Statistics for Making Decisions

Nicholas T. Longford

• Publisher: Chapman and Hall/CRC
• Year: 2021
• Print ISBN: 9780367342678
• eBook ISBN: 9781000347609
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Making decisions is a ubiquitous mental activity in our private and professional or public lives. It entails choosing one course of action from an available shortlist of options. Statistics for Making Decisions places decision making at the centre of statistical inference, proposing its theory as a new paradigm for statistical practice. The analysis in this paradigm is earnest about prior information and the consequences of the various kinds of errors that may be committed. Its conclusion is a course of action tailored to the perspective of the specific client or sponsor of the analysis. The author's intention is a wholesale replacement of hypothesis testing, indicting it with the argument that it has no means of incorporating the consequences of errors which self-evidently matter to the client.

The volume appeals to the analyst who deals with the simplest statistical problems of comparing two samples (which one has a greater mean or variance), or deciding whether a parameter is positive or negative. It combines highlighting the deficiencies of hypothesis testing with promoting a principled solution based on the idea of a currency for error, of which we want to spend as little as possible. This is implemented by selecting the option for which the expected loss is smallest (the Bayes rule).

The price to pay is the need for a more detailed description of the options, and eliciting and quantifying the consequences (ramifications) of the errors. This is what our clients do informally and often inexpertly after receiving outputs of the analysis in an established format, such as the verdict of a hypothesis test or an estimate and its standard error. As a scientific discipline and profession, statistics has a potential to do this much better and deliver to the client a more complete and more relevant product.

Nicholas T. Longford is a senior statistician at Imperial College, London, specialising in statistical methods for neonatal medicine. His interests include causal analysis of observational studies, decision theory, and the contest of modelling and design in data analysis. His longer-term appointments in the past include Educational Testing Service, Princeton, NJ, USA, de Montfort University, Leicester, England, and directorship of SNTL, a statistics research and consulting company. He is the author of over 100 journal articles and six other monographs on a variety of topics in applied statistics.

1

First steps

1.1 What shall we do?

Making decisions is a common mental activity in our everyday lives. In any one instance, we contemplate the courses of action, or options, that are available to us. We inquire and deliberate about the likely profit or advantage that we would gain, or the harm, damage or injury that we might incur, or what we would lose, by choosing either of these options. The gains and losses usually depend on the actions taken by other people and institutions, as well as on the current, recent past and future states of the nature and the environment in which we live and operate. Our information about these states and actions is incomplete—we are not privy to all the intentions of others and circumstances and other details related to the options available to us. In some settings, we have some limited means of learning more about the circumstances, and thus informing the decision—help us to choose the option that is likely to fit best our desires, goals or remits and cause least harm, damage or loss.

This general description covers a wide range of scenarios, from the mundane to the momentous. When crossing a busy road we pay attention to the traffic, observe the direction in which the vehicles are moving and assess their speed. We set off to cross the road when we judge it safe to do so, even allowing for something untoward to happen, such as being slowed down by a stumble or impeded by an obstacle, and allow ample time and distance for the drivers of approaching vehicles to react if necessary.

As another example, a proposal for marriage follows (or should follow) a careful deliberation of the match with one’s acquaintance, how he or she is likely to develop over the years of cohabitation, how well you would support one another and how you would cope with the vicissitudes, illnesses, hardships, disappointments and discord you are likely to encounter in the future that is full of optimistic plans and hopes, but also entails a lot of uncertainty. And, until a divorce or death, it precludes marrying someone else.

Some decisions are difficult to make because they involve comparisons of apparently incomparable values. One such comparison is ‘life or money’—how much are you willing to pay to (possibly) extend or improve your life, or the life of a patient. It entails many imponderables. The claim that a particular course of action, such as a costly or inconvenient treatment, or a complex surgery, would extend the life may be in balance even after carefully weighing the potential advantages and drawbacks. For example, a surgery on a patient’s vital organ is highly likely to improve the functioning of the organ but may adversely affect other organs, the heart and brain in particular. We are rarely in full control of our immediate environment or of events affecting us (our fate). Our understanding is usually limited, and so our deliberation is full of ‘maybes, ifs and buts’.

While deciding on one matter, other matters are not waiting patiently for their turn in an orderly queue, nor are newly arising matters obediently joining the end of the queue. They have to be dealt with at the time of their calling, placing constraints on our resources: time, money, mental capacity, distracting us from activities we are currently pursuing or would prefer to pursue. The choices we have to make are structured: some are in a time sequence and depend on the choices made earlier, some fall into natural groups (how often do you cross the road, at a variety of locations?) or packages (driving a car and conducting a job interview). For some it is essential to define a set of rules or a policy, or to adopt a policy formulated by others (the Driving Code and instructions and advice of a coach given to an athlete), because decisions have to be made instantly, permitting no deliberation or consultation. In brief, we have a limited capacity to handle the many decision-related tasks that are constantly pressing on us. We might be well served by a strategy for distributing our resources so that we get the best deal from a package of decisions, even if some problems could have been dealt with better, at the expense of poor responses to the others, due to not paying full attention to them. Part of the strategy may be investment in our decision-making capacity (such as by education and training), possibly a digression in the short term that may be amply rewarded later. Our capacity to deal with these tasks is not static; it is evolving gradually as we accumulate experience, information and insight. Our priorities and value judgements are also evolving and they influence our decisions by our appraisal of the consequences of the options available to us.

This volume carves out a small segment of this general class of problems and treats them with statistical methods that are not novel but have fallen into disuse in the mainstream statistical practice because they can be meaningfully applied only after thorough introspection and assessment, combined with careful calculation and exploration. One may be unwilling to do or take part in this, finding it unpleasant, too demanding, vexing or contentious, and disclosure of the results may cause some discomfort. The approach proposed runs counter to the prevailing way of conducting statistical evaluations, in which a dataset is analysed objectively, in detachment from and without any interference by the party that has a stake in the outcome of the analysis. In contrast, we want to integrate in the analysis this party’s perspective, which includes their value judgements, priorities and remit.

We assume that every analysis is associated with such a party, called the client, who sponsors or provides some other form of incentive for the analysis and supplies the purpose—to assist in the process of promoting or achieving a specified goal (agenda) in production, provision of services, pursuit of peace, comfort and happiness, academic research and exploration, or some other gainful activity. A relevant analysis responds to this client-specific purpose, and we do not cons...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• Author
• 1 First steps
• 2 Statistical paradigms
• 3 Positive or negative?
• 4 Non-normally distributed estimators
• 5 Small or large?
• 6 Study design
• 7 Medical screening
• 8 Many decisions
• 9 Performance of institutions
• 10 Clinical trials
• 11 Model uncertainty
• 12 Postscript
• References
• Solutions to exercises
• Index