# In a longitudinal study, should I impute the outcome Y, measured at time 2, for individuals who were lost to follow-up?

I have repeat measures at 2 times points in a sample of people. There are 18k people at time 1, and 13k at time 2 (5000 lost to follow-up).

I want to regress an outcome Y measured at time 2 (and the outcome is not able to be measured at time 1) on set of predictors X measured at time 1. All variables have some missing data. Most of it appears relatively random, or the missingness seems well described by the observed data. However, the vast majority of the missingness in the outcome Y is due to the loss-to-follow up. I will use multiple imputation (R::mice), and will use the full dataset to impute values for X, but I have recieved 2 pieces of conflicting advice regarding the imputation of Y:

1) Impute Y from X and V (V = useful auxiliary variables) in the full sample of 18k.

2) Do not impute Y in indivividuals lost to follow-up (and thus drop them from any subsequent regression modelling).

The former makes sense because information is information, so why not use it all; But the latter makes also makes sense, in a more intuitive way - it just seems wrong to impute the outcome for 5000 people based on Y ~ X + V, to then turn around and estimate Y ~ X.

Which is (more) correct?

This previous question is useful, but doesn't directly address missingness due to loss to follow-up (though perhaps the answer is the same; I don't know).

Multiple imputation for outcome variables

• This seems contradictory to me--can you explain?: "Most of it appears relatively random, or the missingness seems well described by the observed data." – rolando2 May 26 '14 at 22:05
• Multiple imputation and most other imputation procedures require that your data be missing at random (MAR). It would necessary to understand the mechanism of attrition in your study. I would suspect that in your follow-up studies, however, your missing values are likely not MAR or MCAR. – StatsStudent Feb 3 '15 at 6:30

## 2 Answers

I think this is an instrumentation case. You want a missing X, not a missing Y.

Y~X


But X is frequently missing or mismeasured.

X~Z and Z does not impact Y- except through X.


Then you can run:

 X~Z
Y~Predicted(X)


And require some adjustment for the standard errors.

You also may want to look at the Heckmann 2 step procedure if you have a lot of sample attrition. http://en.wikipedia.org/wiki/Heckman_correction

I would argue that neither is most appropriate.

Imputation is generally not appropriate when data are not MAR or MCAR and data rarely occur that way. When imputing your $X$ values, that may be a reasonable assumption to make, but certainly not for your $Y$ data.

Dropping all of the missing data from your data causes your parameters to become biased (if the data are not MCAR, see above) and significantly reduces the precision of your estimates. This is a "complete-case" analysis and is inadvisable.

I would suggest reviewing the survival analysis methods out there. These are methods designed to analyze your data given that some of your $Y$ outcomes are unobserved due to censoring. There are models that will take this into account if you can identify which observations are censored.