Chapter Preview. Actuaries, like other business professionals, communicate quantitative ideas graphically. Because the process of reading, or decoding, graphs is more complex than reading text, graphs are vulnerable to abuse. To underscore this vulnerability, we give several examples of commonly encountered graphs that mislead and hide information. To help creators design more effective graphs and to help viewers recognize misleading graphs, this chapter summarizes guidelines for designing graphs that show important numerical information. When designing graphs, creators should

1. (1) Avoid chartjunk.

2. (2) Use small multiples to promote comparisons and assess change.

3. (3) Use complex graphs to portray complex patterns.

4. (4) Relate graph size to information content.

5. (5) Use graphical forms that promote comparisons.

6. (6) Integrate graphs and text.

7. (7) Demonstrate an important message.

8. (8) Know the audience.

Some of the guidelines for designing effective graphs, such as (6), (7), and (8), are drawn directly from principles for effective writing. Others, such as guidelines (3), (4), and (5), come from cognitive psychology, the science of perception. Guidelines (1) and (2) have roots both in effective writing and in graphical perception. For example, the writing principle of brevity demonstrates how eliminating pseudo three-dimensional perspectives and other forms of chartjunk improve graphs. As another example, the writing principle of parallel structure suggests using small multiple variations of a basic graphical form to visualize complex relationships across different groups and over time.

To underscore the scientific aspect of graphical perception, we examine the process of communicating with a graph, beginning with a sender's interpretation of data and ending with a receiver's interpretation of the graph.

Chapter Preview. This chapter continues our study of time series data. Chapter 7 introduced techniques for determining major patterns that provide a good first step for forecasting. Chapter 8 provides techniques for detecting subtle trends in time and models to accommodate these trends. These techniques detect and model relationships between the current and past values of a series using regression concepts.

Autocorrelations

Application: Inflation Bond Returns

To motivate the introduction of methods in this chapter, we work in the context of the inflation bond return series. Beginning in January 2003, the U.S. Treasury Department established an inflation bond index that summarizes the returns on long-term bonds offered by the Treasury Department that are inflation indexed. For a Treasury inflation-protected security (TIPS), the principal of the bond is indexed by the (three-month-lagged) value of the (non-seasonally-adjusted) consumer price index. The bond then pays a semiannual coupon at a rate determined at auction when the bond is issued. The index that we examine is the unweighted average of bid yields for all TIPS with remaining terms to maturity of 10 or more years.

Monthly values of the index from January 2003 through March 2007 are considered, for a total of T = 51 returns. Atime series plot of the data is presented in Figure 8.1. This plot suggests that the series is stationary, and so it is useful to examine the distribution of the series through summary statistics that appear in Table 8.1.

Chapter Preview. Statistical reports should be accessible to different types of readers. Such reports inform managers who desire broad overviews in nontechnical language and analysts who require technical details to replicate the study. This chapter summarizes methods of writing and organizing statistical reports. To illustrate, we will consider a report of claims from third-party automobile insurance.

Overview

The last relationship has been explored, the last parameter has been estimated, the last forecast has been made, and now you are ready to share the results of your statistical analysis with the world. The medium of communication can come in many forms: you may simply recommend to a client to “buy low, sell high” or you may give an oral presentation to your peers. Most likely, however, you will need to summarize your findings in a written report.

Communicating technical information is difficult for a variety of reasons. First, in most data analyses, there is no one “right” answer that the author is trying to communicate to the reader. To establish a right answer, one need only position the pros and cons of an issue and weigh their relative merits. In statistical reports, the author is trying to communicate data features and the relationship of the data to more general patterns, a much more complex task. Second, most reports written are directed at a primary client or audience. In contrast, statistical reports are often read by many different readers whose knowledge of statistical concepts varies extensively; it is important to take into consideration the characteristics of this heterogeneous readership when judging the pace and order in which the material is presented.

