from the posterior samples.
 
Again, the generic function \texttt{nextBest} will be used together with this rule object to derive the next best dose:

<<nextbest-NextBestMaxGainSamples>>=
doseRecGainSamples <- nextBest(GainsamplesNextBest,
                               doselimit=max(data4@doseGrid),
                               model=newDLTmodel,
                               samples=DLTpostsamples,
                               Effmodel=newEffmodel,
                               Effsamples=Effpostsamples,
                               data=data4)
@
The list of numerical results given in the output will be the same as those given using \texttt{NextBestMaxGain} class object which includes the next dose suggested, the current estimates of TD30, TD35 and Gstar and their corresponding dose levels at dose Grid. We can also see the plot:
<<nextbest-NextBestMaxGainSamplesplot,fig=TRUE>>=
doseRecGainSamples$plot
@

In this plot, the posterior distribution of Gstar is shown as a histogram. The vertical lines on the plot show all current estimates of TD30, TD35 and Gstar. In addition, the next dose and the maximum allowable dose are also given in blue and red lines, respectively.


Next, we will introduce some further \texttt{Stopping} rules that can be applied to the above two classes of escalation rules. 
After the escalation based on two responses and two separate pseudo DLT and efficacy models, we will select one dose, which is the minimum of the estimate of TD30 (TDtargetEndOfTrial) and the optimal gain dose (Gstar)  as the recommended dose for potential further clinical trials. The main feature of these stopping rules is that the trial could be stopped if the current estimates of this selected quantity is 'accurate' enough. In particular, we will also consider the ratio of the 95\% credibility interval bounds of its current estimate. The smaller this ratio, the more accurate is the estimate. 

For example, we would like to stop our trial if the ratio is less than or equal to 5. The functions \texttt{StoppingGstarCIRatio} is  used for this purpose: 

<<StoppingGstarCIratio>>=
myStopping7 <- StoppingGstarCIRatio(targetRatio = 5,
                                    targetEndOfTrial=0.3)

myStopping8 <- myStopping7 | StoppingMinPatients(72)

@

To note here, at the moment the class \texttt{StoppingGstarCIRatio} cannot be used together with other \texttt{Stopping} class rules with the "and" operator \texttt{\&} and the ``or'' operator \verb-|- (this is still under development).

Similarly, the \texttt{stopTrial} function can then be used in order to determine if the rule has been fulfilled:
<<StopTrial-stoppingGstar>>=

stopTrial(stopping=myStopping7,dose=doseRecGain$nextdose,model=newDLTmodel,
          data=data4, Effmodel=newEffmodel)

stopTrial(stopping=myStopping7,
          dose=doseRecGainSamples$nextdose,
          samples=DLTpostsamples,
          model=newDLTmodel,
          data=data4,
          TDderive=function(TDsamples){
            quantile(TDsamples,prob=0.3)},
          Effmodel=newEffmodel,
          Effsamples=Effpostsamples,
          Gstarderive=function(Gstarsamples){
            quantile(Gstarsamples,prob=0.5)})
@
% For cases when either no or DLT and efficacy samples are involved. 

Next, we will now look at how to construct the design objects. We will also start with an empty data set, the object \texttt{data3} of the \texttt{DataDual} class introduced in earlier examples.

There are two functions which we can used. The \texttt{DualResponsesDesign} can be used without MCMC samples, while \texttt{DualResponsesSamplesDesign} can be used when MCMC samples are involved. For example, we use the object \texttt{Effmodel} of the \texttt{Effloglog} class specified earlier as the efficacy model in the following code:

<<DualResponseDesign>>=
design1 <- DualResponsesDesign(nextBest=GainNextBest,
                               model=DLTmodel,
                               Effmodel=Effmodel,
                               data=data2,
                               stopping=myStopping7,
                               increments=myIncrements1,
                               cohortSize=mySize,
                               startingDose=25)

design2 <- DualResponsesSamplesDesign(nextBest=GainsamplesNextBest,
                                      model=DLTmodel,
                                      Effmodel=Effmodel,
                                      data=data2,
                                      stopping=myStopping8,
                                      increments=myIncrements1,
                                      cohortSize=mySize,
                                      startingDose=25)
@

