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10 - Likelihoods and confidence likelihoods
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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Summary
The likelihood function is the bridge between the data and the statistical inference when the model is chosen. The dimension of the likelihood might be reduced by profiling, conditioning or otherwise to allow a confidence distribution for a parameter of interest to emerge. We shall now turn this around and develop methods for converting confidence distributions for focus parameters into likelihoods, which we call confidence likelihoods. This is particularly useful for summarising the most crucial part of the inference in a manner convenient for further use, for example, in meta-analyses, as discussed further in Chapter 13. We shall in particular discuss normal conversion methods along with acceleration and bias corrected modifications, and illustrate their uses. Connections to bootstrapping of likelihoods are also examined.
Introduction
The likelihood function is the link between data as probabilistically modelled and statistical inference in terms of confidence distributions or otherwise. This chapter prepares the ground for Chapter 13 on meta-analysis. In a meta-analysis the results of several studies that address a set of related research hypotheses are combined. In its simplest form, the problem is to combine estimates of a common parameter from independent studies, which is traditionally solved by a weighted average with weights obtained from the respective standard errors. This would be optimal if the estimators of the individual studies were normally distributed. From the confidence distributions or confidence intervals of the studies, it might be seen that the normal assumption is far-fetched. Rather than just averaging, it would be better if a pseudo-likelihood could be constructed from the information provided in each of the reports, and combine these by multiplication to a collected pseudo-likelihood. We will call a pseudo-likelihood obtained from a confidence distribution or a confidence interval a confidence likelihood (c-likelihood), and methods for such constructions are developed in the text that follows.
Fisher (1922) introduced the likelihood function as the primary tool for statistical analysis of data. This and the wealth of new concepts, new points of views and new theory constitute the third revolution in parametric statistical inference (Hald, 1998, p. 1), after the first when Laplace (1774) introduced the method of inverse probability and that of Gauss and Laplace, who introduced the least squares method and linear models for normally distributed data in 1809–1812. The likelihood function is a minimal but sufficient reduction of the data.
Preface
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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Shocks sometimes lead to new ideas. One of us was indeed shocked when Robert Wolpert (1995) pointed out that the Raftery et al. (1995) approach of Bayesian synthesis of two independent prior distributions for a parameter was flawed, due to the so-called Borel paradox. The very positive comments he had prepared for the discussion at the Joint Statistical Meetings in 1994 in Toronto were hard to bring forward (Schweder, 1995). The paper under discussion concerned Bayesian estimation of the abundance of bowhead whales off Alaska. The method and resulting estimate had just been accepted by the Scientific Committee of the International Whaling Commission (IWC). The Borel paradox became a central issue at the next IWC meeting, along with associated problems of combining different information sources for the same parameters. A distributional estimate of bowhead abundance in place of the Bayesian posterior was clearly needed. This led to the idea of achieving a distribution from all the confidence intervals obtained by varying the confidence level. It also led to the collaboration of the two authors of the present book, from our paper (Schweder and Hjort, 1996) on the Borel paradox and likelihood synthesis and onwards, via papers tying together the general themes of confidence, likelihood, probability and applications.
Posterior distributions without priors?
Constructing distributions for parameters from the set of all confidence intervals was a new and very good idea, we thought, but it turned out to be not so new after all. Cox (1958) mentions the same idea, we later on learned, and the original discovery of distribution estimators not obtained by a Bayesian calculation from a prior distribution dates back to Fisher (1930) and his fiducial argument. Like most contemporary statisticians we were badly ignorant of the fiducial method, despite its revolutionary character (Neyman, 1934). The method fell into disrepute and neglect because of Fisher's insistence that it could do more than it actually can, and it disappeared from practically all textbooks in statistics and was almost never taught to statisticians during the past fifty years. Fiducial probability was said to be Fisher's biggest blunder. But Efron (1998), among others, expresses hope for a revival of the method, and speculates that Fisher's biggest blunder might be a big hit in our new century.
