EM Algorithm and Pattern Mixture Modelling - Is this a correct understanding? I'm quite new to both the EM algorithm and pattern mixture models (for NMAR data). I am hoping someone can confirm my understanding is correct, and if not, let me know how to do it correctly. 
The model
The formulation is that we have a single response variable $y_{ij}$ following a standard regression
$$Y = X\beta + Z u + \epsilon_{ij}$$
with $y_{ij}$ possibly missing, but all $x_{ij}$ are observed, along with binary variable $M_{ij}$ where $M_{ij} = 1$ means $y_{ij}$ is missing. We assume $P(M_{ij} = 1) = \pi_{ij}$, and 
$$\log \left( \frac{\pi_{ij}}{1-\pi_{ij}} \right) = \mu + \gamma y_{ij} + \alpha x_{ij}$$
We assume that the joint distribution of $Y,M$ can be factored as
$$P(Y,M|X,\theta,\eta) = P(Y|X,M,\theta).P(M|X,\eta)$$
where $\theta$ and $\eta$ are disjoint parameter vectors.
Estimation
I would like to estimate both parameter vectors $\theta , \eta$ using the EM-algorithm. The advantage of the above factorisation is that, since $\theta, \eta$ are disjoint, the likelihood can be maximised seperately since the parameter vectors are disjoint.
As I understand the procedure, it goes as follows:


*

*Obtain initial estimates for $\theta_0, \eta_0$. For example, using only the observed data we might fit the models as normal

*Using each of these estimates $\theta_0, \eta_0$, we calculate the expected log-likelihoods:
$$\mathbb{E}_{\theta_0} \log L(\theta |M,X,Y_{observed})$$
$$\mathbb{E}_{\eta_0} L(\eta| M,X,Y_{observed})$$


*Maximise the above expected log-likelihoods to obtain new estimates $\theta_1, \eta_1$.

*Repeat steps 2-3 until parameter estimates have converged.
Is this correct?
 A: A couple of points:


*

*When you want to work with missing not at random (MNAR) models, it is useful to use the notation $y_{ij}^o$ and $y_{ij}^m$ to denote the observed and missing components of the complete response outcome $y_{ij}$.

*You wrote that you want to work with pattern mixture models, in which indeed the decomposition of the joint distribution of the outcome and missingness process is $$p(Y, M \mid X; \eta, \theta) = p(Y \mid M, X; \theta) \, p(M \mid X; \eta).$$ However, this is in contrast to the logistic regression you wrote, because there you postulate that the probability of a specific response $y_{ij}$ being missing depends on $y_{ij}$. This means that this logistic regression model specifies the distribution $p(M \mid Y, X; \eta)$, which is what you do in selection models not pattern mixture models.

*Pattern mixture models and selection models are more easy to conceptualize when you have dropout, and not general intermittent missing data.

*In pattern mixture models, you need to specify a model for the complete response vector $\{y_i^o, y_i^m\}$ per dropout pattern. In each pattern, you can only identify from the observed data the marginal distribution of $y_i^o$. The conditional distribution $[y_i^m \mid y_i^o]$ is unidentifiable from the data, and you only proceed by making specific assumptions for this distribution. A general principle is that these assumptions should only define $[y_i^m \mid y_i^o]$, but they should not affect the distribution $[y_i^o]$. Even though there is some debate on whether this should always be the case.

*In selection models, the likelihood is: $$\ell(\theta, \eta) = \prod_{i = 1}^n \int p(y_i^o, y_i^m; \theta) \; p(M_i \mid y_i^o, y_i^m; \eta) \; dy_i^m.$$
