# EM algorithm to impute missing value for one variable

This is from Robert Hogg's Introduction to Mathematical Statistics 6th, exercise 6.6.5. p366, It says, Suppose $X_1$, $X_2$, $X_{n1}$, are a random sample from a $N(\theta,1)$ distribution. Suppose $Z_1$, $Z_2$,... $Z_{n2}$, are missing observations. Show that the first step EM estimate is:

$\hat\theta^{(1)}=\frac{n_1\bar{x}+n_2\hat\theta^{(0)}}{n}$

where $\hat\theta^{(0)}$ is an initial estimate of $\theta$ and $n=n_1+n_2$. Note that if $\hat\theta^{(0)}=\bar{x}$ then $\hat\theta^{(k)}=\bar{x}$ for all k.

I can solve this problem and get $\hat\theta=\bar{x}$.

My question is then what are the imputed data? Should they are:

$x_1$, $x_2$, ...$x_{n1}$, $\bar{x}$, $\bar{x}$,..., $\bar{x}$ (with $n_2$ $\bar{x}$s).

In fact, this question is related a recent published paper in Lancet.

Prestmo, A., et al. (2015). "Comprehensive geriatric care for patients with hip fractures: a prospective, randomised, controlled trial." Lancet 385(9978): 1623-1633.

In the statistical analysis the authors stat that:

We used single imputation with the expectation maximation algorithm for individual missing items on questionnaires and performance tests, with scores from the same timepoint as predictors.

Should the authors tell readers that what kind of distributions they assumed for the single imputation? otherwise there can be many different results for imputed data (from different distributions) I think.

Thank you very much.

Without more details about what they assumed, you couldn't reproduce what they did, so in that sense at least (i.e. for their work to be reproducible) then you'd need to know information like the assumed model, yes.

Whether that constitutes a "should" is really dependent on what normative criteria we're applying (are we addressing the editorial standards of the journal as our criterion for 'should', for example?).

From your quote it sounds like they might have applied some kind of regression model for their imputation, but that's not very precise.

• I think "single imputation with the expectation maximation algorithm" is really confusing me. Jul 14 '15 at 3:43
• Filling in (imputing) one or more missing values once and in effect treating them as data after that (thereby ignoring their uncertainty), compare multiple imputation. Jul 14 '15 at 3:49

Yes, they should have. As Glen_b has noted, it is difficult if not impossible to actually reproduce their analysis based on the vague description of how they performed their imputation. My guess is that the distribution they used is a multivariate normal distribution, but that is just that - a guess.

However, it is absolutely something they can get away with in the field. While the paper is a few years old, Klebanoff and Cole published an analysis in the American Journal of Epidemiology looking at the data included in that journal as well as Epidemiology, International Journal of Epidemiology and Annals of Epidemiology, which might be considered some of the more methodologically sophisticated journals in the field, and found very few papers gave a full reporting of what they did. Part of their conclusion:

Authors who utilize multiple imputation or a similar method should state the fraction or number of observations deleted from the unimputed analysis because of missing data and the fraction or number recovered by imputation. The variables used to impute missing data should be stated. Revealing the set of variables upon which the missing-at-random assumption rests is akin to revealing the set of confounders upon which the assumption of no unmeasured confounding rests. We were surprised that eight of the 12 papers using multiple imputation stated the number of data sets imputed, while only seven stated the variables used to perform the imputations. Finally, we suggest that authors provide primary results from their imputed and complete case analyses, along with the corresponding confidence intervals. We understand that the few investigators routinely incorporating these procedures might view this recommendation as a “step backward” in methodological sophistication. However, we see this suggestion as lending a bridge between current and future practice.