Frontmatter
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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6 - The fiducial argument
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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To obtain a distribution representing the inferred uncertain knowledge about the parameter directly from the data without access to any prior distribution was the glorious goal that Fisher succeeded in reaching with his fiducial distribution for a scalar parameter, but for vector parameters he got into trouble. He regarded the fiducial distribution as an ordinary probability distribution subject to the usual Kolmogorovian laws for sigma-additive probability measures. Fisher did not develop any mathematical theory of fiducial probability, but chose to illustrate his thoughts by examples. It was soon found that even in Fisher's examples with more than one parameter, there were no unique fiducial distributions, and inconsistencies and paradoxes were identified. After re-visiting Fisher's ideas over 1930–1935, which underlie our confidence distributions, we summarise and discuss the big debate over the fiducial argument, which died out only after Fisher's death in 1962, leaving the statistical community to regard it as badly flawed and Fisher's biggest blunder. The chapter ends with three attempts at saving the fiducial argument. Despite the potential problems with multivariate fiducial distributions, their marginals are often exact or approximate confidence distributions.
The initial argument
Fisher introduced the fiducial argument in 1930. In the introduction to Fisher (1930) he indicates that he had solved a very important problem that had escaped “the most eminent men of their time” since Bayes introduced his theorem and the method of inverse probability was established by Laplace. Later he stated that he really had a solution to “more than 150 years of disputation between the pros and cons of inverse probability [that] had left the subject only more befogged by doubt and frustration” (Fisher discussing Neyman in Neyman [1934, p. 617]). This was a view he held throughout his life.
Fisher saw the fiducial probability as a probability measure over the parameter inherited from the probability model of the sampling experiment and the observed data. His early interpretation of fiducial probability was very similar to Neyman's coverage probability for intervals. Later he thought of fiducial distributions as a representation of the information in the data as seen on the background of the statistical model. It would then serve as an epistemic probability distribution for rational people. This is close to the Bayesian understanding of a posterior distribution, at least for people agreeing on the prior distribution.
13 - Meta-analysis and combination of information
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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Summary
The impact of a scientific report is a function both of the presentation and the originality and the quality of the results. To achieve maximal impact, new empirical results should be reported in a form that enables readers to combine them effectively with other relevant data, without burdening readers with having to redo all the analyses behind the new results. Meta-analysis is a broad term used for methods analysing a set of similar or related experiments jointly, for purposes of general comparison, exhibiting grander structure, spotting outliers and examining relevant factors for such, and so on. As such the vital concept is that of combining different sources of information, and often enough not based on the full sets of raw data but on suitable summary statistics for each experiment. This chapter examines some natural and effective approaches to such problems, involving construction of confidence distributions for the more relevant parameters. Illustrations include meta-analysis of certain death-after-surgery rates for British hospitals. Yet other applications involving meta-analysis methods developed here are offered in Chapter 14.
Introduction
Meta-analysis, the art and science of combining results from a set of independent studies, is big business in medicine, and is also an important tool in psychology, social sciences, ecology, physics and other natural sciences. The yearly number of medical meta-analyses has increased exponentially in recent years, and rounded 2000 by 2005 (Sutton and Higgins, 2008).
The first issue to resolve in a meta-analysis is to define the parameter, possibly a vector, to study. This might not be as easy as it sounds because related studies might vary with respect to definition and they will also usually vary in their experimental or observational setup and statistical methods. Along with the definition, the population of related potential studies is delineated. In rather rare cases is it clear that the value of the parameter is exactly the same in related studies. Then a fixed effects model may be appropriate. When the parameter varies across related studies, or when there are unobserved heterogeneities across studies, the statistical model to be used in the meta-analysis should include random effects.
The next problem is to determine the criterion and algorithm for selecting studies for the meta-analysis. Ideally the collection should be a random sample from the population of related potential studies.
8 - Exponential families and generalised linear models
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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Contents
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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Overview of examples and data
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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A generous number of real datasets are used in this book to illustrate aspects of the methodology being developed. Here we provide brief descriptions of each of these real data examples, along with key points to indicate which substantive questions they relate to. Some of these datasets are small, partly meant for simpler demonstrations of certain methods, whereas other are bigger, allowing also more ambitious modelling for reaching inference conclusions. Key words are included to indicate the data sources, the types of model we apply and for what inferential goal, along with pointers to where in our book the datasets are analysed.
Lifelength in Roman era Egypt
In Spiegelberg (1901) the age at death has been recorded for 141 Egyptian mummies, 82 male and 59 female, dating from the Roman period of ancient Egypt from around year 100 B.C. These lifelengths vary from 1 to 96 years, and Pearson (1902) argued that these can be considered a random sample from one of the better-living classes in that society, at a time when a fairly stable and civil government was in existence. These data are analysed by Claeskens and Hjort (2008, pp. 33–35), in which nine different parametric models for hazard rates are compared and where the Gompertz type models are found to be best.