Chapter Preview. Regression analysis is a statistical method that is widely used in many fields of study, with actuarial science being no exception. This chapter provides an introduction to the role of the normal distribution in regression, the use of logarithmic transformations in specifying regression relationships, and the sampling basis that is critical for inferring regression results to broad populations of interest.

What Is Regression Analysis?

Statistics is about data. As a discipline, it is about the collection, summarization, and analysis of data to make statements about the real world. When analysts collect data, they are really collecting information that is quantified, that is, transformed to a numerical scale. There are easy, well-understood rules for reducing the data, through either numerical or graphical summary measures. These summary measures can then be linked to a theoretical representation, or model, of the data. With a model that is calibrated by data, statements about the world can be made.

Statistical methods have had a major impact on several fields of study:

• In the area of data collection, the careful design of sample surveys is crucial to market research groups and to the auditing procedures of accounting firms.

• Experimental design is a subdiscipline devoted to data collection. The focus of experimental design is on constructing methods of data collection that will extract information in the most efficient way possible.

• […]

Chapter Preview. A regression analyst collects data, selects a model, and then reports on the findings of the study, in that order. This chapter considers these three topics in reverse order, emphasizing how each stage of the study is influenced by preceding steps. An application, determining a firm's characteristics that influence its effectiveness in managing risk, illustrates the regression modeling process from start to finish.

Studying a problem using a regression modeling process involves a substantial commitment of time and energy. One must first embrace the concept of statistical thinking, a willingness to use data actively as part of a decision-making process. Second, one must appreciate the usefulness of a model that is used to approximate a real situation. Having made this substantial commitment, there is a natural tendency to “oversell” the results of statistical methods such as regression analysis. By overselling any set of ideas, consumers eventually become disappointed when the results do not live up to their expectations. This chapter begins in Section 6.1 by summarizing what we can reasonably expect to learn from regression modeling.

Models are designed to be much simpler than relationships among entities that exist in the real world. A model is merely an approximation of reality. As stated by George Box (1979), “All models are wrong, but some are useful.” Developing the model, the subject of Chapter 5, is part of the art of statistics. Although the principles of variable selection are widely accepted, the application of these principles can vary considerably among analysts.

Chapter Preview. When modeling financial quantities, we are just as interested in the extreme values as in the center of the distribution; extreme values can represent the most unusual claims, profits, or sales. Actuaries often encounter situations where the data exhibit fat tails, or cases in which extreme values in the data are more likely to occur than in normally distributed data. Traditional regression focuses on the center of the distribution and downplays extreme values. In contrast, the focus of this chapter is on the entire distribution. This chapter surveys four techniques for regression analysis of fat-tailed data: transformation, generalized linear models, more general distributions, and quantile regression.

Introduction

Actuaries often encounter situations in which the data exhibit fat tails, meaning that extreme values in the data are more likely to occur than in normally distributed data. These distributions can be described as “fat,” “heavy,” “thick” or “long” as compared to the normal distribution. (Section 17.3.1 will be more precise on what constitutes fat tailed.) In finance, for example, the asset pricing theories assume normally distributed asset returns. Empirical distributions of the returns of financial assets, however, suggest fat-tailed distributions rather than normal distributions, as assumed in the pricing theories (see, e.g., Rachev, Menn, and Fabozzi, 2005). In health care, fat-tailed data are also common. For example, outcomes of interest such as the number of inpatient days or inpatient expenditures are typically right skewed and heavy tailed as a result of a few high-cost patients (Basu, Manning, and Mullahy, 2004).

Chapter Preview. This chapter introduces regression applications of pricing in credibility and bonus-malus experience rating systems. Experience rating systems are formal methods for including claims experience into renewal premiums of short-term contracts, such automobile, health, and workers' compensation. This chapter provides brief introductions to credibility and bonus-malus, emphasizing their relationship with regression methods.

