Re: missing data items

From: LSheiner <lewis@c255.ucsf.edu> Subject: Re: missing data items Date: Mon, 11 Sep 2000 16:49:00 -0700 All, Mats' suggsetion is quite close to my #2 (integrate out the missing data), which is easier to implement in NONMEM (modulo the difficulties Mats notes) than in more restricted environments. There, it requires a lot of math to figure out what the marginal likelihood is. Taking a very simple example where the primary response is a linear function of a covariate, and they were not measured at the same time (as sometimes is the case in PK/PD), the code for Mats' suggestion might look like the followIng (the data item TYPE identifies the DV as being the primary response (TYPE=1) or the covariate (TYPE=2)) $DATA ID TIME DV TYPE ;.............DV = Y WHEN TYPE=1; DV = RX WHEN TYPE = 2 $PRED ; "MODEL" THE COVARIATE XX = THETA(1) + ETA(1) YY = THETA(2) + ETA(3)*XX IF(TYPE.EQ.1) THEN Y = XX + EPS(1) ELSE Y = YY + EPS(2) ENDIF Them above is equivalent to what Nick offered, and, contrary to his note, is not equivcalent to imputing the data. It is joint modeling. Imputing means substituting somes prediction of the covariate for the covariate wherever it is missing, and then using the prediction as though it had been measured. The control file would then look just as it would if the covariate hadn't ever been missed. Although popular, this method can produce bias (as can any missing data method if the missingness mechanism is not ignorable), and will always produce somewhat wrong standard errors of parameter estimates, as one is making up some data but not acknowledging it. There is a way to use this method that avoids the latter problem, called multiple imputation. It involves 2 differences. First, the expected value of the covariate (e.g, median of observed values, or mean, or a regression prediciton) is not filled-in, but rather that value *plus a random error* with variance equal to the residual error variance of the model used to predict the covariate. Second, one does the analysis K=5 (or so) times, each with a different set of imputed values for the missing data, and computes the final parameter estimates from the set of K estimates and standard errors from the K analyses. A reference is Rubin, D. B. (1996). Multiple imputation after 18+ years. J Amer Stat Assoc 91: 473-489. The method I suggested as my #2 differs from the above slightly in that it attempts to conditionon the oberved values where they exist, and only integrate where necessary. For the same example as above, the code to do it that way, might look like this: $DATA ID TIME RX DV TYPE ;.............RX IS THE VALUE X IF MEASURED OR 999 IF NOT ;................DV = Y WHEN TYPE=1; DV = RX WHEN TYPE = 2 ;*** NOTE: AN RX ENTRY OF SOME KIND APPEARS ON ALL RECORDS, WHILE ;*** DV = RX APPEARS ONLY ON SOME RECORDS (ONE FOR EACH TIME X IS MEASURED) $PRED ; INTEGRATE OUT THE MISSING COVARIATE IF(RX.EQ.999.OR.TYPE.EQ.2) THEN XX = THETA(1) + ETA(1) ELSE XX = RX ENDIF YY = THETA(2) + ETA(3)*XX IF(TYPE.EQ.1) THEN Y = YY + EPS(1) ELSE Y = XX + EPS(2) ENDIF LBS. -- _/ _/ _/_/ _/_/_/ _/_/_/ Lewis B Sheiner, MD (lewis@c255.ucsf.edu) _/ _/ _/ _/_ _/_/ Professor: Lab. Med., Bioph. Sci., Med. _/ _/ _/ _/ _/ Box 0626, UCSF, SF, CA, 94143-0626 _/_/ _/_/ _/_/_/ _/ 415-476-1965 (v), 415-476-2796 (fax)