Applied statistics

The term applied statistics is used both to denote the work of professional statisticians, who are responsible for the production and presentation of statistics, and to refer to the methods of statistical interpretation that are undertaken by professional statistitions, by professional users of statistics and by the general public. It is an activity that makes use of the results of statistics theory but is distinct from the activity of developing that theory. It has made major contributions to scientific research and to the conduct of operational activities in a variety of fields. It has provided both a familiar source of information and a notorious source of misinformation. Mistakes of interpretation commonly arise from misplaced confidence in intuitive judgements, including errors by medical and legal professionals that have led to false diagnoses and wrongful imprisonment. Deliberate misinterpretation of statistics by politicians and marketing professionals is so much a popular perception that its genuine use is often treated with suspicion. However, its use is generally unavoidable, and its misinterpretation can be avoided given a grasp of elementary mathematics and of the readily understood theorems of verbal logic.

Overview: the basics

Statistics are observations that are recorded in numerical form. It is essential to their successful handling to accept that statistics are not facts and therefore incontrovertible, but observations about facts and therefore fallible. The reliability of the information that they provide depends not only upon their successful interpretation, but also upon the accuracy with which the facts are observed and the extent to which they truly represent the subject matter of that information. An appreciation of the means by which statistics are collected is thus an essential part of the understanding of statistics and is least as important as a familiarity with the tools that are used in its interpretation.

The basic laws of chance from which much of statistics theory has been derived are little more than a formalisation of intuitive concepts, and the use of the resulting algorithms for the solution of many everyday statistical problems should require only a grasp of basic mathematical principles. Failures of interpretation by professional users suggest, however, that "probability blindness" is an inherent characteristic of the human brain that prevents the effective employment of intuition for that purpose.

The Statistics profession

The American Statistical Association describes a professional statistician as an expert in

• producing trustworthy data;
• analyzing data to make their meaning clear; and.
• drawing practical conclusions from data..

Professional statisticians are normally qualified in mathematics and are normally familiar with the underlying mathematics of statistics theory but their professional activity involves them in the use of the results that theory, rather than in the development of those results.

The collection of statistics

The methodology adopted for the collection of observations has a profound influence upon the problem of extracting useful information from the resulting statistics. That problem is at its easiest when the collecting authority can minimise disturbing influences by conducting a "controlled experiment". A range of more complex methodologies (and associated software packages) referred to as "the design of experiments" is available for use when the collecting authority has various lesser degrees of control. The object of the design in each case is to facilitate the testing of an hypothesis by helping to exclude the influence of factors that the hypothesis does not take into account. When extraneous influences cannot be eliminated by physical exclusion, their effects can sometimes be estimated by purely analytical techniques such as the "analysis of variance" For example, the attitudes of the statistics authorities to extraneous influences has usually been passive, and the testing of economic hypotheses has used a version of the analysis of variance termed "econometrics" (sometimes confused with economic modelling, which is a purely deterministic technique).

The taking of samples reduces the cost of collecting observations and increases the opportunities to generate false information. One source of error arises from the fact that every time a sample is taken there will be a different result. That source of error is readily quantified as the sample's "standard error", or as the "confidence interval" within which the mean observation may be expected to lie. That source of error cannot be eliminated, but it can be reduced to an acceptable level by increasing the size of the sample. The other source of error arises from the likelihood that the characteristics of the sample differ from those of the "population" that it is intended to represent. That source of error does not diminish with sample size and cannot be estimated by a mathematical formula. Careful attention to what is known about the composition of the "population" and the reflection of that composition in the sample is the only available precaution. The composition of the respondents to an opinion poll, for example, is normally chosen to reflect as far as possible the composition of the intended "population" as regards sex, age, income bracket etc. The remaining difference is referred as the "sample bias"'.

The use by statisticians of the term "population" refers, not to people in general, but to the category of things or people about which information is sought. A precise definition of the target population is an essential starting point in a statistical investigation, and also a possible source of misinformation. Difficulty can arise when, as often happens, the definition has to be arbitrary. If the intended population were the output of the country's farmers, for example, it might be necessary to draw an arbitrary dividing line between farmers and owners of smallholdings such as market gardens. Any major change over time in the relative output of farm products by the included and excluded categories might then lead to misleading conclusions. Technological change, such as the change from typewriters to word processors has sometimes given rise to serious difficulties in the construction of the price indexes used in the correction of GDP for inflation. Since there is no objective solution to those problems, it is inevitable that national statistics embody an element of judgement exercised by the professional statisticians in the statistics authorities.