In Example 1.4 we fit a simple exponential model to the lifelengths of the male to motivate the concepts of deviance functions and confidence curves. In Example 3.7 we find the confidence distribution for the ratio of hazard rates for female to that for the men (in spite of Karl Pearson's comment, “in dealing with [these data] I have not ventured to separate the men and women mortality, the numbers are far too insignificant”). A certain gamma process threshold crossing model is used in Exercise 4.13, providing according to the Akaike information criterion (AIC) model selection method a better fit than the Gompertz. For Example 9.7 we compute and display confidence bands for the survival curves, for the age interval 15 to 40 years. Then in Example 10.1 the data are used to illustrate the transformation from confidence distribution to confidence likelihood, for the simple exponential model, following Example 1.4, whereas Example 11.1 gives the confidence density for the cumulative probability F(y0), that is, the probability of having died before y0, again comparing male with female.
Dedication
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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Subject index
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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Appendix: Large-sample theory with applications
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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Name index
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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7 - Improved approximations for confidence distributions
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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Summary
Previous chapters have developed concepts and methodology pertaining to confidence distributions and related inference procedures. Some of these methods take the form of generally applicable recipes, via log-likelihood profiles, deviances and first-order large-sample approximations to the distribution of estimators of the focus estimands in question. Sometimes these recipes are too coarse and are in need of modification and perfection, however, which is the topic of the present chapter. We discuss methods based on mean and bias corrected deviance curves, t-bootstrapping, a certain acceleration and bias correction method, approximations via expansions, prepivoting and modified likelihood profiles. The extent to which these methods lead to improvements is also briefly illustrated and discussed.
Introduction
Uniformly most powerful exact inference is in the presence of nuisance parameters available only in regular exponential models for continuous data and other models with Neyman structure, as discussed and exemplified in Chapter 5. Exact confidence distributions exist in a wider class of models, but need not be canonical. The estimate of location based on the Wilcoxon statistic, for example, has an exact known distribution in the location model where only symmetry is assumed; see Section 11.4. In more complex models, the statistic on which to base the confidence distribution might be chosen on various grounds: the structure of the likelihood function, perceived robustness, asymptotic properties, computational feasibility, perspective and tradition of the study. In the given model, with finite data, it might be difficult to obtain an exact confidence distribution based on the chosen statistic. As we shall see there are various techniques available for obtaining approximate confidence distributions and confidence likelihoods, however, improving on the first-order ones worked with in Chapters 3–4.
Bootstrapping, simulation and asymptotics are useful tools in calculating approximate confidence distributions and in characterising their power properties. When an estimator, often the maximum likelihood estimator of the interest parameter, is used as the statistic on which the confidence distribution is based, bootstrapping provides an estimate of the sampling distribution of the statistic. This empirical sampling distribution can be turned into an approximate confidence distribution in several ways, which we address in the text that follows.
9 - Confidence distributions in higher dimensions
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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This chapter is concerned with confidence distributions for vector parameters, defined over nested families of regions in parameter space. For a given degree of confidence the region with that confidence is a confidence region in the sense of Neyman. Such regions are often called simultaneous confidence regions. They form level sets for the confidence curve representing the confidence distribution. Analytic confidence curves for vector parameters, and also for parameters of infinite dimensions, for example, all linear functions of the mean parameter vector in the normal distribution, are available in some cases. Some of these are reviewed and certain generalisations are discussed.
Introduction
Fisher's general concept of fiducial distribution is difficult in higher dimensions. Joint fiducial distributions are not subject to ordinary probability calculus. Marginals and other derived distributions need in fact not be fiducial; cf. also Chapter 6. For vector parameters one must therefore settle for a less ambitious construct to capture the inferential uncertainty. Pitman (1939, 1957) noted that the fiducial probabilities in higher dimensions must be restricted to sets of specific forms, to avoid inconsistencies. One consequence of this insight is that only certain types of dimension reduction by integrating the higher dimensional fiducial density are valid. See Chapter 6 for examples of inconsistencies when not observing these restrictions, which usually are hard to specify.
Neyman (1941) was more restrictive. He looked only for a confidence region of specific degree. We lean towards Neyman and define confidence distributions in higher dimensions as confidence distributed over specified nested families of regions.
