Rense Nieuwenhuis » hierarchical

R-Sessions 32: Forward.lmer: Basic stepwise function for mixed effects in R

Rense Nieuwenhuis — Fri, 13 Feb 2009 10:59:03 +0000

Intended to be a customized solution, it may have grown to be a little more. forward.lmer is an early installment of a full stepwise function for mixed effects regression models in R-Project. I may put in some work to extend it, or I may not. Nevertheless, in a ‘forward sense of stepwise’, I think it can be pretty useful as it is. Also, it has an interesting take on the stepwise concept, I think.

Most stepwise functions (as far as I know) take a base model and a bunch of variables, and then iteratively adds and/or subtracts some variables, according to various criteria, to come to the best fitting regression model. All very interesting, but how to deal with interaction variables? And moreover: most existing functions do not work with mixed effects models ((I use the term ‘mixed effects model’ to describe this stepwise function to refer to what is often referred to as hierarchical or multilevel regression models, as well)).

Built around the lme4 package in R, forward.lmer provides a forward stepwise procedure to mixed effects models. Also, it allows the user not only to enter single variables to models, but also to do the same with blocks of variables. This opens up many options: users can add the complete interactions at once (i.e. both the original and the multiplicative terms), or add these consequetively. Future development will focus on additional selection criteria for interactions, such as the criterium that at least the multiplicative term needs to be statistically significant.

The user provides a starting model and a set of variables to evaluate. The procedure then updates the starting model with the addition of every single variable (or block of variables). The models are ordered based on their LogLikelihood (other criteria, i.e. BIC and AIC following soon), after which the best fitting model is evaluated against one of two criteria. The first criterium is that at least one of the added parameters is statistically significant. The other criterium is that the addition of the parameters together is statistically significant.

There are several parameters to be specified:

start.model: The starting model the procedure starts with. This can be a null-model, or a model already containing several variables. All lmer-models (i.e. logistic, poisson, linear) are supported.
blocks: a vector of variable names (as character strings) to be added to a model. Several variables can a concatenated within the same character string, so that these are added as a block of variables, instead of a single variables at once.
max.iter: The maximum number of variables that are evaluated. If max.iter is reached, the procedure stops without adding more variables.
sig.level: This is the p-value against which it is tested whether the new model fits better than a base model. Either sig.level or zt needs to be specified, but not both at once.
zt: This is either the T or Z value that is used to test whether (at least) one of the added variables is statistically significant. T values are used for linear regression, Z values for binary response models.
print.log: Should a log be printed? The log contains information on which variables (and on which criteria) were added in each step.

The forward.lmer function returns the best fitting model (according to the given criteria). Of course, one can use this resulting model as a starting model for a new stepwise procedure.

forward.lmer <- function( start.model, blocks, max.iter=1, sig.level=FALSE, zt=FALSE, print.log=TRUE) {


	# forward.lmer: a function for stepwise regression using lmer mixed effects models

	# Author: Rense Nieuwenhuis
	# Initialysing internal variables

	log.step <- 0

	log.LL <- log.p <- log.block <- zt.temp <- log.zt <- NA

	model.basis <- start.model
	# Maximum number of iterations cannot exceed number of blocks

	if (max.iter > length(blocks)) max.iter <- length(blocks)
	# Setting up the outer loop

	for(i in 1:max.iter)

		{
		models <- list()
		# Iteratively updating the model with addition of one block of variable(s)

		# Also: extracting the loglikelihood of each estimated model

		for(j in 1:length(blocks))

			{

			models[[j]] <- update(model.basis, as.formula(paste(". ~ . + ", blocks[j])))

			}
		LL <- unlist(lapply(models, logLik))
		# Ordering the models based on their loglikelihood.

		# Additional selection criteria apply

		for (j in order(LL, decreasing=TRUE))

			{
			##############

			############## Selection based on ANOVA-test

			##############
			if(sig.level != FALSE)

				{

				if(anova(model.basis, models[[j]])[2,7] < sig.level)

					{
					model.basis <- models[[j]]
					# Writing the logs

					log.step <- log.step + 1

					log.block[log.step] <- blocks[j]

					log.LL[log.step] <- as.numeric(logLik(model.basis))

					log.p[log.step] <- anova(model.basis, models[[j]])[2,7]
					blocks <- blocks[-j]
					break

					}

				}
			##############

			############## Selection based significance of added variable-block

			##############	
			if(zt != FALSE)

				{

				b.model <- summary(models[[j]])@coefs

				diff.par <- setdiff(rownames(b.model), rownames(summary(model.basis)@coefs))

				if (length(diff.par)==0) break

				sig.par <- FALSE
				for (k in 1:length(diff.par))

					{

					if(abs(b.model[which(rownames(b.model)==diff.par[k]),3]) > zt)

						{

						sig.par <- TRUE

						zt.temp <- b.model[which(rownames(b.model)==diff.par[k]),3]

						break

						}

					}					
				if(sig.par==TRUE)

					{

					model.basis <- models[[j]]
					# Writing the logs

					log.step <- log.step + 1

					log.block[log.step] <- blocks[j]

					log.LL[log.step] <- as.numeric(logLik(model.basis))

					log.zt[log.step] <- zt.temp

					blocks <- blocks[-j]
					break

					}

				}

			}

		}
	## Create and print log

	log.df <- data.frame(log.step=1:log.step, log.block, log.LL, log.p, log.zt)

	if(print.log == TRUE) print(log.df, digits=4)
	## Return the 'best' fitting model

	return(model.basis)

	}

As always, you're invited to use this function, or to adapt it and use that. However, it is required to make mention of this function and its author. Additionally, since I intend to continue working on this function (perhaps even evolve it to a 'package' on CRAN), I would love to hear about any experiences in using it.

