mixEMM: A mixed-effects model for analyzing cluster-level...

In mixEMM: A Mixed-Effects Model for Analyzing Cluster-Level Non-Ignorable Missing Data

Description

This function fits a mixed-effects model for clustered data with cluster-level missing values in the outcome.

Usage

mixEMM(Ym, Xm, Zm, gamma, maxIter = 100, tol = 0.001)

Arguments

Ym	is an N by p outcome data from N clusters/batches/experiments; p is the number of samples within each cluster. The first sample within each cluster is assumed to be a reference sample with different error variance. Missing values are coded as NAs.
Xm	is a covariate array of dimension N by k by p, where k is the number of covariates.
Zm	is a design array for random-effects, with a dimension of N by h by p, where h is the number of variables with random effects.
gamma	is the parameter for the missing-data mechanism. The missingness of the outcome in cluster i depends on the mean of the outcome. The missing probability is modelled as exp(-gamma0 - gamma*mean(y)). The parameter gamma can be estimated by borrowing information across outcomes and finding the common missing-data patterns in the high-dimensional data. For example, by estimating the relationship the observed average value of \bar\mathbf{y}_{i} and the missing rate, or the parameter can be selected by the log-likelihood profile (see the Reference). If gamma = 0, the missingness is ignorable. The parameter gamma0 does not affect the estimation of the EM algorithm, and is mostly determined by the missing rate. So it is set as 0 in the estimation here.
maxIter	the maximum number of iterations in the estimation of the EM algorithm.
tol	the tolerance level for the absolute change in the observed-data log-likelihood function.

Details

The model consists of two parts, the outcome model and the missing-data model. The outcome model is a mixed-effects model,

\mathbf{y}_{i} = \mathbf{X}_{i}\boldsymbol{α}+\mathbf{Z}_{i}\boldsymbol{b}_{i}+\mathbf{e}_{i},

where \mathbf{y}_{i} is the outcome for the i-th cluster, \mathbf{X}_{i} is the covariate matrix, \boldsymbol{α} is the fixed-effects, \mathbf{Z}_{i} is the design matrix for the random-effects \mathbf{b}_i, and \mathbf{e}_{i} is the error term.

The non-ignorable batch-level (or cluster-level) abundance-dependent missing-data model (BADMM) can be written as

\textrm{Pr}≤ft(M_{i}=1|\mathbf{y}_{i}\right)= \mathrm{exp}≤ft(-γ_{0} - γ \bar\mathbf{y}_{i} \right),

where M_{i} is the missing indicator for the i-th cluster, and \bar\mathbf{y}_{i} is the average of \mathbf{y}_{i}. If M_{i}=1, the outcome of the i-th cluster \mathbf{y}_{i} would be missing altogether. The estimation of the mixEMM model is implemented via an ECM algorithm. If γ \neq 0, i.e., the missingness depends on the outcome, the missing-data mechanism is missing not at random (MNAR), otherwise it is missing completely at random (MCAR) for the current model. The parameter γ can be estimated by borrowing information across outcomes and finding the common missing-data patterns in the high-dimensional data. For example, by estimating the relationship the observed average value of \bar\mathbf{y}_{i} and the missing rate, or the parameter can be selected by the log-likelihood profile (see the Reference).

Value

A list containing

alpha.hat	the estimated fixed-effects.
alpha.se	the standard errors for the estimated fixed-effects.
sigma0.hat, sigma2.hat	the estimated sample error variances. It returns the variances for the first (reference) sample and the other samples within each cluster/batch.
D	the estimated covariance matrix for the random-effects.
RE	the estimated random-effects.
loglikelihood	the observed-data log-likelihood values.

References

Chen, L. S., Wang, J., Wang, X., & Wang, P. (2017). A mixed-effects model for incomplete data from labeling-based quantitative proteomics experiments. The Annals of Applied Statistics, 11(1), 114-138. doi: 10.1214/16-AOAS994

Examples

data(sim_dat)

Z = sim_dat$X[, 1, , drop = FALSE]
fit0 = mixEMM(Ym = sim_dat$Ym, Xm = sim_dat$X, Zm = Z, gamma = 0.14)

Example output

mixEMM documentation built on May 2, 2019, 9:31 a.m.