We first look at confidence distributions for the mean vector μ in a multinormal distribution obtained from a sample, of dimension p, say. When the covariance matrix is known, the confidence distribution is simply, as will be seen, a multivariate normal distribution about the sample mean. This is obtained from a natural multivariate pivot, yielding the cumulative distribution function C(μ) assigning confidence to intervals {m: m ≤ μ﹜ in Rp. Thus confidence is also assigned to intervals and rectangles in Rp. The fiducial debate clarified that fiducial probability and hence confidence cannot be extended to the Borel sets. The confidence density for μ is thus not particularly useful, although directly obtained. Though integration over intervals yields valid confidence statements, integration over other sets might not do so.
12 - Predictions and confidence
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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Summary
The previous chapters have focussed on confidence distributions and associated inference for parameters of statistical models. Sometimes the goal of an analysis is, however, to make predictions about as yet unobserved or otherwise hidden random variables, such as the next data point in a sequence, or to infer values of missing data, and so forth. This chapter discusses and illustrates how the concept of confidence distributions may be lifted to such settings. Applications are given to predicting the next observation in a sequence, to regression models, kriging in geostatistics and time series models.
Introduction
In earlier chapters we have developed and discussed concepts and methods for confidence distributions for parameters of statistical models. Sometimes the goal of fitting and analysing a model to data is, however, to predict as yet unobserved random quantities, like the next observation in a sequence, a missing data point in a data matrix or inferring the distribution for a future Y0 in a regression model as a function of its associated covariates x0, and so on. For such a future or onobserved Y0 we may then wish to construct a predictive distribution, say Cpred(y0), with the property that Cpred(b)−Cpred(a) may be interpreted as the probability that a ≤ Y0 ≤ b. As such intervals for the unobserved Y0 with given coverage degree may be read off, via [C−1pred(α),C−1pred(1−α)], as for ordinary confidence intervals.
There is a tradition in some statistics literature to use ‘credibility intervals’ rather than ‘confidence intervals’, when the quantity in question for which one needs these intervals is a random variable rather than a parameter of a statistical model. This term is also in frequent use for Bayesian statistics, where there is no clear division in parameters and variables, as also model parameters are considered random. We shall, however, continue to use ‘confidence intervals’ and indeed ‘confidence distributions’ for these prediction settings.
We shall start our discussion for the case of predicting the next data point in a sequence of i.i.d. observations, in Section 12.2. Our frequentist predictive approach is different from the Bayesian one, where the model density is being integrated over the parameters with respect to their posterior distribution.
5 - Invariance, sufficiency and optimality for confidence distributions
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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Summary
The previous two chapters have dealt with the basic concepts, motivation and machinery for confidence distributions and confidence curves. Here we first discuss ways of reducing problems to simpler ones via the paths of invariance and sufficiency, involving also the concepts of loss functions, risk functions and power functions for confidence distributions. A Rao–Blackwell type result is reached about lowering the expected confidence loss via sufficiency. We furthermore provide a Neyman–Pearson type optimality result valid for certain confidence procedures, applicable in particular to the exponential class of distributions and to the natural parameters of generalised linear models.
Confidence power
Let C(ψ) be the confidence distribution function for some one-dimensional focus parameter ψ. The intended interpretation of C is that its quantiles are endpoints of confidence intervals. For these intervals to have correct coverage probabilities, the cumulative confidence at the true value of the parameter must have a uniform probability distribution. This is an ex ante statement. Before the data have been gathered, the confidence distribution is a statistic with a probability distribution, often based on another statistic through a pivot.
The choice of statistic on which to base the confidence distribution is unambiguous only in simple cases. Barndorff-Nielsen and Cox (1994) are in agreement with Fisher when emphasising the structure of the model and the data as a basis for choosing the statistic. They are primarily interested in the logic of statistical inference. In the tradition of Neyman and Wald, emphasis has been on inductive behaviour, and the goal has been to find methods with optimal frequentist properties. In exponential families and in other models with Neyman structure (see Lehmann 1959, chapter 4), it turns out that methods favoured on structural and logical grounds usually also are favoured on grounds of optimality. This agreement between the Fisherian and Neyman–Wald schools is encouraging and helps to reduce the division between these two lines of thought.