Risk Classification and Experience Rating

Risk classification is a key ingredient of insurance pricing. Insurers sell coverage at prices that are sufficient to cover anticipated claims, administrative expenses, and an expected profit to compensate for the cost of capital necessary to support the sale of the coverage. In many countries and lines of business, the insurance market is mature and highly competitive. This strong competition induces insurers to classify risks they underwrite to receive fair premiums for the risk undertaken. This classification is based on known characteristics of the insured, the person, or firm seeking the insurance coverage.

For example, suppose that you are working for a company that insures small businesses for time lost because of employees injured on the job. Consider pricing this insurance product for two businesses that are identical with respect to number of employees, location, age and sex distribution, and so forth, except that one company is a management consulting firm and the other is a construction firm.

Abstract. Proportional-hazards models are frequently used to analyze data from randomized controlled trials. This is a mistake. Randomization does not justify the models, which are rarely informative. Simpler methods work better. This discussion is salient because the misuse of survival analysis has introduced a new hazard in epidemiology: It can lead to serious mistakes in medical treatment. Life tables, Kaplan-Meier curves, and proportional-hazards models, aka “Cox models,” all require strong assumptions, such as stationarity of mortality and independence of competing risks. Where the assumptions fail, the methods also tend to fail. Justifying those assumptions is fraught with difficulty. This is illustrated with examples: the impact of religious feelings on survival and the efficacy of hormone replacement therapy. What are the implications for statistical practice? With observational studies, the models could help disentangle causal relations if the assumptions behind the models can be justified.

In this chapter, I will discuss life tables and Kaplan-Meier estimators, which are similar to life tables. Then I turn to proportional-hazards models, aka “Cox models.” Along the way, I will look at the efficacy of screening for lung cancer, the impact of negative religious feelings on survival, and the efficacy of hormone replacement therapy.

What are the conclusions about statistical practice? Proportional-hazards models are frequently used to analyze data from randomized controlled trials.

Abstract. The “salt hypothesis” is that higher levels of salt in the diet lead to higher levels of blood pressure, increasing the risk of cardiovascular disease. Intersalt, a cross-sectional study of salt levels and blood pressures in fifty-two populations, is often cited to support the salt hypothesis, but the data are somewhat contradictory. Four of the populations (Kenya, Papua, and two Indian tribes in Brazil) do have low levels of salt and blood pressure. Across the other forty-eight populations, however, blood pressures go down as salt levels go up–contradicting the hypothesis. Experimental evidence suggests that the effect of a large reduction in salt intake on blood pressure is modest and that health consequences remain to be determined. Funding agencies and medical journals have taken a stronger position favoring the salt hypothesis than is warranted, raising questions about the interaction between the policy process and science.

It is widely believed that dietary salt leads to increased blood pressure and higher risks of heart attack or stroke. This is the “salt hypothesis.” The corollary is that salt intake should be drastically reduced. There are three main kinds of evidence: (i) animal experiments, (ii) observational studies on humans, and (iii) human experiments. Animal experiments are beyond the scope of the present chapter, although we give a telegraphic summary of results. A major observational study cited by those who favor salt reduction is Intersalt (1986, 1988).

Abstract. Regression models have been used in the social sciences at least since 1899, when Yule published a paper on the causes of pauperism. Regression models are now used to make causal arguments in a wide variety of applications, and it is perhaps time to evaluate the results. No definitive answers can be given, but this chapter takes a rather negative view. Snow's work on cholera is presented as a success story for scientific reasoning based on nonexperimental data. Failure stories are also discussed, and comparisons may provide some insight. In particular, this chapter suggests that statistical technique can seldom be an adequate substitute for good design, relevant data, and testing predictions against reality in a variety of settings.

Introduction

Regression models have been used in social sciences at least since 1899, when Yule published his paper on changes in “out-relief” as a cause of pauperism: He argued that providing income support outside the poorhouse increased the number of people on relief. At present, regression models are used to make causal arguments in a wide variety of social science applications, and it is perhaps time to evaluate the results.

