options(width=80) # out.format <- knitr::opts_knit$get("out.format") # img_template <- switch( out.format, # word = list("img-params"=list(fig.width=6, # fig.height=6, # dpi=150)), # { # # default # list("img-params"=list( dpi=150, # fig.width=6, # fig.height=6, # out.width="504px", # out.height="504px")) # } ) # # knitr::opts_template$set( img_template ) knitr::opts_chunk$set(echo = F,fig.width=7,fig.height = 3,dpi=300) pF = params$pF pE = params$pE p0 = params$p0 p1 = params$p1 shape1F = params$shape1F shape2F = params$shape2F shape1E = params$shape1E shape2E = params$shape2E n.range_1 = params$n.range_1 n.range_2 = params$n.range_2 n.range_4 = params$n.range_4 cF = params$cF cE.range_1 = params$cE.range_1 cE = params$cE firstInterim.at = params$firstInterim.at furtherInterims.at = params$furtherInterims.at nfinal = params$nfinal nfinal.vec = params$nfinal.vec ptrue.range_1 = params$ptrue.range_1

- Uninteresting and target response rate: $p_0$=
`r p0`

, $p_1$=`r p1`

- Beta prior for futility criterion: Beta(
`r shape1F`

,`r shape2F`

) -
Beta prior for efficacy criterion: Beta(

`r shape1E`

,`r shape2E`

) -
First interim at:

`r firstInterim.at`

- Second and potentially further interims at (selected based on logistic considerations):
`r furtherInterims.at`

- $p_F$=
`r pF`

and $c_F$=`r cF`

, i.e. futility criterion $P(p>$`r pF`

$|Data) <$`r cF`

- $p_E$=
`r pE`

and $c_E$=`r cE`

, i.e. efficacy criterion $P(p>$`r pE`

$|Data) \geq$`r cE`

n.range_1=as.numeric(n.range_1) par(mfrow=c(1,2),cex=.7,cex.main=.9) #cex.main=.7,cex.lab=.8,cex.axis=.7, plotBDP2(x="n",y="Prob0Successes",n=c(n.range_1[1],n.range_1[2]),p0=p0,p1=p1) plotBDP2(x="n",y="PostProb0or1Successes",n=c(n.range_1[1],n.range_1[2]),pF=pF,shape1F=shape1F,shape2F=shape2F)

The left figure shows true (for $p_{true}=p_0$, in green) and false (for $p_{true}=p_1$, in red) stopping probability. False stopping probability should be bounded to an acceptable limit, e.g. 5%. Any n with false stopping probability below this limit can be used as timing of first interim. The actual choice may be guided by logistical considerations.

From left figure, false stopping probability was evaluated and first interim chosen to be performed at `r firstInterim.at`

patients.

From right figure read P(p>$p_F$| 0 of n), in red, and P(p>$p_F$| 1 of n), in green, for the selected timing of first interim, i.e. for `r firstInterim.at`

patients.
The value of $c_F$ can be chosen anywhere between these two numbers and
will ensure that the trial is stopped at first interim if no successes have been observed so far.

From right figure $c_F$ was chosen: $c_F$ = `r cF`

.

Note: If it is not intended to stop the trial only after a run of treatment failures but also in case that a low number of successes is observed, the timing of first interim and $c_F$ can be chosen on basis of the operating characteristics and expected patient number without evaluating these plots.

Timings of second and potentially further interims are selected based on logistical considerations. The choice was to perform (a) further potential interim(s) at `r furtherInterims.at`

patients.

cE.range_1=as.numeric(cE.range_1) cEaccuracy=0.001 cEvec=seq(from=cE.range_1[1],to=cE.range_1[2],by=cEaccuracy) interims.at= c(firstInterim.at,as.numeric(unlist(strsplit(furtherInterims.at," ")))) n=nfinal par(cex.lab=.9,cex.axis=.9,mar=c(5.1, 4.1, 0.5, 2.1)) res=plotBDP2(x="cE",y="PEcall",n=n,interim.at=interims.at,pF=pF,cF=cF,pE=pE,cE=cEvec,p0=p0,p1=p1, shape1F=shape1F,shape2F=shape2F,shape1E=shape1E,shape2E=shape2E,col=c("green","red"),cex.sub=.8) abline(v=cE,col="gray",lty="dashed")

The plot shows power (in red) and type I error (in green) as function of $c_E$.

Threshold $c_E$ should be selected such that type I error probability can be tolerated and power is maximized.

