The MCMC Procedure

Data are said to be MCAR if the probability of a missing value (or the failure of observing a value) does not depend on any other observations in the data set, regardless of whether they are observed or missing. That is, the observed and unobserved values are independent of each other: if is missing, it is MCAR if the probability of observing is independent of other (and other covariates ) in the data set. Under this assumption, both the observed and unobserved data are random samples of all the data; hence, fitting a model based only on the observed data does not introduce any biases. This type of analysis is called a complete-case analysis. To carry out a complete-case analysis, you must specify MISSING=CC in the PROC MCMC statement.

Data are said to be MAR if the probability of a missing value can depend on some observed quantities but does not depend on any unobserved data. For example, suppose that are completely observed for all observations and some are missing. MAR states that the probability of observing is independent of other missing (values that could have been observed) and that it depends only on (and, potentially, observed ).

The MAR assumption states that the missing are no longer random samples and that they need to be modeled (via the likelihood specification of the missing values). At the same time, the independence assumption of the missing values on the unobserved quantities states that the missing mechanism (usually an binary indicator variable such that if is missing and otherwise) can be ignored and does not need to be taken into account. Hence, MAR is sometimes referred to as ignorably missing. It is not the missing values that can be ignored, it is the missing mechanism that can be ignored.

By default, PROC MCMC treats the missing data as MAR (this assumes that you do not input a binary indicator variable and model it specifically): each missing value becomes an extra parameter and PROC MCMC updates it in every iteration. PROC MCMC assumes that both the missing values and observed values arise from the same distribution (which is specified in the MODEL statement),

where consists of observed () and missing () values, and is the likelihood function with parameters .

You can use the MODEL statement to model missing covariates. Using multiple MODEL statements enables you to specify, for example, a marginal distribution for missing values in covariate x and a conditional distribution for the response variable y given x as follows:

model x ~ normal(alpha, var=s2_x);
model y ~ normal(beta * x, var=s2_y);

In each iteration, PROC MCMC draws samples for every missing value in variable x, then every missing value in variable y, conditional on the drawn values of the x variable.

Data are said to be MNAR if the probability of a missing value depends on unobserved data (or data that could have been observed): the probability that is missing depends on the missing values of other . This is a very general scenario that assumes that the missing mechanism is no longer ignorable (it is sometimes referred to as nonignorably missing) and that a model for the missing mechanism is required in order to make correct inferences about the model parameters.

Let be the missing value indicator for , where if is missing and otherwise. This is usually part of an input data set where you preprocess the response variable and create this missing value indicator variable. Modeling MNAR data implies that you must specify a joint likelihood function over , where represents the covariates and represents the model parameters. This joint distribution can be factored in two ways: a pattern-mixture model and a selection model.

The selection model factors the joint distribution and into a marginal distribution for and a conditional distribution for ,

where , is usually a binary model with a logit or probit link that involves regression parameters , and is the sampling distribution that generates with model parameters .

The pattern-mixture model factors the opposite way, a marginal distribution for and a conditional distribution for ,

where .

You can use PROC MCMC to fit either model by specifying multiple MODEL statements: one for the marginal distribution and one for the conditional distribution. Suppose that the variable r is the missing data indicator, which is modeled using a logit model, and that the response variable y is a Poisson regression that includes the missing variable indicator as one of its covariates. The following statements are a PROC MCMC program that fits a pattern-mixture model:

pi = logistic(alpha * x1);
model r ~ binary(pi);
mu = beta0 + beta1 * x2 + beta3 * r;
model y ~ poisson(exp(mu));

The first MODEL statement uses a binary model with logit link to model the missing mechanism, and the second MODEL statement models the response variable with a Poisson regression that includes the missing value indicator as one of its covariates. Each of the two sets of regression has its covariates and regression coefficients. If this hypothetical data set contained missing values in covariates x1 and x2, you could add two more MODEL statements to handle each variable as follows:

model x1 ~ normal(mu1, var=s2_x1);
pi = logistic(alpha * x1);
model r ~ binary(pi);
model x2 ~ normal(mu2, var=s2_x2);
mu = beta0 + beta1 * x2 + beta3 * r;
model y ~ poisson(exp(mu));