That is what I read the other day. For calculation of descriptive statistics, values below the LLOQ (lower limit of quantification)  were set to…. Then I wondered, wasn’t there a trick in JAGS to incorporate the presence of missing data while estimating parameters of a distribution. How would that compare with standard methods such as imputing LLOQ at 0, half LLOQ, LLOQ or just ignoring the missing data? A simulation seems to be called for.

Bayes estimation

A bit of searching gave me a way function to estimate the mean and standard deviation, here wrapped in a function so resulting data has a nice shape.
library(R2jags)
library(plyr)
library(ggplot2)

Bsensnorm <- function(yy,limit) {
  isCensored <- is.na(yy)
  model1 <- function() {
    for ( i in 1:N ) {
      isCensored[i] ~ dinterval( yy[i] , limit )
      yy[i] ~ dnorm( mu , tau ) 
    }
    tau <- 1/pow(sigma,2)
    sigma ~ dunif(0,100)
    mu ~ dnorm(0,1E-6)
  }
    datain <- list(yy=yy,isCensored=1-as.numeric(isCensored),N=length(yy),limit=limit)
  params <- c('mu','sigma')
  inits <- function() {
    list(mu = rnorm(0,1),
        sigma = runif(0,1))
  }
  
  jagsfit <- jags(datain, model=model1, inits=inits, 
      parameters=params,progress.bar=”gui”,
      n.chains=1,n.iter=7000,n.burnin=2000,n.thin=2)
  data.frame( system=’Bayes Uninf’,
      mu_out=jagsfit$BUGSoutput$mean$mu,
      sigma_out=jagsfit$BUGSoutput$mean$sigma)
}

Somewhat informative prior

BsensInf <- function(yy,limit) {
  isCensored <- is.na(yy)
  model2 <- function() {
    for ( i in 1:N ) {
      isCensored[i] ~ dinterval( yy[i] , limit )
      yy[i] ~ dnorm( mu , tau ) 
    }
    tau <- 1/pow(sigma,2)
    sigma ~ dnorm(1,1) %_% T(0.001,)
    mu ~ dnorm(0,1E-6) 
  }
  datain <- list(yy=yy,isCensored=1-as.numeric(isCensored),N=length(yy),limit=limit)
  params <- c('mu','sigma')
  inits <- function() {
    list(mu = abs(rnorm(0,1))+.01,
        sigma = runif(.1,1))
  }
  
  jagsfit <- jags(datain, model=model2, inits=inits, parameters=params,progress.bar="gui",
      n.chains=1,n.iter=7000,n.burnin=2000,n.thin=2)
  data.frame( system=’Bayes Inf’,
      mu_out=jagsfit$BUGSoutput$mean$mu,
      sigma_out=jagsfit$BUGSoutput$mean$sigma)
}

Simple imputation

L0sensnorm <- function(yy,limit) {
  yy[is.na(yy)] <- 0
  data.frame(system=’as 0′,mu_out=mean(yy),sigma_out=sd(yy))
}
Lhsensnorm <- function(yy,limit) {
  yy[is.na(yy)] <- limit/2
data.frame(system=’as half LOQ’,mu_out=mean(yy),sigma_out=sd(yy))
}
Llsensnorm <- function(yy,limit) {
  yy[is.na(yy)] <- limit
  data.frame(system=’as LOQ’,mu_out=mean(yy),sigma_out=sd(yy))
}
LNAsensnorm <- function(yy,limit) {
  yy <- yy[!is.na(yy)]
  data.frame(system=’as missing’,mu_out=mean(yy),sigma_out=sd(yy))
}

Simulations

limitest <- function(yy,limit) {
  yy[yy
  do.call(rbind,list(
    Bsensnorm(yy,limit),
    BsensInf(yy,limit),
    L0sensnorm(yy,limit),
    Lhsensnorm(yy,limit),
    Llsensnorm(yy,limit), 
    LNAsensnorm(yy,limit)))
}
simtest <- function(n,muin,sdin,limit) {
  numnotna<- 0
  while(numnotna
    yy <- rnorm(n,muin,sdin)
    numnotna <- sum(yy>limit)
  }
  ll <- limitest(yy,limit)
  ll$n_na <- sum(yy
  ll
}
datain <- expand.grid(n=c(5,6,8,10,12,15,20,50,80),muin=seq(1,3.5,.05),sdin=1,limit=1)
dataout <- mdply(datain,simtest)

Results for means

After all simulations, a difference is calculated between the simulation mean and the estimated means. This made interpreting the plots much more easy.
dataout$dif <- dataout$mu_out-dataout$muin

