Chapter 1

Introduction

Collecting, analysing and drawing inferences from data are central to research in the medical and social sciences. Unfortunately, for any number of reasons, it is rarely possible to collect all the intended data. The ubiquity of missing data, and the problems this poses for both analysis and inference, has spawned a substantial statistical literature dating from 1950s. At that time, when statistical computing was in its infancy, many analyses were only feasible because of the carefully planned balance in the dataset (for example, the same number of observations on each unit). Missing data meant the available data for analysis were unbalanced, thus complicating the planned analysis and in some instances rendering it unfeasible. Early work on the problem was therefore largely computational (e.g. Healy and Westmacott, 1956; Afifi and Elashoff, 1966; Orchard and Woodbury, 1972; Dempster et al., 1977).

The wider question of the consequences of nontrivial proportions of missing data for inference was neglected until a seminal paper by Rubin (1976). This set out a typology for assumptions about the reasons for missing data, and sketched their implications for analysis and inference. It marked the beginning of a broad stream of research about the analysis of partially observed data. The literature is now huge, and continues to grow, both as methods are developed for large and complex data structures, and as increasing computer power and suitable software enable researchers to apply these methods.

For a broad overview of the literature, a good place to start is one of the recent excellent textbooks. Little and Rubin (2002) write for applied statisticians. They give a good overview of likelihood methods, and give an introduction to multiple imputation. Allison (2002) presents a less technical overview. Schafer (1997) is more algorithmic, focusing on the EM algorithm and imputation using the multivatiate normal and general location model. Molenberghs and Kenward (2007) focus on clinical studies, while Daniels and Hogan (2008) focus on longitudinal studies with a Bayesian emphasis.

The above books concentrate on parametric approaches. However, there is also a growing literature based around using inverse probability weighting, in the spirit of Horvitz and Thompson (1952), and associated doubly robust methods. In particular, we refer to the work of Robins and colleagues (e.g. Robins et al., 1995; Scharfstein et al., 1999). Vansteelandt et al. (2009) give an accessible introduction to these developments. A comparison with multiple imputation in a simple setting is given by Carpenter et al. (2006). The pros and cons are debated in Kang and Schafer (2007) and the theory is brought together by Tsiatis (2006).

This book is concerned with a particular statistical method for analysing and drawing inferences from incomplete data, called Multiple Imputation (MI). Initially proposed by Rubin (1987) in the context of surveys, increasing awareness among researchers about the possible effects of missing data (e.g. Klebanoff and Cole, 2008) has led to an upsurge of interest (e.g. Sterne et al., 2009; Kenward and Carpenter, 2007; Schafer, 1999a; Rubin, 1996).

Multiple imputation (MI) is attractive because it is both practical and widely applicable. Recently developed statistical software (see, for example, issue 45 of the Journal of Statistical Software) has placed it within the reach of most researchers in the medical and social sciences, whether or not they have undertaken advanced training in statistics. However, the increasing use of MI in a range of settings beyond that originally envisaged has led to a bewildering proliferation of algorithms and software. Further, the implication of the underlying assumptions in the context of the data at hand is often unclear.

We are writing for researchers in the medical and social sciences with the aim of clarifying the issues raised by missing data, outlining the rationale for MI, explaining the motivation and relationship between the various imputation algorithms, and describing and illustrating its application to increasingly complex data structures.

Central to the analysis of partially observed data is an understanding of why the data are missing and the implications of this for the analysis. This is the focus of the remainder of this chapter. Introducing some of the examples that run through the book, we show how Rubin's typology (Rubin, 1976) provides the foundational framework for understanding the implications of missing data.

1.1 Reasons for missing data

In this section we consider possible reasons for missing data, illustrate these with examples, and draw some preliminary implications for inference. We use the word ‘possible’ advisedly, since with partially observed data we can rarely be sure of the mechanism giving rise to missing data. Instead, a range of possible mechanisms are consistent with the observed data. In practice, we therefore wish to analyse the data under different mechanisms, to establish the robustness of our inference in the face of uncertainty about the missingness mechanism.

All datasets consist of a series of units each of which provides information on a series of items. For example, in a cross-sectional questionnaire survey, the units would be individuals and the items their answers to the questions. In a household survey, the units would be households, and the items information about the household and members of the household. In longitudinal studies, units would typically be individuals while items would be longitudinal data from those individuals. In this book, units therefore correspond to the highest level in multilevel (i.e., hierarchical) data, and unless stated otherwise data from different units are statistically independent.

Within this framework, it is useful to distinguish between units where all the information is missing, termed unit nonresponse and units who contribute partial information, termed item nonresponse. The statistical issues are the same in both cases, and both can in principle be handled by MI. However, the main focus of this book is the latter.

Example 1.1 Mandarin tableau Figure 1.1, which is also shown on the cover, shows part of the frontage of a senior mandarin's house in the New Territories, Hong Kong. We suppose interest focuses on characteristics of the figurines, for example their number, height, facial characteristics and dress. Unit nonresponse then corresponds to missing figurines, and item nonresponse to damaged—hence partially observed—figurines.

Figure 1.1 Detail from a senior mandarin's house front in New Territories, Hong Kong. Photograph by H. Goldstein.

1.2 Examples

We now introduce two key examples, which we return to throughout the book.

Example 1.2 Youth Cohort Study (YCS) The Youth Cohort Study of England and Wales (YCS) is an ongoing UK government funded representative survey of pupils in England and Wales at school-leaving age (School year 11, age 16–17) (UK Data Archive, 2007). Each year that a new cohort is surveyed, detailed information is collected on each young person's experience of education and their qualifications as well as information on employment and training. A limited amount of information is collected on their personal characteristics, family, home circumstances, and aspirations. Over the life-cycle of the YCS, different organisations have had responsibility for the structure and timings of data collection. Unfortunately, the documentation of older cohorts is poor. Croxford et al. (2007) have recently deposited a harmonised dataset that comprises YCS cohorts from 1984 to 2002 (UK Data Archive Study Number 5765). We consider data from pupils attending comprehensive schools from five YCS cohorts; these pupils reached the end of Year 11 in 1990, 1993, 1995, 1997 and 1999. We explore relationships between Year 11 educational attainment (the General Certificate of Secondary Education) and key measures of social stratification. The units are pupils and the items are measurements on these pupils, and a nontrivial number of items are partially observed.

 

Example 1.3 Randomised controlled trial of patients with chronic asthma We consider data from a 5-arm asthma clinical trial to assess the efficacy and safety of budesonide, a second-generation glucocorticosteroid, on patients with chronic asthma. 473 patients with chronic asthma were enrolled in the 12-week randomised, double-blind, multi-centre parallel-group trial, which compared the effect of a daily dose of 200, 400, 800 or 1600 mcg of budesonide with placebo. Key outcomes of clinical interest include patients' peak expiratory flow rate (their maximum speed of expiration in litres/minute) and their Forced Expiratory Volume, FEV1, (the volume of air, in litres, the patient with fully inflated lungs can breathe out in one second). In summary, the trial found a statistically significant dose-response effect for the mean change from baseline over the study for both morning peak expiratory flow, evening peak expiratory flow and FEV1, at the 5% level. Budesonide treated patients also showed reduced asthma symptoms and bronchodilator use compared with placebo, while there were no clinically significant differences in treatment related adverse experiences between the treatment groups. Further details about the conduct of the trial, its conclusions and the variables collected can be found elsewhere (Busse et...