Introduction to Mixed (Hierarchical) models for biologists using R (IMBR01)
https://www.prstatistics.com/course/introductiontomixedhierarchicalmodelsforbiologistsusingrimbr01/
14 May 2018  18 May 2018
Delivered by Prof Subhash Lele.
This course will be held at Orford Musique, 3165 Chemin du Parc, Orford, QC J1X 7A2, Canada and can be reached directly by Montreal airport shuttle
Course overview: Mixed models, also known as hierarchical models and multilevel models, is a useful class of models for many applied sciences, including biology, ecology and evolution. The goal of this course is to give a thorough introduction to the logic, theory and most importantly implementation of these models to solve practical problems in ecology. Participants are not expected to know mathematics beyond the basic algebra and calculus. Participants are expected to know some R programming and to be familiar with the linear and generalized linear regression. We will be using JAGS (Just Another Gibbs Sampler) for Markov Chain Monte Carlo (MCMC) simulations for analyzing mixed models. The course will be conducted so that participants have substantial handson experience.
Monday 14th Linear and Generalized linear models To understand mixed models, the most important ﬁrst step is to thoroughly understand the linear and generalized linear models. Also, when conducting the data analysis, it is useful to ﬁt a simpler ﬁxed eﬀects model before trying to ﬁt a more complex mixed eﬀects model. Hence, we will start with a very detailed review of these models. We are assuming that the participants are familiar with these models and hence we will emphasize some important, but not commonly covered, topics. This will also give us an opportunity to unify the notation, review the basic R commands and ﬁll out any gaps in knowledge and understanding of these topics. 1. We will show the use of nonparametric exploratory techniques such as classiﬁcation and regression trees (CART) for learning about important covariates and possible nonlinearities in the relationships. 2. We will emphasize graphical and simulation based methods (e.g. Gelman and Hill, 2006) to understand and explore the implications of the ﬁtted model. 3. We will discuss graphical tools such as marginal and conditional plots that are useful for conveying the results of a multiple regression model to a lay person. 4. We will emphasize the use of graphical tools to conduct regression diagnostics and appropriateness of the model. 5. We will discuss the important concepts of confounding, eﬀect modiﬁcation and interaction. These are particularly important to conduct causal, not just correlational, inference using observational studies.
Tuesday 15th Computational inference Many of the topics that will be covered involve the use of matrix algebra and calculus. While these mathematical techniques are essential tools for a mathematical statistician who is trying to understand the theory behind the methods, they can be avoided in practice by using simulation based techniques. The builtin functions such as the ’lm’ and ’glm’ to ﬁt the regression models use the method of maximum likelihood to estimate the parameters and conduct statistical inference. We will discuss the use of JAGS (Just Another Gibbs Sampler) and the R package ’dclone’ to ﬁt the same models. We will use a diﬀerent statistical philosophy, namely the Bayesian inference, to ﬁt these models. We will show how the Bayesian approach can be tricked into giving frequentist answers using data cloning (Lele et al. 2007, Ecology Letters). We will also discuss the rudiments of frequentist and Bayesian inference although we will not go into the pros and cons of them at this time. That will be covered during sessions 3 and 4 of the ﬁfth day (and, over beer afterwards). 1. What makes an inference statistical inference? 2. What do we mean by probability of an event? 3. How do we quantify uncertainty in an inferential statement in the frequentist framework? 4. How do we quantify uncertainty in an inferential statement in the Bayesian framework? We will then discuss the simulation based methods to quantify uncertainty. 1. Parametric bootstrap to quantify frequentist uncertainty 2. Markov Chain Monte Carlo to quantify Bayesian uncertainty 3. Fitting LM and GLM using JAGS and Bayesian approach
Wednesday 16th Linear Mixed Models Historically, linear mixed models arose in the study of quantitative genetics and heritability issues. They were successfully applied in animal breeding and led to the ’white’ revolution with abundance of milk supply for the developing world. They were, also, used in horse racing and other such fun areas. The other situation where linear mixed eﬀects models were developed were in the context of growth curves. We will follow this historical trajectory of mixed models, paying tribute to the great statisticians R. A. Fisher, C. R. Rao and Jerzy Neyman, and study linear mixed models ﬁrst. The questions they tried to solve were: Deciding the genetic value of a sire and/or a dam, studying heritability of traits, studying coevolution of traits etc. These can be answered provided we assume that the sires and dams in our experiment or sample are merely a sample from a superpopulation of sires and dams. In growth curve analysis, we need to take into account that each individual is unique in its own way but is also a part of a population. How do we discuss both individual level and population inferences? In modern times, linear mixed eﬀects models have arisen in the context of small area estimation in survey sampling where one is interested in inferring about a census tract based on county or state level data. These models arise also in the context of combining remote sensed data from diﬀerent resolutions and types. The main issues that we will be discussing are: 1. What is a random eﬀect? What is a ﬁxed eﬀect? How do we decide if an eﬀect is random or ﬁxed? 2. How do we modify a linear regression model to accommodate random eﬀects? 