We can use the function \texttt{DualResponsesSamplesDesign} to specify a design when an efficacy model is specified under the \texttt{EffFlexi} class object. For example, we use the object \texttt{Effmodel2} of the  \texttt{EffFlexi} class specified in earlier examples here:
<<DualResponsesDesign-Flexi>>=
design3 <- DualResponsesSamplesDesign(nextBest=GainsamplesNextBest,
                                      model=DLTmodel,
                                      Effmodel=Effmodel2,
                                      data=data2,
                                      stopping=myStopping8,
                                      increments=myIncrements1,
                                      cohortSize=mySize,
                                      startingDose=25)
@
 
We specified the above three designs using all previous rules for nextBest (the escalation rule), stopping, increments and cohort size. 

Next, we have to specify the scenarios for simulations. For example, for simulations using the DLT model and efficacy model from the \texttt{LogisticIndepBeta} and \texttt{Effloglog} objects, respectively, we can specify the scenario as below:


<<trueDLTtrueEff>>=
myTruthDLT<- function(dose)
{ DLTmodel@prob(dose, phi1=-53.66584, phi2=10.50499)
}

myTruthEff<- function(dose)
{Effmodel@ExpEff(dose,theta1=-4.818429,theta2=3.653058)
}

myTruthGain <- function(dose)
{return(myTruthEff(dose) * (1-myTruthDLT(dose)))}
@

The true DLT, efficacy and gain curves can be obtained. We can see the corresponding curves as
<<Truecurves, fig=TRUE>>=
TruthTD<-function(prob){DLTmodel@dose(prob, phi1=-53.66584, phi2=10.50499)}
GAIN<-function(xi){-(-4.8218429+3.653058*log(xi))/(1+exp(-53.66584+10.50499*xi))}
Txi<-(optim(1,GAIN,method="BFGS")$par)
maxg<-(optim(1,GAIN,method="BFGS")$value)
gstar<-exp(Txi)
td30<-TruthTD(0.3)
td35<-TruthTD(0.35)
DoseLevels<-seq(2,300,1)
plot(DoseLevels,myTruthDLT(DoseLevels), col='red',type='l',lwd=3,ylab='Values',
  ylim=c(0,max(1,max(myTruthEff(DoseLevels)))))
points(td30,0.3,col='violet',pch=15,cex=2)
points(td35,0.35,col='violet',pch=16,cex=2)
lines(DoseLevels,myTruthEff(DoseLevels),col='blue',type='l',lwd=3)
lines(DoseLevels,myTruthGain(DoseLevels),col='green3',type='l',lwd=3)
points(gstar,-maxg,col='green3',pch=17,cex=2)
legend('topright',bty='n',cex=1.2,c('p(DLT)=0.3','p(DLT)=0.35','Max gain','p(DLTs)',
'efficacy','gain'),text.col=c('violet','violet','green3','red','blue','green3'),
pch=c(15,16,17,NA,NA,NA),lty=c(NA,NA,NA,1,1,1),col=c('violet','violet','green3','red','blue','green3'))

@

Using all the above commands, we can obtained the DLT (red), efficacy(blue) and gain (green) curves and also their corresponding true values for the TD30 (TDtargetEndOfTrial), TD35 (TDtargetDuringTrial) and the Gstar. In addition, the above scenario for DLT and efficacy can be used for both cases (modal estimates or MCMC samples). 