A crude four-point scale may be useful:

1. Regression usually works, although it is (like anything else) imperfect and may sometimes go wrong.

2. Regression sometimes works in the hands of skillful practitioners, but it isn't suitable for routine use.

3. Regression might work, but it hasn't yet.

4. Regression can't work.

Abstract. Making sense of earthquake forecasts is surprisingly difficult. In part, this is because the forecasts are based on a complicated mixture of geological maps, rules of thumb, expert opinion, physical models, stochastic models, and numerical simulations, as well as geodetic, seismic, and paleoseismic data. Even the concept of probability is hard to define in this context. For instance, the U.S. Geological Survey developed a probability model according to which the chance of an earthquake of magnitude 6.7 or greater before the year 2030 in the San Francisco Bay Area is 0.7 ± 0.1. How is that to be understood? Standard interpretations of probability cannot be applied. Despite their careful work, the USGS probability estimate is shaky, as is the uncertainty estimate.

Introduction

What is the chance that an earthquake of magnitude 6.7 or greater will occur before the year 2030 in the San Francisco Bay Area? The U.S. Geological Survey estimated the chance to be 0.7 ± 0.1 (USGS, 1999). In this chapter, we try to interpret such probabilities.

Making sense of earthquake forecasts is surprisingly difficult. In part, this is because the forecasts are based on a complicated mixture of geological maps, rules of thumb, expert opinion, physical models, stochastic models, numerical simulations, as well as geodetic, seismic, and paleoseismic data.

Abstract. The “Huber Sandwich Estimator” can be used to estimate the variance of the MLE when the underlying model is incorrect. If the model is nearly correct, so are the usual standard errors, and robustification is unlikely to help much. On the other hand, if the model is seriously in error, the sandwich may help on the variance side, but the parameters being estimated by the MLE are likely to be meaningless–except perhaps as descriptive statistics.

Introduction

This chapter gives an informal account of the so-called “Huber Sandwich Estimator,” for which Peter Huber is not to be blamed. We discuss the algorithm and mention some of the ways in which it is applied. Although the chapter is mainly expository, the theoretical framework outlined here may have some elements of novelty. In brief, under rather stringent conditions the algorithm can be used to estimate the variance of the MLE when the underlying model is incorrect. However, the algorithm ignores bias, which may be appreciable. Thus, results are liable to be misleading.

To begin the mathematical exposition, let i index observations whose values are yi. Let θ ∈ Rp be a p × 1 parameter vector. Let y → fi (y|θ) be a positive density. If yi takes only the values 0 or 1, which is the chief case of interest here, then fi (0|θ) > 0, fi (1|θ) > 0, and fi (0|θ) + fi (0|θ) + fi (1|θ) = 1. Some examples involve real- or vector-valued yi, and the notation is set up in terms of integrals rather than sums.

Abstract. A number of algorithms purport to discover causal structure from empirical data with no need for specific subject-matter knowledge. Advocates claim that the algorithms are superior to methods already used in the social sciences (regression analysis, path models, factor analysis, hierarchical linear models, and so on). But they have no real success stories to report. The algorithms are computationally impressive and the associated mathematical theory may be of some interest. However, the problem solved is quite removed from the challenge of causal inference from imperfect data. Nor do the methods resolve long-standing philosophical questions about the meaning of causation.

Causation, Prediction, and Search by Peter Spirtes, Clark Glymour, and Richard Scheines (SGS) is an ambitious book. SGS claim to have methods for discovering causal relations based only on empirical data, with no need for subject-matter knowledge. These methods–which combine graph theory, statistics, and computer science–are said to allow quick, virtually automated, conversion of association to causation. The algorithms are held out as superior to methods already in use in the social sciences (regression analysis, path models, factor analysis, hierarchical linear models, and so on).