A Simple HLM: The Dyestuff Example

Hierarchical Linear Models (HLMs) have their own terminology that challenges a lot of what is taught as basic statistics. We have random effects, random coefficients, and mixed-models added on to what we already know about random error terms and fixed effects. It will help to start with a simple data set that so we can introduce new terms one at a time.

The lme package in the R programming language has a simple data set that provides a good starting point. If you have R running on your machine and the lme package installed, typing help(Dyestuff) produces the following description of the data set:

The Dyestuff data frame provides the yield of dyestuff (Naphthalene Black 12B) from 5 different preparations from each of 6 different batchs of an intermediate product (H-acid). 

The Dyestuff data are described in Davies and Goldsmith (1972) as coming from “an investigation to find out how much the variation from batch to batch in the quality of an intermediate product (H-acid) contributes to the variation in the yield of the dyestuff (Naphthalene Black 12B) made from it. In the experiment six samples of the intermediate, representing different batches of works manufacture, were obtained, and five preparations of the dyestuff were made in the laboratory from each sample. The equivalent yield of each preparation as grams of standard colour was determined by dye-trial.”

Notice that the batches are not randomly chosen out of some universe of manufactured batches. They are whatever we were able to "obtain".  Just to make things concrete, here are the first few lines of the file printed by the R head() command (the > before the head command is the R prompt):

> head(Dyestuff)

  Batch Yield

1     A  1545

2     A  1440

3     A  1440

4     A  1520

5     A  1580

6     B  1540

At first, one might be tempted to run an analysis of variance (ANOVA) on this data since the stated intention is to analyze variation from batch to batch. However, there are issues presented by this simple data set that help us understand the added complexity and potential benefits of hierarchical modeling.

In general, the standard approach is to move directly from a data set to some kind of estimator. In the next few posts, I will argue that the standard approach misses a number of important opportunities.

The 'hlmmc' package

From July 1 through August 15, I taught Sociology S534 Advanced Sociological Analysis: Hierarchical Linear Models at the University of Tennessee in Knoxville. A syllabus for the course is available here. Over the next few months I will serialize the lectures and a user guide for the software package ('hlmmc') used in the course. 'hlmmc' is a software package written in the R programming language for studying Hierarchical Linear Models (HLMs) using Monte Carlo simulation. A manual for the software package is available here. It explains how to download the software and how each function in the package can be used (also, see NOTE below).

The approach I used in the course was to write out the equations for each linear model as a function in the R programming language. The functions were designed so that data for the study could be simulated for each observation in the sample. This is a useful exercise because it forces you to confront a number of difficult issues. What statistical distributions will be used to not only simulate the distributions of the error terms but also the distributions of the independent variables? Typically, a normal distribution is chosen for the error terms and the distributions of the independent variables are ignored in planning a study. Monte Carlo Simulation forces the researcher to confront these issues in addition to explicitly stating the functional form of the model, the actual parameters in the model and a likely range for the parameter values that might be encountered in practice.

In the next posting I'll start with a very simple hierarchical data set and develop an R function for simulating data from a model that could be used to generate similar data. The Monte Carlo function frees us from the sample data and lets us explore more general issues raised by the model. The pitch to my students was that the Monte Carlo approach would provide a deeper understanding of the issues raised by hierarchical models.

The basic feedback from students was that, in the short time available for the class, they were able to get a deeper understanding of HLMs than if the class had be structured as a conventional statistics class (math lectures and exercises?). One student comment in the course evaluations that s/he "...wished all stat classes were taught this way." For better or worse, my classes always have been.

If you are interested in the topic and want to follow the serialization in future postings, I will try to follow my lectures, which were live programming demonstrations, as closely as possible keeping in mind some of the questions and problems my students had when confronting the material.

NOTE: Assuming you have R installed on your machine and you have downloaded the files LibraryLoad.R and HLMprocedures.R to some folder on you local machine, you can load the libraries and procedures using the following commands in R:

> setwd(W)

> source(file="LibraryLoad.R")

> source(file="HLMprocedures.R")

where > is the R prompt and W is the absolute path to the working directory where you downloaded the files LibraryLoad.R and HLMprocedures.R. The Macintosh and Windows version of R have  'Source' selection in the File menu that will also allow you to navigate your directory to the folder where these two files have been downloaded. When you source the LibraryLoad.R file, error messages will be produced if you have not installed the supporting libraries listed in the manual (here).