3. Why bother ﬁtting a mixed eﬀects models? What do we gain? 4. How to modify the JAGS linear models program to ﬁt a linear mixed eﬀects model using JAGS? 5. What is the diﬀerence between a Bayesian and a frequentist inference? 6. What is a prior? What is a noninformative prior? 7. How do we interpret the results of a linear mixed eﬀects model ﬁt? Graphical and simulation based methods 8. How do we do model selection with mixed eﬀects models? 9. How do we do model diagnostics in mixed eﬀects models? 10. Parameter identiﬁabilty issues in linear mixed models As we discuss these applications, we will discuss some subtle computational issues involved in using MCMC. In my recollection (which may be biased as it has been about 25 years since the quote), Daryl Pregibon said: MCMC is the crack cocaine of modern statistics; it is addictive, seductive and destructive. Hence, it is important for a practitioner to understand these issues in order not to misuse the MCMC technique. 1. What is a Markov Chain Monte Carlo method? Why is it necessary for mixed models? 2. What are the subtleties in implementing MCMC?: Convergence of the algorithm, Mixing of the chains. 3. Pros and cons of using MCMC
Thursday 17th Generalised Linear Mixed Models We will again start the discussion of GLMM in its historical context. One of the initial uses of mixed models were in the context of over dispersion in count data. Zero inﬂated count data was another important example. The example that drove the current revolution in the use of GLMM was in the context of spatial epidemiology. Clayton and Caldor (1989, Biometrics) showed that one can use spatial correlation to improve the prediction in mapping disease rates. This was also an example of the application of Empirical Bayes methods that allow one to pool information from diﬀerent spatial areas (or, studies, or, scales, and so on). 1. Zero inﬂated data In many practical situations, we observe that there are many locations where there are zero counts, far in excess of what would be expected under the Poisson regression model. This can be eﬀectively modelled using a mixed model framework. The mixed models framework allows us to use much more complex and realistic models. 2. Over dispersion in GLM, Spatial GLM, Spatiotemporal GLM The Poisson regression model assumes that the mean and variance are equal. This is, often, not true in practice. Generally the variance in the data exceeds the mean. One can show that such overdispersion can be modelled using a mixed eﬀects model. These models also arise in the context of capturerecapture sampling where capture probabilities vary across space or time or individuals. 3. Longitudinal or panel data with discrete response variable Many times we have data on diﬀerent individuals where within the individual there is temporal dependence but individuals are independent of each other. Cluster sampling is another situation where we have dependence within a cluster but independence between clusters. Such data needs to take into account the innate variation between individuals before one can discuss the eﬀect of interesting covariates or risk factors. Such data are eﬀectively modelled as GLMM. 4. Measurement error, missing data Missing data and measurement error are ubiquitous in ecological studies. Mixed models provide a convenient way to take into account these diﬃculties and infer about the underlying processes of interest. We will discuss these issues in the context of Population Viability Analysis, Spatial population dynamics and sourcesink analysis, Occupancy and abundance surveys. These also arise while doing usual linear and generalized linear models if the covariates are measured with error. 5. Additional topics depending on the interest of the participants. These may include, for example, discussion of Species Distribution Models, Resource Selection Functions and Animal movement models. 6 Computational issues: Advanced topics
Friday 18th Mixed Models in a Bayesian Framework MCMC is not the only approach to analyse mixed models. We will brieﬂy discuss Laplace approximation based techniques (INLA, in particular) along with approximate techniques such as Composite likelihood and Approximate Bayesian Computation. Because of the mathematical nature, this discussion will be somewhat limited, only giving the basics and hinting at the important issues. 7 Philosophical issues: Sophie’s choice 1. What are the philosophical problems with using the frequentist quantiﬁcation of uncertainty? 2. What are the philosophical problems with using the Bayesian quantiﬁcation of uncertainty? 3. Sophie’s choice?
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