Introduction

In this chapter we consider experiments on people and animals. Examples are clinical trials, testing educational strategies, psychological experiments, animal nutrition experiments, and trials on ambient conditions in the workplace. There are some issues peculiar to such experiments: these are discussed in Sections 7.2 and 7.7–7.10. On the other hand, the plot structure of such experiments is typically not very complicated, usually being one of those covered in Chapters 6, 4 or 2. These plot structures are specialized to experiments on people and animals in Sections 7.3, 7.4 and 7.5 respectively.

What is an observational unit in such an experiment? Usually it is a person or animal for a set time-period. A measurement may be taken before the experiment starts. This measurement is sometimes called a baseline measurement. It may be used to block the people or animals. It may be used as a covariate in the analysis of the data. Thirdly, the data to be analysed may be the differences between the post-treatment measurement and the baseline measurement.

Sometimes measurements are taken at several different times without changing the treatments. This can be enforced on animals or on hospital patients, but if non-hospitalized patients are asked to come back for measurement too often they may simply drop out of the trial.

What is the experimental unit? It may be a person or animal for the duration of the trial. It may be a group of people or animals for the duration of the trial if treatments can be administered only to whole groups.

Introduction

In the preceding chapters we have met all the principles of orthogonal designs (which will be defined in Section 10.12). The plots may be unstructured, as in Chapter 2. There may be a single system of blocks, as in Chapter 4. If there are two systems of blocks then we may have each block of one system meeting each block of the other system, as in Chapter 6 and Section 7.3, or each block of one system may contain several blocks of the other system, as in Section 8.3. The experimental units may be the same as the observational units, or each experimental unit may contain several observational units, as in Section 8.1.

The treatments also may be unstructured, as in Chapter 2. They may be divided into different types, such as control and new treatments, as in Chapter 3. They may consist of all combinations of two or more treatment factors, as in Chapter 5. The different treatment effects may all be estimated in the same stratum, or they may be estimated in two or more strata, as in Sections 8.2–8.4. We may even assume that some interactions are zero, as in Chapter 9.

The purpose of the present chapter is to give a unifying framework that not only encompasses all the designs we have met so far but also permits the construction and analysis of infinitely many more. Once this framework is understood there is absolutely no need to memorize the structure of any individual named design.

The reader may wish to omit proofs on a first reading of this chapter.

This textbook on the design of experiments is intended for students in their final year of a BSc in Mathematics or Statistics in the British system or for an MSc for students with a different background. It is based on lectures that I have given in the University of London and elsewhere since 1989. I would like it to become the book on design which every working statistician has on his or her shelves.

I assume a basic background in statistics: estimation, variance, hypothesis testing, linear models. I also assume the necessary linear algebra on which these rest, including orthogonal projections and eigenspaces of symmetric matrices. However, people's exposure to these topics varies, as does the notation they use, so I summarize what is needed at various points in Chapter 2. Skim that chapter to see if you need to brush up your knowledge of the background.

My philosophy is that you should not choose an experimental design from a list of named designs. Rather, you should think about all aspects of the current experiment, and then decide how to put them together appropriately. Think about the observational units, and what structure they have before treatments are applied. Think about the number and nature of the treatments. Only then should you begin to think about the design in the sense of which treatment is allocated to which experimental unit.

To do this requires a notation for observational units that does not depend on the treatments applied. The cost is a little more notation; the gain is a lot more clarity.

Types of block

If the plots are not all reasonably similar, we should group them together into blocks in such a way that plots within each block are alike.

There are three main types of block.

Natural discrete divisions

These divisions between the experimental units are already present. If the experimental units are new-born animals then litters make natural blocks. In an experiment on people or animals, the two sexes make obvious blocks. In testing tags on cows' ears, the ears are the experimental units and cows are the blocks. In an industrial process, a block could be a batch of chemical or ore used for part of the process.

Example (Insect repellent) Midges in Scotland are a severe irritant in July and August. A researcher wants to try out some insect repellents, which are applied to people's skin. Twelve people volunteer for the experiment. It is known that people vary widely in their attractiveness to midges, so the researcher uses people as blocks, applying different repellents to each arm. After a fixed period of exposure to midges, the number of midge bites on each arm of each volunteer is recorded.

Sometimes there is more than one type of natural discrete block. If the experimental units are half-leaves of tobacco plants then whole leaves make one sort of block while the plants make another. In a consumer experiment, such as Example 1.11, testers and weeks are both natural blocks. In an experiment in the laboratory, technicians, benches and days may all be blocks.

If an experiment is carried out on plots that had previously been used for another experiment then you should consider whether to deem the previous treatments to be blocks.

Randomization

Although randomization is generally regarded as essential, there is less agreement on how to carry it out and on how to verify whether any proposed randomization procedure achieves its objectives. Here I discuss some possibilities.

Random sampling

Many statistics textbooks give the impression that every collection of observations is a random sample. This is rarely the case for designed experiments. Experimental resources are too valuable for us to choose a random subset and ignore the rest. We use the experimental units which are to hand in the laboratory or the field, or we use all suitable volunteers in a clinical trial. In a small trial it is more important that the experimental units be reasonably similar than that they be random, and in a large trial it may be more important that they be representative. If the experimental units are to be working farms, then stratified random sampling can give a representative sample, but in practice we can still use only those farms whose farmer is willing to participate in the experiment.

