From: "Balasubramani, G.K."
Subject: [NMusers] Missing Covariate values 
Date: Tue, October 26, 2004 8:43 am

Hello all,
I have a multivariate data with outcome is a continuous measure
and missing occurs in both outcome as well as in covariates and
the missing outcome measure  follows the assumption of MNAR. We
are trying to do the pattern mixture model approach, but I had
some difficulty of doing that, since one or more values in the
covariates are missing.  The data structure I have is of the
following four patterns:
1. Non- missing outcome(continuous measure) with all the values of the covariates are available 
2. Non-missing outcome with one or more vaues of the covariates are missing 
3. Missing outcome with  all the values of the covariates are available 
4. Missing outcome with one or more vaues of the covariates are missing. 
I modelled on non-missing outcome using the first two groups and estimated
the parameters and using these estimated parameters, I predict  the missing
outcome, but for only with all the values of the covatiate. I cannot be able
to find out the values for the missing outcome with missed covariate values. 
In this type of situation, how to model this case using the data patterns in 3
and 4. Is there any method available for the missing covarites values for imputation. 
Any suggestion.
Thanks in advance.

From: "Nick Holford"  
Subject: RE: [NMusers] Missing Covariate values 
Date: Wed, October 27, 2004 1:05 pm 


Methods for dealing with missing covariates include:
1. Naive: Impute the missing value with some central statistic such as the median
2. Multiple Imputation: Estimate the parameters of the covariate distribution e.g.
using a multivariate normal. Sample from the MVN for the missing parameters to
create datasets with imputed values. Fit several such data sets and use the average
of the resulting parameters describing the outcome data.
3. Joint Model: Model the covariates as if they were dependent variables
('outcomes'). This is essentialy equivalent to method 2 but integrates over the
covariate distribution for the missing values and does not have to explicitly impute
(i.e. make up and use a value as if it was real data). The joint model can be done
simultaneously with the outcome data or in a two stage approach with explicit
imputation from the first stage (model for the covariates alone).
Take a look at this thread which illustrates code for the joint model approach.
This is an example of it's application:
Mould DR, Holford NH, Schellens JH, Beijnen JH, Hutson PR, Rosing H, et al.
Population pharmacokinetic and adverse event analysis of topotecan in patients with
solid tumors. Clinical Pharmacology & Therapeutics. 2002;71(5):334-48

The MEM likelihood approach for modelling outcomes when they are not complete for
all subjects makes model based assumptions about the trajectory of the outcomes. If
you don't mind making some model based assumptions this is better than making absurd
assumptions such as Last Observation Carried Forward. See
Jonsson EN, Sheiner LB. More efficient clinical trials through use of scientific
model-based statistical tests. Clin Pharmacol Ther 2002;72(6):603-14 and
Mallinckrodt CH, Clark SW, Carroll RJ, Molenbergh G. Assessing response profiles
from incomplete longitudinal clinical trial data under regulatory considerations. J
Biopharm Stat 2003;13(2):179-90

You can also try modelling the missing data mechanism and test if it is Missing
Completely At Random, Missing at Random or Not Missing at Random. See Hu C, Sale ME.
A joint model for nonlinear longitudinal data with informative dropout. J
Pharmacokinet Pharmacodyn 2003;30(1):83-103. This does not help to estimate the
outcome data model parameters themselves but could be useful for simulating clinical
trials with a model for the missingness mechanism.
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand tel:+64(9)373-7599x86730 fax:373-7556