How to impute values below the LOD as random values in a log-normal distribution of 0-LOD using MLE

[kpberger]

Hi all. I have a data set with measured concentrations of biomarkers in serum. For values below the limit of detection (LOD), I would like to impute random values from the distribution of 0-LOD using maximum likelihood estimation. The LOD changes with each observation (based on amount of serum available). I've read some SAS blog posts on computing MLEs but this is all a bit over my head (statistics does not come easily to me!).

Here is some example data. "abovelod' is a binary indicator of if the concentration is above the LOD or not. "lod" is the LOD value for each person. "concentration" is the value I'd like to impute.

data biomarkers; input id abovelod lod concentration;
datalines;
1 1 0.5	0.6
2 0	0.6	.
3 1 0.4 1.2
4 1 0.8	0.9
5 0 0.2 .
6 0 0.7	.
7 1 0.4 1.5
8 1 0.3 0.8
9 1 0.2 1.1
10 1 0.5 0.7
;
run;
data biomarkers; set biomarkers;
ln_concentration=log(concentration);
run;

Any direction appreciated, I know this is a larger ask.

Also, I would appreciate keeping the discussion away from how to handle values below the LOD. Thank you!

Edit: Variable "concentration" can be assumed to be log normal.

[ballardw]

Since you have different limits for each subject, and at least as shown, a very small sample of LOD values I'm not sure what you would actually mean by "mle" in this case.

From the range 0 to LOD is easy:

data biomarkers; 
   input id abovelod lod concentration;
   if not(abovelod) then do;
      concentration = rand('uniform');
      if concentration > lod then do until (concentration le lod);
         concentration = rand('uniform');
      end;
   end;
datalines;
1 1 0.5	0.6
2 0 0.6	.
3 1 0.4 1.2
4 1 0.8	0.9
5 0 0.2 .
6 0 0.7	.
7 1 0.4 1.5
8 1 0.3 0.8
9 1 0.2 1.1
10 1 0.5 0.7
;

[mkeintz]

Back in the reptilian part of my brain, I thought there was more to "rand('uniform'".  I checked. And there is: you can use the 2nd and 3rd argument of "rand('uniform',...,...)" to replace the do group

   if not(abovelod) then do;

with

  id not(abovelod) then concentration=rand('uniform',0,lod);