


 I was attempting to use the residual resampling approach to generate 999 bootstrap samples of alpha and beta and find their confidence intervals. However, I keep getting the error message:Error in nls(resample.mp ~ cases/(alpha + (beta * cases)), start = init.values,  :         singular gradientafter R has only produced a few bootstraps.Could anyone suggest where I am going wrong? Would greatly appreciate it!Mary manpower<-read.table("http://www.creem.st-and.ac.uk/len/classes/mt3607/data/manhours_surgical.dat", header=TRUE)#attach dataattach(manpower)B<-999#number of data pointsn<-dim(manpower)[1]#alpha level to use for the confidence limitsalpha<-0.05#matrix that's going to contain the bootstrap coefficientsboot.coef<-matrix(NA, B+1, 2)#fit the initial modelinit.values<-c(alpha=20,beta=0)model<-nls(manhours~cases/(alpha+(beta*cases)), start=init.values, trace=TRUE)pred<-predict(model)resid<-resid(model)#do the bootstrapfor (i in 1:B){  #resample the residuals  resample.index<-sample(1:n,n,replace=T)  resample.mp<-pred+resid[resample.index]  #refit the model  init.values<-c(alpha=20,beta=0)  new.model<-nls(resample.mp~cases/(alpha+(beta*cases)),start=init.values, trace=TRUE)  #extract the parameter estimates  boot.coef[i,]<-coef(new.model)}#add the original parameter estimates boot.coef[B+1,]<-coef(model)#calculate confidence intervalsfor(i in 1:2){  ci<-quantile(boot.coef[,i],probs=c(alpha/2,(1-alpha/2)))  cat("residual bootstrap confidence intervals for parameter",i,"are",ci,"\n")}

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