Example 14.1 (Small trial on volunteers) In Question 2.1, the psychologist needs to choose as his experimental units eleven of the students listed in Table 2.7. For such a small trial, he should choose people of the same sex and approximately the same age. A random sample of eleven people would almost certainly be more variable, giving him less power to detect any treatment difference.

The chief situation in which random sampling plays a role in experimentation is when observational units are much smaller than experimental units.

In this and subsequent chapters we shall discuss the nature and utility of some 25 varieties of negative binomial regression that are useful for modeling count response data. In addition, we also examine certain models that are related to the negative binomial family of models. This chapter will primarily be devoted to an examination of the derivation of the negative binomial model and to the two foremost methods of its estimation. We also consider how the probabilities generated from a negative binomial model differ from the Poisson, as well as how they differ among various negative binomial models based on both mean and ancillary parameters.

Varieties of negative binomial

I mentioned that the basic negative binomial model can be enhanced to allow for the modeling of a wide range of count response situations. The Poisson can likewise be enhanced to adjust for data that violate, for instance, its essential distributional assumptions. In fact, many of the same distributional problems face both Poisson and negative binomial models. We therefore find similar approaches to the handling of such data for both the Poisson and negative binomial. These include models such as zero-inflated Poisson (ZIP), which is directly related to the zero-inflated negative binomial (ZINB). Other models without a specific negative binomial correlate are also discussed, e.g. generalized Poisson, which has a heterogeneity parameter like the NB-2 model, but which can also be used to model underdispersion.

Two general methods are used to estimate count response models: (1) iteratively re-weighted least squares algorithm based on the method of Fisher scoring, and (2) a maximum likelihood Newton–Raphson type algorithm. Although the maximum likelihood approach was first used with both the Poisson and negative binomial, we shall discuss it following our examination of IRLS. We do this for strictly pedagogical purposes, as will become evident as we progress.

Derivation of the IRLS algorithm

The traditional generalized linear models (GLM) algorithm, from the time it was implemented in GLIM (generalized linear interactive modeling) through its current implementations in Stata, S-Plus, and other GLM software, uses some version of an IRLS estimating algorithm. This method arises from Fisher scoring, which substitutes the expected Hessian matrix for the observed Hessian matrix in a Taylor series defined updating step for a solution of the estimating equation. The resulting Newton–Raphson or updating equation for the regression coefficients may be written in terms of ordinary least squares (OLS) due to the simplification afforded by Fisher scoring. The reason for its initial development had much to do with the difficulty of modeling individual GLM models using full maximum likelihood algorithms. In the late 1960s and early 1970s, statistical software was limited to mainframe batch runs. That is, one wrote an estimating algorithm in a higher programming language such as FORTRAN, tied it together with data stored on cards, and submitted it to a mainframe, which usually resided at a remote site.

This is the first text devoted specifically to the negative binomial regression model. Important to researchers desiring to model count response data, the procedure has only recently been added to the capabilities of leading commercial statistical software. However, it is now one of the most common methods used by statisticians to accommodate extra correlation – or overdispersion – when modeling counts. Since most real count data modeling situations appear to involve overdispersion, the negative binomial has been finding increased use among statisticians, econometricians, and researchers who commonly analyze count response data.

This volume will explore both the theory and varieties of the negative binomial. It will also provide the reader with examples using each type of major variation it has undergone. However, of prime importance, the text will also attempt to clarify discrepancies regarding the negative binomial that often appear in the statistical literature. What exactly is a negative binomial model? How does it relate to other models? How is its variance function to be defined? Is it a member of the family of generalized linear models? What is the most appropriate manner by which to estimate parameters? How are parameters to be interpreted, and evaluated as to their worth? What are the limits of its applicability? How has it been extended to form more complex models? These are important questions that have at times found differing answers depending on the author.

Negative binomial regression has traditionally been used to model otherwise overdispersed count or Poisson data. It is now considered to be the general catch-all method used when Poisson data are found to be overdispersed, particularly when the source of overdispersion has not been identified. When we can identify that which gives rise to extra correlation, and hence overdispersion, the basic Poisson and negative binomial algorithms may themselves be further adjusted or enhanced to directly address the identified source of the extra correlation. For example, when overdispersion results from an excess of zero counts in the response, an appropriate strategy is to model the data using either a zero-inflated Poisson (ZIP) or zero-inflated negative binomial (ZINB). Employing a hurdle model may also result in a better fit. On the other hand, if the response is structured such that zero counts are not possible, such as in hospital length of stay data, a zero-truncated Poisson (ZTP) or zero-truncated negative binomial (ZTNB) model may be appropriate.

A variety of alternative models have been developed to address specific facts in the data that give rise to overdispersion. Models dealing with an excess or absence of zeros typically define a mixture that alters the distributional variance of the Poisson distribution. Other models are constructed to alter not the probability and log-likelihood distributions, but rather the Poisson and negative binomial variance functions. We discuss these types of models in this chapter.