Statistical inference

Although statistics is sometimes thought of as a branch of mathematics, some of its findings can be successfully interpreted by verbal inference, and there are others that only require the use of a few simply-expressed rules (such as those set out in paragraph 1 of the tutorials subpage). However, there is evidence to suggest that most people confidently prefer an intuitive approach, unaware of the "probability blindness" that is characteristic of the human brain.

The development of statistics theory that has enabled more complex statistical problems to be tackled, has been the work of mathematical geniuses such as Bernoulli, Laplace and Pascal, but the skills required for the effective use of statistics are different from those required for the understanding of the mathematical derivation of their theorems. Statistics theory is mainly concerned with the special circumstances under which events are governed solely by chance, without any influence from human action or recognisable natural forces, whereas applied statistics is usually concerned with its use under less restricted circumstances. Also, the terminology of statistical theory attaches precise meanings to some everyday words that may not always be applicable under less restricted circumstances. When, for example, a degree of statistical confidence is expressed in the statement that the strength of a type of steel will not fall below a stated level, that degree of confidence may apply only to certain circumstances - and not, for example, under conditions of widely varying temperatures. Thus the successful user of statistics has to combine an awareness of the theoretical tools of inference that are available with an appreciation of the extent to which they can safely be applied to a particular problem - if indeed they can be so applied, bearing in mind the financial disasters that have resulted from the mistaken reliance upon statistics in situations containing deterministic risks. . The user who plans to employ those tools for the analysis of data must also be prepared to spend a good deal of time acquiring a grasp of the relevant theorems of statistics theory, and mastering the intricacies of the free statistical software that is available for that purpose. Managers who supervise such work, and users of its application, may seek to be excused from such expenditure of effort, but cannot escape responsibilty for acquiring an understanding of statistical concepts that is at least sufficient for an awareness of the limitations of such analysis. And, as the statistician, M J Moroney has emphasised, there can never be any question of making a decision solely on the basis of a statistical test: an engineer doing a statistical test must remain an engineer, an economist must remain an economist, a pharmacist a pharmacist.

A major contribution of statistics theory to the interpretation of results obtained from a sample is the quantification of the intuitive concept of "significance" in a way that enables an objective answer to be given to the question of how likely it is that, what might appear to be information, is really only a matter of chance (although the way that question is usually put by statisticians is "whether the result is significant at the 5 per cent level"). If - and only if - it can be established by other methods that the sample used was not biassed, then one of a variety of statistical tests can be used to answer that question. When established, the conclusion is best reported in jargon-free English, using a phrase such as "this result could arise by chance once in twenty trials". Equally useful is its quantification of the concept of the term "confidence", enabling an objective answer to be given to a question such as "how confident are we that the toxicity of a drug does not exceed a stipulate level" (or that a structure can bear at least a stipulate load, or that a variable lies within a stipulated confidence interval). Among the most powerful of the techniques of statistical analysis is the use of "correlation" (sometimes termed "regression") to explore a relationship when the available evidence is subject to errors that are attributable solely to chance. It could be used, for example, to estimate the average constant of proportionality in a (hypothesised) linear relation between IQs and examination marks. In a more complex case it is used to explore the relationship between household income and household saving, taking account of other factors believed to affect that relationship. But, besides being very powerful, regression methods are especially prone to the production of false information. Success in interpreting the data requires the tackling of problems that are often more difficult than the statistical manipulation.

Statistical misinterpretation

Probability blindness

It is well known that most people find it difficult to draw the right conclusion from statistical evidence , but misinterpretation of statistics by acknowledged experts is also not uncommon.

Medical misinterpretation

Research fallacies

It has been authoritatively claimed that the results of most medical research are flawed because of statistical misinterpretation.

The false positive fallacy

When the following question was put to students and staff of the Harvard Medical School If a test of a disease that has a prevalence rate of 1 in 1000 has a false positive rate of 5%, what is the chance that a person who has been given a positive result actually has the disease. - 45 per cent gave the intuitive answer of 95% when the true answer is 2% (see paragraph 2 of the tutorials subpage).

Legal misinterpretation

There have also been many cases of statistical interpretation by lawyers, some of which have resulted in serious injustice.

The prosecutors' fallacy

Wrongful convictions in rape cases have arisen from a confusion between the probability of effect given cause and that of effect given ,known as the "prosecutor's fallacy" (see paragraph 3 of the tutorials subpage).

The Sally Clark case

In 1999 the British Solicitor, Sally Clark was convicted of murdering her two babies on the evidence of an expert witness who told the jury that the odds against the deaths being natural were one in 73 million; a figure he obtained by misusing the multiplication rule and taking the square of the odds against a single cot death - overlooking the possibility that the two deaths had a common cause.