Selected was $c_E$ = `r cE`

. This leads to a type I error of `r round(res$y.p0[which.min(abs(res$x.p0-cE))[1]],3)`

at $p_0$=`r p0`

,
and a power of `r round(res$y.p1[which.min(abs(res$x.p1-cE))[1]],3)`

at $p_1$=`r p1`

.

n.range_2=as.numeric(n.range_2) nvec=c(n.range_2[1]:n.range_2[2]) interims.at= c(firstInterim.at,as.numeric(unlist(strsplit(furtherInterims.at," ")))) par(mfrow=c(1,2),cex.main=1,cex.sub=.9,cex=.6,cex.lab=1.1,cex.axis=1.1) plotBDP2(x="n",y="PFstopEcall",n=nvec,interim.at=interims.at,pF=pF,cF=cF,pE=pE,cE=cE,ptrue=p0, shape1F=shape1F,shape2F=shape2F,shape1E=shape1E,shape2E=shape2E, progress=T,#cex.legend=.8, main="Type I error and probability of true stopping \nfor varying n") plotBDP2(x="n",y="PFstopEcall",n=nvec,interim.at=interims.at,pF=pF,cF=cF,pE=pE,cE=cE,ptrue=p1, shape1F=shape1F,shape2F=shape2F,shape1E=shape1E,shape2E=shape2E,progress=T,#cex.legend=.8, main="Power and probability of false stopping \nfor varying n")

The left plot shows the probabilities for declaration of efficacy and cumulative stopping for futility at final for $p_{true}=p_0$. The right plot shows the same quantities for $p_{true}=p_1$.

nvec= as.numeric(unlist(strsplit(nfinal.vec," "))) pvec=seq(ptrue.range_1[1],ptrue.range_1[2],by=.01) interims.at= c(firstInterim.at,as.numeric(unlist(strsplit(furtherInterims.at," ")))) par(mfrow=c(1,2),cex=.9,mar=c(5.1, 4.1, 0.5, 2.1)) n=nvec[1] vn.int=interims.at[interims.at<n] plotBDP2(x="ptrue",y="PEcall",n=n,interim.at=vn.int,pF=pF,cF=cF,pE=pE,cE=cE,ptrue=pvec, shape1F=shape1F,shape2F=shape2F,shape1E=shape1E,shape2E=shape2E,col=1,cex.sub=.7) abline(v=p0,col="grey") abline(v=p1,col="grey") for (jj in 2:length(nvec)) { n=nvec[jj] vn.int=interims.at[interims.at<n] plotBDP2(x="ptrue",y="PEcall",n=n,interim.at=vn.int,pF=pF,cF=cF,pE=pE,cE=cE,ptrue=pvec, shape1F=shape1F,shape2F=shape2F,shape1E=shape1E,shape2E=shape2E,add=TRUE,col=jj) } legend("bottomright",title="final analysis at ",legend=nvec,text.col = 1:length(nvec),cex=.8) n=nvec[1] vn.int=interims.at[interims.at<n] plotBDP2(x="ptrue",y="ExpectedNumber",n=n,interim.at=vn.int,pF=pF,cF=cF,pE=pE,cE=cE,ptrue=pvec,shape1F=shape1F,shape2F=shape2F,col=1,ylim=c(0,max(nvec)),cex.sub=.7) abline(v=p0,col="grey") abline(v=p1,col="grey") for (jj in 2:length(nvec)) { n=nvec[jj] vn.int=interims.at[interims.at<n] plotBDP2(x="ptrue",y="ExpectedNumber",n=n,interim.at=vn.int,pF=pF,cF=cF,pE=pE,cE=cE,ptrue=pvec,shape1F=shape1F,shape2F=shape2F,col=jj,add=TRUE) } legend("bottomright",title="final analysis at ",legend=nvec,text.col = 1:length(nvec),cex=.8)

vn.int=firstInterim.at n=n.range_4[2] plotBDP2(x="n",y="bFbE",n=n,pF=pF,cF=cF,pE=pE,cE=cE, shape1F=shape1F,shape2F=shape2F,shape1E=shape1E,shape2E=shape2E,col=c("red","green"),cex.sub=.8,cex.main=.9)

The plot shows the decision boundaries for the selected trial design in terms of number of successes per number of enrolled patients.

The trial is stopped for futility if the observed number of successes among enrolled patients is in the red part of the plot, including the boundary. Efficacy can be called if the observed number of successes among patients at final analysis is in the green part of the plot, including the boundary.

The range of final sample size should be increased to the maximal feasible number to check for contradictory results, i.e. for overlap between red and green areas.

Kopp-Schneider, A., Wiesenfarth, M., Witt, R., Edelmann, D., Witt, O. and Abel, U. (2018). Monitoring futility and efficacy in phase II trials with Bayesian
posterior distributions - a calibration approach.
*Biometrical Journal*, to appear.

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