Uninformed prior did not perform with very little data

With very little data, the standard deviation gets large and the mean is low. This may just be because there is no information stating they are not large and small respectively, and the chain just wanders of. Anyway, reason enough to find that with only few samples it is not a feasible method. Adding information about the standard deviation much improves this.  

ggplot(dataout[dataout$n %in% c(5,6,8) & dataout$system %in% c(‘Bayes Uninf’,’Bayes Inf’),],aes(x=muin,y=dif))  + #group=Source,
    geom_smooth(method=’loess’) +
    geom_point() + 
    facet_grid(system ~n  , drop=TRUE) +
    scale_x_continuous(‘simulation mu’) +
    scale_y_continuous(‘difference observed and simulation mu’) 

At low number of observations variation of the mean swamps bias.  

The variation in estimates is rather large. Even so, there is a trend visible that removing missing data and imputing LLOQ give too high estimates when mu is low and hence the number of missings is high. When mu is high then all methods obtain the same result.

ggplot(dataout[dataout$n %in% c(5,6,8) & dataout$system!=’Bayes Uninf’,],aes(x=muin,y=dif))  + #group=Source,
    geom_smooth(method=’loess’) +
    geom_point() + 
    facet_grid(system ~n  , drop=TRUE) +
    scale_x_continuous(‘simulation mu’) +
    scale_y_continuous(‘difference observed and simulation mu’) 

At intermediate number of observations method choice gets more important 

Missing and imputation of LLOQ are clearly biased. Half LLOQ and 0 do much better. At 15 observations the informed prior is fairly similar to uninformed prior. 

ggplot(dataout[dataout$n %in% c(10,12,15) & dataout$system!=’Bayes xUninf’,],aes(x=muin,y=dif))  + #group=Source,
    geom_smooth(method=’loess’) +
    geom_point() + 
    facet_grid(system ~n  , drop=TRUE) +
    scale_x_continuous(‘simulation mu’) +
    scale_y_continuous(‘difference observed and simulation mu’) 

At large number of observations bias gets far too large

At this point the missing option is not plotted any more, it is just not competitive. Setting at LLOQ gets a big positive bias, while 0 gets a negative bias, especially for 80 observations. There is no point in setting up an informed prior. Half LLOQ seems least biased out of the simple methods.

ggplot(dataout[dataout$n  %in% c(20,50,80) & dataout$system!=’as missing’,],aes(x=muin,y=dif))  + #group=Source,
    geom_smooth(method=’loess’) +
    geom_point() + 
    facet_grid(system ~n  , drop=TRUE) +
    scale_x_continuous(‘simulation mu’) +
    scale_y_continuous(‘difference observed and simulation mu’) 

Standard deviation is negative biased for simple imputation

The uninformed prior estimation is too bad to plot and retain reasonable axes (and setting limits on y means the data outside those limits get eliminated from the smoother too, so that is not an option). Especially setting missing as LLOQ or ignoring them gets too low estimates of standard deviation. The informed prior gets a bit too large standard deviation.

ggplot(dataout[dataout$n%in% c(5,6,8) & dataout$system!=’Bayes Uninf’,],aes(x=muin,y=sigma_out))  + #group=Source,
    geom_smooth(method=’loess’) +
    geom_point() + 
    facet_grid(system ~n  , drop=TRUE) +
    scale_x_continuous(‘simulation mu’) +
    scale_y_continuous(‘estimated sigma’) 

ggplot(dataout[dataout$n%in% c(10,12,15) & dataout$system!=’Bayes Uninf’,],aes(x=muin,y=sigma_out))  + #group=Source,
    geom_smooth(method=’loess’) +
    geom_point() + 
    facet_grid(system ~n  , drop=TRUE) +
    scale_x_continuous(‘simulation mu’) +
    scale_y_continuous(‘estimated sigma’) 
With a large number of observations half LLOQ results in a negative bias on the standard deviation.

ggplot(dataout[dataout$n %in% c(20,50,80) & dataout$system!=’as missing’,],aes(x=muin,y=sigma_out))  + #group=Source,
    geom_smooth(method=’loess’) +
    geom_point() + 
    facet_grid(system ~n  , drop=TRUE) +
    scale_x_continuous(‘simulation mu’) +
    scale_y_continuous(‘estimated sigma’) 

Conclusion

At low number of observations using a Bayesian model and a good prior on the standard deviation is the best way to obtain descriptive statistics. Lacking those setting
Finally, setting data to

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