Example 5.1 Upper endpoint of a uniform distribution
Assume Y1,…,Yn are independent from the uniform distribution on [0, θ], and consider both Vn = Y(n) and Un = Y(n−1), the largest and second largest among the observations. Then both Vn/θ and Un/θ are pivots, hence each giving rise to confidence distributions.
11 - Confidence in non- and semiparametric models
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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References
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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4 - Further developments for confidence distribution
- Tore Schweder, Universitetet i Oslo, Nils Lid Hjort, Universitetet i Oslo
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Confidence distributions were introduced, developed and broadly discussed in the previous chapter, with emphasis on typical constructions and behaviour in smooth parametric models. The present chapter considers various extensions and modifications for use in less straightforward situations. These include cases in which the parameter range of the focus parameter is bounded; the Neyman–Scott problem with a high number of nuisance parameters; the Fieller problem with a ratio of two normal means, and other cases of multimodal likelihoods; Markov chain models; and hazard rate inference.
Introduction
The likelihood machinery is and remains a very powerful and versatile toolbox for theoretical and practical statistics. Theorems 2.2 and 2.4, along with various associated results and consequences, are in constant use, for example, qua algorithms in statistical software packages that use the implied approximations to normality and to chi-squaredness. The confidence distribution methods developed in Chapter 3 also rely in part on this machinery. Along with further supplements and amendments using modified profile deviances or bootstrap techniques for improved accuracy, as investigated in Chapters 7 and 8, this may lead to broadly applicable algorithms implemented in standard statistical software packages.
In the present chapter we pursue the study of stylised and real data examples further, beyond the clearest and cleanest cases dealt with in the previous chapter. We illustrate how certain difficulties arise in cases with bounded parameters or bounded confidence, or with multimodal log-likelihoods, and extend the catalogue of confidence distributions to situations involving Markov chains and time series, hazard rate models with censored data and so forth.
Bounded parameters and bounded confidence
In Chapter 3 we illustrated the use of confidence distributions in fairly regular situations, where the distributions in question in particular have a full, natural range, with C(ψ) starting somewhere at zero and growing smoothly to one. There are important situations in which this picture needs modification, however, and we shall discuss two such general issues here. The first concerns cases where the focus parameter lies in an interval with a natural boundary, as with variance components or in situations where a priori concerns dictate that one parameter must be at least as large as another one.
1 - Confidence, likelihood, probability: An invitation
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This chapter is an invitation to the central themes of the book: confidence, likelihood, probability and confidence distributions. We sketch the historical backgrounds and trace various sources of influence leading to the present and somewhat bewildering state of ‘modern statistics’, which perhaps to the confusion of many researchers working in the applied sciences is still filled with controversies and partly conflicting paradigms regarding even basic concepts.
Introduction
The aim of this book is to prepare for a synthesis of the two main traditions of statistical inference: those of the Bayesians and of the frequentists. Sir Ronald Aylmer Fisher worked out the theory of frequentist statistical inference from around 1920. From 1930 onward he developed his fiducial argument, which was intended to yield Bayesian-type results without the often ill-founded prior distributions needed in Bayesian analyses. Unfortunately, Fisher went wrong on the fiducial argument. We think, nevertheless, that it is a key to obtaining a synthesis of the two, partly competing statistical traditions.
Confidence, likelihood and probability are words used to characterise uncertainty in most everyday talk, and also in more formal contexts. The Intergovernmental Panel on Climate Change (IPCC), for example, concluded in 2007, “Most of the observed increase in global average temperature since the mid-20th century is very likely due to the observed increase in anthropogenic greenhouse gas concentrations” (Summary for Policymakers, IPCC, 2007). They codify ‘very likely’ as having probability between 0.90 and 0.95 according to expert judgment. In its 2013 report IPCC is firmer and more precise in its conclusion. The Summary for Policymakers states, “It is extremely likely that more than half of the observed increase in global surface temperature from 1951 to 2010 was caused by the anthropogenic increase in greenhouse gas concentrations and other anthropogenic forcings together” (IPCC, 2013, p. 17). By extremely likely they mean more than 95% certainty.
We would have used ‘confidence’ rather than ‘likelihood’ to quantify degree of belief based on available data. We will use the term ‘likelihood’ in the technical sense usual in statistics.
Confidence, likelihood and probability are pivotal words in the science of statistics. Mathematical probability models are used to build likelihood functions that lead to confidence intervals. Why do we need three words, and actually additional words such as credibility and propensity, to measure uncertainty and frequency of chance events?