If the \texttt{EffFlexi} class object will be used for the simulations. Using the same DLT scenario and a new efficacy scenario will be specified such that

<<TruecurveswithFlexi>>=
myTruthEff1<- c(-0.5478867, 0.1645417,  0.5248031,  0.7604467,  
               0.9333009  ,1.0687031,  1.1793942 , 1.2726408 , 
               1.3529598 , 1.4233411 , 1.4858613 , 1.5420182)

d1 <- data2@doseGrid
myTruthGain1 <- myTruthEff1 * (1-myTruthDLT(d1))
@


The corresponding curves can also be plotted as:

<<TruecurvesFlexi,fig=TRUE>>=
maxg1<-max(myTruthGain1)
gstar1 <- data2@doseGrid[which.max(myTruthGain1)]
DoseLevels1<-seq(1,300,1)
TruthTD<-function(prob)
  {DLTmodel@dose(prob, phi1=-53.66584, phi2=10.50499)}
td30<-TruthTD(0.3)
td35<-TruthTD(0.35)
plot(DoseLevels1,myTruthDLT(DoseLevels1), col='red',type='l',
  lwd=3,ylab='Values',ylim=c(0,max(1,max(myTruthEff1))))
points(td30,0.3,col='violet',pch=15,cex=2)
points(td35,0.35,col='violet',pch=16,cex=2)
lines(d1,myTruthEff1,col='blue',type='l',lwd=3)
lines(d1,myTruthGain1,col='green3',type='l',lwd=3)
points(gstar1,maxg1,col='green3',pch=17,cex=2)
legend('topright',bty='n',cex=1.2,c('p(DLT)=0.3','p(DLT)=0.35',
'Max gain','p(DLTs)','efficacy','gain'),text.col=c('violet','violet',
'green3','red','blue','green3'),pch=c(15,16,17,NA,NA,NA),
lty=c(NA,NA,NA,1,1,1),col=c('violet','violet','green3','red','blue','green3'))
@


Similarly, we also get the DLT, efficacy and gain values and the corresponding real values of the TD30, TD35 and Gstar.

Then after establishing the real scenarios, we can simulate the trials. First, we will look at two examples when the \texttt{Effloglog} class object is used as the efficacy model. We will show first an example when no MCMC samples are involved:
<<simulateDualResponseDesign,results=hide>>=
Sim1 <- simulate(object=design1,
                 args=NULL,
                 trueDLE=myTruthDLT,
                 trueEff=myTruthEff,
                 trueNu=1/0.025,
                 nsim=10,
                 seed=819,
                 parallel=FALSE)
@


The \texttt{simulate} function is used in all cases to simulate trials with specified scenarios. From the above, we specified the true precision (\texttt{trueNu}) of the efficacy responses be 1/0.025. In other words, we used a value of 0.025 as the true variance of the efficacy responses in this simulation. For the arguments \texttt{args},\texttt{nsim}, \texttt{seed} and \texttt{parallel}, please refer to earlier examples for details for their specification and description details. 


When MCMC samples are used, we can also specify the simulations in a similar way with an additional argument \texttt{mcmcOptions} such that we have

<<simulateDualResponsesamplesDesign,results=hide>>=
Sim2 <- simulate(object=design2,
                 args=NULL,
                 trueDLE=myTruthDLT,
                 trueEff=myTruthEff,
                 trueNu=1/0.025,
                 nsim=10,
                 seed=819,
                 mcmcOptions=options,
                 parallel=FALSE)
@


When the \texttt{EffFlexi} class object is used as the efficacy model, we will generate the simulations as follows:
<<simulateDualResponseFlexiDesign,results=hide>>=
Sim3<-simulate(object=design3,
               args=NULL,
               trueDLE=myTruthDLT,
               trueEff=myTruthEff1,
               trueSigma2=0.025,
               trueSigma2betaW=1,
               mcmcOptions=options,
               nsim=10,
               seed=819,
               parallel=FALSE)
@


For the specification of the arguments \texttt{object}, \texttt{args}, \texttt{trueDLE}, \texttt{trueEff}, \texttt{mcmcOptions},\texttt{nsim},\texttt{seed},\texttt{parallel} please refer to earlier examples for details. In addition, two arguments have to be used when the \texttt{EffFlexi} class efficacy model is used for the simulations: First,  \texttt{trueSigma2} has to be specified as the true variance of the efficacy responses and \texttt{trueSigma2betaW} as the true variance of the random walk model to be used in the simulation.