R-Sessions 25: Book – Mixed Effects Models in S and S-PLUS (Pinheiro & Bates, 2000)

Rense Nieuwenhuis — Wed, 01 Oct 2008 10:00:51 +0000

Despite the reference to S and S-PLUS in the title of this book, it offers an excellent guide for the nlme-package in R-Project. Reason for this is the close resemblance between R and S. The nlme-package, available in R-Project for estimation of both linear and non-linear multilevel models, is written and maintained by the authors of this book.

The book is not an introduction to R. Basic knowledge of R-Project (or S / S-PLUS) is required to get the most out of it, as well as some knowledge on multilevel theory. Although the book forms a thorough introduction to multilevel modeling, addressing both some theory, the mathematics and of course the estimation and specification in R-Project (or S / S-PLUS), the learning curve it offers is quite steep. The authors are not shunned to apply matrix algebra and specify exactly the used estimation procedures.

Not only the specification of basic models is described, but many other subjects are brought up. A specific grouped-data object is considered, as well as ways to visualize hierarchical data and multilevel models. Heteroscedasticity, often a violation of assumptions, can be caught in the models easily, as is described clearly in one of the chapters. Finally, not only linear models are tackled, but non-linear models as well.

All in all, this book is an excellent addition for those who have prior knowledge of both R-Project and multilevel analysis. Using real-data examples and by providing tons of output, the authors accomplish to make clear the necessity of the more complex models and thereby invite the reader to invest time for the more fundamental aspects of multilevel analysis.

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R-Sessions is a collection of manual chapters for R-Project, which are maintained on Curving Normality. All posts are linked to the chapters from the R-Project manual on this site. The manual is free to use, for it is paid by the advertisements, but please refer to it in your work inspired by it. Feedback and topic requests are highly appreciated.
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R-Sessions 23: Book: Data Analysis Using Regression and Multilevel/Hierarchical Models — Gelman & Hill (2007)

Rense Nieuwenhuis — Tue, 23 Sep 2008 10:00:05 +0000

Data Analysis Using Regression and Multilevel/Hierarchical Models

Andrew Gelman is known for his expertise on Bayesian statistics. Based on that knowledge he wrote a book in multilevel regression using R and WINbugs. This book aims to be a thorough description of (multilevel) regression techniques, implementation of these techniques in R and bugs, and a guide on interpreting the results of your analyses. Shortly put, the books excels on all three subjects.

Admittedly, this review has been written based on first impressions on the book. But, a sunny day in the park reading this book (literally) left me to believe that I have some understanding on what this book is trying to achieve. I bought this book in order to have an overview on fitting multilevel regression models using R. Starting to read the book, I soon found out that that is indeed what it has to offer me, but it offers me a lot more. After some introductory chapters, the book starts off with an introduction to both linear regression as well as introducing the reader to R software, by showing how to fit linear regression models in R. This is readily expanded to logistic regression and generalized regression models. All is illustrated lushly with many examples and illustrations.

Before these ‘basic’ regression models are extended to multilevel models, Bayesian statistics are introduced. Based on simulation techniques, causal inferences, based on regression models, are made. The multilevel section of the book is set up similarly. First, ‘basic’ multilevel regression models are introduced. Throughout the book, the lmer function is used. This function is not only able to fit simple multilevel models, but logistic and generalized models as well. It can even estimate non-nested models. All in all, this forms a thorough introduction to multilevel regression analysis in itself, but the book continues here as well to introduce the reader to Bayesian statistics.

All above-mentioned models, as well as more complicated models, are fitted using WINbugs as well. This very flexible method allows the reader to estimate a greater variety of (multilevel) models. Causal inference on multilevel models, using Bayesian statistics, is described as well. The third main part of the book elaborates on the skills the reader uses to ‘just’ fitting models. It learns the reader to really think about what it going on. Topics such as ‘understanding and summarizing the fitted models’, ‘sample size and power calculations’, and most of all ‘model checking and comparison’ each receive their own chapter of the book. In this we can see that the authors of this book aimed higher than just writing instructions on how to let R fit (multilevel) regression models. The aim of this book, is to teach the reader how to analyze data the proper way. Much attention is paid to assumptions, testing theory, and interpretation of what you’re doing. To quote the authors: “If you show something, be prepared to explain it”.

This philosophy seemed to be a guideline for the authors while writing this book, as well as flexibility. The book starts off with some examples of the authors’ own research. These examples return throughout the book, resulting in some degree of familiarity with the data by the reader. Due to this, the concepts, models and/or analyses described are certainly more easy to be understood. As a reader, you start to think along with the author, when a new problem is described. The relative worth of the techniques, as well as their drawbacks, are made perfectly clear. The use of R software, as well as WINbugs, pays of well in the sense that it requires some more effort to master these programs, but in that process the reader learns to think deeply about what he really want to do and how it is done properly.

I found it not an easy book, but thanks to the many examples throughout the book it can be fully understood by people with some prior knowledge in regression techniques. All of the examples in the book can be tried yourself, since the data and syntax are available on the author’s website on the book. This helps the reader to get some feel for the more difficult subjects of the book. All in all, this seems to me as a great book for every applied researcher that has basic prior understanding of regression analysis. Due to its focus on one set of techniques, a great depth of understanding can be derived from this book.