# Marginal and Conditional Multiple Inference for Linear Mixed Model Predictors

## TL;DR

This paper develops a comprehensive framework for multiple inference in linear mixed models, providing valid confidence sets for both marginal and cluster-specific predictors, with practical applications demonstrated through simulations and a Covid-19 mortality study.

## Contribution

It introduces a novel approach for cluster-specific multiple inference in linear mixed models, including confidence sets valid under both marginal and conditional laws.

## Key findings

- Confidence sets are valid for both marginal and conditional inference.
- Marginal confidence sets are asymptotically valid for conditional inference.
- Method allows hypothesis testing without re-sampling techniques.

## Abstract

In spite of its high practical relevance, cluster specific multiple inference for linear mixed model predictors has hardly been addressed so far. While marginal inference for population parameters is well understood, conditional inference for the cluster specific predictors is more intricate. This work introduces a general framework for multiple inference in linear mixed models for cluster specific predictors. Consistent confidence sets for multiple inference are constructed under both, the marginal and the conditional law. Furthermore, it is shown that, remarkably, corresponding multiple marginal confidence sets are also asymptotically valid for conditional inference. Those lend themselves for testing linear hypotheses using standard quantiles without the need of re-sampling techniques. All findings are validated in simulations and illustrated along a study on Covid-19 mortality in US state prisons.