Furthermore, we can also plot, summarize and plot the summary of the simulated results using \texttt{plot} and \texttt{summary} function:

<<plotsimulationresults1,fig=TRUE>>=
plot(Sim1)
@
<<plotsimulationresults2,fig=TRUE>>=
plot(Sim2)
@
<<plotsimulationresults3,fig=TRUE>>=
plot(Sim3)
@

The plots give an overview of the final dose recommendations and trial trajectories. In addition, they also give a summary of the efficacy variance and also the random walk model variance when the \texttt{EffFlexi} class object is used for the efficacy model.

Then, the summary and the plot of the summary of the simulations can be obtained by:

<<summarysimulationresults1, fig=TRUE>>=
Sum1 <- summary(Sim1,
                trueDLE=myTruthDLT,
                trueEff=myTruthEff)
Sum1
print(plot(Sum1))
@

<<summarysimulationresults2,fig=TRUE>>=
Sum2 <- summary(Sim2,
                trueDLE=myTruthDLT,
                trueEff=myTruthEff)
Sum2
print(plot(Sum2))
@

<<summarysimulationresults3,fig=TRUE>>=
Sum3 <- summary(Sim3,
                trueDLE=myTruthDLT,
                trueEff=myTruthEff1)
Sum3
print(plot(Sum3))
@


In the first simulation, \texttt{Sim1} the trial will only stop when the ratio of the 95\% credibility interval bounds of the current estimate of the minimum between TD30(TDtargetEndOfTrial) and Gstar is less than or equal to 5. The last two simulations, \texttt{Sim2} and \texttt{Sim3}, use trials which only stop either when a maximum of 72 patients has been treated or when the ratio of the 95\% credibility interval is less than or equal to 5. We can see that in all of the above simulations all trials require a total of around 60 patients for a study. 

As a reminder, for the dual endpoint dose escalation design which uses two separate models to describe the dose-responses relationship, the gain function is used to determine the next best dose and the final recommended dose at the end of a trial. More specifically at the end of a trial, we will recommend the dose level closest below  which is the minimum of final estimate of the TD30 (TDtargetEndOfTrial) and Gstar.

The DLT and efficacy scenario that we used for the first, \texttt{Sim1} and the second simulations, \texttt{Sim2} are the same. The real TD30 (TDtargetEndOfTrial) is given in the summary and is 152.6125 mg and the dose level at doseGrid which is closest and below this real TD30 is 150 mg. The real Gstar is 130.0097 mg and the dose level in the dose grid closest to this Gstar is 125 mg. 

In this case, the real Gstar is less than the real TD30 and we will expect most of the recommendations to be made at a dose level close to the real Gstar. In other words, under this scenario, we will expect most of the recommendations made at 125 mg. We can see that the simulated results agrees to what we are expecting. From the summaries and the plots of the summaries, 125 mg is the dose level which is selected most often in both of these simulations.  

For the scenario of last simulation, \texttt{Sim3}, we have the same real TD30 and the real Gstar is 125 mg. Since the real TD30 is greater than the real Gstar, we will also expect recommendations should be made close to the real Gstar under this scenario. We can see that from the simulated results in the summary or the plot of summary, the procedure also recommends 125 mg most often in the simulations, which agrees with our real scenario.

Now, we will also look at the fitted dose-DLT and dose-efficacy curves obtained under all these three simulations. From the plots of summaries, 
we can see that in all cases, the fitted dose-DLT curves (solid-red curve) do not approximate very well to the real dose-DLT curve (solid-black curve). The 95\% credibility interval of the DLT curve (broken-red curves) is also given when MCMC samples are involved in the simulation. In contrast, we can see that the fitted efficacy curve (solid-blue curve) gives a very good fit to the real efficacy curve (solid-black) in all cases. The approximation to the real efficacy curve is better when the linear linear log-log model, \texttt{Effloglog} is used, compared to when the flexible form , \texttt{EffFlexi} is used. In addition, we can also see the 95\% credibility interval of the efficacy curve (broken-blue line) when MCMC sampling of the efficacy responses is involved.   



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