Sources of statistics

Published statistics are widely used for research into subjects such as economics, health and demography. A large volume of time-series statistics is published by national statistics authorities. In the United States, for example, statistical series are published regularly by over 70 federal agencies. There are also a number of databases and collations of national statistics published by international agencies such as the United Nations and the OECD , most of which are available on the internet . Although a number of attempts have been made to introduce common standards, nationally-published statistics are seldom directly comparable, except after extensive adjustments .

Many of the published series involve the quarterly or monthly collection of data concerning a specified topic such as an industry, a disease or a category of crime or of economic activity, often by a comprehensive survey (and sometimes by sample surveys) of holders of the relevant data. Another type of source, that has been of value in medical, social and criminological research, has been the occasional collection of comparative data from a group of people born in a particular place on a particular date, by a technique termed "birth-cohort study" (such as the study of the people born in Britain in one week in April 1970 , the study of people born in Philadelphia in 1958, and others)

Major applications of statistics

Statistical analysis has been widely used for scientific research, drawing upon both published statistics and the results of surveys and laboratory experiments, and it has enabled major scientific advances to be made in subjects as diverse as medicine and criminology. It has also had operational uses in the fields of business decision-making. , climatology, metrology , cost/benefit analysis, and finance. For those operational applications it has often had to draw upon theoretical knowledge and professional judgement, as well as upon records of past experience. As the distinguished economist Frank Knight observed in 1921, "business decisions ... deal with situations which are far too unique, generally speaking, for any sort of statistical tabulation to have any value for guidance. The conception of an objectively measurable probability or chance is simply inapplicable." - and drew a distinction between risk, which can sometimes be estimated objectively; and uncertainty, which never can. Business decisions necessarily involve the use of subjective estimates, which can be combined using a "decision tree" to produce a probability estimate for each of the range of possible outcomes, enabling the values of those outcomes to be combined to yield a "net present expected value". The estimation of net present value in that way, is an essential feature of cost/benefit analysis, and in those and other applications it may be necessary to base decisions upon a combination of "aleatory" (chance-related) and "epistemic" (cognitive) factors. In the practice of risk management, in particular, it is important not to neglect epistemic factors. The data used for statistical quality control are usually the results of tests on samples drawn from a batch of the products of a manufacturing process and its purpose is to decide whether to accept that batch. The normal criteria for acceptance are designed to ensure, at a specified level of confidence that the quality of the batch does not depart by more than a specified percentage from a target level (but in cases where the costs of departures depend upon their magnitude the more complex Taguchi criteria is sometimes adopted). Statistical analysis has also been used to devise economical sampling methods

Adjustments, reliability, accuracy and bias

Those who regard statistics as being on a par with the proverbial "damned lies" and those who believe them always to be totally objective estimates, are both mistaken. The concept of an objective statistic is a mathematical abstraction and although some, such as some financial statistics, approximate to that ideal, most published and unpublished statistics embody some element of subjective judgement. The embodiment of subjective adjustments in published statistics for the purpose of increasing their usefulness is the responsibility of professional statisticians whose competence and objectivity is usually subject to oversight by an independent body (politicians in the major democracies have seldom had any influence upon the collection and publication of national statistics, and many countries have sought to allay suspicions to the contrary by delegating those functions to public bodies that are free from possible government influence.).

It has seldom been feasible to assess the accuracy of published statistics, and indications of their reliability are available only when a variable is estimated from more than one source or when revisions are made. The substantial discrepancies between police-reported and victim-reported crime rates , for example, are presumably the result of differences in the reliability of the two methods, and the differences between income-based and expenditure-based estimates of GDP may be similarly attributable. The major revisions that have been made to early estimates of GDP also provide an indication of the reliability to be expected of such estimates.

The commonest cause of bias in published statistics has been faulty sampling. An early example was the "Lanarkshire Milk Trial", which yielded a perverse effect of giving milk to school children because teachers gave the milk to sickly rather than healthy children. Similar errors that did not produce perverse results may have gone undetected. A more recent example was the BZW equity-gilt study of financial returns which covered only firms existing at the end of the period under review, thereby excluding those that had failed in the interim (an error known as "survivor bias"). Survivor bias has also marred studies of medical treatment that have used samples in which patients who died early were under-represented. The most far- reaching example, however, was the practice of major credit-rating agencies of basing their risk assessment upon data from a period of exceptionally low financial volatility. even during the latter stages of what was widely seen as a "bubble" . Reliance upon such risk estimates to the exclusion of the exercise of expert judgment is generally held to have made a significant contribution to the financial crash of 2008.